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Time-dependent inverse box-model for the estuarine circulation and primary productivity in the Strait… Riche, Olivier 2011

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Time-dependent Inverse Box-model For The Estuarine Circulation and Primary Productivity in The Strait of Georgia by Olivier Riche Licence et Maˆıtrise, Universit´ e du Havre, 1997 D.E.S.S, Universit´ e de Toulon et du Var, 1998 M.Sc., Universit´ e du Qu´ ebec a ` Rimouski, 2002 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy in The Faculty of Graduate Studies (Oceanography) The University Of British Columbia (Vancouver) September, 2011 c Olivier Riche 2011  Abstract During 2002–2006, a comprehensive set of observations covering physical, biological, radiative and atmospheric parameters was obtained from the southern Strait of Georgia (SoG), Western Canada by the STRATOGEM program. Monthly time series of estuarine layer transports over 2002–2005 were estimated using a time-dependent 2-box model in a formal inverse approach. These transports are then consistent with the temperature and salinity fields, as well as riverine freshwater inflow (R) and atmospheric heat fluxes. Uncertainty was analyzed by resampling observations using bootstrap methods. The transport time series were then combined with observations of nutrient concentrations to construct monthly time series of nutrient uptake for nitrate, phosphate, and silicic acid. Analysis of these time series suggests that the SoG estuarine circulation is not very sensitive to the seasonal changes of R. Comparison of the surface layer transport (U1 ) and R yields the first observational relationship between the SoG estuarine circulation and R. This relationship (U1 =2.68 m2 s−2/3 × 103 R1/3 ) is consistent with estuarine theories. Although the flows change slightly with the freshet, a 5-fold change in R results only in a 40% change in U1 . Based on the calculated sink of near-surface nutrients, net primary productivity is estimated to be 212 gC m−2 yr−1 , which is similar to values obtained differently in similar estuaries. Comparison of the nitrate and phosphate uptake rates suggests that the primary productivity (PP) is mainly new PP during spring and summer. ii  Abstract Thus, PP is mainly controlled by the upwelling supply of nutrients through deep inflow and entrainment. The uptake of silicic acid (Si) is almost two times larger than the uptake of nitrate during diatom spring blooms, while it is similar during the summer blooms. Such a high Si uptake suggests that spring diatoms form heavier frustules or that heterotrophic silicoflagellates compete with diatoms for Si. Speculative considerations based on comparison of the estimated production rate of near-surface oxygen and new PP also suggest that the regenerated PP is small. In addition, the summer heterotrophic respiration might be in excess by as much as 2 gO m−2 d−1 relative to the net PP.  iii  Preface The author’s contribution to this work was: a) the collection of some of the data in a field program designed by others, and b) the analysis of the entire dataset described here. The author will be the lead author in manuscripts to be submitted.  iv  Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iv  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  v  Preface  List of Tables  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  x  Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xiii  Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii Dedication  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix  1 Introduction  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  1.1  Physical Oceanography  . . . . . . . . . . . . . . . . . . . . . . . . .  5  1.2  Biological Oceanography . . . . . . . . . . . . . . . . . . . . . . . . .  11  1.3  Two-layer Model and Governing Equations  14  1.4  Objectives, Approach, Thesis Contributions and Plan  . . . . . . . . . . . . . . . . . . . . . .  17  2 Inverse Methods and Box Models . . . . . . . . . . . . . . . . . . . .  19  v  Table of Contents 2.1  Introduction: Mathematical Framework  . . . . . . . . . . . . . . . .  19  2.2  Inverse Problem: Estimating the Circulation of the Strait of Georgia  29  2.3  Solution Uncertainty and Residuals . . . . . . . . . . . . . . . . . . .  46  2.4  Forward Problem: Estimating the Net Primary Productivity . . . . .  49  3 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  54  3.1  Introduction  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  54  3.2  Data Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  54  3.2.1  STRATOGEM . . . . . . . . . . . . . . . . . . . . . . . . . .  54  3.2.2  JEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  56  3.2.3  Freshwater Inflow, Surface Heat and Air-sea Fluxes . . . . . .  57  3.3  Box Model Inputs  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3.3.1  Separation Depth  . . . . . . . . . . . . . . . . . . . . . . . .  3.3.2  Spatial Averaging and Hypsography  3.3.3  58 59  . . . . . . . . . . . . . .  60  Time Dependence  . . . . . . . . . . . . . . . . . . . . . . . .  61  3.3.4  Input Time Series  . . . . . . . . . . . . . . . . . . . . . . . .  61  3.3.5  Air-Sea Oxygen Flux  . . . . . . . . . . . . . . . . . . . . . .  71  3.3.6  Riverine Inputs . . . . . . . . . . . . . . . . . . . . . . . . . .  71  4 Circulation and Transports . . . . . . . . . . . . . . . . . . . . . . . .  79  4.1  Introduction  4.2  Results  4.3  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  79  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  81  4.2.1  Estimates of Estuarine Transports and Mixing Exchange . . .  81  4.2.2  Sensitivity Analysis  . . . . . . . . . . . . . . . . . . . . . . .  85  4.2.3  Residuals of the Conservation Equations . . . . . . . . . . . .  96  Discussion  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101  vi  Table of Contents 4.3.1  General Circulation in the Box Model  . . . . . . . . . . . . . 101  4.3.2  Seasonal Transports . . . . . . . . . . . . . . . . . . . . . . . 105  4.3.3  Comparison With Transports in the Strait of Juan de Fuca  4.3.4  Net Outflow from the Strait of Georgia  4.3.5  Circulation Sensitivity to the Freshwater Inflow . . . . . . . . 115  4.3.6  Annual Variability of the Circulation . . . . . . . . . . . . . . 117  4.3.7  Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121  . 107  . . . . . . . . . . . . 113  5 Nutrients Uptake and Primary Productivity . . . . . . . . . . . . . 125 5.1  Introduction  5.2  Results  5.3  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129  5.2.1  Supply and Sink Rates of Nutrients  5.2.2  Supply and Sink Rates of Oxygen  5.2.3  Uptake Ratios  5.2.4  Estimates of Net Primary Productivity  5.2.5  Estimates of Net Community Productivity . . . . . . . . . . . 157  Discussion  . . . . . . . . . . . . . . 129 . . . . . . . . . . . . . . . 138  . . . . . . . . . . . . . . . . . . . . . . . . . . 144 . . . . . . . . . . . . 149  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160  6 Discussion and Conclusion  . . . . . . . . . . . . . . . . . . . . . . . . 183  6.1  Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 183  6.2  Seasonality and Variability of Water Transports in Estuaries . . . . . 185  6.3  Estimates and Variability of Net Primary Productivities in Estuaries  6.4  Recommendations for Future Studies . . . . . . . . . . . . . . . . . . 201  195  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210  vii  List of Tables 2.1  Total Depths and Volumes of the Model Boxes . . . . . . . . . . . . .  30  2.2  Inversion Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . .  46  2.3  Coefficients of Variation in the Parametric Bootstraps . . . . . . . . .  48  4.1  Analysis of Variance and F-tests for TD and QSS Transports . . . . .  82  4.2  A Priori and Estimated Values of the Equations Residuals . . . . . .  99  4.3  SoG Transports from Previous Studies . . . . . . . . . . . . . . . . . 111  5.1  Averages of Surface Supply and Sink Rates of Nitrate (mol s−1 ) in the Euphotic Zone of the SoG. . . . . . . . . . . . . . . . . . . . . . . . . 139  5.2  Averages of Surface Supply and Sink Rates of Phosphate (mol s−1 ) in the Euphotic Zone of the SoG . . . . . . . . . . . . . . . . . . . . . . 140  5.3  Averages of Surface Supply and Sink Rates of Silicic Acid (mol s−1 ) in the Euphotic Zone of the SoG. . . . . . . . . . . . . . . . . . . . . 141  5.4  Averages of Surface Supply and Sink Rates of O2 (mol s−1 ) in the Euphotic Zone of the SoG. . . . . . . . . . . . . . . . . . . . . . . . . 142  5.5  Estimates of the Average Excess of Heterotrophic Respiration . . . . 160  5.6  Comparison of Si-replete and Iron-replete Bloom in Coale et al. [2004] and SoG Spring Bloom . . . . . . . . . . . . . . . . . . . . . . . . . . 177  viii  List of Tables 5.7  Comparison of Si-depleted and Iron-replete Bloom in Coale et al. [2004] and SoG Summer Bloom . . . . . . . . . . . . . . . . . . . . . 177  6.1  Ranges of PP rates in Temperate Estuaries . . . . . . . . . . . . . . . 197  6.2  Macronutrient Limitation During Summer . . . . . . . . . . . . . . . 198  ix  List of Figures 1.1  Geography of the Strait of Juan de Fuca/Haro Strait/Strait of Georgia  1.2  Detailed Sampling Area of STRATOGEM and Other Sampling Important Locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1.3  4  6  Along-channel Cross-section of the Strait of Juan de Fuca/Haro Strait/Strait of Georgia System . . . . . . . . . . . . . . . . . . . . . . .  8  1.4  Fraser River Discharge . . . . . . . . . . . . . . . . . . . . . . . . . .  11  2.1  Physical Fluxes and Processes in the Box Model . . . . . . . . . . . .  31  2.2  Biogeochemical Fluxes and Processes in the Box Model . . . . . . . .  32  2.3  Chart of the Inversion Procedure . . . . . . . . . . . . . . . . . . . .  48  3.1  Vertical Profiles of Salinity and Temperature . . . . . . . . . . . . . .  62  3.2  Hypsography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  63  3.3  Salinity and Freshwater Time Series . . . . . . . . . . . . . . . . . . .  65  3.4  Temperature and Surface Net Heat Flux Time Series . . . . . . . . .  66  3.5  Heat Flux Observations and Estimates . . . . . . . . . . . . . . . . .  68  3.6  Surface Heat Budget . . . . . . . . . . . . . . . . . . . . . . . . . . .  69  3.7  Phosphate Concentration Time Series . . . . . . . . . . . . . . . . . .  72  3.8  Nitrite+Nitrate Concentration Time Series . . . . . . . . . . . . . . .  73  3.9  Silicic Acid Concentration Time Series . . . . . . . . . . . . . . . . .  74  x  List of Figures 3.10 Dissolved Oxygen Time Series . . . . . . . . . . . . . . . . . . . . . .  75  3.11 Air-sea Oxygen Flux Time Series . . . . . . . . . . . . . . . . . . . .  76  3.12 River Biogeochemical Inputs Time Series . . . . . . . . . . . . . . . .  78  4.1  Transport Estimates and Their Errors . . . . . . . . . . . . . . . . .  86  4.2  Transport Estimates and Errors in Quasi-steady State . . . . . . . . .  89  4.3  Circulation Sensitivity to Separation Depth  . . . . . . . . . . . . . .  91  4.4  Circulation Sensitivity to the Trade-off Parameter α=γ s−1 . . . . . .  94  4.5  Circulation Sensitivity to the a Priori Parameter β  97  4.6  Residuals of the Mass, Salt and Heat Equations . . . . . . . . . . . . 102  4.7  Comparison Between Li et al. [1999]’s Exchange Flow and U1 . . . . . 108  4.8  Advective Transports in HS Box Model . . . . . . . . . . . . . . . . . 110  4.9  Surface Seaward Transport Plotted With Respect to Freshwater Inflow 118  . . . . . . . . . .  4.10 Residuals of the Fit of U1 as a Power of R . . . . . . . . . . . . . . . 119 4.11 Annual Mean of SoG Transports and Freshwater Inflow . . . . . . . . 122 4.12 Annual Variability of SoG Transports and Freshwater Inflow . . . . . 123 5.1  Surface Box Nitrate Supply and Sink Rates . . . . . . . . . . . . . . . 130  5.2  Surface Box Phosphate Supply and Sink Rate . . . . . . . . . . . . . 131  5.3  Surface Box Silicic Acid Supply and Sink Rates . . . . . . . . . . . . 132  5.4  Surface Box O2 Supply and Sink Rates . . . . . . . . . . . . . . . . . 133  5.5  Surface Nitrate and Phosphate Uptake Rates . . . . . . . . . . . . . . 150  5.6  Surface silicic acid and Phosphate Uptake Rates . . . . . . . . . . . . 151  5.7  Surface Dissolved O2 Release and Phosphate Uptake Rates . . . . . . 152  5.8  NPP Rate Estimates based on Net Biological Uptake Rates of N and P158  5.9  Estimate of the Chl-a-normalized NPP Rate . . . . . . . . . . . . . . 161  xi  List of Figures 5.10 Comparison of the Chl-a Normalized NPP Rate With Averaged PAR over the Mixing Layer . . . . . . . . . . . . . . . . . . . . . . . . . . 162 5.11 Net Community Production Rate Estimates Based on Net Biological Production Rates of O2 . . . . . . . . . . . . . . . . . . . . . . . . . . 163 5.12 Estimates of the Excess of Regenerated NPP rate and Heterotrophic Respiration Based on NCP and New NPP Rates . . . . . . . . . . . . 164 5.13 Estimates of the Diatom and Total Autotrophic Biomass . . . . . . . 169 5.14 SoG Nitrate Concentrations at 0 m, Average at 0 m and Over 0–30 m 171 5.15 SoG N and Si Average Concentrations at 0 m and Over 0–30 m . . . 173 5.16 Estimate of the Net Biological Uptake Rate of Si . . . . . . . . . . . 174 5.17 Estimates of the Si:N and Si:P Ratios Based on the Net Biological Uptake Rates of Si, N and P . . . . . . . . . . . . . . . . . . . . . . . 175 6.1  Comparison of STRATOGEM and IOS SoG Temperature Box Averages189  6.2  Comparison of STRATOGEM and IOS SoG Salinity Box Averages . 190  6.3  Comparison of STRATOGEM and IOS SoG Nitrate Box Averages . . 191  6.4  Histogram of FW Input in the SoG over 1912-2008 . . . . . . . . . . 194  6.5  Mean Monthly FW Input in the SoG . . . . . . . . . . . . . . . . . . 196  xii  Glossary α  Trade-off parameter  A′  Matrix A scaled by W−1  A  Data matrix of m rows and n columns  b  Data vector of m rows and 1 column, containing internal and external sources  c  Data vector b associated with z  Chl-a Chlorophyll-a pigments D′r  Diagonal matrix with diagonal coefficients d′i (α2 ), r rows and columns  d  Separation depth between SoG top and bottom boxes  d′i (α2 ) ith Diagonal coefficient of D′ r (α2 ), equals to εˆ  ˆ Estimated residuals associated with x  ε  Residuals vector of m rows and 1 column  E  Net upward entrainment transport  F  Heat flux  FSW  SW component of F  λ′ 2i λ′ 2i +α2  xiii  Fall  Fall starts in August and ends in October  FW  FreshWater  GPP Gross primary production HS  Haro Strait  Ik  Square identity matrix, k rows and columns  J  ˆ Objective function to minimize to find x  k  Effective rank ≤r and λk ≪ λi for i > k  λi  ith Singular value of the SVD  Λ′  Matrix of the singular values of A′ , m rows and n columns  Λ−1  Inverse matrix associated with Λ, n rows and m columns  Λ  Matrix of the singular values of A, m rows and n columns  Λ′ −1 Inverse matrix associated with Λ′ , n rows and m columns λ′ i  ith Singular values of A′  LLS  Linear Least Squares method  LW  LongWave  M  Mixing exchange rate due to vertical turbulence  m  Number of rows of a matrix  N  Nitrogen based on nitrate and nitrite xiv  n  Number of columns of a matrix  NCP Net community production NPP (Total) net primary production NPPn New net primary production NPPr Regenerated net primary production ωi  ith Diagonal element of W  O2  Dissolved dioxygen gas  P  Phosphate  PP  Primary production  QSS  Quasi-Steady State  R  FW input rate in the SoG  r  Rank of the matrix A with r≤ min(m, n)  Ra  Autotrophic (phytoplankton) respiration  Rh  Heterotrophic (zooplankton and bacteria) respiration  S  Colum-scaling diagonal matrix, n rows and n columns  s  Value of each diagonal element of S  Si  Silicic acid  SoG  Strait of Georgia xv  SoJdF Strait of Juan de Fuca Spring Since the earliest spring blooms can occur in February, spring is defined as February-April Summer Summer starts in May and ends in July SVD Singular Vector Decomposition SVDM Singular Vector Decomposition Methodology SW  ShortWave  Tu  Data resolution matrix, m rows and m columns  Tv  Solution resolution matrix, n rows and n columns  T  Operator of matrix transposition  TD  Time Dependence  U′  Square matrix of the data singular vectors of A′ , m rows and columns  ui  ith Column of U  Uk  Rectangular matrix of the k first data singular vectors, m rows and k columns  U  Square matrix of the data singular vectors of A, m rows and columns  u  Vector of the flow speed  U1  Seaward surface horizontal transport of the estuarine circulation  U2  Landward deep horizontal transport of the estuarine circulation  xvi  U01 , U02 , W01 , and W02 A priori values of U1 , U2 , W1 , and W2 V′  Square matrix of the solution singular vectors of A′ , n rows and columns  vj  j th Column of V  Vk  Rectangular matrix of the k first solution singular vectors, n rows and k columns  V  Square matrix of the solution singular vectors of A, n rows and columns  V1  SoG top box volume  V2  SoG bottom box volume  VH  HS box volume  W  Row-scaling diagonal matrix, m rows and m columns  W1  Downward vertical transport of the estuarine circulation  Winter Winter starts in November and ends in January, before the earliest spring blooms starts ˆ x  Estimated solution vector  xA  A priori solution vector of n rows and 1 column  xU  Vector of the estimated transports associated with the estuarine circulation  x  True solution vector of n rows and 1 column  x  ˆ Mean of x  z  Solution vector centered around xA xvii  Acknowledgments I wish to first thank professor Rich Pawlowicz for giving me the opportunity to do research in field oceanography in Beautiful British Columbia. I also wish to thank him for his longstanding academic and financial support during both the field sampling phase and the thesis writing phase of my PhD. I would also like to thank the professors who have been or are still on my supervisory committee: professors Susan Allen, Maite Maldonado, Doug Oldenburg, Philippe Tortell, and finally Grant Ingram. They provided guidance during the long process of shaping my research. Professors Allen and Maldonado also helped reviewing several versions of my thesis and provided very valuable insights. Finally, I would like to thank the people, too many to be named, who helped in the collection and analysis of the data used in my research.  xviii  xix  Chapter 1 Introduction Estuaries are very productive marine ecosystems. Marine organisms living in an estuary can either migrate in and out (e.g., salmon, large marine mammals) or reside there for most of their life cycle (phytoplankton and zooplankton). Phytoplankton and zooplankton are food sources for higher levels in the food web. An estuarine ecosystem represents an important asset for local economy and for recreational tourism. In an estuary, the economical and recreational exploitation of the ecological resources can greatly contribute to the dynamics of the ecosystem when the exploitation becomes excessive, e.g. sport fishing [Peterman and Steer 1981]. Monitoring the ecological condition of an estuary (e.g. biomass of phytoplankton and growth rate, nutrient concentrations and uptake rates as estimated in Chapter 5) is an important step in understanding the dynamics. Better understanding of the ecosystem dynamics can help develop a sustainable estuary in the long term. The motivation of this thesis is to study the recent conditions of the Strait of Georgia (SoG) and to provide estimates of the estuarine circulation of the SoG (chapter 4), the primary productivity of the SoG ecosystem, and its variations over seasonal and interannual timescales (chapters 5 and 6). In the rest of this section, I will introduce the concepts of estuary, estuarine circulation and box model. In the next sections, I will introduce the physics (section 1.1), the biology (section 1.2) of the SoG and the mathematical equations (section 1.3), and define the objectives, the approach and the significant contributions of the thesis 1  Chapter 1. Introduction (section 1.4). An estuary is “semi-enclosed body of water where freshwater and seawater masses mix or overlap forming vertical and horizontal gradients along their main channel” [Cameron and Pritchard 1963]. Real-life estuaries that fit this definition can greatly differ in geomorphology and hydrological regimes [Dyer 1973]. Studies typically focus on shallow, partly-mixed estuaries under a temperate climate, e.g. the Chesapeake Bay estuary [Harding et al. 2002]. Apart from partlymixed estuaries, the possible salinity structures of an estuary [Dyer 1973] can be highly-stratified salt-wedge estuaries (e.g. Fraser River mouth [Halverson and Pawlowicz 2008]), fjord-like estuaries (e.g. SoG [Thomson 1994, Pawlowicz et al. 2007]) characterized by a deep basin separated from the shelf by a sill near the mouth, and well-mixed estuaries (e.g. lower Delaware estuary [Moore et al. 2009], Gwangyang Bay [Shaha et al. 2010]). The literature contains descriptions of estuaries over a wide geographical range, in: arctic regions [Sorokin and Sorokin 1996], subtropical regions [Wua and Chou 2003], and subpolar regions [Kristiansen et al. 2001]. An estuary tends to get narrower landwards and quickly widen seawards. This geomorphological feature varies from estuary to estuary [Gay and O’Donnell 2009] and plays a role in the amplification of the seaward flow, as discuss later in this chapter. An estuary is a dynamic physical system. Horizontal and vertical transports can change the properties of the water column. Organisms can be transported vertically to shallow or deep waters, and horizontally into and out of the estuary. The transports can also alter factors necessary for growth and survival of organisms (nutrients, light, food abundance, etc.). Thus, the physics and the biology of an estuary are coupled processes. Since they vary over many time and space scales, these processes are difficult to sample, analyze and understand. 2  Chapter 1. Introduction The Strait of Georgia Ecosystem Monitoring (STRATOGEM) program was an example of an extensive study of the Strait of Georgia (SoG), an important estuarine system in the coastal waters of British Columbia [see Figs 1.1, 1.2, and Pawlowicz et al. 2007]. One of the goals of STRATOGEM was to understand the variability of estuarine circulation and primary productivity over the entire SoG using 3 years of data [Pawlowicz et al. 2007]. This thesis fulfils this goal of STRATOGEM by providing time series and averages of the mass transports and the primary production (PP) rates. Other important goals were to understand the physical factors that control the timing of the phytoplankton bloom [Collins et al. 2009], to estimate and analyze secondary production and the associated secondary productivity [Sastri and Dower 2009], and its interaction with primary production [El-Sabaawi et al. 2009, 2010]. An estuary is a complex and dynamic system, but it can be simplified for study. A basic approach to study an estuary is to reduce it into a two-layer system [MacCready and Geyer 2010] supplied with freshwater (FW) and seawater by external sources, and exchanging heat and gas (e.g. O2 and CO2 ) with the atmosphere. Such a simple approach, a so-called box model, can give one useful insights. A combination of boxes and equations can be used to fit a particular system: e.g., Strait of Georgia [Pawlowicz et al. 2007], Salish Sea (formerly known as Strait of Georgia/Haro Strait/Juan de Fuca system) [Li et al. 1999, Pawlowicz 2001], St. Lawrence system [Bugden 1981, Savenkoff et al. 2001], Chesapeake Bay [Hagy et al. 2000, Austin 2002]). This approach usually works well even though estuaries greatly differ in freshwater (river flow, evaporation/precipitation, ice sheet formation), tide and wind regimes all of which may affect stratification and mixing differently. This approach also works well even though estuaries differ in geomorphology (e.g. bathymetry, coastline and connection to the ocean). 3  Chapter 1. Introduction  Figure 1.1: Geography of the Strait of Juan de Fuca/Haro Strait/Strait of Georgia. Northern and Southern Entrances of the SoG are indicated: Johnstone Strait (JS) and Boundary Pass (BP), respectively. The dashed squares indicates the sampling area of the STRATOGEM and JEMS (Joint Effort to Monitor the Strait of Juan de Fuca) programs described in Chapter 3.  4  Chapter 1. Introduction Using a box model allows one to quantify aspects of the estuarine circulation and the ecosystem productivity. Transports and tracer fluxes in, out and within an estuary are basic parameters of interest as they help one to quantify whether freshwater, plankton and particles remain in, are exchanged within, or whether they leave the estuary [Thomson 1994]. The estuarine circulation will be introduced in further detail in section 1.1. The knowledge of the estuarine circulation allows one to make estimates of biogeochemical and biological quantities by accounting for advection.  1.1  Physical Oceanography  The oceanography of an estuary is always influenced by its geomorphology. The SoG is a deep basin isolated by sills from seawater sources. It is 220-km long and oriented in a northwest-southeast direction. Its width ranges between 25 and 55 km [ Thomson 1981, 1994 and Fig. 1.1]. On average, it is about 155 m deep, but its central and northern areas can reach 400 m and deeper (Fig. 1.3). Its horizontal area ranges from 3.5×109 m2 , at 155 m, to 7×109 m2 , at the surface (see hypsography, Chapter 3, Fig. 3.2). The total volume of the SoG is 1.1 ×1012 m3 (Chapter 2, Table 2.1). The main passage for seawater is through the Southern Entrance, mainly through Boundary Pass. The Southern Entrance represents 93% of the mass transport into the SoG versus 7% for the Northern Entrance [Thomson 1994]. Through Boundary Pass, the connection is restricted by a sill that is located near 160 m depth [Davenne and Masson 2001]. Thus, the SoG is like a fjord, isolated from open ocean sources. The Fraser River plume is one of the characteristic oceanographic features of the SoG. It is easily visible on satellite imagery during the peak of the Freshet when the load of sediments is the highest of the year [Stronach 1981]. The plume can carry freshwater into the SoG farther than the Fraser River mouth. The plume dynamics 5  Chapter 1. Introduction  Figure 1.2: Detailed Sampling Area of STRATOGEM and Other Important Sampling Locations. STRATOGEM water sampling stations are the 9 open diamonds (S1 to S5). The important locations other than the STRATOGEM stations are the triangles. The sampling at the STRATOGEM stations and at other important locations is described in Chapter 3.  are complex to study because, among other aspects, they can vary over different timescales: e.g. semidiurnal to annual timescales [Halverson and Pawlowicz 2008]. The surface of the SoG within and near the plume is fresher than the rest of the  6  Chapter 1. Introduction SoG. Average plume practical salinity ranges between 15 and 26 [Halverson and Pawlowicz 2008]. The plume can move across the SoG, and along the east coast of the SoG northward and southward, but tends to remain in the southern SoG. On average the rest of the SoG is strongly stratified in the upper 20 m (see section 3.3.1 and Pawlowicz et al. [2007]) because of freshwater input from the rivers. Below 20 m, the water column is weakly stratified because of mixing of surface and deep SoG water and entrainment of deep water. I will discuss the vertical structure of salinity and temperature in further detail later in this chapter and chapter 3. On large timescales ( 1 month), the overall circulation of the SoG is an estuarine circulation driven by freshwater (FW) input, entrainment and tidal mixing (Fig. 1.3). I will discuss the estuarine circulation and deep water advection in further detail in the next paragraphs. Note that previous studies suggest that the deep flow in the Central SoG is on average geostrophic and cyclonic: i.e northward flow on the eastern side of the SoG, and southward on the western side [Stacey et al. 1991, Marinone and Pond 1996]. In addition, the surface circulation can be affected by the wind on small timescales (≪1 month) [St. John et al. 1993, D′ Asaro and Dairiki 1997]. Winds are driven by the dominant pressure system (Aleutian low or Pacific high) on larger scales ( 1 month) [Thomson 1994, Marinone and Pond 1996]. The wind can drive the surface circulation by drift, advection and mixing [Thomson 1994]. The wind varies from overall strong winds blowing to the north and northwest in winter (due to the Aleutian low) to overall weaker winds blowing to the south and southeast in summer (due to the Pacific high). The wind is also affected by the local coastal topography that channels the wind [Thomson 1981]. The hydrology of the SoG is also an important factor in the estuarine circulation. Fig. 1.4 shows the Fraser River discharge over the 2000s. The river discharge has a very marked seasonal cycle. However, the magnitude of the freshet peak varies from 7  Chapter 1. Introduction  Figure 1.3: Along-channel Cross-section of the Strait of Juan de Fuca/Haro Strait/Strait of Georgia (SoJdF/HS/SoG) System. This is a schematic view of the SoJdF/HS/SoG system. The straight arrows indicate the surface and deep transports in the SoJdF and SoG. The round arrows indicate the vertical mixing in the SoJdF, HS and SoG. The arrow size indicates the relative magnitude of the mixing (see text). The light grey area shows the representative along-channel bathymetry with respect to the distance from the western end of the SoJdF. It also shows the STRATOGEM and JEMS (Joint Effort to Monitor the Strait of Juan de Fuca) sampling areas.  8  Chapter 1. Introduction year to year. In particular, the STRATOGEM sampling period (2002–2005), captured both the largest and the smallest flows of the 2000s: the 2002 freshet peak has the largest discharge, and the 2004 freshet peak has the smallest. The Fraser river is the largest single contributor to the FW discharge into the SoG, about 63% of the total FW inflow according to linear regression coefficients from the literature [Pawlowicz et al. 2007]. Usually the total discharge is of the order of 103 m3 s−1 in winter and 104 m3 s−1 in summer. However, in winter local large rainfalls could provide more freshwater than predicted by estimates based on the regressions. The freshwater mass from the Fraser River and other rivers is almost always mixed with salt within the riverbed before it enters the SoG itself because of the intrusion of salt wedges through the river mouth [Geyer and Farmer 1989, Halverson and Pawlowicz 2008]. The Fraser river can also supply heat and nutrients to the SoG, in particular nitrate and silicic acid, both essential to siliceous phytoplankton like diatoms. Fig. 1.3 shows a schematic view of the SoG general circulation. On a seasonal time scale, the SoG estuarine circulation is mainly forced by the FW inflow and the vertical mixing. Dense deep seawater enters the SoG from Haro Strait (HS) at Boundary Pass, mixes with deep SoG seawater, and sinks to intermediate depth (50– 200 m) or near the bottom (200–400 m) [Waldichuk 1957, LeBlond et al. 1991, Masson 2002, Pawlowicz et al. 2007]. The depth to which it sinks is controlled by the density of the oceanic seawater and the vertical turbulent mixing modulated by the neapspring cycle [LeBlond et al. 1991]. The magnitude of the vertical turbulent mixing in the Strait of Juan de Fuca (SoJdF), HS and SoG are qualitatively represented by the size of the curved arrows in Fig. 1.3. Compiled data on the vertical eddy diffusivity suggests that HS has the largest values while the SoG has the smallest [Li et al. 1999, Masson 2002]. The surface SoG water is a mixture of deep SoG water and freshwater from the rivers. 9  Chapter 1. Introduction Deep seawater entering the SoG originally comes from the Pacific Ocean. Seawater intrusions start entering into the SoJdF at depth of 100–200 m [Pawlowicz et al. 2007]. Seawater is transported through SoJdF with little change in its oceanic characteristics [Pawlowicz 2001, see Figs 5 a-d]. However, when Pacific seawater is upwelled or downwelled along the coast of Vancouver Island, the properties of the seawater intrusions in the SoJdF can change. Upwelling and downwelling are seasonal coastal processes with interannual variations. Thus, properties of the oceanic intrusions in the SoG, like temperature, salinity, dissolved oxygen (O2 ) and nutrients, can change from year to year [Masson 2002]. Previous studies have estimated the water transports at various locations inside and close to the SoG with various approaches. All these studies provide reasonably consistent average estimates of water transports. The approach used in this thesis enables one to estimate the seasonal changes of the water transports besides to estimate their average values. Godin et al. [1981]’s estimates were based on the integration of current measurements across two channels: SoJdF and Johnstone Strait (JS). England et al. [1996] estimated the transports indirectly by first estimating flushing time with different techniques of mixing box-model. Marinone and Pond [1996] used a sophisticated and complex 3D model of the SoG. Li et al. [1999] built a prognostic time-dependent box-model of SoJdF/HS/SoG system with parametrized mixing. Pawlowicz [2001] used inverse modelling and a box-model framework, and applied it to SoJdF/HS/SoG system. Pawlowicz et al. [2007] used a hierarchy of mixing box-models to determine both flushing times and transport magnitudes in the SoG. Using a box-model approach enables one to compare estuaries. Later in Chapters 4, 5 and 6, the SoG and other other estuaries (Chesapeake Bay and St. Lawrence) will be compared.  10  Chapter 1. Introduction 1  0.9  Fraser River discharge (× 104 m3 s−1)  0.8  0.7  0.6  0.5  0.4  0.3  0.2  0.1  0 Jan  Jul 2001 Jul  02  Jul  03  Jul  04  Jul  05  Jul  06  Jul  07  Jul  08  Jul  Figure 1.4: Fraser River Discharge. This is a time series of the Fraser River discharge measured in 104 m3 s−1 , at Hope, about 150 km, east of Vancouver. The grey-shaded time segment shows the STRATOGEM time frame.  1.2  Biological Oceanography  In the SoG, primary production (PP), the biomass of phytoplankton, is available as food for zooplankton. Phytoplankton is the lowest link in the SoG food web and thus, the survival of organisms higher in the food web may depend on PP. In the SoG, PP varies seasonally and peaks in spring [Harrison et al. 1983]. Although the 11  Chapter 1. Introduction spring bloom usually occurs in mid or late March, in 2005 it occurred in February. During later blooms, in summer and in fall, PP is usually smaller than during spring blooms. Note that hereafter we will define the spring season as the period which starts in February and finishes in April. The other seasons are defined accordingly (see Glossary, p. xiii). Summer and fall blooms produce less PP because of limitation due to nutrients (summer) and light (fall) [Harrison et al. 1983]. Episodic (spring and summer) grazing by zooplankton and sinking of, most likely, diatoms can reduce the biomass, thus it can balance a large rate of phytoplankton growth during favourable light and nutrient conditions. The spring maximum in PP coincides with an intense uptake of nutrients and eventually leads to the surface minimum of nutrients in summer [Masson 2006]. All phytoplankton species require dissolved nitrogen (e.g., nitrate and ammonium; nitrite when in large concentration) and phosphate for growth and cellular maintenance. However, diatoms, a dominant group of siliceous phytoplankton during SoG spring bloom, require silicic acid to build frustules, their outer shells. Other phytoplankton groups, like silicoflagellates (cosmopolitan marine phototrophic flagellates able to form an external siliceous skeleton [Henriksen et al. 1993]), also require silicic acid, and competition can happen when surface level runs low, for instance, after a large spring bloom. Deeper, in the aphotic zone, nutrients remain at about relatively constant level [Masson 2006] because of lack of PP. Through vertical tidal mixing and discrete wind-induced mixing events, deep nutrients are transported to the surface to replenish the surface nutrient pool [Yin et al. 1996, 1997]. Some studies suggest [Mackas and Harrison 1997] or assume [Li et al. 2000] that the nutrient supplies may be a limiting factor during intensive uptake events (spring and summer blooms). Previous studies have analyzed the composition of SoG phytoplankton communi12  Chapter 1. Introduction ties, and provided estimates of the SoG average PP, seasonality and average primary productivity (PP rate). These estimates are broadly consistent with each other, although they used different techniques of measurement or estimation that can lead to different definitions of the PP and the corresponding PP rate. I will discuss this matter in detail in Chapter 5. Harrison et al. [1983] reviewed the SoG biology including seasonal PP estimates and cycles. Mackas and Harrison [1997] provided a complete budget of new PP based on the sources of nitrate in the SoG. Li et al. [2000] used a biology-physics coupled box-model of the SoG to estimate PP and its variability assuming nutrient limitation in summer. Pawlowicz et al. [2007] estimated new PP. In the SoG, phytoplankton biomass is dominated by diatoms during spring blooms. The most abundant diatom species is the chain-forming Skeletonema costatum [Harrison et al. 1983]. Generally in spring and summer there is a succession of diatom species:Thalassiossira sp. are the most abundant diatoms in early spring, Skeletonema costatum blooms overlap with Thalassiossira sp. blooms or closely follow them, and finally blooms of Chaetoceros sp. and Nitzschia sp. occur in late spring and summer [Harrison et al. 1983]. Diatoms are known to have a resting stage [McQuoid and Hobson 1996]. Diatoms move into their resting stage when the availability of nutrients and light become unfavourable to growth. The resting cells of diatoms could provide the seed population for the following year blooms [Harrison et al. 1983, McQuoid and Hobson 1996]. Diatoms have to compete for nutrients and light with other phytoplankton groups. In particular, in spring and summer they compete for nutrients with silicoflagellates (e.g. Ebria tripartita), autotrophic flagellates (e.g. Gymnodinium sp.), and autotrophic ciliates (e.g. Myrionecta rubra, previously known as Mesodinium rubrum). Gymnodinium sp. are more abundant during summer, but can also bloom during spring [Harrison et al. 1983]. In the SoG, mesozooplankton graze on phytoplankton. Mesozooplankton include 13  Chapter 1. Introduction copepods, dominated by Neocalanus plumchrus, and euphausiids, dominated by Euphausia pacifica. Some copepods can shift their diet like Metridia pacifica, an omnivorous copepod species, while other zooplankton species, like Neocalanus plumchrus can not do this [El-Sabaawi et al. 2009, 2010]. Copepods and euphausiids coexist throughout the year because their abundance is strongly influenced by temperature: copepods are more abundant during spring and summer, euphausiids during winter [Harrison et al. 1983]. Neocalanus plumchrus overwinter in a dormant stage and dominate the deep zooplankton community during winter [Campbell et al. 2004]. All higher levels in the foodweb of the SoG are not well-known yet. However, links between mesozooplankton and their predators, e.g. salmon sp. [Garay and Soberanis 2008] and herring [Therriault et al. 2009], and between fish and their predators, e.g. seabirds, have been investigated [Therriault et al. 2009].  1.3  Two-layer Model and Governing Equations  In order to write a two-layer model of the SoG, it is necessary to make several assumptions about the circulation. First, many physical processes, e.g. wind events like storms, the flux of correlated variations due to turbulent mixing (see Eq. 2.57 and section 2.2 for further detail), diurnal, semi-diurnal and spring-neap tides are all lumped together as vertical transports in the box model and only their contribution over long timescales affect the vertical transports. This is a useful assumption since the STRATOGEM dataset generally contained monthly cruises only. Second, the observations of temperature and salinity, and the associated box averages used in the two-layer model are assumed to be in a mass, heat and salinity balance over long timescales. The conservation equations of three physical tracers (mass, heat and salinity) are written to estimate the SoG circulation which satisfies the dynamics of 14  Chapter 1. Introduction the SoG over long timescales: Mass conservation: − div(ρu)  =  ∂ρ ∂t  (1.1)  ∂ρCp T ∂t  (1.2)  ∂ρS ∂t  (1.3)  Heat conservation: − div (ρCp T u − K ∇(ρCp T ))=  Salinity conservation: − div (ρSu − K ∇(ρS))  =  where ∇() represents the gradient of the bracketed quantity and div() its divergence: ∇() = [  ∂ ∂ ∂ T (), (), ()] ∂x ∂y ∂z  (1.4)  div() =  ∂ ∂ ∂ () + () + () ∂x ∂y ∂z  (1.5)  where x, y and z represent displacements along the horizontal and vertical directions (denoted by x-, y- and z-direction), ρ the density, T the temperature, S the salinity, u the flow speed, Cp the water specific heat capacity, and K represents the diffusivity either due to molecular diffusion (KM ) or due to eddy diffusion (KE ). Note that Cp the specific heat capacity varies less than 5% over the range of temperature and salinity of seawater [IOC, SCOR and IAPSO 2010], so that it can be assumed to be constant in Eq. 1.2 and the equations derived from Eq. 1.2. Note that the Practical Salinity is assumed to be proportional to the Absolute Salinity, which is reasonable 15  Chapter 1. Introduction in estuarine waters [IOC, SCOR and IAPSO 2010]. Derivation of the box-model equations from Eqs 1.1–1.3 are detailed in Chapter 2. Using the two-layer paradigm, the SoG model is defined with two boxes and the conservation of the mass, heat, salinity (Eqs 1.1–1.3), nutrients and dissolved oxygen (collectively represented as q) are written in a two-layer system. These tracer equations can be written as: Vj  ∂qj = (uj )q + (φj )q + εj ∂t  (1.6)  where qj is the average of q over the box j (j =1, top box, and 2, bottom box), (uj )q the sum of all the advective transports of tracer q into and out of the box j, (φj )q the sum of the internal sources and sinks of tracer q in box j, and (Vj ∂qj /∂t) the time rate of change of the total amount of tracer q in box j. The rightmost term εj represents the estimated error due to measurement error, assumptions and neglected physical and biological processes. Storage occurs when the time derivative is positive. On the other hand, drawdown occurs when the time derivative is negative. Any net biological uptake of nutrients (e.g., nitrate, phosphate and silicic acid) and net biological production of dissolved O2 are represented by φ terms. The net uptake of nutrients represented by (φ1 )N for nitrate and (φ1 )P for phosphate will be used as an indirect estimate of the primary productivity. Their scaling into organic carbon currency using either an estimated Redfield ratio (see section 5.2.3) or the expected Redfield ratio [Redfield et al. 1963] will provide an estimate of the net primary productivity in various forms: total, new, and regenerated (defined in section 5.1).  16  Chapter 1. Introduction  1.4  Objectives, Approach, Thesis Contributions and Plan  The primary objective of this thesis is to quantify the estuarine circulation and the nutrient supplies and sinks over monthly to interannual time scales over 2002–2005. Then, I use this quantitative knowledge to estimate the nutrient uptake rates, and, by carefully scaling, net primary productivity (NPP rate). My approach is to infer the monthly estuarine transports using inverse modelling in a 2-box model. Then, one can infer the nutrient supplies and sinks using a forward box model in the same 2-box model. I now summarize the contributions of this thesis on the understanding of the SoG circulation and its primary productivity. In chapter 2, a formal mathematical framework and a time-dependent inverse two-box model of the SoG are constructed in order to infer the monthly variability of the SoG circulation. This is a more rigorous approach than it has been used previously in the literature. A careful analysis of the approximations made in the derivation of the budget equations of physical and biogeochemical tracers is carried out in sections 2.2 and 2.3. Chapter 3 contains a detailed description of the oceanography of the SoG as observed during STRATOGEM. In chapter 4, transport time series of the SoG estuarine circulation are estimated over three years with a monthly time resolution, for the first time. This inference is based on consistency of the transports with observations of salinity and FW input, temperature and surface heat fluxes. Using these estimated SoG transports, I provide an observational study (one of the few observational studies based on a rigorous mathematical framework) of the relationship between FW input (R) and surface seaward transport (U1 ) in the SoG and possibly  17  Chapter 1. Introduction for any estuary. The analysis of the transport time series provides some novel findings. It suggests that the seasonality of the total upward transport (W2 ) is very small, even when the seasonality of the riverine inflow (R) is large. Based on 2002–2005 data, the seasonality of surface seaward transport (U1 ) is weakly linked to (R), while the seasonality of other transports is not linked to R. The transport analysis is discussed in more detail in section 6.2. In chapter 5, I estimate the seasonal and annual average rates, with error estimation, of the net primary production (NPP rates) based on SoG nutrient budgets over three years. When the estimated NPP rates are compared with previous estimates in the SoG and in other estuaries, I show that the estimated NPP rates are reasonable and typical of NPP rates in a temperate estuary (comparison in section 6.3). The analysis of the nutrient budgets suggests that, as proposed first by Mackas and Harrison [1997] for N, the estuarine entrainment is the largest supply of N, P and Si. The comparison of the estimated NPP rates based on N and P budgets suggests that the average f-ratio is large (average f=0.88, see section 6.3). These results are discussed in more detail in section 6.3. Recommendations for future work are given in section 6.4.  18  Chapter 2 Inverse Methods and Box Models 2.1  Introduction: Mathematical Framework  The thesis objectives lead to two mathematical problems. First, I define a mathematical problem associated with the estuarine circulation of the matrix form: Ax = b.  (2.1)  In chapter 1, the equations of the conservation of the mass, heat and salt in the top and bottom boxes of the form of Eq. 1.6, can be written into Eq. 2.1, as will be done in section 2.2. I have a direct knowledge of salinity and temperature whose values are used in the definition of A and whose time-derivatives are used in the definition of b in Eq 2.1. I also have a direct knowledge of freshwater input and heat fluxes (also used in b). But I have no knowledge of the advective transports (x, unknown term) associated with the estuarine circulation. The problem is then to derive an estimate of the advective transports x from the observations A and b where A and b both have errors associated with them. The second mathematical problem is more straightforward mathematically, but involves more assumptions. It is associated with the primary productivity. It has the matrix form: AxU − b = x.  (2.2)  19  Chapter 2. Inverse Methods and Box Models In chapter 1, the equations of the conservation of the nutrients and dissolved O2 in the top and bottom boxes of the form of Eq. 1.6, can be written into Eq. 2.2, as it will be done in section 2.4. I have a direct knowledge of the concentration of nutrients and dissolved oxygen (represented by A in Eq. 2.2), the associated influxes, and the time rate of their concentration changes (b). I also have an estimation of the transports from the estuarine circulation problem (xU , the transports associated with the circulation which are assumed to be known in this second problem). From this knowledge, it is possible to quantify the sink terms (x, unknown term) which are assumed to be associated with primary productivity, and after further analysis estimate the new primary productivity associated with some of the nutrients. I can also qualify the net community productivity associated with the dissolved oxygen balance and analyze the variability of the estimated net biological uptake rate associated with the silicic-acid concentration. To be able to apply the appropriate methodology to solve these mathematical problems, I first need to define the notions of forward and inverse problems. These notions are connected to the notion of well-posed and ill-posed problems. According to Wunsch [1996] a well-posed problem is a mathematical problem that has a unique “well-behaved” solution. Note that “well-behaved” means that this solution is stable to perturbations on the boundaries, on the initial conditions and on the sources. By convention, such problems correspond to textbook cases or classic problems in mathematics and physics [Wunsch 1996]. An example of such a classic problem is Dirichlet’s problem of Laplace’s equation ∇2 φ=ρ where φ is the unknown field [Wunsch 1996]. Such well-posed problems are often called “forward” problems. However, many real-life problems are ill-posed. An ill-posed problem can have multiple, irregular solutions which may be sensitive to the data. An irregular solution can be for instance a fast-changing solution with at least a jump between two values. An ill20  Chapter 2. Inverse Methods and Box Models posed problem can also be created from a well-posed problem by interchanging the unknown quantities of the classic problem with some of the known quantities. These problems can often be described as inverse problems because they involve taking the inverse of a matrix. In this chapter and the following chapters, I will define the forward problem as the mathematical problem that consists of finding the unknown sink terms in the conservation equations by addition and subtraction of the advection terms. This problem is similar to Eq. 2.2. Its solution is unique by construction. In the particular case of the nutrients in the SoG, the sink term represents a net biological uptake rate or net sink. I will define an inverse problem as finding the unknown transports when no simple way is available to compute the solution by addition or subtraction of the known terms. This problem is similar to Eq. 2.1. In the case of the estuarine circulation in the SoG, these problems can be written more precisely in matrix form as: Ax = b + ε.  (2.3)  where A is a matrix of m rows and n columns (m-by-n) that contains known coefficients (including spatially-averaged temperature and salinity) of the mass, heat and salt equations, b is an m-by-1 vector of all internal and external sources, and x is an n-by-1 vector that contains the unknown advective transports. The net advective flux of mass, heat and salt is the vector Ax. In the particular case of a unique and exact solution of Eq. 2.3, the residuals ε must equal 0. Finding a unique and exact solution x is not always possible. This becomes a general rule with real-life problems. Since solutions are not exact in real-life problems, there are residuals ε(=0) in Eq. 2.3: ε = Ax − b.  (2.4) 21  Chapter 2. Inverse Methods and Box Models There is generally some uncertainty introduced in Eq. 2.3 by sampling errors in the observations because the sampling errors can have propagated to A and b. If any important process is missing from the model defined by Eq. 2.3 a misfit error is also added to the uncertainty ε. As it is not clear whether the solution x is stable to perturbations in the inverse problem (and whether the solution in the forward problem is stable by propagation of the errors of the transports) it is necessary to carry out an error estimation (detailed in section 2.3) and a sensitivity analysis (discussed in chapter 4) for any solution. The inverse problem can be solved by a Singular Value Decomposition (SVD) Methodology (SVDM) [Wunsch 2006]. The SVD of any m-by-n matrix A is a decomposition: A = UΛVT  (2.5)  The matrix Λ is a m-by-n diagonal matrix [Wunsch 2006]. The r non-zero coefficients on the diagonal are called the singular values (λi ) of the SVD. The singular values are arranged in decreasing order (λ1 > λ2 >...> λr ). The rank of A is also equal to r. By definition, the rank of a matrix is the maximum number of linearly independent columns (or equivalently rows) of the matrix. Thus, the rank r is bounded by: r ≤ min(m, n)  (2.6)  The rest of Λ is padded with zeros. The columns of matrices U and V form two sets of orthonormal vectors: m vectors ui and n vectors vi . The vectors ui and vi are called singular vectors of A. They can be used to decompose the elements of b and x, respectively. The SVDM is based on this generalized diagonalization (Eq. 2.5) of m-by-n matrices (square or non-square). The SVDM can be used to find a solution of any type of linear system of equations: just-determined (when m=n), overdetermined (when 22  Chapter 2. Inverse Methods and Box Models m>n) and underdetermined (when m<n). Unlike the Linear Least-Squares approach (LLS), the SVDM can work for a linear system with rank deficiency. Rank deficiency happens when r< min(m,n). With a rank-deficient system of linear equations, the LLS cannot distinguish between certain equations in Eq. 2.3 or elements of vector x [Wunsch 2006]. The SVDM helps to make appropriate choices. It still leads to a reasonable estimated solution x ˆ. The SVDM also provides additional tools to analyze the system and the solution. The inverse problem to be solved is formally overdetermined because 6 conservation equations will be used to determine the 4 unknown transports. With both LLS and SVDM, the “best” solution of this inverse problem, x ˆ is found by minimizing the norm of the residual ε. This norm is the objective function J defined by: J = εT ε.  (2.7)  The LLS solution, the so-called “pseudo-inverse”, x ˆ (in Eq. 2.3) exists if the system is full-rank. It minimizes Eq. 2.7. It is defined by: x ˆ = (AT A)−1 AT b  (2.8)  x ˆ = VΛ−1 UT b,  (2.9)  However, the SVD solution:  which can be rewritten as  r  x ˆ= i=1  (uT i b) vi , λi  (2.10)  exists even if the system is rank-deficient. The matrix Λ−1 is the inverse of Λ defined as a n-by-m diagonal matrix with λ−1 on the diagonal. The rest of the matrix is i padded with zeros. Eq. 2.8 (when it exists) and Eq. 2.9 are equivalent and this follows from the decomposition of A in Eq. 2.5. With the estimated solution x ˆ the 23  Chapter 2. Inverse Methods and Box Models ˆ is: corresponding residual ε m  (uT i b)ui  εˆ = −  (2.11)  i=r+1  which is the part of b unresolved by the range vectors and is spanned by the nullspace vectors ui (i=r+1...n). The range vectors are the vectors ui and vi (i = 1, ..., r) that satisfy: AT ui = λi vi and Avi = λi ui  (2.12)  while any nullspace vectors satisfy: AT ui = 0 and Avi = 0.  (2.13)  To separate the range vectors from the nullspace vectors, it is sometimes useful to define the m-by-r matrix Ur and the n-by-r matrix Vr , the matrices formed by the r first singular vectors ui and vi . The range vectors ui and vi are associated with the singular value λi . By construction, any vector ui or vi that is not a range vector has to be a nullspace vector. Note that in Eq. 2.11 if the system is full-rank and m < n, ˆ = 0. The true solution x is spanned by the range vectors vi and can be written as: ε r  x= i=1  n  (uT i b) vi + γi vi λi i=r+1  (2.14)  where the second right-hand term is the part of x unresolved by the range vectors vi (i=1, ..., r). For this reason γi ’s for j=r+1, ..., n are unknown because the remaining vi (i=r+1, ..., n) are nullspace vectors (i.e. Avi =0). When an objective function J, more sophisticated than Eq. 2.7, is chosen to improve the solution x ˆ, the SVD enables one to understand the effects of such a function [Wunsch 2006]. For instance, J = ε T ε + α 2 zT z  (2.15) 24  Chapter 2. Inverse Methods and Box Models where the solution z is the solution x centered around an a priori average xA so that: z = x − xA and c = b − AxA .  (2.16)  The coefficient α2 is a coefficient that quantifies the compromise made between minimizing the residuals ε and being close to an a priori estimate xA . The corresponding LLS solution is: ˆ z = (AT A + α2 In )−1 AT c  (2.17)  which can be written using the SVD as ˆ z = V(ΛT Λ + α2 In )−1 ΛT UT c.  (2.18)  The SVD solution in turn can be rewritten as r  ˆ z= i=1  λi (uT i c) vi . 2 λi + α 2  (2.19)  By comparing Eq. 2.10 and Eq. 2.19 (setting xA = 0 without loss of generality), it clearly appears that the scaling has “tapered” the coefficients of vi , and thus the norm of the solution x ˆ. The term tapering refers to the weighting down of the coefficients of the vector vi by α2 [Wunsch 2006]. The coefficients of the vector vi (in Eq. 2.19) are different from the values they would have in the simple SVD form (Eq. 2.10). This ensures that there is always an inverse solution even if some of the λi ’s ≪ α (in this case λi 2 + α2 ≃ α2 .). However, this procedure also adds a bias to the equation residuals ε. A bias is a difference between the estimated value and the ˆ is: expected value of any quantity. Using the SVD, the new residual ε r  m  (uT i c)ui  ˆ=− ε i=r+1  − i=1  α2 (ui T c) ui . λi 2 + α 2  (2.20)  25  Chapter 2. Inverse Methods and Box Models The true solution z(= x − xA ) is spanned by the range vectors vi and can be written as  r  z= i=1  n  λi (uT i c) vi + (viT z)vi (2.21) 2 λi + α 2 i=r+1  where the second right-hand term is the part of x unresolved by the range vectors vi (j=1...r). When I applied Eq. 2.10 to get the “pseudo-inverse” solution of the system Eq. 2.3, the inverse procedure gave large month-to-month uncertainty. Instead, it was more useful to apply Eq. 2.19 to get the so-called “optimal” inverse solution x ˆ. The inverse optimal solution x ˆ minimizes an objective function J: J = εT W−2 ε + zT S−2 z.  (2.22)  where W and S are weighting matrices [Wunsch 2006]. This generalization of Eq. 2.15 represents a compromise between the observations and the a priori knowledge. The inverse optimal solution of Eq. 2.22, x ˆ is of the form: x ˆ = xA + (AT W−2 A + S−2 )−1 (AT W−2 )(b − AxA )  (2.23)  where z and c have been expanded using Eqs 2.16, and using the SVD: T  T  T  x ˆ = xA + V′ (Λ′ Λ′ + S−2 )−1 (Λ′ U′ W−1 )(b − AxA )  (2.24)  where Λ′ , U′ and V′ are the matrices of the singular values, data and solution singular vectors when the SVD is applied on the scaled matrix A′ (=W−1A). This inverse procedure has been devised to improve and build on the “pseudoinverse” procedure. It tapers the coefficients of vi when there are very small singular values (associated with unstable solutions) of the matrix A. It also constrains the optimal solution to keep values closer to the positive a priori solution xA . In Eq. 2.22, 26  Chapter 2. Inverse Methods and Box Models several new matrices and vector have to be defined: the matrices W and S (6-by6 matrix and 4-by-4 matrix, respectively) and the a priori solution vector xA . The matrices W and S are applied to scale the system equations and the solution, respectively. W normalizes the conservation equations so that all conservation equations have the same weight in the solution (so-called row-scaling) and it has the form:         W=        ω1  0  0  0  0  0  0  ω2  0  0  0  0  0  0  ω3  0  0  0  0  0  0  ω4  0  0  0  0  0  0  ω5  0  0  0  0  0  0  ω6                 (2.25)  In Eq. 2.22, the matrix S normalizes the solution x ˆ so that the scaled version of x ˆ has components with a magnitude of O(1) (so-called column-scaling). S is a square diagonal matrix of identical non-zero coefficients:   s 0    0 s S=   0 0  0 0  0 0      0 0    s 0   0 s  (2.26)  The actual values of the inversion parameters (s, ωi and xA ) are shown later in this chapter (see Table 2.2). The solution (data) range vectors can be used to resolve the solution (data). Their resolution capacity can be measured by the solution (data) resolution matrix 27  Chapter 2. Inverse Methods and Box Models Tv (Tu ). In the case of the simple SVD, these matrices are defined by Tv = Vr VrT  (2.27)  Tu = Ur UT r  (2.28)  and  For instance, one can use the matrix Tv (Tu ) to find the relationship between the ˆ with respect to the true solution x (b): ˆ (data vector b) estimated solution x ˆ = Vr VrT x x  (2.29)  ˆ = Ur UT b b r  (2.30)  and  ˆ = b) is A full rank: r=n (or r=m). In the case of ˆ = x (or b and the condition for x the tapered SVD of the row- and column-scaled matrix A′ , the resolution matrices are defined by: Tv = SVr′ D′r (α2 )Vr′ T S−1  (2.31)  Tu = WU′r D′r (α2 )U′r T W−1  (2.32)  and  where the matrix D′ r (α2 ) is a r-by-r diagonal matrix which non-zero coefficients d′i (α2 ) are of the form: d′ i (α2 ) =  λ′i 2 λ′i 2 + α2  (2.33)  The matrices Vr′ and U′r are the matrices V′ and U′ with r columns. For α = 0, D′r (α2 ) = Ir and Eqs 2.31–2.32 turn back into Eqs 2.27–2.28. Note that when λi ≪ α, d′i (α2 ) ≃0. If the matrices Tv and Tu are different from the identity matrices, some of the r solution (data) range vectors cannot help to completely resolve the solution 28  Chapter 2. Inverse Methods and Box Models (the data). They can be considered as nullspace vectors because they are associated with such small λi ’s that they cannot be useful as range vectors. When the data resolution matrix is not the identity matrix, one could reason that there is not enough information to distinguish some of the equations from each other. An alternative explanation could be that some of the equations have more weight than the others despite the a priori weights ωi ’s. This is similar to a situation where some of the equations are linearly dependent. The data resolution matrix can help build a “data ranking” assessment and determine which observations are the most important [Wunsch 2006]. The diagonal coefficients help to compare the relative weight of each equation. The non-diagonal coefficients show the strength of the linear dependence between each pair of equations.  2.2  Inverse Problem: Estimating the SoG Circulation  The introduction (chapter 1) suggested that the dynamics of the SoG system set up an estuarine circulation. The SoG estuarine circulation allows one to idealize the SoG as a two-box system. Table 2.1 gives the volume and depth of each SoG box. It is also necessary to define the other systems which communicate with the SoG. In this SoG idealization, the only connection to the open ocean is through the Southern Entrance. Haro Strait (HS) is directly connected to the SoG by the Southern Entrance and it is the most important neighbouring system (as shown in chapter 1). Because of strong vertical mixing in HS, the water column is usually quite uniform relative to the SoG, although this does not mean it is perfectly uniform. Table 2.1 also defines HS  29  Chapter 2. Inverse Methods and Box Models Total Depth (m)  Volume ×1011 (m3 )  Upper SoG (V1 )  30  1.9  Lower SoG (V2 )  370  9.1  Haro Strait (VH )  200  1.6  Domain Name  Table 2.1: Total Depths and Volumes of the Model Boxes. Note that the SoG sea surface is about ∼7×109 m2 volume and depth and illustrates why it is represented by a smaller box than the SoG. Fig. 2.1 shows the different processes and fluxes in, out and between the SoG boxes and the different tracers affected by them (Ti , Si , with i=1 representing the surface box, 2 the deep box, H the HS box, R the river). At this point in the description of the box model, it is important to define the separation depth (d) between the SoG boxes. I define d as the depth above which the flow in the SoG is seaward and below which the flow in the SoG is landward. The surface outflow carries freshwater from the rivers and seawater from the SoG, while the deep inflow brings dense seawater from HS. Analyzing the vertical profiles of salinity and temperature in the SoG can help locating these two types of water masses and determining their depth ranges. In section 3.3.1, I will explain how the separation depth d is chosen. Later in chapter 4, I will set d to 30 m. The two-box model is a useful paradigm to study an estuary. This idealization of the SoG into two boxes will enable me to study both the physics (Fig. 2.1) and the biology (Fig. 2.2) in the same domains and carry the estimates of the circulation to the advective transports of nutrients and dissolved O2 . It will also help when it is necessary to make comparison with other systems. Previous studies have used this approach successfully [Li et al. 1999, 2000, Pawlowicz 2001, Johannessen et al. 2003,  30  Chapter 2. Inverse Methods and Box Models  Figure 2.1: Physical Fluxes and Processes in the Box Model. The left-hand boxes represent the SoG, and the right-hand box the Haro Strait. The thick arrows represent the transports in and out the SoG (U1 , U2 , W1 and W2 ). In the model, the surface SoG water enters the surface of HS (U1 ) while the deep HS water enters the deep SoG (U2 ). The arrow thickness approximates the relative magnitude of transports. The upper left arrow represents the freshwater inflow (R). The thick wavy arrow represents the turbulent and radiative heat fluxes (F ). FSW is the shortwave component of F (thin wavy arrow) that penetrates deeper into the SoG than the longwave component.  31  Chapter 2. Inverse Methods and Box Models  Figure 2.2: Biogeochemical fluxes and processes in the Box Model. The same convention for the boxes and the arrows in Fig. 2.1 apply to this figure. The sink terms are denoted by φ1 and φ2 (top and bottom, respectively) and are inside small boxes with inward arrows. Air-sea exchange fluxes are represented by a double wavy arrow.  32  Chapter 2. Inverse Methods and Box Models Pawlowicz et al. 2007] to investigate various aspects of the SoG oceanography. In a two-layered system, a classical approach consists in applying Knudsen’s hydrographic theorem [Dyer 1973]. However, there are several issues when one studies a real system which is necessarily more complex than the simplified estuary used in Knudsen’s theorem. These issues involve the use of multiple tracers, and the degree to which a quasi-steady approximation is valid. Knudsen’s approach can only combine two sets of equations to close the problem: mass and salt conservation equations. The salt conservation equation is the reasonable choice of tracer to estimate the estuarine circulation. However, inferring the water transports based on only the salinity can be an ill-posed problem if there is any inaccuracy or inconsistency in the data. Inaccuracy and inconsistency can lead, in the salt equations, to coefficients indistinguishable from zero within the error bars [Smith and Hollibaugh 1997, Dale et al. 2004]. It can also lead to inaccurate advective transports for other tracers (in particular biogeochemical tracers) that are important in multidisciplinary studies. Using a formal inverse approach enables one to add other tracers, for instance temperature, and improves the reliability of the estimated transports [Roson et al. 1997, Pawlowicz and Farmer 1998, Pawlowicz 2001]. One more tracer adds two equations to solve: one for the surface box, and another for the bottom box. The formal inverse approach enables one to handle a system with more equations than unknowns. Time dependence may be an important element of the SoG modelling, especially at the surface where changes can have a short timescale in spring and fall. To compute the time derivatives, the numerical scheme described later in section 3.3.3 is used. The salinity can change quickly during the early freshet (spring). Heat and salt can be stored in the SoG and released later. Thus, the salt and heat inflows and outflows  33  Chapter 2. Inverse Methods and Box Models need not to be balanced all the time. On the other hand, the time rate of volume change can be shown to be negligible over periods of a couple of months. For instance, Godin et al. [1981] estimated a net inward transport of 350 m3 s−1 based on longterm sea-level change (as high as 19 cm) measured from tide gauges during April to June 1973. Assuming that this time rate of change remains reasonable for the smaller STRATOGEM study area and different analysis period, this net inward transport can be considered to be negligible because it is smaller than the largest surface mass inflow (greater than 1165 m3 s−1 according to a linear regression from the literature, section 3.2.3), the freshwater discharge. The net inward transport is smaller than the freshwater discharge by one order of magnitude on average, and by two orders of magnitude in summer. Note that there could be also a net barotropic transport around the Vancouver Island (e.g. SoG inflow at Johnstone Strait and outflow at Juan de Fuca Strait), but sea level observations cannot be used to study this. This is because an inflow and an outflow of the same magnitude produce no change in sea level. The inverse problem in Eq. 2.3 is a linear form of the differential equations of conservation of mass, heat or salt, which were introduced in Chapter 1 as Eqs 1.1–1.3. There is no internal source of heat or salt. The sources of freshwater, seawater and heat have been identified only at the boundaries of the SoG, for this reason they will be applied later as boundary conditions. Eqs 1.1–1.3 can be simplified by assuming that the water is incompressible. Thus, Eqs 1.1–1.3 can be rewritten in the  34  Chapter 2. Inverse Methods and Box Models form: Continuity equation: − div(u)  = 0  (2.34)  Heat conservation: − div (T u − K ∇T ) =  ∂T ∂t  (2.35)  ∂S ∂t  (2.36)  Salt conservation: − div (Su − K ∇S) =  where ∇() represents the gradient of the bracketed quantity and div() its divergence: ∇() = [  ∂ ∂ T ∂ (), (), ()] ∂x ∂y ∂z  (2.37)  div() =  ∂ ∂ ∂ () + () + () ∂x ∂y ∂z  (2.38)  where x, y and z represent displacements along the horizontal and vertical directions (denoted by x-, y- and z-direction), ρ the density, T the in-on-site temperature, S the salinity (measured as practical salinity), u the flow speed, and K represents the diffusivity either due to molecular diffusion (KM ) or due to eddy diffusion (KE ). When discussing the salt and heat budget equations in this section, we will show that KM ≪ KE and thus K ≃ KE . The diffusivity K represents diffusion in all directions and it depends on the gradient of T (or S). Note that Cp the specific heat capacity varies less than 5% over the range of temperature and salinity of seawater [IOC, 35  Chapter 2. Inverse Methods and Box Models SCOR and IAPSO 2010], so that it can be assumed to be constant in Eq. 2.35 and the equations derived from Eq. 2.35. The quantities from Eqs 2.34–2.36 are now integrated over the SoG boxes. The continuity equation (Eq. 2.34) can be integrated over the top and bottom boxes of volumes V1 and V2 , respectively (see Table 2.1) to obtain the first two equations of the inverse problem. According to the divergence theorem, if the surrounding boundaries of the boxes are called a1 and a2 , the integrals of the divergence over the volumes V1 and V2 in Eq. 2.34 can be replaced by the fluxes through the surfaces a1 and a2 . Continuity equation: u · da1  =  0  (2.39)  u · da2  =  0  (2.40)  a1  a2  The inner product of the vector u and dai is represented by a dot inside the integrals. The vectors da1 and da2 represent elements of the surfaces a1 and a2 , respectively, and are oriented outward from the volumes. This orientation is chosen so that outward flow is positive. Noting that at the surface there is a river inflow R and at depth water intrusions U2 , a negligible inflow at the northern connection (see chapter 1), and no flow through the bottom and through the walls of the SoG boxes, Eqs 2.39–2.40  36  Chapter 2. Inverse Methods and Box Models become: Mass budget: − U1 − W1 + W2 = − R + ε1  U2 + W1 − W2 =  (2.41)  ε2  (2.42)  The residuals ε1 and ε2 in Eqs 2.41–2.42 represent the estimation errors on u. The transports in the top and bottom boxes U1 , U2 , W1 and W2 in Eqs 2.41–2.42 can be closely represented by integrals: u · da1 = U1  (2.43)  u · da2 = − U2  (2.44)  u · da1 = W1 − W2  (2.45)  surface outward transport: SC1  deep inward transport  : SC2  vertical transport  : SB  =−  u · da2 SB  where SC1 and SC2 defined the cross-section of the southern connection (see chapter 1) in the top and bottom boxes while SB is the separation boundary (at depth d) between the top and bottom boxes. Although the vertical advective transport (Eq. 2.45) is on average over the year a net upward transport, its magnitude or direction can change seasonally and over the pycnocline. Eq. 2.45 summarizes these cases by introducing the average upward and downward advectives transports W2 and W1 . W1 =  u · da1  (2.46)  SB ′  37  Chapter 2. Inverse Methods and Box Models W1 is the downward transport and SB ′ represents the locations of SB where the speed is outward (with respect to the boundaries of the top box), u · da1 >0 and W2 = −  u · da1  (2.47)  SB ′′  W2 is the upward transport and SB ′′ represents the locations of SB where the speed is inward, u · da1 <0. This decomposition of the vertical transport is important later when the vertical advective transports of heat and salt have to be estimated. Then, the next step is to obtain the heat and salt budget equations by estimating the advective transports of heat and salt, and integrating the conservation equations of heat and salt, Eqs 2.35–2.36, over the top and bottom boxes. The molecular diffusion (diffusivity KM ) for heat and salt can be neglected in Eqs 2.35–2.36 after scaling it against eddy diffusion (diffusivity KE ) following Pond and Pickard [1978]. A typical molecular diffusivity for heat and salt is 10−7 and 10−9m2 s −1 , respectively [Pond and Pickard 1978]. The eddy diffusivity depends on the properties of the mean flow and it has the scale of a speed times a length: − q′ v′ = KE  ∂q ∂y  (2.48)  where q′ and v′ represent fluctuations about quantity q and horizontal speed v along y-direction, respectively. The overlined quantities q′ v′ and q are time-averages and represent fluxes of correlated variations over a timescale shorter than the circulation timescale. So, to scale the eddy diffusion, it is necessary to estimate the speed associated with the SoG circulation and the characteristic lengthscales. The characteristic horizontal and vertical lengthscales in the SoG are about 104 and 102 m, respectively. The order of magnitude of the horizontal speed ranges from 0.1 to 1 m s−1 [Thomson 1994, 1981]. Based on these scales and the continuity equation (Eq. 2.34), the scale of the vertical speed is at least two orders of magnitude 38  Chapter 2. Inverse Methods and Box Models smaller than the scale of the horizontal speed. These numbers lead to horizontal and vertical eddy diffusivities of 0.1–1×104 and 0.1–1 m2 s−1 , respectively. These values are consistent with the maxima given by Pond and Pickard [1978], 105 and 10−1 m2 s−1 , respectively, suggesting weaker horizontal mixing in the SoG and vertical mixing of similar magnitude in the SoG. However, these values are large enough to prevail over molecular diffusivity. The flow speed and the tracer quantity (q represents either T or S) are written in the form of a mean and fluctuations about that mean at any location in the SoG: u = u + u′  (2.49)  q = q + q′  (2.50)  where the mean is defined over a timescale characteristic of the circulation timescale. Thus, the advective fluxes in the conservation equations represent estuarine transports. Since the molecular diffusion is neglected, K ≃ KE . Using Eqs 2.48, 2.49 and 2.50, the conservation of the quantity q (either T or S) can be written: − div(q u + q′ u′ ) =  ∂q +e ∂t  (2.51)  where e is the error due to the assumption on the molecular diffusion and the simplification by the terms qu′ , q′ u, q′ u′ , and  ∂q′ ∂t  over the circulation timescale. The term  q′ u′ represents any flux of correlated variations. Similarly to obtain the equations of the mass budget, Eqs 2.41–2.42 are integrated over the volumes V1 and V2 and yields the advective transports using the divergence theorem: (q u + q′ u′ ) · da1 =  −  V1  a1  (q u + q′ u′ ) · da2 =  − a2  V2  ∂q dV1 + e1 ∂t ∂q dV2 + e2 ∂t  (2.52) (2.53)  39  Chapter 2. Inverse Methods and Box Models The time-derivative integrals  1 Vi  Vi  ∂q/∂t dVi are exactly equal to the time deriva-  tive of the box averages q1 (corresponding to T1 or S1 ) and q2 (T2 or S2 ) of the quantity q since the volumes V1 and V2 do not change significantly with time. (q u + q′ u′ ) · da1 =  − a1  (q u + q′ u′ ) · da2 =  − a2  ∂q1 + e1 ∂t ∂q2 + e2 V2 ∂t V1  (2.54) (2.55)  The means u and q can be further approximated. If the spatial variations of the speed are neglected, the speed mean can be replaced by the mean over the surface, ua . Thus, the surface integral of the mean speed leads to one of the transports defined by Eqs 2.43–2.45 and 2.46–2.47, depending on the surface considered (U1 , U2 , W1 or W2 ). The tracer mean over the surface, q can be replaced by the tracer mean over the volume qv , further simplifying the form of the conservation equations. Once these approximations are made, the integral advective flux becomes: (q u + q′ u′ ) · da = a  a  qv ua · da +  q′ u′ · da + e  (2.56)  a  where e takes into account the abovementioned approximations on q and on u. In Eq. 2.56, since only correlated fluctuations of q′ and u′ will contribute to the second right-hand integral (  a  q′ u′ · da), this integral is likely to describe the turbulent fluxes  due to entrainment and mixing exchange. However, the timescale of these fluctuations are smaller than the changes that the observations can resolve. Thus, this term is unknown in this problem although only its monthly average would be required. If the fluctuations occur on a larger timescale than expected, they will contribute to the error term e. If this surface integral is neglected Eq. 2.56 becomes: (q u + q′ u′ ) · da = a  qv ua · da + equ  (2.57)  a  40  Chapter 2. Inverse Methods and Box Models where equ is the total error due to the approximations on qu made up to this point. The surface integral is applied to the model boxes and simplified by inspecting the transports through the surface, the bottom, the separation boundary, the northern and southern connections, and the walls: (q u + q′ u′ ) · da1 = q1 U1 + eSC1  (2.58)  (q u + q′ u′ ) · da2 = − qH U2 + eSC2  (2.59)  (q u + q′ u′ ) · da1 = q1 W1 − q2 W2 + eSB  (2.60)  SC1  SC2  SB  =−  (q u + q′ u′ ) · da2 SB  (2.61) where qH is the average in HS box, the other advective and turbulent fluxes over the surface, the bottom, the walls are zero and the fluxes through the northern connection are assumed to be negligible compared to the ones through the southern connection. Similarly to the continuity equations, the equations of the advective and turbulent transports (Eqs 2.58–2.60) and conservation equations (Eqs 2.54–2.55) lead to the  41  Chapter 2. Inverse Methods and Box Models budget equations of heat and salt in the top and bottom boxes: Heat budget: − T1 U1 − T1 W1 + T2 W2 =  V1  ∂T1 − TR R ∂t  (2.62)  d  a FSW ρ0 Cp ∂T2 TH U2 + T1 W1 − T2 W2 = V2 ∂t a FSW − ρ0 Cp  k e−kz dz −  −  0  a (F − FSW ) + ε3 ρ0 Cp (2.63)  ∞  k e−kz dz + ε4 d  Salt budget: − S1 U1 − S1 W1 + S2 W2 = SH U2 + S1 W1 − S2 W2 =  ∂S1 + ε5 ∂t ∂S2 + ε6 V2 ∂t V1  (2.64) (2.65)  The quantities Ti , Si , Ui , Wi , εj , R, F , FSW and k variables are time-dependent scalars (where  i  is defined as in Fig. 2.1,  j  as in Eqs 2.41–2.42 and 2.63–2.65).  The quantities εj , R, F , FSW and k represent the equation residuals (as defined in the introduction), the freshwater inflow, the net surface heat flux and its shortwave (hereafter SW) component and a light attenuation coefficient (in m−1 ). In Eqs 2.62 and 2.63, F and FSW fluxes are converted into ◦ C m3 s−1 by the factor a/(ρ0 Cp ) where a is the SoG surface area (Table 2.1), ρ0 a reference density and Cp the water specific heat capacity. I assume a single-band light attenuation of FSW , the SW component of F, or “blue” component [Kara et al. 2005]. It penetrates deeper than the longwave component and some fraction can potentially enter the lower box. It decays according to k, the light attenuation (over the characteristic distance k−1 , 42  Chapter 2. Inverse Methods and Box Models average 3.6 m), based on an estimated 1% photosynthetically available radiation (PAR) level in the SoG (at an average depth of 15 m), using STRATOGEM data. I will refer to Eqs 2.41–2.42 and 2.62–2.65 as the “flux form” of the conservation equations by opposition to the following “tracer difference form” equations: Mass budget (unchanged): − U1 − W1 + W2 U2 + W1 − W2  = − R + ε1  (2.66)  =  ε2  (2.67)  =  V1  Heat budget: (T2 − T1 )W2  ∂T1 + (T1 − TR )R ∂t  a − FSW ρ0 Cp ∂T2 (TH − T1 )U2 + (T1 − T2 )W2 = V2 ∂t a FSW − ρ0 Cp  d  k e−kz dz − 0  (2.68) a (F − FSW ) + ε3 ρ0 Cp (2.69)  ∞  k e−kz dz + ε4 d  Salt budget: (S2 − S1 )W2  =  (SH − S1 )U2 + (S1 − S2 )W2 =  ∂S1 + S1 R + ε 5 ∂t ∂S2 V2 + ε6 ∂t  V1  (2.70) (2.71)  There is an advantage to using Eqs 2.66–2.71 instead of Eqs 2.41–2.42 and 2.62– 2.65. A numerical issue that becomes relevant when inversion is attempted is that the advective terms (lefthand side of Eqs 2.62–2.65) are usually significantly larger than the other terms (righthand terms). Thus, the advective terms can dominate 43  Chapter 2. Inverse Methods and Box Models these smaller terms, during the inversion procedure. The inversion then tends to show only mass conservation which is a weak constraint on the circulation. However, the differences of the advective terms, in Eqs 2.66–2.71, are of the same order of magnitude as the other sink terms and the forcing terms. This makes the inversion procedure more likely to produce tracer conservation, as well as mass conservation. The “tracer difference form” is obtained by combining Eqs 2.62–2.65 with Eqs 2.41– 2.42. Adding Eqs 2.41–2.42 to Eqs 2.62–2.65 does not change the formal validity of Eqs 2.62–2.65. In Eqs 2.68–2.71, I chose to express all the conservation equations in terms of U2 and W2 , while W1 and U1 only appear in Eqs 2.41–2.42. The terms U1 and W1 mathematically result from the addition of U2 and R, the freshwater flow in the case of U1 , and from the difference between vertical upwelling W2 and deep landward flow U2 in the case of W1 . The “tracer difference form” of the conservation equations shows that the heat content and the salt content of the SoG boxes are controlled by forcings, sources, vertical upwelling and water intrusions. Upward and downward transports (W2 and W1 ) are convenient mathematical parametrizations of the circulation. But, in physical terms, it is more appropriate (and efficient) to separate entrainment processes associated with a net unidirectional mass flux from turbulent mixing processes, in which no net mass flux occurs but a tracer flux does occur. The parameter E (=W2 -W1 ) can be defined as the entrainment while M (± W1 ) as the rate of turbulent mixing (mixing exchange for short). On average W2 > W1 in positive estuaries since upwelling should occur more often than downwelling. In this case, M vanishes completely from Eqs 2.41–2.42, which is consistent with the properties of turbulent mixing. Thus, the quantities E and M enable one to interpret the mathematical transports U1 , U2 , W1 , W2 in physical terms. The next step of the conservation equations consists of writing these constraints 44  Chapter 2. Inverse Methods and Box Models in a matrix equation of the form of Eq. 2.3. This form of the conservation equations will be convenient for applying the LLS or the SVDM. The matrix A and the vectors x and b are defined as         A=        −1 0  −1 1  0  1  1  0  0  0  0  TH − T1 0  0  0  0  SH − S1 0  0       −1   T2 − T1    T1 − T2    S2 − S1   S1 − S2 (2.72)  x = U1 ,  U2 ,    W1 , −R  W2 ]T     0   ∂T1  V1 ∂t + (T1 − TR ) R − ρ0aCp (H1 FSW + F − FSW ) b=  2  − ρ0aCp H2 FSW V2 ∂T ∂t   1  V1 ∂S + S1 R ∂t  2 V2 ∂S ∂t  (2.73)               (2.74)  where H1 (=0.99 for d=30 m) and H2 (=0.01 for d=30 m) are the integral coefficients appearing in Eqs 2.68–2.69, respectively. Table 2.2 shows the actual values of the inversion parameters (s, ωi and xA ). Horizontal and vertical transports in the vector x ˆ were scaled by s, 5×104 m3 s−1 , as this is known to be a reasonable scale for magnitude [Godin et al. 1981, Li et al. 1999, Pawlowicz et al. 2001, Masson and Cummins 2004, Pawlowicz et al. 2007]. The scaling 45  Chapter 2. Inverse Methods and Box Models Parameter  W row-scaling  S Column-  xA a Priori Transports  scaling Diagonal/Vector  ω1 , ω2  ω3 , ω4  ω5 , ω6  s  U01 , U02 , W01 , W02  1.6, 1  2.6, 1.5  4.7, 0.4  5  ×104  ×105  ×105  ×104  ×104  C m3 s−1  psu m3 s−1  m3 s−1  m3 s−1  Coefficients Scale  Unit  m3 s−1  ◦  4.5,  4,  2.1,  Table 2.2: Inversion Parameters coefficients ωi are obtained by estimating the residuals of the conservation equations assuming that the magnitude of the transports were about 5×104 m3 s−1 , and the sources and forcings at their absolute maximum value. The a priori estimates of xA elements was taken as the approximate average of the “pseudo-inverse” (Eq. 2.9) for the whole time series, but is also consistent with previous estimates (see chapter 1 and Table 4.3). The xA components reflect the pattern of the estuarine circulation in the SoG where vertical turbulent mixing M(= W01 ) is usually smaller than the other transports. The entrainment E(= W02 − W01 ) and the landward transport U02 are close and the seaward transport U01 is greater because of the contribution of the river inflow R (maximum at about 104 m3 s−1 ).  2.3  Solution Uncertainty and Residuals  ˆ is known with an uncertainty εxˆ . Eq. 2.3 is known with an error ε. The solution x In the standard inverse approach, these uncertainties can be estimated as a function of uncertainties in the observations b. However, there are also uncertainties in the  46  6.2  Chapter 2. Inverse Methods and Box Models matrix A, as its coefficients are derived from observations of T and S. In the standard inverse method (described in the introduction), only small changes in b are taken into account to estimate the error in the inverse solution. More sophisticated procedures known as the total least-squares or the total inversion [Wunsch 2006] can be used to take into account the small changes in both b and A. However, these procedures are non-linear. The use of non-parametric and parametric bootstraps [Efron and Tibshirani 1993] enables one to use a linear procedure. ˆ . When bootFig. 2.3 shows the different steps that lead to the inverse solution x strap procedures are applied to the input data of the top and bottom boxes, the coefficients of the replicates of the vector b and the matrix A are changed by an amount ∆b and ∆A, respectively. This implies that the replicates of the solution x ˆ and the residuals εi ’s, through the inverse procedure, are also changed by an amount ∆ˆ x and ∆εj . This shows that the bootstrap can provide one with an estimate of the error of all the knowns (b and A) and unknowns (ˆ x and εj ) of the problem. The latter is based on small changes not only of the elements of b but also of the coefficients of A. The method of non-parametric and parametric bootstraps is based on Efron and Tibshirani [1993]. In the case of the observations from the different hydrographic stations, it consists of resampling the vertical profiles in a given cruise with repetition allowed and generating replicates of the box averages with these new profile sets. In the case of observations obtained at one particular location (forcings and sources), a parametric bootstrap was applied assuming a reasonable coefficient of variation (ratio of standard error over average, see Table 2.3). The non-parametric and parametric bootstraps are combined to produce enough bootstrap replicates (200 replicates) for all the known quantities. It is then possible to estimate the residuals εj of Eqs 2.66– 2.71, the error in x ˆ the solution of the inverse problem, and the errors ∆A and ∆b on 47  Chapter 2. Inverse Methods and Box Models  Figure 2.3: Chart of the Inversion Procedure. The data are input on the lefthand side. After averaging and time derivation, they yield the solution, on the righthand side.  Variables  Coefficient of Variation (%)  N, P and Si concentrations  15  TH and SH  2.5  F and FSW  15  TR and R  15 and 10, respectively  Table 2.3: Coefficients of Variation Used in the Parametric Bootstraps. They are applied to the riverine inputs, the biogeochemical tracer concentrations, the heat fluxes and HS salinity and temperature. N is nitrate+nitrite, P phosphate and Si Silicic acid.  48  Chapter 2. Inverse Methods and Box Models equations of the biogeochemical tracers (see next section).  2.4  Forward Problem: Estimating the Net Primary Productivity  To solve the forward problem, I will keep the same idealization of the SoG into a two-box model (Fig. 2.2) with time dependence and the same separation depth d. The solution is, however, more simple than in the case of the inverse problem, as it does not require the inverse procedure. Fig. 2.2 lays out the boxes, the biogeochemical fluxes and processes and the biogeochemical tracers monitored in the SoG. Phosphate (P), nitrate (N), silicic acid (Si) and dissolved oxygen (O) exchanges in the two SoG boxes are represented by the following Eqs 2.77–2.84 of conservation in the “tracer difference form” (for consistency with the physical box model). These equations result from integrations similar to the integration of Eq. 2.51 where q can be the tracer N, Si, P or O. For instance, the “Flux form” of the conservation equations for P are written as: − P1 U1 − P1 W1 + P2 W2 = PH U2 + P1 W1 − P2 W2 =  ∂P1 − PR R − (φ1 )P + ε′1 ∂t ∂P2 − (φ2 )P + ε′2 V2 ∂t V1  (2.75) (2.76)  where surface processes, for instance for P, are net upwelling (P2 W2 − P1 W1 ), advective export (−P1 U1 ), river inflow (PR R), the net biological uptake (φ1 )P and the surface storage/drawdown rate term (V1 ∂P1 /∂t), and deep processes are net upwelling (P2 W2 − P1 W1 ), deep intrusions (PH U2 ), the net biological uptake (φ2 )P and the deep storage term (V2 ∂P2 /∂t). Note that we will use the words “storage rate term” 49  Chapter 2. Inverse Methods and Box Models or “storage rate” instead of “storage/drawdown rate term”. Storage occurs when the time derivative is positive, so that the storage rate is positive too. In the other hand, drawdown occurs when the time derivative is negative, so that the storage rate is negative in this case. Any net biological uptake of nutrient and net biological production of dissolved O2 are represented by φ terms. The general equations rewritten  50  Chapter 2. Inverse Methods and Box Models in the “tracer difference form” are: P budget: (P2 − P1 )W2  =  V1  (PH − P1 )U2 + (P1 − P2 )W2  =  V2  (N2 − N1 )W2  =  V1  (NH − N1 )U2 + (N1 − N2 )W2  =  V2  (Si2 − Si1 )W2  =  V1  (SiH − Si1 )U2 + (Si1 − Si2 )W2  =  V2  =  V1  ∂P1 + (P1 − PR )R − (φ1 )P + ε′1 ∂t (2.77) ∂P2 − (φ2 )P + ε′2 ∂t  (2.78)  similarly for the other nutrients and O2 N budget: ∂N1 + (N1 − NR )R − (φ1 )N + ε′3 ∂t (2.79) ∂N2 − (φ2 )N + ε′4 ∂t  (2.80)  Si budget: ∂Si1 + (Si1 − SiR )R − (φ1 )Si + ε′5 ∂t (2.81) ∂Si2 − (φ2 )Si + ε′6 ∂t  (2.82)  O2 budget: (O2 − O1 )W2  ∂O1 + (O1 − OR )R − (φ1 )O2 + ε′7 ∂t (2.83)  − kO2 Osaturation − Osurf ace (OH − O1 )U2 + (O1 − O2 )W2  =  V2  ∂O2 − (φ2 )O2 + ε′8 ∂t  (2.84)  51  Chapter 2. Inverse Methods and Box Models One difference between these equations and the inverse problem equations is that each equation has a sink term (φ1 or φ2 ) of unknown size. In the case of O2 , φ1 is a net biological production term, that is a source term. The forward solution for φ1 or φ2 by any of the 4 biogeochemical tracers can readily be obtained by addition of all the known quantities. At this stage, the transports (in particular U2 and W2 ) have been obtained through the resolution of the inverse problem. Thus, the contribution of the advective transports to the biogeochemical conservation equations can be estimated. For instance, the sink term (φ1 )N from the surface N budget is determined by the equation: ∂N1 − (N2 − N1 )W2 + (N1 − NR )R. (2.85) ∂t The air-sea exchange flux of O2 can be estimated once the piston velocity kO2 , (φ 1 )N = V 1  and the difference between saturation level and surface level are known (see further detail in section 3.2.3). Based on the budget made by Mackas and Harrison [1997], the atmospheric deposition of N as well as the anthropogenic sources have been neglected with respect to the other terms in the box model. The atmospheric deposition and anthropogenic sources of P and Si have been assumed to be negligible. Turbulent diffusion terms are combined mathematically to advective transports (for instance (N2 − N1 )W2 and (NH − N1 )U2 for Eqs 2.79–2.80). This could be an additional source of errors in the estimation of the sink terms (φ1 )N , (φ1 )P and (φ1 )Si because the turbulent component of the circulation, u′ , and the corresponding correlated variations of the biogeochemical tracers, q′ , are unknown. For instance, the transports of N in the top box V1 has the form: (N u + N ′ u′ ) · da1 = N1 U1 + eSC1  (2.86)  (N u + N ′ u′ ) · da1 = N1 W1 − N2 W2 + eSB  (2.87)  SC1  and SB  52  Chapter 2. Inverse Methods and Box Models Eq. 2.85 is known with an equation error of ε′1 . I can estimate the ε′j ’s by applying the estimation method in section 2.3. Further analysis is necessary on the sink terms φ1 or φ2 of Eqs 2.77–2.84 to be able to estimate the primary productivity. This is the goal of sections 5.2.3 and 5.2.4 and in Chapter 5.  53  Chapter 3 Observations 3.1  Introduction  This chapter describes the time series of the observed variables needed to find the solution of the inverse and forward problems in chapter 2. First, I introduce the sampling method used to collect the observations, and the data averaging used to compute the time series. Time Series of freshwater, salt, heat and temperature are used to estimate the parameters of the physical dynamics, and time series of nitrate, phosphate, silicic acid and dissolved O2 are used to estimate the parameters of the biogeochemical cycles. Then, I discuss the information from the time series that reflect surface and deep characteristics of the SoG.  3.2  Data Sources  3.2.1  STRATOGEM  Using a hovercraft (CCGH Siyay), the STRATOGEM program was able to sample 9 stations (Fig. 1.2) over a 240-km long track in about 9 hours (including both sampling and steaming times). The stations were located in the central and southern regions of the SoG, between 48◦ 55.0’N and 49◦ 21.5’N. The surveys started early in the morning (8-9 am) near the beginning of every month from April 2002 to  54  Chapter 3. Observations June 2005, with more frequent sampling around the spring bloom. This amounts to 47 cruises. In addition, another 9 cruises with a field sampling design different from the standard STRATOGEM were carried out. This different sampling design helped to investigate the Fraser River plume and stations different from the standard STRATOGEM stations. CTD (conductivity-temperature-depth) casts provided continuous vertical profiles of physical (pressure, temperature and conductivity) and biogeochemical tracers. The CTD instrument was equipped with additional sensors to measure chlorophyll-a fluorescence, dissolved O2 (oxygen), photosynthetically available radiation (PAR), and beam transmissivity. Profiles were made at the front of the hovercraft (limiting ship mixing) to sample at high vertical resolution the entire water column down to within 15 m of the bottom. The data were binned (over 1 m) to obtain the continuous profiles from the sensors and combined to obtain additional variables: e.g. salinity and density. Note that salinity measurements are reported on the Practical Salinity Scale PSS-78 [UNESCO 1981 a,b, 1983]. The vertical profiles of the variables of interest for this study (temperature, salinity and dissolved O2 ) are accurate to ±0.003◦ C, ±0.01 psu and ±0.2 mL L−1 , respectively. Water samples were also collected to measure macronutrients (nitrate/nitrite, phosphate and silicic acid) in the water column, to identify and count phytoplankton cells, and to provide a baseline for the correction of and comparison with CTD measurements (dissolved O2 , chlorophyll-a concentration and salinity). They were taken at depths of 0, 5, 10 and 30 meters, from the front of the hovercraft, using 5 and 8 L Niskin bottles. In addition, in order to estimate the SoG deep profile of macronutrients, samples were collected at 250 m at station 2-2 and 50, 100, 200, 300 and 390 m at station 4-1. Water samples for macronutrients were sub-sampled on deck from the Niskin 55  Chapter 3. Observations bottles into acid-washed cups, filtered through a 0.7 µm GF/F, and then stored and frozen in acid-washed vials for lab analysis (within 2 months). Water samples (for the other biogeochemical measurements) were sub-sampled on deck from the Niskin bottles into containers pre-rinsed with seawater. The error on the macronutrient concentration was estimated to be equal or less than ± 15% or 0.5 µM for nitrate (short for both nitrate and nitrite), ± 8% for phosphate and silicic acid. Macronutrient measurements on water samples that had not been frozen (analyzed within a few days after sampling) showed that the accuracy could be significantly improved. But logistical issues prevented this from being done on a regular basis. Water samples for measurement of dissolved O2 and chlorophyll-a were kept fresh and analyzed within 24 hours. Dissolved oxygen was estimated from water samples using Winkler titration [Parsons et al. 1984, Culberson 1991] with an accuracy close to or larger than 5%. Chlorophyll-a was estimated from smaller water sub-samples (100–200 mL). The samples were vacuum filtered through polycarbonate membranes of 0.2, 2 and 20 µm. Biomass of chlorophyll-a pigments was measured using a Turner Designs 10AU fluorometer [Parsons et al. 1984]. Replicates suggest an uncertainty of ± 10%. Water samples (100 mL) for taxonomy were preserved using Lugols solution in dark glass bottles and analyzed quantitatively for microplankton following the Utermohl method [Hasle 1978]. Estimates of carbon biomass per cell were based on determinations of cell biovolume [Haigh et al. 1992] and subsequent conversion of biovolume into carbon biomass [Montagnes and Franklin 2001, Strathman 1967].  3.2.2  JEMS  The Joint Effort to Monitor the Strait of Juan de Fuca (JEMS) program collected similar continuous profiles and water samples at three hydrographic stations in HS  56  Chapter 3. Observations (Fig. 1.1). Based on the available JEMS information [Newton et al. 2002], the methods of sampling and the accuracy of the measurements are similar to that of STRATOGEM.  3.2.3  Freshwater Inflow, Surface Heat and Air-sea Fluxes  This section contains a description of the various sources of freshwater, heat, and air-sea exchanges that contribute to surface changes of temperature, salinity and dissolved O2 , respectively. Freshwater (FW) inflow estimates (see Fig. 1.4) are based on the Fraser River discharge measured at Hope by Environment Canada (water station 08MF005). A linear regression (y-intercept 1165 m3 s−1 and slope 1.66) is used to estimate the total freshwater inflow from the Fraser River discharge [Pawlowicz et al. 2007]. We assume other measurements (temperature, macronutrients, and dissolved O2 ), necessary for the computation of the river fluxes into the whole SoG, to be similar to those measured in the Fraser River at Hope. Thus, the observations used to measure the temperature, the macronutrients and the dissolved O2 in the rivers are based on the observations from the Fraser River at Hope. The surface heat budget is composed of both a measured heat flux (ShortWave radiation or SW) and estimated fluxes (LongWave radiation or LW, sensible and latent heats). The SW heat flux is measured by a SW-band downwelling-radiation sensor located 10 m above the ground at Totem Field station on the University of British Columbia (UBC) campus (maintained by the Bio-Met program). The other heat fluxes (LW, sensible and latent heats) were estimated using weather observations at either Vancouver International Airport station (LW radiation) or Halibut Bank buoy (sensible and latent heats), and the methods of Pawlowicz et al. [2001]. The air-sea flux of dissolved O2 is estimated using a Fickian formulation of the  57  Chapter 3. Observations flux [Woolf 2005, flux proportional to saturation level], an empirical formulation of the gas piston velocity [Wanninkhof and McGillis 1999], and the observed surface and estimated saturation levels of dissolved O2 . The wind observations used for the gas piston velocity come from the Halibut Bank buoy, while the dissolved O2 observations come from STRATOGEM surface observations (between 1–2 m). The sampling intervals of all these fluxes are different from the sampling period of the observations from STRATOGEM. In general, these time series have a 30-minute (SW heat flux) or 60-minute (all the others) intervals. This high time resolution was kept to estimate the different surface source terms in the conservation equations. Just before being substituted into the conservation equations, they were averaged over a one-month moving window and interpolated to the STRATOGEM sampling dates. This is a good practise to ensure that all the input data have approximately the same characteristic timescale.  3.3  Box Model Inputs  Observations provide the necessary information to compute the unknown terms occurring in the conservation equations in the physical and biogeochemical box models (Eqs 2.66–2.71 and Eqs 2.77–2.84). The equation coefficients are the box-averaged observed temperature, salinity, nutrients and dissolved O2 concentrations near the surface (0–30 m) and at depth (30–400 m) in the SoG and in the entire water column of HS. The terms in the equations are the various surface sources of mass, heat, nutrients or dissolved O2 .  58  Chapter 3. Observations  3.3.1  Separation Depth  The separation depth d defines the vertical separation between the surface outflow and the deep inflow in the SoG box model. The surface outflow carries freshwater from the rivers and seawater from the SoG, while the deep inflow brings dense seawater from HS. Analyzing the vertical profiles of salinity and temperature in the SoG enables one to identify these two types of water masses and to determine their depth ranges. The separation depth should roughly correspond to the lower boundary of the pycnocline, below which the water column contains mainly seawater from HS. In estuaries, salinity mainly controls the density, and determining the depth of the halocline is equivalent to determining the pycnocline. Additional information on the vertical structure of the water column can be obtain from looking at the temperature profile and determining the depth of the thermocline. A detailed analysis of the vertical profiles (not shown) of temperature and salinity suggested an appropriate separation depth would be below a level between 15 m and 30 m, based on the depth of the halocline and the thermocline. These variations of the halocline and thermocline are seasonal variations. In winter, both wind-induced mixing and convection of cooling water tend to deepen the mixed layer close to 30 m or down to 50 m. In summer, stratification due to both salinity (river freshwater) and temperature (surface heating) tends to shallow the mixed layer above 30 m. Further analysis of the vertical profiles at station S4-1, traditionally used to represent the SoG, suggested a deeper separation depth, below the 50-m level [Pawlowicz et al. 2007]. The separation depth has to define not only the top box of the circulation box model (section 2.2, Fig. 2.1), but also the top box of the primary production box model (section 2.2, Fig.2.2). A shallow separation depth would underestimate the PP and the associated productivity, while a deeper value would not affect the estimated  59  Chapter 3. Observations PP and its productivity. A detailed analysis was carried out using vertical profiles of chlorophyll-a fluorescence (not shown), a proxy for primary production (PP), and PAR (not shown), a variable to determine the depth range where photosynthesis occurs. SoG PP was found to be located at or above 30 m where 1% and more of PAR reaches the water column. The 30-m level corresponds to the deeper separation depth suggested by the physical analysis in the previous paragraph. During Summer however, the 1% level can be significantly shallower, and even <1 m in the Fraser Plume. Thus, the analysis of SoG vertical profiles suggested that the separation depth should be between 30 m (this analysis) and 50 m (Pawlowicz et al. [2007]). To keep the box-model simple, a constant separation depth was chosen to be 30 m (see also discussion in section 4.3.1). Note that the change of flow should happen in a gradual transition layer between the inflowing and the outflowing water masses. The two-layer box model ignores the thickness of the transition layer and assumes a discrete change from a surface seaward flow to a deep landward flow at the separation depth.  3.3.2  Spatial Averaging and Hypsography  To determine the best average of the observations from STRATOGEM over the surface and bottom boxes, it is necessary to respect the SoG geomorphology and to use all available data. In order to get the box averages for various tracers, a mean vertical profile is first found by averaging all station data at a particular depth. Box averages are then hypsographically weighted averages of the mean profiles over the depth range represented by each box. The hypsography (Fig. 3.2) shows that the area of horizontal sections of the SoG decreases roughly proportionally to the depth. Thus, assuming a separation depth of  60  Chapter 3. Observations 30 m, the volume of the surface box is smaller than, but not negligible compared to, the volume of the bottom box: the surface box volume is about 21% of the bottom box volume, and 17% of SoG volume. In the case of HS, the box average is a simple average of the tracer over the water column. Since the JEMS and STRATOGEM sampling dates are different it is necessary to interpolate HS box averages to the STRATOGEM sampling dates.  3.3.3  Time Dependence  The transports and the primary productivity are expected to be time-dependent (section 2.2). Thus, we need to calculate the time-derivative terms of physical and biogeochemical inputs. In my box model, the time-derivative terms are computed by estimating the derivative of the observation averages, in each box and for each tracer. The derivative scheme is based on a 5-point parabolic fit which reasonably represents the actual time derivative of the tracers. The reason for this choice is to remove the sampling noise that appears from survey to survey in the temperature and salinity time-derivatives. Time derivatives at the beginning and the end of the tracer time series are handled by applying a 2- or 3-point time-difference scheme. The time derivatives are part of the “known terms” like the surface sources and the forcing terms.  3.3.4  Input Time Series  Salinity In summer, average surface salinity rapidly decreased (Fig. 3.3a). This is due to the addition of freshwater into the SoG surface water by the local rivers, and the 61  0  0  10  10  20  20  30  30  40  40  depth (m)  depth (m)  Chapter 3. Observations  50  50  60  60  70  70  80  80  90  90  100  default d  maximum d  100 5  10 15 20 25 salinity (psu)  30  5  10 15 temperature (°C)  20  Figure 3.1: Vertical Profiles of Salinity and Temperature during STRATOGEM. The envelope of temperature and salinity vertical profiles is shown at the 9 regular locations. Each square is an average value of data binned over a 1 m. The dashed lines show d–D, the range of the separation depth. mixing of freshwater and SoG surface water during the freshet (a Glossary with the definitions of the seasons can be found on p. xiii). The freshwater addition and water  62  Chapter 3. Observations  0 surface volume: 1.9 × 1011 m3 50  11  bottom volume: 9 × 10  3  m  100  depth (m)  150 surface area: ∼7 × 109 m2 200  250  300  350  400 0  1  2  3 4 5 section area (×109 m2)  6  7  8  Figure 3.2: Hypsography of the Whole SoG. The 0-m section area is extrapolated from the 1-m and 2-m values. In addition, the volumes of the top and bottom boxes are indicated in m3 .  63  Chapter 3. Observations mixing are fast processes with respect to the sampling period (1-2 weeks during the early freshet, otherwise 1 month). HS water enters and mixes with deep SoG water as seawater intrusions (Fig. 3.3b). From spring until summer (or fall), HS salinity increases as it mixes with oceanic seawater coming from the Strait of Juan de Fuca (SoJdF). At the same time, SoG deep water mixes with denser HS water and it is entrained upward bringing more salt to the SoG surface. The SoG deep salinity maximum was reached later than HS salinity maximum because it takes time for the water to be transported into the SoG and then mixed with the SoG deep water [Pawlowicz et al. 2007]. Temperature The average surface temperature follows the surface net heat flux with a delay of 1 to 2 months (Fig. 3.4a). The surface net heat flux is the net budget of SW, LW, sensible and latent heat fluxes. Eq. 2.68 shows the different processes that can contribute to the changes of surface temperature: e.g., the summer heat influx, the outflow transport of SoG water and the vertical exchange of surface and deep SoG waters. The yearly maximum temperature is reached in summer. In spring and summer, the well-mixed HS water enters and brings heat into deep SoG (Fig. 3.4b). HS heat input peaks in late-summer when HS temperature is close to the summer maximum temperature (similarly with the surface of the SoG). During this same period, the SoG bottom water receives heat from HS and SoG surface. Since heat transport and diffusion in SoG bottom occur through vertical water mixing, SoG bottom average increases steadily until late August -early September after the yearly temperature maximum occurs in HS (a delay of 1 to 2 months). In 2003 and 2004, the temperature average of the SoG bottom box levelled off after September.  64  Chapter 3. Observations SoG 0−30 m FW  29.5  input rate  29  4  28  6  27.5  8  27  10  26.5  12  26  14  25.5  16  25  18  24.5 Apr  Freshwater × 103m3s−1  2  28.5 Salinity (psu)  0  20 Jul  Oct 2003 Apr  Jul  Oct  04  Apr  Jul  Oct  05  Apr  32.5 SoG 30−400 m HS 0−200 m  Salinity (psu)  32  31.5  31  30.5  30 Apr  Jul  Oct 2003 Apr  Jul  Oct  04  Apr  Jul  Oct  05  Apr  Figure 3.3: Salinity and Freshwater Time Series. In both panels, the left axis indicates the Box averages of the Salinity (psu). Each marker is a box average from a single cruise. In the upper panel, the right axis indicates the surface FW input rate (m3 s−1 ). Observations, Estimations and Budget of Heat Fluxes The observed SW radiation (Fig. 3.5a) and the estimated LW radiation, latent and sensible heats (Figs 3.5b,c and d, respectively) are monthly averaged as explained in 65  Chapter 3. Observations  14  15  13  −2  Temperature ( ° C)  11  5  10 0  9 8 7  Flux (× 10 Wm )  10  12  a)  −5  SoG 0−30 m Net Heat Flux in SoG  6 Apr  −10 Jul  Oct 2003 Apr  Jul  Oct  04  Apr  Jul  Oct  05  Apr  Oct  04  Apr  Jul  Oct  05  Apr  10.5 b)  Temperature ( ° C)  10  SoG 30−400 m HS 0−200 m  9.5 9 8.5 8 7.5 Apr  Jul  Oct 2003 Apr  Jul  Figure 3.4: Temperature and Surface Net Heat Flux Time Series. In both panels, the left axis indicates the Box Averages of the Temperature (◦ C). Each marker is a box average from a single cruise. In the upper panel, the right axis indicates the surface net heat flux (× 10 W m−2 ). The study of phase lags of between the SoG and HS deep temperatures have been studied by Pawlowicz et al. [2007].  66  Chapter 3. Observations section 3.2.3. During summer, the maximum of the monthly averaged SW heat flux (about 260 W m−2 ) is about 25% of the observed maximum (about 1010 W m−2 ) because of the alternation of day and night. At the same time, both the estimated LW heat flux and latent heat reach a minimum: about -80 and -60 W m−2 , respectively. The minimum of averaged sensible heat is very small (about -20 W m−2 ) during this period. The estimated sensible heat flux in the SoG suggests that the heat exported during winter and the heat imported during the summer were about the same. It also suggests that the sensible heat flux is small compared to the other heat fluxes. The observed and estimated surface heat fluxes were combined to determine the net surface heat flux into the SoG (Fig. 3.6). The net heat flux shows heating (130 W m−2 ) in summer, and cooling (-60 W m−2 ) in fall-winter. A minimum net heat flux of -90 W m−2 was observed in Jan 2004 and 2005, but winter 2003 was warmer (closer to 0 with a January minimum of -50 W m−2 ). Nutrients The surface box averages of the nutrient concentration exhibit a typical seasonal cycle of depletion during the spring-summer and replenishment during winter with occasional fall depletion (Fig. 3.7: phosphate, Fig. 3.8: nitrate and Fig. 3.9: silicic acid). Depletion periods were associated with the uptake of nutrients by phytoplankton during spring (between 50–75% of nutrients depleted), summer and fall blooms [Harrison et al. 1983]. During depletion periods, the box average nutrient concentration never gets close to zero, but examination of the raw observations shows that near surface nutrients can be completely depleted. It is generally thought that replenishment corresponds to nutrient upwelling and low primary production. Although nitrate and phosphate levels are at a minimum in summer, silicic acid  67  Chapter 3. Observations  a) SW flux  Wm  −2  1000  500  0 Apr  Wm  −2  0  Jul  Oct 2003 Apr  Jul  Oct  04  Apr  Jul  Oct  05  Apr  Jul  Oct  04  Apr  Jul  Oct  05  Apr  Jul  Oct  04  Apr  Jul  Oct  05  Apr  Jul  Oct  04  Apr  Jul  Oct  05  Apr  b) LW flux  −50 −100 Apr  Wm  −2  100  Jul  Oct 2003 Apr  c) Latent heat  0 −100 −200 Apr  Jul  Oct 2003 Apr  d) Sensible heat  Wm  −2  50 0 −50 −100 Apr  Jul  Oct 2003 Apr  Figure 3.5: Heat Flux Observations and Estimates Into the SoG. SW stands for shortwave radiation, while LW stands for longwave radiation. SW flux is observed, while LW flux, sensible and latent heats are estimated. The gray lines are the heat flux data. The dashed black lines are the monthly averages. See section 3.2.3 for further detail.  68  Chapter 3. Observations  300  SW Sensible Latent LW  250  net heat fllux  200  W m−2  150  100  50  0  −50  −100 Apr  Jul  Oct 2003 Apr  Jul  Oct  04  Apr  Jul  Oct  05  Apr  Figure 3.6: Surface Heat Budget. The net heat flux is the sum of the shortwave (SW), longwave (LW), latent and sensible heat fluxes.  69  Chapter 3. Observations level is often at a minimum in spring, recovering somewhat in summer. A possible contribution to this replenishment is the river input of silicic acid (see section 3.3.6) [Harrison et al. 1991]. This occurs after the spring bloom unlike the replenishment of nitrate and phosphate. SoG bottom average and HS average time series show that nutrients are abundant and continuously available at depth. Note that the SoG surface average of nutrients also reached these deeper values during the end of the replenishment. The SoG bottom average of phosphate has a slight increasing trend over the whole time series that is not due to any error in the analysis of water samples. Apart from phytoplankton uptake and nutrient upwelling, the phosphate concentration can be changed by the adsorption of phosphate on sinking particles and its aggregation on organic matter. However, our data set did not allow us to investigate the sinking of particles or organic matter. As previously observed in the Western North Pacific deep water from 1968 to 1998 [Ono et al. 2001], another explanation to the increase of the SoG phosphate concentration could be the non-local increase of the phosphate concentration of the source water coming from the Pacific Ocean and entering in the SoG. Bottom averages of silicic acid have a definite seasonal cycle with a maximum timing that coincides with surface replenishment. Dissolved Oxygen The surface box average of the dissolved O2 (Fig. 3.10) increases with the increased solubility level during winter and spring (Fig. 3.11) and with the high primary production during spring blooms, as phytoplankton photosynthesized O2 in excess of respiration needs (spring maximum about 7 mL L−1 ). The surface and bottom box averages followed closely the same seasonal cycle, but the bottom range was markedly  70  Chapter 3. Observations lower than surface range: between 1.5 and 3 mL L−1 less than the top box average. SoG bottom box average reached a minimum of about 3 mL L−1 in fall. In 2003, it reached the minimum 1–2 months after the HS minimum. HS box average shows that large amounts of dissolved O2 were exported during the fall-winter into the deep SoG (HS maximum around 5 mL L−1 ).  3.3.5  Air-sea Oxygen Flux  The air-sea O2 flux was estimated as explained in section 3.2.3, using the surface observed O2 and the estimated saturation level (Fig. 3.11). The flux was negative (exporting from SoG) when the observed level exceeded the saturation level, and the flux was positive in the other case (importing from the atmosphere). The maximum flux of dissolved O2 exported from the SoG always occurred during the spring bloom, with a large amount of oxygen produced. For instance, the highest average flux occurred in spring 2004, reached around 1.2×104 mol s−1 , and arose from an oxygen excess that ranges between 2.7 and 3.1 mL L−1 (39–47% above surface saturation).  3.3.6  Riverine Inputs  Inputs of nutrients and dissolved O2 were estimated as explained in section 3.2.3, from observations at Hope in the Fraser River (section 3.2.3). The freshwater input used to estimate the inputs of nutrients has been described in section 3.2.3 using Figs 1.4 and 3.3a. The silicic-acid input was by far the most significant nutrient input from the river by comparison with the amount of upwelled nutrients. Assuming a net upwelling transport of 5×104 m3 s−1 (from the a priori estimates in chapter 2) and using the bottom box averages, one can show that the river flux of P and N nutrients (maximum 2 and 75 mol s−1 , respectively) are much smaller than the estimated 71  Chapter 3. Observations  2.5  Phosphate µM  2  1.5  1  SoG 0−30 m HS 0−200 m SoG 30−400 m  0.5  0 Jul  Oct 2003 Apr  Jul  Oct  04  Apr  Jul  Oct  05  Apr  Figure 3.7: Phosphate Concentration Time Series. Each marker is a box average of the phosphate concentration (µM) in the SoG or in the HS from a single cruise.  72  Chapter 3. Observations  35  30  Nitrite+Nitrate µM  25  20  15  10 SoG 0−30 m HS 0−200 m SoG 30−400 m  5  0 Jul  Oct 2003 Apr  Jul  Oct  04  Apr  Jul  Oct  05  Apr  Figure 3.8: Nitrite+Nitrate Concentration Time Series. Each marker is a box average of the nitrite+nitrate concentration (µM) from a single cruise.  73  Chapter 3. Observations  60  50  Silicic acid µM  40  30  20 SoG 0−30 m HS 0−200 m SoG 30−400 m 10  0 Jul  Oct 2003 Apr  Jul  Oct  04  Apr  Jul  Oct  05  Apr  Figure 3.9: Silicic Acid Concentration Time Series. Each marker is a box average of the silicic-acid concentration (µM) from a single cruise.  74  Chapter 3. Observations  9  8  7  dissolved O2 mL L  −1  6  5  4  3  SoG 0−30 m  2  HS 0−200 m SoG 30−400 m 1  0 Jul  Oct 2003 Apr  Jul  Oct  04  Apr  Jul  Oct  05  Apr  Figure 3.10: Dissolved Oxygen Time Series. Each marker is a box average of the dissolved O2 concentration (mL L−1 ) from a single cruise.  75  Chapter 3. Observations  10  1.5  observed O (1−2 m) 2  O2 saturation level (1−2 m) air−sea O flux (1−month moving average) 2  9.5 1 9  −1  dissolved O2 (mL L )  0.5 8  7.5  0  7 −0.5  DO flux into SoG (×104mol×s−1)  8.5  6.5  6 −1 5.5  5  −1.5 Jul  Oct 2003 Apr  Jul  Oct  04  Apr  Jul  Oct  05  Apr  Figure 3.11: Air-sea Oxygen Flux Time Series  76  Chapter 3. Observations upwelling input (100 and 1250 mol s−1 , respectively), while it is close in the case of the silicic acid: 700-1000 mol s−1 from the river is smaller but not zero compared to 2500 mol s−1 from the upwelling transport. In the case of oxygen, the river input (1000-4000 mol s−1 ) was of similar order of magnitude to the upwelled dissolved O2 (8900 mol s−1 , still assuming a transport of 5×104 m3 s−1 ) or the amount of oxygen due to air-sea exchange (seasonal average from -4000 to 2000 mol s−1 , see Table 5.4, but maximum can reach 1.2×104 mol s−1 from previous section).  77  Chapter 3. Observations  5.5  10 P N Si O  2  5  4.5  7  4  6  3.5  5  3  4  2.5  3  2  2  1.5  1  1  2  2  O River Flux (× 103 mol s−1)  8  −1  Nutrient River Flux (× 10 mol s )  9  0 Apr  0.5 Jul  Oct 2003 Apr  Jul  Oct  04  Apr  Jul  Oct  05  Apr  Figure 3.12: River Biogeochemical Inputs Time Series. Each curve indicates the estimated river influx (mol s−1 ). Phosphate (P), nitrate (N), and silicic-acid (Si) rates are indicated on the left axis, while dissolved-O2 rate is indicated on the right axis.  78  Chapter 4 Circulation and Transports 4.1  Introduction  In this chapter, the objective is to estimate the advective transports in the SoG. The estimation procedure extensively uses the SVD mathematical framework and the SoG idealization as a two-box model described in chapter 2. Under this idealization of the SoG, temperature and salinity are averaged over each domain represented by the boxes. Water, heat and salt enter the deep box, are exchanged between the two boxes and circulate in and out of the surface box. In the box model, these exchanges are due to the estuarine circulation and the vertical diffusion. I will apply the SVD method to compute an inverse solution for the transports from the input data defined in chapter 3, solving, as closely as possible, the conservation equations Eqs 2.66–2.71. The inverse solution (similar to Eq. 2.24) provides an estimate of each transport (U1 , U2 , W1 , W2 ) and the derived physical parameters E(=W2 − W1 ), net upward entrainment, and M (=W1 ), vertical rate of turbulent exchange of water parcels (mixing exchange for short). Each transport is associated with an estimated error ∆ˆ xi and each conservation equation is solved with an estimated uncertainty εj . The transport estimates can be affected by the parametrization of the box model and the inversion procedure. Time dependence alters the seasonal variability of these estimates, and it increases the strength of the relationship between their seasonality 79  Chapter 4. Circulation and Transports and the freshwater inflow seasonality. This aspect will be investigated and the alternative results (without the time dependence) are presented for later comparison and discussion (chapter 6). I will also investigate the sensitivity of the estimates to the depth of separation (d) between the boxes and the sensitivity to the parametrization of the inversion (trade-off parameter α, a priori transports xA ). I will determine the robustness of the estimates to realistic changes in the parametrizations. Finally, the residuals εj ’s are analyzed and discussed to investigate their consistency with the a priori error estimates of the conservation equations. In the discussion section, I consider the general circulation of the SoG described by the transport estimates and make comparisons with previous studies. The freshwater inflow is an important forcing of the estuarine circulation. I will discuss the freshwater influence on the circulation seasonality and the sensitivity of the circulation seasonality to the freshwater (FW) inflow. Finally, I will investigate the interannual variability of the circulation over the three years of observations. Here, I will highlight the main contributions of this chapter: 1. first inference of transport time series of the SoG estuarine circulation with a monthly time resolution and over three years.  2. inference of SoG transports based on consistency with observations of salinity and freshwater input, temperature and surface heat fluxes.  3. one of the few observational studies of the link between freshwater input (R) and surface seaward transport (U1 ).  80  Chapter 4. Circulation and Transports 4. the finding that only small changes of U1 occur even with large changes of R.  5. the finding that there are only very small changes of W2 even with large changes of R.  6. the finding that, based on 2002-2005 data, annual changes of U1 are linked to annual changes of R.  4.2 4.2.1  Results Estimates of Estuarine Transports and Mixing Exchange  Fig. 4.1 shows the seaward (U1 ) and net upward entrainment (E) transports, and the mixing exchange (M) as defined in Fig. 2.1. The estimates of U2 (landward transport) and W2 (total upwelling), and the corresponding errors are not shown because U2 is not significantly different from E and there is little change in W2 time series. Each band around the estimates represents the estimated standard error σbi using bootstrap replicate statistics. Since the distribution of the transport replicates for a single cruise was not always symmetric, lower and upper errors have been computed using the difference between the 16th , 50th and 84th percentiles of the 200 replicates (see Fig. 4.1) with σbi the average of lower and upper errors. TD columns of Table 4.1 show the values of σb , the mean of the σbi ’s, taken for xˆ=U1 , E, M and W2 . In  81  Chapter 4. Circulation and Transports xˆ  TD  QSS  σe  σb  σ  f  p-value  σe  σb  σ  f  p-value  U1  5.8  4.9  1.1  1.40  0.06  4.4  4.6  0.9  0.91  0.64  E  5.2  5.3  1.1  0.96  0.55  2.9  5.1  0.9  0.32  1  M  8.4  10  1.9  0.70  0.92  5.5  9.3  1.6  0.35  1  W2  4  4.8  0.9  0.69  0.93  3.2  4.6  0.8  0.48  1  Table 4.1: Analysis of Variance and F-tests for Time Dependence (TD) and quasisteady state (QSS) Transports. Columns σb , σe , and σ contain estimated standard errors and variability of the transports U1 , E, M and W2 . Columns f and p-value contain test statistics f and probability levels of the F-test on the estimated transports U1 , E, M and W2 . All values of σ’s are ×103 m3 s−1 . The QSS columns will be used later in section 4.2.2 addition, an overall mean value x and its corresponding error σ can be calculated: 1 x= l  l  xˆi .  (4.1)  i=1  The error σ depends on σb the sampling and estimation error and σe the seasonal variability of xˆ contain in the time series xˆi : σ=  σb 2 + σe 2 l  (4.2)  where σe is: σe =  1 l−1  l  xˆi − x  2  (4.3)  i=1  with l=47 the number of cruises: one cruise per month, except during March and April 2003 and 2004, 2 or 3 cruises per month. The values of σe and σ are given in 82  Chapter 4. Circulation and Transports Table 4.1, TD columns. Is there a seasonal cycle? A simple test can be derived by taking a statistic f=σe 2 /σb 2 (f values of TD in Table 4.1). This f statistic has an approximate F distribution with 46 and 199 degrees of freedom [Harris 2001]. The larger the statistic, the more likely it is that the variability in the time series is greater than its noise level. This can be quantified with the associated one-sided p-value. The p-values go to zero as the seasonal variability gets larger than the noise, and they approach 1 as time series variation becomes smaller than the noise. These p-values are quite large for almost all of the parameters (ˆ xi ’s in Table 4.1), suggesting that, if there is any seasonal variability, it is too small to be determined by the F-test. However, U1 shows a seasonal cycle (p=0.06, a 94% or less confidence level). Note that the test is somewhat conservative because it does not take into account any autocorrelation in the time series. If autocorrelation were taken into account the degrees of freedom associated with the time series and σe would be smaller than 46, thus σe would be larger than it is in Table 4.1 and f and p would be larger. Table 4.1 shows that U1 is not constant with a reasonable small p-value (=0.06), although the variance of the estimates for E, M and W2 is large enough that it hides any likely seasonality. Note that p<0.05 is more commonly used as a significant test. In Fig. 4.1, the general shape of U1 and E indicates that there is seasonality but its magnitude is relatively small (σe about 5×103 m3 s−1 ) compared to the average value (4–4.5×104m3 s−1 ). The F-test suggests that the variability of U1 is statistically significant, while one cannot make any conclusion about E. The average values are (4.5±0.1)×104 m3 s−1 (1 σ) for U1 and (4.0±0.1)×104 m3 s−1 (1 σ) for E, while the individual transports estimates range between 3.4×104 m3 s−1 and 6.2 ×104 m3 s−1 for U1 , and 2.8×104 m3 s−1 and 5×104 m3 s−1 for E (both representing a ratio of about 1.8). On average, the estimated error (1 σb ) in U1 and E is around 5×103 m3 s−1 . 83  Chapter 4. Circulation and Transports The magnitude of the seasonality of the mixing exchange M is as large as its average value. The magnitude of M is on average lower than the other transports, about (1.9±0.3)×104 m3 s−1 (1 σ), while it ranges between 0.8×104 m3 s−1 and 4.4×104 m3 s−1 (a factor of 5). On average, the estimated error in M is about 1.0×104 m3 s−1 (1 σb ). This is consistent with the result of the F-test that suggests that the variability of M is hidden by its sampling and estimation error. Note that the magnitude of the seasonality of the total upward transport (W2 = E + M, not shown) is very small (σe =4×103 m3 s−1 ) compared to the average magnitude of W2 (6.2×104 m3 s−1 ) and similar to the sampling and estimation error of U1 and E. Again, the F-test suggests that any seasonal variability of W2 is hidden by the variance. Fig. 4.1 also shows the freshwater inflow R (defined in chapter 3). There is a close similarity between the circulation variability (both U1 and E) and the freshwater variability. The circulation transports increase at the same time as the freshwater input increases and vice versa. The relationship is not a simple scaling. Typically, the freshwater inflow R varies by a factor of 5, but U1 by a factor of only 1.4. The sensitivity of the circulation transport U1 relative to change in R is further discussed in section 4.3.5. Similar relationships between R and the transports can be seen in the interannual variability (see section 4.3.6). On the other hand, there is a mirror-symmetry between the variability of M and the variability of R. M decreases when R increases and vice versa. However, the F-test indicates that the variance of M could hide the seasonality magnitude of M and thus, the relationship between M and R could be hidden by the variance of M as well. Oscillations, of large magnitude and of approximate two-month period, are visible around March-April 2003 and 2004 (and possibly around May 2005) when the 84  Chapter 4. Circulation and Transports sampling period usually is a week or two weeks. The model could be aliasing the fortnightly tidal signal in the transports and the box averages. However, an analysis of the phase between these oscillations and the tidal current (estimated at Active Pass, east side of Vancouver Island) or the sea surface level (at Sand Head) in MarchApril 2003 and 2004 show no clear relationship with the fortnightly tides. Instead, these oscillations could also be associated with the model misfit defined in chapter 2, section 2.2, Eqs 2.56–2.57 when the temperature and salinity averages at the separation depth are approximated by the box averages. Before the freshet period, the approximation error is the largest and it could propagate to the estimated transports.  4.2.2  Sensitivity Analysis  Time Dependency or Quasi-steady State The purpose of this section is to determine the influence of time dependence on the transport estimates in the budget equations (Eqs 2.66–2.71). Fig. 4.2 shows estimates of the transports when one assumes quasi-steady state, while time dependence was assumed in the previous Fig. 4.1. There is no major change in the general appearance of the transports between Time Dependence (TD) and Quasi-Steady State (QSS). In both cases, the mean values of the transports are very similar to each other, as are the average errors. Fig. 4.2 shows the mean values x and the average error (1σ) with TD and in QSS while Table 4.1 shows the errors σe , σb and σ with TD and in QSS. Fig. 4.2 shows that the mean value of each QSS transport is close to the corresponding TD transport. Table 4.1 also shows that the seasonality magnitude σe of the QSS transports is smaller than of the TD transports. The F-test applied to the QSS transports suggests that their variance would hide any seasonality. This indicates that even if there is some seasonality in the QSS transports it is difficult 85  Chapter 4. Circulation and Transports  4  7  x 10  U  1  E 6  M Freshwater  5  mean U1  4  3  m s  −1  mean E  3  mean M 2  1  0 Jul  Oct 2003 Apr  Jul  Oct  04  Apr  Jul  Oct  05  Apr  Figure 4.1: Transport Estimates and Their Estimated Errors (σbi ) . Transports are the lines inside the strips, error ranges (σbi ) are represented by the strips (see legend). The markers and the bars, at the right hand side, represent the overall means (x) and the errors (σ), respectively (U1 diamond, E triangle, and M circle) over the 47 surveys.  86  Chapter 4. Circulation and Transports to distinguish it from the error in the time series. A detailed analysis shows that there are minor differences between the TD and QSS transports each year from November to January. QSS transports U1 and E are slightly higher than TD transports during that period: on average 3% of x for both U1 and E. QSS transports M and W2 are lower than TD transports during the same period: on average 12% of x for W2 and 45% for M. As a consequence, the magnitudes of U1 , E and M with TD overlap with each other between October and February. The magnitudes of U1 , E and M in QSS are distinct between December and February but overlap with each other earlier in fall between October and December. During March-April 2003 and 2004, the two-month oscillations are larger with TD than with QSS. However, when they occur in the case of TD the magnitude of U1 overlaps with the magnitude of M within the uncertainties. This is also true between E and M. This suggests that mixing exchange in the TD case could be as large as the net upward entrainment and the horizontal transports. On the other hand, the QSS estimates have smaller oscillations and only the magnitude of U1 tends to overlap with E. In both cases, during March-April 2003 and 2004, more cruises were carried out per month. Thus, possible short timescale processes may have been sampled and more probably undersampled. This may have introduced noise in the transports estimates. In the TD case, short timescale variations are introduced in the transport estimates. Such a difference between transports over circulation timescales and transports over shorter timescales has been previously observed in Chesapeake Bay: oceanic exchange rate has been estimated to be on average (8±2.3)×103 m3 s−1 [Austin 2002], while mass transports have been estimated between -2×104 m3 s−1 and 4×104 m3 s−1 when they were induced by short-timescale (2–3 days) local and remote wind events [Wong and Valle-Levinson 2002]. Note that,  87  Chapter 4. Circulation and Transports in Wong and Valle-Levinson [2002], local winds could force net mass transports to be landward temporarily. This analysis shows that the seasonality of the transports increases when the time dependence is included in the budget equations. The difference between TD and QSS is likely significant in the case of the surface outflow U1 according to the results of the F-test (section 4.2.1). Separation Depth In this section, the influence of the separation depth (d) is investigated by using average transports x and the corresponding error σ over a range of d. A realistic range of d is between 15 and 50 m. Within this range, the choice of a particular d has no significant effect on the transports U1 , U2 , E and M. The values taken by the transports are similar within the uncertainties (Fig. 4.3). The circulation transports take values around 4.5×104 m3 s−1 for U1 and 4×104 m3 s−1 for U2 and E. These values x change by less than σ with any change of d within 15– 50 m (Fig. 4.3): U1 , U2 and E change by about or less than 3% of x, while M changes by 10%. The circulation transports (U1 , U2 and E) reach a weak maximum when d is in the range 20–30 m (close to the default depth), but the difference between minimum and maximum values in that range of d is of the same magnitude as the error (σ). Analysis and comparison of the actual transport time series indicate only small changes in the seasonal variability of the circulation transports when d is changed from 15 m to 30 m. As already observed in the case of d=30 m (section 4.2.1), U2 and E (=W2 − W1 ) remain significantly close to each other for any other chosen value of d. The average x and the associated error σ of the mixing exchange M and the total  88  Chapter 4. Circulation and Transports  4  7  x 10  U1 E 6  M Freshwater  5  mean U  4  mean E  3  m s  −1  1  3  2 mean M  1  0 Jul  Oct 2003 Apr  Jul  Oct  04  Apr  Jul  Oct  05  Apr  Figure 4.2: Transport Estimates and Their Estimated Errors in QSS. Overall means (x) and overall errors (σ) are the vertical lines and markers, at the right hand side, (U1 diamond, E triangle and M circle). QSS overall means and overall average errors are represented by the large markers and the thick lines. 89  Chapter 4. Circulation and Transports upwelling W2 (=E + M) increase proportionally to the increase of d. However, the changes are small. A linear approximation of the sensitivity suggests a 1 m change in d causes M to change by only 102 m3 s−1 . Doubling (halving) d from 30 m on average increases M by only +16% (or -8%). The linear trend indicates that even with a small d (a few meters) there is still mixing exchange occurring: about 1.3×104 m3 s−1 for d=2 m. The slope of the other transports drops faster than the slope of M between 2 and 5 m, but it seems unlikely that the actual d is this shallow (Fig. 3.1). It is more realistic to expect the shallowest reasonable value of d to be about 15 m. Overall, this analysis indicates that the transports are relatively insensitive to changes of d in the range 15–50 m. The default choice of d=30 m leads to only small departure between the estimated transports and the transports for the true d, whatever it may be. Therefore, even if the true value of d differs from the default there should be little effect on the analysis done here. Inversion Parameters In chapter 2, the concept of a trade-off parameter (α) was introduced (Eq. 2.15) and generalized (Eq. 2.22). In this section, the influence of this parameter on the transports is investigated. Fig. 4.4 shows that bias and variability are linked. However, as shown next, the sensitivity of the results to changes in α are small. The trade-off between bias and variability is relatively balanced. This is the reason why the default value of α turns out to be a reasonable choice. In the default solution, the value of α was set to 2×10−5 m−3 s, that is s−1 (Table 2.2, Chapter 2). Now we rewrite the trade-off parameter as α=γ × s−1 where γ is a scaling factor of the default parameter (s−1 ). To show this scaling factor in the  90  Chapter 4. Circulation and Transports  4  7  x 10  W2 6  3 −1  volume transport (m s )  5 U1 E, U2  4  3  M  2  1  Default depth 30 m  0 2  5  10  15  20 25 30 35 separation depth (m)  40  45  50  Figure 4.3: Circulation Sensitivity to Separation Depth. Overall averages and errors of U1 , E, U2 , M and W2 over 47 surveys are the vertical lines and markers. Averages and errors of U2 and E have been shifted right and left, respectively. This prevents 91 vertical bars showing errors from overlapping.  Chapter 4. Circulation and Transports equations, Eq. 2.22 is rewritten as: J = εT W−2 ε + γ 2 zT S−2 z = εT W−2 ε + zT (γ −1 S)−2 z.  (4.4)  Eq. 2.22 can be equivalently rewritten as: J′ = γ −2 εT W−2ε + zT S−2 z = εT (γW)−2 ε + zT S−2 z.  (4.5)  These equations show that it is mathematically equivalent to scale down the matrix S by γ −1 and scale up the matrix W by γ. Note that the scale difference between J and J′ is not important when finding the minimum. In turn, the scaling of the inversion parameters changes the size of the estimated transports. In Fig. 4.4, transports over the whole time series have been averaged at a given value of γ. Fig. 4.4 shows that the transports increase with decreasing γ and they approach the values found for the simple SVD, case γ →0. The variability (vertical bars) tends to increase more markedly than the average. The transports can be mathematically represented by a vector form of Eq. 2.24 while introducing the scaling factor γ: r  x ˆ = xA + i=1  −1 A λ′i u′ T i W (b − Ax ) ′ vi . λ′ i 2 + γ 2 s−2  (4.6)  Eq. 4.6 shows the behaviour of the transports with decreasing γ. When γ decreases the coefficients in front of the vi′ ’s (range vectors of the solution) increase, as well as the norm of x − xA (the deviation of the transports about the a priori averages). When γ is small enough (i.e. when γ ≪ λ′i × s) Eq. 4.6 approaches Eq. 2.9 (the simple SVD equation) because the cost function becomes: J = εT W−2 ε  (4.7)  When γ is large enough (i.e. when γ ≫ λ′i × s), the second right-hand term of Eq. 4.6 is down-weighted by γ −2 and the first right-hand term, the a priori solution xA , 92  Chapter 4. Circulation and Transports prevails, leading to transports very close to the a priori averages. This last behaviour is also clearly shown in Fig. 4.4 in the magnitude of both transport averages and overall variations. This shows that choosing the scaling factor is equivalent to trading between bias and variability. When γ becomes large (γ >1), the transports are more biased toward the a priori averages and have less variability. Reciprocally, when γ becomes small (γ <1), the transports are less biased toward the a priori averages and have more variability. However, note that the curves in Fig 4.4 are roughly horizontal. This suggest that the a priori transports are close to the true transports, but might also imply that the data and the inversion procedure provide little information about transports. Thus, the optimal solution reverts to the a priori transports. In order to determine whether this is the case, we will now deliberately vary the a priori transports to see the effects on the solution. A Priori Transport Averages This section investigates the influence of the a priori averages (xA with components xA i ’s equal to U01 , U02 , W01 or W02 in the default case, Table 2.2) on the transports. The analysis of Fig. 4.5 shows that when the magnitude of the a priori averages decreases, the transports tend to have large variability. They also tend to have a larger magnitude than the a priori averages. Reciprocally, when the magnitude of the a priori averages increases, the transports tend to have a small variability. In addition, they tend to take values close to the a priori averages but smaller than the a priori averages. The chosen a priori averages lead to transports with which the two aforementioned effects are relatively well-balanced. Thus, the data has an effect on  93  Chapter 4. Circulation and Transports  4  8  x 10  Simple SVD case γ=0  7  W  2  Transport (m3 s−1)  6  5 U1 E, U  4  2  3  2  M  Default case γ=1, α=s−1  1 0.1  0.2  0.5  γ  1  2  5  10  Figure 4.4: Circulation Sensitivity to the Trade-off Parameter α=γ s−1 . Overall averages and errors of U1 , E, U2 , M and W2 over 47 surveys are the vertical lines and markers. 94  Chapter 4. Circulation and Transports the solution, confirming the a priori estimates. In more detail, the change of xA is introduced by the scale parameter β such that the new a priori averages are β xA instead of xA . Eq. 4.6 (with default α) can thus be written:  r  x ˆ = β xA + i=1  −1 A λ′ i u′ T i W (b − β Ax ) ′ vi λ′ i 2 + s−2  (4.8)  where β appears in both right-hand terms. Fig. 4.5 shows the ratio between the transports and the a priori averages (i.e. the ratios xˆi /(β xA ˆi equals to U1 , i ) with x E, M, and W2 ) as a function of β. Both transport average and variability magnitude (vertical bars) of the transports change with β. Eq. 4.8 suggests that the transports take values somewhere between the true and the a priori estimates. If the transport −1 estimates were only based on the data, the ratios xˆi /(β xA . i ) would only vary as β  The ratios at β=2 would be close to half the ratios at β=1. If the transport estimates were only based on the a priori information, the ratios xˆi /(β xA i ) would be close to 1. Observing the curves in Fig. 4.5, neither of these situations occurs. The ratio −1 xˆi /(β xA nor always close to 1. i ) is not proportional to β  In Fig. 4.5, when β ≫1 the ratios xˆi /(β xA i ) tend to be smaller than 1 within error bars. It is subtle for both U2 and E, but very marked for U1 and M. Since ˆ lie between the true estimates and the a priori estimates, when the the estimates x a priori values becomes greater than the true values the ratios xˆi /(β xA i ) have to become smaller than 1. When β ≪1 the ratios xˆi /(β xA i ) tends to be greater than 1. When β decreases, all the ratios xˆi /(β xA i ) increase. It is not as clear as when the situation β ≫ 1 (especially M) because the error bars are larger when β ≪ 1. Since ˆ lie between the true estimates and the a priori estimates, when the the estimates x a priori values become less than the true values the ratios xˆi /(β xA i ) have to become greater than 1. 95  Chapter 4. Circulation and Transports Thus as suggested by this analysis of the sensitivity to β, in the inversion procedure where the default a priori averages are used (β=1, Fig. 4.5), the transports estimates are close to both the a priori estimates and the true estimates. The analysis above suggests that the data information and the inversion procedure provide enough information to approach the true estimates. This indicates that the curves in Fig. 4.4 are roughly horizontal because the a priori estimates are similar to the true values.  4.2.3  Residuals of the Conservation Equations  Figs. 4.6a–c show the residuals ε1 , ..., ε6 of the budget equations (Eqs 2.66–2.71) and the associated bootstrap variation. In the inverse procedure, the weighting scales, ω1 , ..., ω6 , are the a priori values of ε1 , ..., ε6 . The weighting scales have been defined in section 2.1 and Table 2.2 so that all the equations could be ranked against each other’s absolute magnitude and be scaled to have the same weight. In addition, the ωi ’s give an idea of the sizes of the residuals when all the source and forcing terms in ˆ ) are the budget equations take on large values. If the actual residuals (when x = x smaller than the ωi ’s, one can expect that the inversion procedure worked reasonably well. As suggested above, at the end of the sensitivity analysis (section 4.2.2), the estimated transports (ˆ x) are close to both the a priori solutions (xA ) and the true solutions (x). This should be also true of the actual residuals. The mathematical framework in chapter 2 enables one to verify this criterion and to write the residuals (ˆ ε) with respect to the residuals associated with the a priori solution (εA ).  96  Chapter 4. Circulation and Transports  3 U1  U2  E  M  Ratio between transport and a priori average  2.5  2  1.5  E, U2 1 U1 M  0.5 0.1  0.2  0.5 1 β (log scale)  2  5  10  Figure 4.5: Circulation Sensitivity to the A Priori Parameter β . β is the ratio A between the new a priori average β xA i and the corresponding default value xi ,  (Table 2.2, U01 , U02 , W01 , and W02 ). Overall averages and errors of U1 , E, U2 , M 97 and W over 47 surveys are the vertical lines and markers. 2  Chapter 4. Circulation and Transports The actual residuals, using Eq. 2.20 and Eq. 2.32, can be written as follows: m  r T  (u′ i W−1 εA )Wu′ i +  ˆ= ε i=r+1  i=1  −1 A s−2 (u′ T i W ε ) Wu′ i ′2 −2 λi + s  (4.9)  where εA = AxA -b, u′ i the range vectors of the SVD of W−1 A, W the weighting ˆ . Note that the a matrix (see Eq. 2.25), the coefficient s that scales the solution x priori residuals (εA i ’s) are easily rewritten: m T  A  (u′ i W−1εA )Wu′ i  ε =  (4.10)  i=1  Using Eq. 2.11, the residuals associated with the “pseudo-inverse” can be rewritten: m T  (u′ i W−1εA )Wu′ i  ε=  (4.11)  i=r+1  Thus, Eq. 4.9 suggests that the residuals (ˆ εi ’s) cannot be as small as one would require. They are smaller than the a priori residuals (εA i ’s, Eq. 4.10), and bigger than the residuals associated with the “pseudo-inverse” (Eq. 4.11). An analysis of the sum of the squared scaled residuals (last row in Table 4.2), the quantity that is minimized in the inverse procedure (see chapter 2 and Eq. 2.15), indicates that the information in the observations is useful and leads to an improvement of the estimated transports relative to the a priori transports. The overall appearance of the residuals (Fig. 4.6a–c and Table 4.2) indicates that the actual residuals (ε′j s) are smaller than or close to the a priori residuals (ωi ’s). This suggests that the residuals (εj ’s) are consistent. As suggested earlier in this chapter (section 4.2.1), the useful information provided by the observations is the seasonality of the transports. In the mass equations (Fig. 4.6a), ε1 (top box) is one order of magnitude smaller than ω1 (on average 3.1% of ω1 and maximum 11%), while ε2 (bottom box) is one to two orders of magnitude smaller than ω2 (on average 1.1% of ω2 and maximum 98  Chapter 4. Circulation and Transports Equation  A priori Residuals  Residuals with x=x ˆ average  Residuals with  (in % of ωj )  x = xA  (in % of ωj )  maximum  average  maximum  Top Mass  ω1 =1.6×104  3.1  11 14  69  Bottom Mass  ω2 =1×104  1.1  4.4  10  10  Top Heat  ω3 =2.7×105  10.5  28 11  31.5  Bottom Heat  ω4 =1.5×105  30  72 27.5  71  Top Salt  ω5 =4.7×105  10  25 9  29  Bottom Salt  ω6 =3.5×104  8  33 73  251  Summed scaled  squared εi ’s 0.54 1.1  6.9  m 2 i=1 (εi /ωi )  0.16  Table 4.2: A Priori and Estimated Values of the Residuals of the Conservation Equations in the SoG Box Model . The last line provides the average and maximum sums of the squared scaled residuals with x = x ˆ and x = xA . 4.4%). The residual ε1 is usually a lot larger than ε2 (on average 3 times larger). The residual ε2 is usually close to zero because the residual curve lies within the error bars (Fig. 4.6). Note that the residuals ε1 and ε2 are smaller when x = x ˆ than when x = xA (Table 4.2, last column). This means that even with a constant solution x = xA , the residuals are smaller than the weighting scales ω1 and ω2 . In section 2.2, Eqs 2.41– 2.42, associated with the residuals ε1 and ε2 , are exact because the transports U1 , 99  Chapter 4. Circulation and Transports U2 , W1 and W2 are the surface integrals of horizontal and vertical flow speeds as defined by Eqs 2.43–2.47. As a consequence, only the estimation error associated with the transports can affect the residuals. The error associated with the external sources is either very small (in the case of the freshwater) or zero (in the bottom box). Further analysis showed that ε2 is more sensitive to change of the inversion parameters than ε1 , but they both remain one order of magnitude smaller than the expected residuals. The difference between ε1 and ε2 is more strongly marked than what the order between ω1 and ω2 suggested: ε1 ≃3 ε2 instead of ω1 ≃2 ω2 . The residuals of the heat equations (Fig. 4.6b) are also smaller than or close to the weighting scales ω3 and ω4 . ε3 (top box) is on average 10.5% of ω3 and maximum 28%, while ε2 (bottom box) is on average 30% of ω4 and maximum 72%. The bottom heat residual has a clear seasonal variability with maximum in summer and minimum in winter. These minimum and maximum are smaller but close to the a priori residuals. All these values are close to the residuals found with the a priori transports. This could suggest either a larger error than expected in estimation (section 2.2) or in model misfit (Eq. 2.4). The residuals of the salt equations (Fig. 4.6c) are smaller than or close to the weighting scales ω5 and ω6 . ε5 (top box) is on average 10% of ω5 and maximum 25%, while ε6 (bottom box) is on average 8% of ω6 and maximum 33%. The bottom salt residual takes values close to zero within the error bars, but these error bars are large. These minimum and maximum are smaller but close to the a priori residuals. As suggested above this could be due to estimation error or model misfit. Assuming a model misfit ∆ A, a solution error ∆ˆ x and a source error ∆ b, the residuals in Eq 2.4 become: ε = (A + ∆ A)(ˆ x+∆x ˆ) − (b + ∆ b)  (4.12) 100  Chapter 4. Circulation and Transports or ε = (A + ∆ A)∆ x ˆ+∆Ax ˆ−∆b  (4.13)  with Aˆ x = b for an exact solution x ˆ. In the case of residuals ε1 and ε2 , ∆ A is zero because the equations are exact. Since the external sources are well-known (ε1 ) or there is none (ε2 ), ∆ b is negligible. Eq. 4.13 shows that the residuals ε1 and ε2 depend only on the estimation error ∆ˆ x. Since the solutions have been found by an inverse procedure where the tapered solutions are biased, the bias can propagate to the residuals (Eq. 2.20). This bias is a linear combination of the components of the vector b, which contains the external sources. Summarizing the analysis of the residuals, the optimal solution is consistent with the a priori solution. Overall, it minimizes the residuals similarly to the a priori solution. However, the optimal solution provides an improvement over the a priori solution as suggested by Eq. 4.9. It minimizes the residuals of the mass budgets and the bottom salt budget further more than the a priori solution (Table 4.2).  4.3 4.3.1  Discussion General Circulation in the Box Model  In the previous section, we obtained 4 time series of the circulation (Fig. 4.1) in a box model representation of the SoG system. In this representation, surface outflow has a significant seasonal variation according to a conservative analysis of variance (Table 4.1). The sensitivity analysis showed that the estimated transports were relatively insensitive to the separation depth between the model boxes, but that time dependence was a necessary assumption to conserve a significant seasonal signal in the time series. The analysis of sensitivity to the inversion parameters and the a 101  Chapter 4. Circulation and Transports  3  m3s−1  2  Top mass Eq., a priori ε1 = 1.6×104  ×103  Bottom mass Eq., a priori ε2 = 104  (a)  1 0 −1 −2 Jul  Oct 2003 Apr  Jul  Oct  04  Jul  Oct  05  Apr  Top heat Eq., a priori ε3 = 2.7×105  5  1.5  Apr  x 10 (b)  Bottom heat Eq., a priori ε4 = 1.5×105  °Cm3s−1  1 0.5 0 −0.5 −1 Jul  Oct 2003 Apr  Jul  Oct  04  x 10(c)  3 −1  Oct  05  Apr  Bottom salt Eq., a priori ε6 = 3.5×104  1 m s  Jul  Top salt Eq., a priori ε5 = 4.7×105  5  1.5  Apr  0.5 0 −0.5 −1 −1.5 Jul  Oct 2003 Apr  Jul  Oct  04  Apr  Jul  Oct  05  Apr  Figure 4.6: Residuals of the Mass, Salt and Heat Equations of the SoG Box Model . Although there is a seasonal variability in b), its magnitude is well below the a priori scale of 1.5×105 . 102  Chapter 4. Circulation and Transports priori averages showed that the estimated transports were likely close to the true transports. Finally, the residuals analysis showed that the residuals in the budget equations were, overall, consistent with the a priori residuals. Although transport estimates of the SoG circulation have been previously made in various ad-hoc ways [England et al. 1996, Pawlowicz 2001, Pawlowicz et al. 2007], this is the first set of estimates based on a rigorous inverse procedure. The monthly timescale of these estimates can help to understand the net effect of all processes, including tides on the estuarine circulation. In addition, physical analysis of the estuarine circulation and historical freshwater data (later in section 4.3.5) suggest that, one can, from an estimate of the freshwater discharge, readily obtain estimates of the estuarine circulation for any period outside the 2002–2005 window. The time series in section 4.2.1 provide monthly information (biweekly during spring 2003 and 2004) on the circulation based on the 3 years of observations. However, the use of a timescale limited to a monthly resolution prevents one from analyzing the detailed influence of fortnightly tides contributing to the vertical mixing (unlike Li et al. [1999]). This would require a smaller time resolution, for instance a daily time resolution. The inverse box model is constrained by, not only the conservation of mass and salt, but also conservation of heat [Pawlowicz and Farmer 1998], in contrast to Li et al. [1999] and Pawlowicz et al. [2007]. Adding heat conservation is useful because the estimated transports have to be consistent with an additional set of constrains, which should improve accuracy. The heat budget may also be important when evaporation or temperature stratification are important (e.g., Roson et al. [1997]), although it is not the case here. Previous studies of the SoG [England et al. 1996, Li et al. 1999, Pawlowicz 2001, Pawlowicz et al. 2007] have also estimated the surface outflow and deep inflow trans103  Chapter 4. Circulation and Transports ports (see Table 4.3 for detail about the averages and ranges of the SoG transport estimates and the approach used). In Table 4.3, overall the average transport estimates, from 3.6±1.1 to 5.5±2 m3 s−1 , and the winter-summer outflow range are consistent with each other across these studies. However, the estimates of Pawlowicz et al. [2007] are slightly higher (5–5.5 m3 s−1 ) because their approach allows a greater seasonal variability by introducing deep water renewal inflow (0.4–4 m3 s−1 ) that combines with the deep inflow (4 m3 s−1 ). Note that a more detailed comparison of my estimates with those of Li et al. [1999] is carried out in section 4.3.2 and a discussion about the use of additional boxes is in section 6.2. From the outflow and inflow transports one can calculate a difference. The difference can be used to estimate a net rate of export of a tracer outside of the SoG. It indicates a net outflow if the difference is significantly larger than 0, and an exchange rate if the difference is not. Thus, it is useful to keep the outflowing and inflowing transports independently constrained by the observations, as in Pawlowicz [2001], Pawlowicz et al. [2007], and unlike, for instance, Li et al. [1999] who assumes that the outflow equals the inflow. However, the estimation of the net outflow is a difficult task to carry out because it usually involves subtracting two large and similar layer transports together [Godin et al. 1981, LeBlond 1983] and the estimated net outflow has the same size as the estimated error. The net outflow from the SoG is discussed later in section 4.3.4. Although I have studied the sensitivity of the the inverse solution to various model parameters, the box model (Fig. 2.1) is itself only an approximation. Two aspects of the real SoG, not reflected in this sensitivity analysis, are the effect of the Northern Entrance and the variability in the separation depth. The layer transports through the Northern Entrance are generally assumed to be smaller than the ones through the Southern Entrance because the cross-sectional 104  Chapter 4. Circulation and Transports area of the channels is markedly smaller. It is not clear from the STRATOGEM and JEMS observations alone if the layer transports through the Northern Entrance are negligible. If there is no significant difference of water properties between Johnstone Strait (JS) and Haro Strait (HS), the layer transports in the SoG box model would be the sum of the layer transports at the Northern and Southern Entrance. Thus, the layer transports through the Northern Entrance could contribute to the net horizontal transport (U1 -U2 ) in the model as much as the layer transports through the Southern Entrance, if they were significantly different from each other. In the SoG box model, the separation depth d between the top and the bottom boxes is the separation depth between the surface seaward transport (U1 ) and the deep landward transport (U2 ). Observations in the SoG suggest that d is likely to change over a year from deep values in winter to shallow values, due to wind-induced mixing and convection of cooling water, in summer, due to stratification by freshwater addition and surface heating, (section 3.3.1, Chapter 3). However, in section 4.2.2, the analysis showed that the transports are relatively insensitive to changes of d, above and below 30 m.  4.3.2  Seasonal Transports  A detailed comparison of my results with those of Li et al. [1999] (Figs 4.7a–d) shows a good general agreement between the models despite a difference in the approach and the model setup. Both transports have the same seasonal pattern with low winter and high summer magnitudes: in winter: 4×104 m3 s−1 for my estimate versus 3×104 m3 s−1 for Li et al. [1999]’s; in summer: 6×104 m3 s−1 for my estimate versus 7×104 m3 s−1 for for Li et al. [1999]’s. The large summer and small winter magnitudes in the case of Li et al. [1999]’s model are due to the additional fortnightly-tide modulation of  105  Chapter 4. Circulation and Transports the estuarine circulation which is not captured in our model (section 4.2.1). The magnitude of the seasonal change is significantly smaller (about ±1×104 m3 s−1 for my estimate and ±2×104 m3 s−1 for [Li et al. 1999]) than the average magnitude of the transports (4.5×104 m3 s−1 for my estimate and 5×104 m3 s−1 for Li et al. [1999]). The good agreement occurs generally during the freshet maximum although there are some differences: a difference in the timing of the freshet maximum of about a month (exactly the end of June in Li et al. [1999]’s and June in ours), secondary peaks are present before and after the freshet maximum in my estimated transport but do not appear in Li et al. [1999]’s. These differences are due to the two different approaches. Li et al. [1999] use an idealized freshwater inflow and idealized deep salinity to force their model while I used observed conditions and forcings in my box model. The relatively small variations of W2 (section 4.2.1), the total upwelling transport (overall seasonality magnitude σe of 0.4×104 m3 s−1 and average transport of 6.2×104 m3 s−1 ), suggests a relatively constant upward transport. This particular property of W2 is useful to estimate the overall residence time in the euphotic and aphotic zones of the SoG. The analysis of Eqs 2.68–2.71 show that the time rate of heat and salt outward advection leads to a residence time τ =volume/outflow: in the top box (T1 and S1 ) τ1 =V1 /(W2 + R)=32±2 days, in the bottom box (T2 and S2 ) τ2 =V2 /W2 =169±8 days. Given the relative invariance of W2 and its large magnitude with respect to R, these residence times are also relatively constant for the SoG box model. The residence times τ1 and τ2 represent characteristic e-folding timescales in the top and the bottom boxes. The salinity and the temperature adjust to 90 % in about 2 × τ1 in the top box (2 × τ2 in the bottom box). For instance, a change of the external source of surface (bottom) salinity from S1 (S2 ) to S0 would take about two τ1 or two months (about two τ2 or a year). In this example, I assumed 106  Chapter 4. Circulation and Transports that the external source remained constant for the sake of simplicity, but in reality the external source would change. The resident times then indicate the lag in the observation of the SoG properties relative to those of the source. This property was used in Pawlowicz et al. [2007]. 4  4  12  x 10  12  (b)  (a) 10  10  8  8  6  6  4  4  2  Li et al. 1999′s exchange flow U1  0 2002F M A M J J A S O N D  x 10  2 0 2003F M A M J J A S O N D  Figure 4.7: Comparison Between Li et al. [1999]’s Horizontal Exchange Transport and Estimated U1 Transport: (a) U1 2002 estimate, (b) U1 2003 estimate  4.3.3  Comparison with Transports in SoJdF  One further step can be taken to validate my estimates with other studies considering the larger system of SoG/HS/SoJdF [Godin et al. 1981, England et al. 1996, Pawlowicz 2001, Masson and Cummins 2004, Thomson et al. 2007, Sutherland et al. 2011]. 107  Chapter 4. Circulation and Transports  4  4  12  x 10  12 (c)  x 10  (d)  10  10  8  8  6  6  4  4  2  2  0 2004F M A M J J A S O N D  0 2005F M A M  Figure 4.7: Comparison Between Li et al. [1999]’s Horizontal Exchange Transport and Estimated U1 Transport: (c) U1 2004 estimate, and (d) U1 2005 estimate An estimate of the inflowing and outflowing transports from HS can be determined by analyzing the recirculation fraction of surface water from the SoG [Pawlowicz et al. 2007]. The recirculation fraction of surface water defines the fraction of the surface water that leaves the SoG surface, mixes with water from the HS and finally reenters into the SoG at depth. The mixed water of HS contains both surface water from the SoG and deep water coming from the Strait of Juan de Fuca (SoJdF). The deep water that comes from the SoJdF has retained the oceanic characteristics of the Pacific Intermediate Water (PIW). The PIW leaves the deep ocean, enters the  108  Chapter 4. Circulation and Transports SoJdF through the Juan de Fuca Canyon, and is advected with little mixing until it enters HS [Pawlowicz 2001, see Figs 5 a-d]. Similarly to Pawlowicz et al. [2007], the recirculation fraction δ can be used to express the mixing of PIW and surface SoG water in HS: qH = δ q1 + (1 − δ) qP IW  (4.14)  where q is the tracer (salinity, temperature, nutrients or O2 ) from either the SoG surface (q1 ), HS (qH ) or PIW (qP IW ). Eq. 4.14 simply expresses that HS water is a mixture of SoG surface water and PIW. The recirculation fraction gives the fraction of q1 that reenters the SoG by seawater intrusions (U2 ). In my box model, the PIW properties are calculated from a climatology of offshore data at depths between 100200 m using CTD and bottle vertical profiles collected by IOS (Institute of Ocean Sciences, Sydney, Canada). Eq. 4.14 is based on the assumption that the dominant processes that change the characteristics of HS water are advection and mixing of water entering and leaving HS. One can apply a simple one-box model to HS (see Fig.  4.8), and write the  equations of conservation of mass and the tracer q as follows: U1 − U2 = U3 − U4  (4.15)  q1 U1 − qH U2 = qH U3 − qP IW U4  (4.16)  where U3 is the surface outflow to the SoJdF and U4 the deep inflow from the SoJdF. When rearranged Eqs 4.15–4.16 lead to Eq.4.14 and δ=  U1 . U1 + U4  (4.17)  In theory, only positive values δ ≤1 correspond to the assumption that advection and mixing dominate the processes affecting any tracer in HS. Otherwise, Eq. 4.14 109  Chapter 4. Circulation and Transports  Figure 4.8: Advective Transports in HS Box Model. The advective transports U1 and U2 are the seaward and landward transports to the SoG from Fig. 2.1, respectively. U3 and U4 are the seaward and landward transports to the SoJdF, respectively. is not true. However, in practice, there are several issues that can lead to δ values outside the range 0≤ δ ≤ 1 and δ values different from the true values. We are using box averages and a climatology to approximate the properties of the water masses from the SoG, HS and the SoJdF. Therefore, one can expect that differences between the estimated and true water properties can occur and yield a few wrong values of δ out of the 47 that we had to estimate. Since we assumed that the advection and  110  Chapter 4. Circulation and Transports Reference  Average Transports  Comment  (×104 m3 s−1 ) This study  Li 1999  Pawlowicz 2001  Formal inverse 2-box model  2002-2005  4.5±0.5(1σb )  outflow  4±0.5(1σb )  inflow and net upward flow  3.4–6.2  winter-summer outflow range  Flows in a forward box model:  from numerical simulations  4.5  mean value  3–7  winter-summer outflow range  Inverse 2-box model:  SoG/HS/SoJdF system  4.6±1.1  outflow (July 1998)  3.6±1.1  inflow (July 1998)  Pawlowicz 2007 3-Box mixing model:  2002-2005  5–5.5 (±2)  annual inflow and outflow  4.4–8  winter-summer inflow range  4.7–9  winter-summer outflow range  4±2  summer deep renewal inflow  0.4±0.02  winter deep water inflow  4±2  constant deep water inflow  Note: Marinone 1996  3D numerical model: flow speed no transport estimates Table 4.3: SoG Transports from Previous Studies 111  Chapter 4. Circulation and Transports mixing processes dominate the processes that affect a tracer, a tracer that necessarily satisfies Eq. 4.14 is a conservative tracer, e.g. salinity. If there are other processes affecting a tracer apart from advection and mixing and if the contribution is close to or larger than the contribution of advection and mixing, larger errors on δ are likely to occur. As a consequence of these issues, the conservative tracers that rely on the most accurate box averages lead to the most reliable estimates of δ. Thus, one would expect that the salinity is the most reliable tracer to use to estimate δ with values within the range 0≤ δ ≤1 and small errors. On the other hand, dissolved O2 is likely to be less reliable than salinity since O2 can be added by photosynthesis near the surface, removed by respiration at any depth by various organisms, and removed at depth by biogeochemical reactions. In the case of temperature, apart from advection and mixing only surface heat fluxes can affect temperature near the surface. When I estimated δ using salinity, δ always satisfied 0≤ δ ≤1. The median of δ is about 0.35±0.1 based on salinity, 0.55±0.35 based on temperature, and 0.7±0.5 based on O2 . The medians of temperature and salinity suggest that δ average is about 0.45, while the medians of the three tracers suggest that δ average is about 0.53. All these estimates of δ are reasonable because they suggest that the HS and SoG outflows have similar order of magnitude. Li et al. [1999] did not estimate this parameter, but their results imply exchange flows between SoG and HS very similar to the exchange flows between HS and SoJdF. Thus, applying Eq. 4.17 with U1 =U4 , an estimated value for Li et al. [1999]’s δ is also about 0.5. Using values at 100 m depth in HS, Pawlowicz et al. [2007] obtained 0.6 based on three different tracers and smaller estimated errors, but their most reliable estimate of δ, based on salinity, is 0.52±0.02. Their fractions obtained when using water-column averages in HS are 0.1 lower and get closer to our values: 0.5, based on the three different tracers; 0.42, 112  Chapter 4. Circulation and Transports based solely on salinity. Pawlowicz et al. [2007]’s smaller estimate errors can be explained by the use of spot measurements for SoG tracers at either station S4-1 or station S5 (see Fig. 1.1). By rearranging Eq. 4.17, one can estimate the order of magnitude of the deep inflow into HS (U4 ): U4 = (δ −1 − 1) U1 .  (4.18)  The range of my estimates of recirculation fraction is 0.35–0.7. The average values around 0.5 suggests that the layer transports at both ends of HS are similar. This is a situation similar to Li et al. [1999]’s where surface outflow and deep inflow are close. The range of the estimated HS layer transports is (1.9–8.4)×104m3 s−1 . These values are smaller than previous estimated layer transports in HS or in the SoJdF [Godin et al. 1981, Pawlowicz 2001, Masson and Cummins 2004, Thomson et al. 2007]: values in HS or the SoJdF are of the order of 10×104 [Thomson et al. 2007, Table 3, column 3, Qin is ∼8.7×104 ] while values on the Pacific end of the SoJdF are higher, about 20×104 . It is reasonable to find layer transports in SoG or in the northern end of HS smaller than or equal to the averaged layer transports in HS or SoJdF. In an estuarine system the magnitude of the layer transports tends to increase towards the mouth of the system because of entrainment. In the case of the SoG/HS/SoJdF the mouth is the Pacific end of the SoJdF.  4.3.4  Net Outflow from the SoG  Data assessment based on the resolution matrix of the inverse problem (Eq. 2.32 in section 2.1) and the small residuals of Eqs 2.66–2.67 (see section 4.2.3) suggest that the relationship between the SoG layer transports (U1 and U2 ) and the freshwater discharge (R) is consistent with: U1 − U2 = R  (4.19) 113  Chapter 4. Circulation and Transports Eq. 4.19 represents the net outflow from the SoG. My results from the SoG box model provide a rough estimate of the net outflow from the SoG, regardless of whether the contribution of the Northern Entrance to the transports is neglected or included in the box-model transports. The estimated net outflow from the SoG is consistent with the net outflow from the SoJdF. The assumptions in Eq. 4.15 are that the net outflow out of SoG and the net outflow out of HS are likely correlated and are both associated with the SoG freshwater forcing. Eqs 4.15 and 4.19 are combined into: U3 − U4 = R  (4.20)  Eq. 4.20 approximates the net outflow from the HS. The next step would be to determined the SoJdF net outflow by estimation the layer transports in the SoJdF. Since the SoG/HS/SoJdF works as a large estuarine system [Thomson 1994, Pawlowicz 2001, Sutherland et al. 2011], due to the upward entrainment of deep water, one expects that the layer transports of HS and SoG would be smaller than those in the SoJdF. Previous studies show that, generally, there is a net outflow in the SoJdF [Godin et al. 1981, Thomson et al. 2007] with a mean magnitude of 2×104 m3 s−1 (about ∼2× R at R freshet peak). Fig. 4.1 shows that the freshwater discharge is likely to be significantly larger than the estimated error of the transports only during the freshet maximum (e.g. 6.2×104 m3 s−1 for U1 , 4.8×104 m3 s−1 for U2 with error bars smaller than their difference in June 2002). Eq. 4.19 implies that one will only observe a net outflow from SoG significantly different from zero during the same interval. The estimated transports in Fig. 4.1 used to estimate the net outflow in Eq. 4.15 are always larger (around 4×104 m3 s−1 ) than the freshwater inflow and they have average errors (±0.5 ×104 m3 s−1 ) of the same size as the average freshwater inflow. Godin et al. [1981] al114  Chapter 4. Circulation and Transports ready suggested that surface outflow and deep inflow were very close: their estimate of the net transport was not significantly different from zero within their estimated error bars. LeBlond [1983]’s justification of this result relies on the argument that net transport comes from the difference of large but close transports. All the other studies either assume exchange transports [Li et al. 1999] or assume that the difference is of the order of the freshwater outflow from the Fraser River [Pawlowicz 2001, Pawlowicz et al. 2007]. Results in Fig. 4.1 indicate that net outflow occurs intermittently. A net outflow is significantly different from 0 only during particularly large freshet discharges compared to the transports errors, e.g. in June 2002.  4.3.5  Circulation Sensitivity to the Freshwater Inflow  In section 4.2.1, it was noted that the circulation transports (U1 and E) and the mixing exchange (M) are correlated with the freshwater inflow (R). There is an overall trend for U1 to increase with increasing R, and for M to decrease with increasing R. But, the trend of U1 is easier to show that the trend of E and M because the larger variance in E and M hides any seasonality (see Table 4.1). An analysis of the correlation between the seasonal variability of U1 and R, and between the seasonal variability of M and R shows a strong and significant correlation for U1 (0.6±0.16 within a 90% confidence interval) and a very weak but significant anti-correlation for M (-0.25±0.23 within a 90% confidence interval). Fig. 4.9 shows the relationship between U1 and R. The overall relationship between U1 and R can be approximated by U1 =(1.1±0.5)R+(4±0.3)×104 m3 s−1 . This linear approximation is close to the the mass equation between the horizontal transports (Eq. 4.19): U1 = E + R  (4.21)  115  Chapter 4. Circulation and Transports with constant E. The seasonality of E (σe ) can be larger than 0.5×104 m3 s−1 , but the F-test shows that most of it is hidden by the uncertainty in the estimates. However, the physical meaning of such a relationship is unclear. It would imply an estuarine circulation in the absence of runoff (i.e. R=0). The linear approximation given above only works because the range of the freshwater R does not cover values lower than 2×104 m3 s−1 . Since the estuarine circulation is driven by the freshwater inflow, one would expect that U1 and E tend toward 0 as R tends toward 0. Previous studies using either analytical models [Chatwin 1976] or numerical models [Hetland and Geyer 2004] of estuaries showed that in theory U1 (or E) varies as a small fractional power of R, e.g. R1/3 . Baker and Pond [1995] were studying the estuarine circulation in Knight Inlet, but found no linear correlation between the surface dewinded transport and the river discharge. Fig. 4.9 shows the data and the curve aR1/3 with a a proportionality constant (a=2.68×103 m2 s−2/3 ). Other curves of the form aR1/n , with n taking various fractional and integer values, have been tried to quantify how tight the fit of U1 to R1/3 is by comparison to other possible powers. Fig. 4.10 shows the plot of the residuals of aR1/n , that is the sum of the squared misfits between the curve aR1/n and the data. Residuals quickly get close to a minimum value when n is 3 or larger. Although it is difficult to determine the exact fractional power, it is clear that the relationship is not consistent with large powers. The mass conservation equation (Eq. 2.66) and the very small variations of W2 (with W2 ) suggest that: U1 = −M + constant  (4.22)  in other words that U1 and M could be anti-correlated. Similar analysis of the relationship between M and R and fits with power func-  116  Chapter 4. Circulation and Transports tions of the form R1/n were attempted. However, the analysis was not conclusive and no satisfying fit could be found. For instance, the analysis showed that even a constant value for M could represent the relationship between M and R as accurately as the functions R1/n with n≥ 1. Given the role of freshwater in controlling the estuarine circulation, one may expect to detect a rapid decrease of U1 as R becomes small in winter. Within the range of observed R, the analysis of U1 shows no such general and rapid decrease of U1 . However, this is not incompatible with the theory. Within the range of observed R, the curve aR1/3 is approximately linear and only starts to steeply curve down when R < 2×103 m3 s−1 . The value of 2×103 m3 s−1 is the smallest value for R found in our data. The analysis of the relationship between U1 and R and the correlation between U1 and M also indirectly suggests that M could depend on R although the variance in M was too large to allow a direct analysis (see F-test in Table 4.1). The basic conclusion is that the estuarine transport is in practise not very sensitive to changes in freshwater input. However, this conclusion is consistent with results from analytical and numerical models [Chatwin 1976, Hetland and Geyer 2004], as well as some observations in an inlet [Baker and Pond 1995].  4.3.6  Interannual Variability of the Circulation  Fig. 4.11 shows, for each year and each transport, the median value x, the average variability σb and the annual range (range between seasonal minimum and maximum). The transport averages do not change a lot from year to year. The mean of R does not have a clear pattern while the mean of the transports reaches an extremum in 2004. Overall, Fig. 4.11 suggests that annual range of all the transports changes from year to year, but these changes are small and it is not clear that they are  117  Chapter 4. Circulation and Transports 4  7  x 10  6  5  1  U ms  3 −1  4  3  2  1  0 0  2  4  6  8 3 3 −1 R ×10 m s  10  12  14  16  Figure 4.9: Surface Seaward Transport Plotted With Respect to Freshwater Inflow . The dashed line represents the theoretical curve aR1/3 (a=2.68×103), the thick line the empirical fit E + R with E=(4±0.3) ×104 m3 s−1 (Eq. 4.21). 118  Chapter 4. Circulation and Transports  4  3.5  4  sum of squares scaled into an average error (×10 )  3  2.5  2  1.5  1  0.5  0 0  1 1 1 1 16 8 4 3  1 2  23 34  1  2 n  3  4  Figure 4.10: Residuals, or Sum of Squares εi 2 , of the fit of U1 as a power of R. The sums of the squares have been scaled into an average error using  47 i=1 εi 2  47  119  Chapter 4. Circulation and Transports correlated to the interannual variability of R. Such a correlation of the transports was expected at least in the case of U1 since the previous section suggests a clear relationship between U1 and R. Note that the data in 2002 and 2005 are limited to the last 8 months and the first 4 months, respectively. The annual averages of 2003 and 2004 estimated with all the data, the last 8 months of the year, and the first 4 months of the year indicate that no large difference occur in the 3 ways to estimate the annual averages. This suggests that the estimated annual averages of 2002 and 2005 are likely close to the true annual averages. Fig. 4.12 shows, for each year and each transport, σe , the seasonal variability. Since the data over 2002 and 2005 do not cover a complete year period, the effect of removing data from the time series has been also investigated. The seasonal variability σe has been recalculated for the years 2003 and 2004 for each layer transport by omitting the first 4 months or the last 8 months. The analysis shows that σe varies by 14% maximum (W1 ) and 8% on average, except for σe of W1 over the last 8 months of 2004 that varies by about 50%. This indicates that the trend found for the annual statistics of the transports are not significantly affected by using shorter annual time series. There is a common pattern between U2 , W1 and W2 suggesting that in 2003 σe is the largest of all the observed years. In the last two years, σe of all the transports has decreased compared to their σe in 2002 and 2003. On the other hand, the σe of R reaches the largest value in 2002, and it is decreasing over 2003–2004. This is consistent with the time series of R in Fig. 4.1: R reached a peak of 1.5×104 m3 s−1 in June 2002. Fig. 4.12 could suggest a delay of about a year in the response of the SoG to change in the freshwater inflow, but this is not realistic. Li et al. [1999] carefully modelled interannual variability of the freshwater inflow and PIW salinity to study the adjustment time of the circulation. They varied the freshet maximum, 120  Chapter 4. Circulation and Transports the salinity maximum, the freshet timing, and the freshet duration time. Li et al. [1999]’s analysis showed that the circulation adjusted within a few months. This is consistent with estimates of residences time for the surface SoG which are a few months (see section 4.3.2). During deep water renewals, residence time of deep water can be two months although deep water renewals occur once a year [Pawlowicz et al. 2007]. Such residence times cannot affect annual averages in Fig. 4.12 and the 2003 maximum of U2 , W1 and W2 is more likely the result of the error σb (Table 4.1), i.e. the sampling and estimation error tends to hide the seasonality of the transports U2 , W1 and W2 . This suggests that only the curve of U1 represents likely interannual variability. However, over 2004–2005, the annual trend differs between U1 (decrease) and R (increase). This suggests that the error σb is still too large in the estimated transport U1 and prevents one from clearly estimating the interannual variability of U1 .  4.3.7  Conclusions  Here, I will summarize the main points of this chapter. First, the statistical analysis of the layer transports of the SoG circulation suggests that the small seasonality of U1 (Fig. 4.1), the surface outflow, is significant, and the seasonality magnitude is sensitive to taking into account time dependence in the equations. Secondly, despite a clear interannual variability of the freshwater input (Fig. 1.4), there is no clear indication of interannual variability of the estuarine circulation (Figs 4.11 and 4.12). These characteristics of the SoG estuarine circulation can be summarized by an observational relationship found between U1 , the surface outflow and R, the freshwater input. The observational relationship is consistent with the theory of estuarine  121  Chapter 4. Circulation and Transports  4  8  x 10  th  th  16 and 84 percentiles median U 1 , △ U 2 and E, o W1 and M, x W2 , and ⋆ R  7  whiskers: min and max  W2 U1  U2 (and E)  5  4  3  volume transport (10 m s  −1  )  6  4  3  W (and M) 1  R 2  1  0 2002  2003  2004  2005  Figure 4.11: Annual Mean of SoG Transports and Freshwater Inflow  122  Chapter 4. Circulation and Transports  U  1  9  U and E 2  W1 and M W  2  R  8  7  W1 (and M)  U  1  U2 (and E)  5  σe  (103 m3s−1)  6  4 R  3 W2 2  1 2002  2003  2004  2005  Figure 4.12: Annual Variability of SoG Transports and Freshwater Inflow  123  Chapter 4. Circulation and Transports physics [Chatwin 1976, Hetland and Geyer 2004] and can be approximated by aR1/n with n≤3 and a=2.68×103 m2 s−2/3 on a physical basis or empirically by E + R with E=(4±0.3) ×104 m3 s−1 (Eq. 4.21). Further examination of the inverse procedure suggests that the layer transports are not sensitive to the separation depth of the box model, the inversion parameters, and the a priori transports (see section 4.2.2). In addition, the residuals of the mass, heat and salt conservation equations (Eq. 2.22) are consistent (Table 4.2). The inverse procedure provides an improvement since the sum of the squared residual of the estimated solution are smaller than the sum of the squared residuals of the a priori solution.  124  Chapter 5 Nutrients Uptake and Primary Productivity 5.1  Introduction  Primary production (PP) is the amount of carbon (C) fixed by primary producers. In the ocean, primary producers are phytoplankton. Phytoplankton use the energy from the sun to fuel biogeochemical reactions that transform inorganic C (CO2 ) into organic C (photosynthesis). A by-product of photosynthesis is O2 . Photosynthesis also requires macronutrients containing inorganic nitrogen (N) and phosphorus (P). Diatoms, a particularly abundant type of coastal phytoplankton also require silicic acid (containing Si and thereafter referred as Si) to build their frustules (siliceous outer structure). In our study, the biological processes associated with the uptake of macronutrients and the production of dissolved O2 are inferred from budget equations of N, P, Si and O2 and from knowledge of the non-biological advective processes (described in Chapter 2 and estimated in sections 5.2.1–5.2.2). The rate at which PP occurs is called primary productivity (PP rate). PP rate can be determined directly by measuring the rate of organic C production, O2 production, or the uptake of nutrients. PP and PP rate are not simple concepts because they can be defined and determined in different ways. Gross PP (GPP) is defined as the total amount of organic C fixed by phytoplankton. From GPP, net PP (NPP) is obtained 125  Chapter 5. Nutrients Uptake and Primary Productivity after subtracting autotrophic respiration (denoted Ra , respiration by phytoplankton only). When the PP rate is estimated over a long period of time, for instance a day, the PP rate of interest is often the NPP rate. In our study, the PP rate is estimated over a month (or a few weeks during freshet). For this reason, the estimated PP rate is an estimated NPP rate. The net community production (NCP) takes into account not only the autotrophic respiration but also the heterotrophic respiration (by bacteria and zooplankton). The NCP can be estimated with the net biological production of O2 . The NPP rate can vary from season to season (section 5.2.4). NPP is usually low in winter low light levels. The first bloom of the year usually corresponds to the largest amount of PP and is usually called the “spring bloom”. Thus, spring is defined here as the period when the spring bloom can occur. In the SoG, spring is therefore the period from early February until late April, as suggested by chlorophyll-a (chla) measurement (shown later in section 5.2.4, Fig. 5.8). The other seasons are defined relative to the SoG spring bloom. Summer is from early May until late July, fall from early August until late October, and winter from November until late January. Nitrogen chemical species like nitrite, nitrate (both hereafter jointly referred as nitrate) or ammonium are taken up by phytoplankton to fix organic C. Depending on which chemical form of nitrogen is used by phytoplankton, NPP is differently identified. The NPP associated with nitrate use is called new NPP since this form of nitrogen is generally replenished by upwelling or input from outside of the euphotic zone, although it is possible to remineralize nitrate in the euphotic zone under low light conditions [Sarmiento and Gruber 2006]. In this study, the new NPP rate can be estimated from the disappearance of near-surface nitrate. This disappearance is assumed to occur because of net biological uptake. The NPP associated with ammonium is called regenerated NPP and it depends on the efficiency of nitrogen 126  Chapter 5. Nutrients Uptake and Primary Productivity regeneration and the amount of sinking of organic nitrogen. The sum of the new and regenerated NPP is called total NPP or simply NPP. One can approximate NPP with new NPP if most of the nitrogen uptake relies on nitrate. In this chapter, I will 1. derive three-year observational estimates of seasonal and annual averages of primary productivity based on nutrient budgets in the SoG (with uncertainty estimation).  2. show that, based on the data, primary production in the SoG is mainly due to new primary production (chapter 5 and further detail in chapter 6).  3. show that the estimated primary production rate over 2002-2005 is typical of NPP rates in temperate estuaries.  4. find that, as suggested first by Mackas and Harrison [1997] for N, estuarine entrainment is also the largest supply of P and Si (as suggested by our analysis of nutrient budgets).  5. find that the annual averages of Si:N and Si:P uptake ratios ( (24.7±2.2):16 and (24.7±2.2):1, respectively ) are larger than the expected ratios [Brzezinski 1985].  In the biological box model (section 2.4), the surface uptake rate of nutrients is represented as a sink rate term in the 3 conservation equations of surface nutrients 127  Chapter 5. Nutrients Uptake and Primary Productivity (negative terms). The surface production rate of O2 is represented as a source rate term (positive term) in the conservation equations of surface O2 . I will collectively refer to these four biological terms as either sink terms or net biological uptake rates because a source rate is merely a negative sink rate. In addition to the biological sink terms, there are advective sink terms in the conservation equations. In the biological box model, I will discuss all the terms appearing in the budget equations to determine which terms are important. In the budget equations, the supply rate terms are the contribution of the net upwelling (advective term), and the riverine inflow (source term), while the sink rate terms are the net biological uptake rate and the advective export (advective term). Since most of these rate terms vary seasonally, an analysis has to be made for each season. Once the terms in the budget equations are determined (using the methods of Chapter 2 and the input data of Chapters 3 and 4), it is possible to estimate primary productivity by scaling the sink rate terms as sources of organic C. The relationships between C, nutrients and O2 have to be determined. The linearity and non-linearity of these relationships are explained in terms of error estimates (N and P) and seasonal variability (Si and O2 ). The average slope of each relationship is compared to the average molar ratios of the elemental composition (C, N, P, Si and O2 ) of phytoplankton in seawater found by Redfield et al. [1963], Brzezinski [1985] and Anderson [1995]. The best estimate of the NPP rate is then chosen. The validity of the estimate is investigated by analysis of the error estimates, comparison with the biomass, and consistency between NPP rate estimates for different nutrients. For the rest of this chapter, we will assume that the new NPP is almost completely equivalent to NPP. In the discussion section 5.3, we will show that this is likely the case on seasonal and annual timescales. Finally, these results are discussed at the end of this chapter.  128  Chapter 5. Nutrients Uptake and Primary Productivity  5.2 5.2.1  Results Supply and Sink Rates of Nutrients  Figs 5.1-5.3 show time series of the surface supply and sink rates of various processes included in the budget equations (section 2.4, Eqs 2.77, 2.79, and 2.81). These include the computed biological sink rate terms and storage rate terms (time derivative terms). In particular, the biological sink rate terms are the terms (φ1 )P , (φ1 )N , and (φ1 )Si defined by:  ∂P1 − PR R + P1 U1 + P1 W1 − P2 W2 + ε′1 ∂t ∂N1 − NR R + N1 U1 + N1 W1 − N2 W2 + ε′3 (φ1 )N = V1 ∂t ∂Si1 (φ1 )Si =V1 − SiR R + Si1 U1 + Si1 W1 − Si2 W2 + ε′5 ∂t (φ1 )P =  V1  (5.1) (5.2) (5.3)  where processes, for instance for P, are net upwelling (P2 W2 − P1 W1 ), advective export (−P1 U1 ), river inflow (PR R), net biological uptake (φ1 )P and the surface storage/drawdown rate term (V1 ∂P1 /∂t). Note that we will use the words “storage rate term” or “storage rate” instead of “storage/drawdown rate term”. The air-sea flux of nutrients is small compared to other processes (based on assumptions made in section 2.4), thus it is neglected with respect to the other terms in the equations. For each of these processes, the general shape of the curves for P, N, and Si is very similar: the maxima and minima occur at the same time. The seasonality of each process is analyzed in further detail later in this section. The net upwelling term (positive term) tends to have the largest magnitude with the smallest variability. This reflects the near-constant upwelling and near-constant levels of nutrients in 129  Chapter 5. Nutrients Uptake and Primary Productivity  2000  1500  1000  mol s−1 (nitrate)  500  0  −500  −1000  −1500  Net Upwelling Rivers Zero Line Net Biological Uptake  −2000  Advective Export Storage Rate  −2500 Apr  Jul  Oct 2003 Apr  Jul  Oct  04  Apr  Jul  Oct  05  Apr  Figure 5.1: Surface Box Nitrate Supply and Sink Rate  130  Chapter 5. Nutrients Uptake and Primary Productivity  150  100 Net Upwelling Rivers Zero Line Net Biological Uptake  50  Advective Export  mol s−1 (phosphate)  Storage Rate  0  −50  −100  −150 Apr  Jul  Oct 2003 Apr  Jul  Oct  04  Apr  Jul  Oct  05  Apr  Figure 5.2: Surface Box Phosphate Supply and Sink Rates  131  Chapter 5. Nutrients Uptake and Primary Productivity  4000  3000  2000  mol s−1 (silicic acid)  1000  0  −1000  −2000  −3000  Net Upwelling Rivers Zero Line Net Biological Uptake  −4000  Advective Export Storage Rate  −5000 Apr  Jul  Oct 2003 Apr  Jul  Oct  04  Apr  Jul  Oct  05  Apr  Figure 5.3: Surface Box Silicic Acid Supply and Sink Rates  132  Chapter 5. Nutrients Uptake and Primary Productivity  4  2  x 10  Net Upwelling Rivers Air−sea O Flux 2  1.5  Net Biological Production Advective Export Storage Rate  1  mol s−1 (O2)  0.5  0  −0.5  −1  −1.5  −2 Apr  Jul  Oct 2003 Apr  Jul  Oct  04  Apr  Jul  Oct  05  Apr  Figure 5.4: Surface Box O2 Supply and Sink Rates  133  Chapter 5. Nutrients Uptake and Primary Productivity the deep water. The P net upwelling increases slightly over 2002–2005 (Fig. 5.2) due to an increase of the P levels in deep waters (about 0.5 µM over 3 years, Fig 3.7) while this was not the case for N and Si (Figs 3.8 and 3.9). There is no satisfactory explanation based on our analysis of the SoG data for the increase of deep P concentration. The explanation may be associated with processes specific to P that we have neglected. In particular, P concentration has been found to change because of adsorption/desorption on particles and sediments [Lebo 1991], possibly modulated by salinity as observed in saline lakes [Clavero et al. 1993] (adsorption rate increases with salinity), on organic matter, iron oxides, aluminium oxides and apatite [Lucotte and D’Anglejan 1983, Lebo 1991, McDowell and Sharpley 2003] (possibly modulated by salinity like in saline lakes), and even on phytoplankton cells as suggested by a recent study [Fu et al. 2005]. Another possible explanation is the large-scale increase of the deep P concentration in the North Pacific. Such a long-term increase of deep P has been previously observed in the Western subarctic Pacific [Ono et al. 2001]. However the increase was only 0.015 µM over 3 years [Ono et al. 2001]. All these processes are beyond the scope of this study. The other supply rate term, the river input rate (positive term), had the smallest magnitude compared to all the other terms. The river terms are especially small for N and P. In the case of Si, the river input rate is larger and reached a significant rate of 1000 mol s−1 in summer (maximum of June 2002, Fig. 5.3), but this is still 50% smaller than the net upwelling term. The advective export rate (negative term) has the largest magnitude of all the sink rate terms. It has a seasonal cycle with high magnitude in late winter-early spring and low magnitude the rest of the year. In contrast with N and P, the Si term had a more complex seasonal cycle of advective export. Fig. 5.3 shows that the magnitude of summer Si advective export can be as high as the following maximum 134  Chapter 5. Nutrients Uptake and Primary Productivity magnitude of late winter-early spring, in some years (for instance summer 2002 and winter 2003). The net biological uptake rate (mostly negative term) has the largest change in magnitude in spring, a magnitude comparable to advective export in summer, and a low magnitude the rest of the year. The net biological uptake rate is expected to remain purely negative. Positive values occur but are generally indistinguishable from zero within the uncertainties. In addition, the positive values of the net biological uptake rate were all of small magnitude:  100 mol N s−1 ,  10 mol P s−1 and  500  mol Si s−1 . After the net biological uptakes, the magnitude of storage rate terms (positive except in spring) is higher than the other sink rate terms during spring, but it is smaller the rest of the year. The budget terms and their relative importance in the budget equations can be made more quantitative by analyzing annual and seasonal averages (Tables 5.1-5.3). The largest annual rate term was, on average, the upwelling supply of nutrients: 1273 mol N s−1 , 104 mol P s−1 , and 2168 mol Si s−1 , with average error bars representing 5% of the upwelling supply rates (Tables. 5.1- 5.3, column 2 row 5). These rates have a very small seasonality (highest around summer, lowest in winter) with seasonal and annual means overlapping within 1 or 2 times the average error bars (±57 mol N s−1 , ±4.5 mol P s−1 and ±94 mol Si s−1 , Tables 5.1-5.3, last row, column 2). Net upwelling was always the largest source of nutrients to the surface SoG. This supply rate of nutrients was almost completely balanced by net biological uptake and advective export. Over the annual timescale, storage rate is zero and river inputs although positive, are relatively small. On average, about a third (P and Si) to a half (N) of the net upwelling supply rate was used for the net biological uptake: -543 mol N s−1 , -38 mol P s−1 , and -765 mol Si s−1 (Table 5.3, column 3 row 5). In winter, the net biological uptake was not significantly different from zero 135  Chapter 5. Nutrients Uptake and Primary Productivity within the uncertainties. The seasonal analysis shows that, in summer, the net biological uptake rates of N and Si were similar (N: -844 mol s−1 , Si: -840 mol s−1 very close to 1:1) and the rates of N and P were close to the Redfield ratio (N: -844 mol s−1 , P: -54 mol s−1 , 15.6:1) [Redfield et al. 1963]. Since these average rates are close to the expected ratios [Brzezinski 1985, Redfield et al. 1963], this may suggest that phytoplankton, in particular, diatoms, a group of siliceous phytoplankton, were blooming during summer although surface nutrient concentrations in summer were at the lowest level of the year (Figs 3.7–3.9). On the other hand, this may imply competition between diatoms and the other phytoplankton groups that are better adapted to low nutrient concentrations as suggested later in Fig. 5.13, section 5.3 [Miller 2004, pp 12–14]. For instance, in summer, C biomass estimates (not shown here) based on phytoplankton taxonomy of water samples suggests that the biomass of silicoflagellates was larger than the biomass of diatoms. Silicoflagellates are nanoplankton equipped with a hollow siliceous skeleton that are competing with diatoms for Si [Takahashi 1987] The seasonal analysis shows that, in spring, the net biological uptake rate of Si is larger than the rate of N (N: -796 mol s−1 , Si: -1460 mol s−1 , about 1.8:1) and the rates of N and P were again close to the Redfield ratio (N: -796 mol s−1 , P: -57 mol s−1 , 14:1). The Si:N ratio of uptake rates (about 1.8:1) and taxonomy data (Fig. 5.13) suggest that spring diatoms were dominant and heavily silicified, compared to summer diatoms. Although Si:N ratio is an average and variations around the average are possible [Brzezinski 1985, Brzezinski et al. 2003a], ratios close or larger than 2:1 are usually unlikely. However, ratios of 2:1 and larger have been recently observed during iron- and Si-enrichment experiments in mesocosms containing Central Equatorial Pacific water [Marchetti et al. 2010]. The spring Si:N and Si:P ratios will be discussed in further detail in section 5.3. 136  Chapter 5. Nutrients Uptake and Primary Productivity The overall importance of the different rate terms in the seasonal analysis was very similar to that of the annual average rates. The largest rate term was usually net upwelling, followed by advective export and net biological uptake. There were a few exceptions to this order because of the different seasonality of different terms. In particular, the rate terms associated with the net biological uptake (N, P and Si) decrease from spring/summer (largest magnitude) to winter (smallest magnitude or negligible). The timing of the seasonal maximum and minimum of N uptake rate was very similar to that of P and Si. Spring and summer rates were similar within the uncertainties for P and N (but not for Si, spring rate about 2 times that of the summer rate). The advective export of N, P and Si had a small seasonality from spring to fall. During winter, advective export of P, N and Si was usually larger (larger negative values). The storage rate terms also had a seasonal pattern. The spring average rate took the largest negative values (P -20 mol s−1 , N -299 mol s−1 , and Si -530 mol s−1 ), indicating a high drawdown during spring blooms while the largest positive value could occur either in fall (P 14 mol s−1 and N 229 mol s−1 ) or winter (Si 172 mol s−1 ). A large positive storage term and the high levels of fall and winter surface nutrients (Figs 3.7–3.9) suggests a net gain of nutrients in the surface layer due to the estuarine entrainment. The N and P storage rates were negligible in summer.  137  Chapter 5. Nutrients Uptake and Primary Productivity  5.2.2  Supply and Sink Rates of Oxygen  Fig. 5.4 shows time series of the terms appearing in the budget equation (section 2.4, Eq. 2.83). In particular, the source rate term is the term (φ1 )O2 defined by: (φ1 )O2 =V1  ∂O1 − OR R + O1 U1 + O1 W1 − O2 W2 ∂t  (5.4)  −kO2 Osaturation − Osurf ace + ε′7 (5.5) where surface box processes are the net upwelling (O2W2 − O1 W1 ), advective export (−O1 U1 ), river inflow (OR R), net biological production (φ1 )O2 , the air-sea exchange flux kO2 Osaturation −Osurf ace , and the surface box storage/drawdown rate term (V1 ∂O1 /∂t). Note that, like in the previous section 5.2.1, we will use the words “storage rate term” or “storage rate” instead of “storage/drawdown rate term”. The O2 time series provide a different picture from nutrient time series (Fig. 5.4). Unlike in the nutrient budgets, in the O2 budget the advective export (negative term) was the largest sink rate terms. The advective export had also one of the largest magnitudes with some seasonality. The largest magnitude occurs during late spring-early summer. There is a slower export during the rest of the year, but its magnitude is never smaller than 0.75×104 mol s−1 . The net biological production of O2 (positive term) was the next largest rate term with large peaks in spring (largest in April 2004, about 2×104 mol s−1 ). The net upwelling (a positive term) has a large magnitude (0.6×104 mol s−1 on average) with a seasonal cycle in opposite phase with advective export. That is, periods of greater upwelling input are also periods of greater advective loss. The storage term (a mostly negative term) and river input (positive term) have similar magnitude but opposite sign and nearly out-of-phase seasonal cycles with values closest to zero near winter. That is, periods of greater O2 138  Chapter 5. Nutrients Uptake and Primary Productivity N  Net  River  Upwelling  Net Biological  Advective  Uptake  Export  Storage  Spring  1266  33  -796  -802  -299  Summer  1303  51  -844  -552  -42 (*)  Fall  1337  17  -446  -678  229  Winter  1183  26  -34 (*)  -1075  99  Mean  1273  33  -543  -770  -7 (*)  ±57  ±4  ± 93  ±56  ±59  Apr’02-Apr’05 Average uncertainty Table 5.1: Averages of Surface Supply and Sink Rates of Nitrate (mol s−1 ) in the Euphotic Zone of the SoG. A positive term represents a supply while a negative term represents a sink rate. Average uncertainties are based on the average standard error of the seasonal and annual values. The terms have been daily interpolated to make it possible to average the terms over a few complete years. The row “Mean Apr’02-Apr’05” provides averages of the term over 3 years starting on and finishing on Apr 1st . (*) These values are not significantly different from zero. river input are also periods of greater O2 negative storage rate (drawdown rate). The magnitude of the air-sea O2 flux was also large. The largest air-sea outflux (negative 139  Chapter 5. Nutrients Uptake and Primary Productivity P  Net  River  Upwelling  Net Biological  Advective  Uptake  Export  Storage  Spring  104  1  -57  -68  -20  Summer  108  2  -54  -54  2 (*)  Fall  107  1  -31  -63  14  Winter  96  1  -6 (*)  -84  7  Mean  104  1  -38  -67  1 (*)  ±4.5  ±0.1  ±6  ±4  ±4  Apr’02-Apr’05 Average uncertainty Table 5.2: Averages of Surface Supply and Sink Rates of Phosphate (mol s−1 ) in the Euphotic Zone of the SoG. A positive term represents a supply rate while a negative term represents a sink rate. Average uncertainties are based on the average standard error of the seasonal and annual values. The terms have been daily interpolated to make it possible to average the terms over a few complete years. The row “Mean Apr’02-Apr’05” provides averages of the term over 3 years starting on and finishing on Apr 1st . (*) These values are not significantly different from zero. term) usually occurs in spring (Fig. 3.11, surface water oversaturated by biological production), while the largest influx (positive term) usually occurs in fall (Fig. 3.11, 140  Chapter 5. Nutrients Uptake and Primary Productivity  Si  Net  River  Upwelling  Net Biological  Advective  Uptake  Export  Storage  Spring  2255  265  -1460  -1590  -530  Summer  2119  596  -840  -1723  153  Fall  2243  230  -657  -1682  133 (*)  Winter  2057  185  -62 (*)  -2007  172  Mean  2168  327  -765  -1748  -18 (*)  ±94  ±38  ± 182  ±122  ±118  Apr’02-Apr’05 Average uncertainty Table 5.3: Averages of Surface Supply and Sink Rates of Silicic Acid (mol s−1 ) in the Euphotic Zone of the SoG. A positive term represents a supply rate while a negative term represents a sink rate. Average uncertainties are based on the average standard error of the seasonal and annual values. The terms have been daily interpolated to make it possible to average the terms over a few complete years. The row “Mean Apr’02-Apr’05” provides averages of the term over 3 years starting on and finishing on Apr 1st . (*) These values are not significantly different from zero.  141  Chapter 5. Nutrients Uptake and Primary Productivity  O2  Net  River  Upwelling  Net Biological Advective  Air-sea  productivity  Export  Flux  Storage  Spring  6260  1599  8965  -12562  -4121  141 (*)  Summer  7124  3386  4619  -13467  -3563  -1901  Fall  5674  1707  3543  -10319  -729 (*)  -125 (*)  Winter  4680  1434  3478  -10726  2224  1091  Mean  5976  2070  5174  -11831  -1639  -249 (*)  ±589  ±220  ± 1190  ±534  ±1042  ±239  Apr’02-Apr’05 Average uncertainty Table 5.4: Averages of Surface Supply and Sink Rates of Dissolved O2 (mol s−1 ) in the Euphotic Zone of the SoG. A positive term represents a source rate while a negative term represents a sink rate. Average uncertainties are based on the average standard error of the seasonal and annual values. The terms have been daily interpolated to make it possible to average the terms over a few complete years. The row “Mean Apr’02-Apr’05” provides averages of the term over 3 years starting and finishing on Apr 1st . (*) These values are not significantly different from zero.  142  Chapter 5. Nutrients Uptake and Primary Productivity spring 2004 and about 0.75×104 mol s−1 in October 2003. The budget terms and their relative importance in the budget equations can be made more quantitative by analyzing annual and seasonal averages (Table 5.4). In Table 5.4, on average, the largest rate term was the advective export (-11831 mol s−1 , column 5). The net upwelling (5976 mol s−1 , column 2) and the net biological production rate (5174 mol s−1 , column 4) were of similar magnitude. Least important were the river input (2070 mol s−1 , column 3) and the air-sea exchange flux (-1639 mol s−1 , column 6). On average, over a few years the storage term was, as expected, negligible within error bars (Table 5.4, Column 7). The overall importance of the different rate terms in the seasonal analysis was very similar to that of the annual analysis. The largest rate term was advective export, followed by net upwelling and net biological production rate, air-sea exchange flux and river input. There were a few exceptions to this order because of the different seasonality of these terms. In particular, the net biological productivity decreases from spring (8965 mol s−1 ) to fall and winter (3478 mol s−1 in winter). The magnitude of the air-sea O2 outflux (negative term) had also a similar seasonality, decreasing from a spring maximum (-4121 mol s−1 ) to a fall minimum (-728 mol s−1 ). In winter, the conditions switched from oversaturation to undersaturation and the air-sea O2 flux became an influx (2224 mol s−1 ). The storage terms also had a seasonal pattern. The largest magnitude occurred in summer (O2 -1901 mol s−1 ). The O2 storage term indicates a maximum drawdown in summer.  143  Chapter 5. Nutrients Uptake and Primary Productivity  5.2.3  Property Uptake Ratios  In order to obtain an estimate of the net primary production rate (NPP rate), the following equation can be used: NPP = GPP − Ra  (5.6)  NPP = NPPn + NPPr  (5.7)  or  where Ra is the autotrophic respiration, and NPPn and NPPr are new NPP and regenerated NPP, respectively. New NPP is a net amount of C fuelled by external sources of nitrogen (N): e.g., nitrate upwelling, N2 fixation. New NPP rate can be computed by using the f-ratio, the ratio of new N uptake over the new and recycled N uptake. This f-ratio can only be estimated when the N cycle is accurately known, i.e. ammonium and nitrate uptake, nitrogen fixation and nitrification [Sarmiento and Gruber 2006]. In upwelling coastal regions, most of the new N is upwelled nitrate, thus the f-ratio can be estimated as the ratio of nitrate uptake over the ammonium+nitrate uptake. Observed f-ratios from different coastal areas were as high as 0.8 [Eppley and Peterson 1979]. Legendre et al. [1999] suggest that f-ratio can be higher than 0.7 when nutrients are mainly supplied by circulation. In addition, the equivalence between NPP and new NPP rate only occurs when NPP is mainly driven by nitrate uptake. Later in this chapter and chapter 6, we will discuss the difference between the estimated new NPP rate and total NPP rate in the SoG. In section 5.2.1, it was noted that siliceous phytoplankton, mainly diatoms, contribute to the NPP. Their contribution to the new NPP rate can be analyzed further by looking at the uptake of silicic acid relative to the other nutrients. 144  Chapter 5. Nutrients Uptake and Primary Productivity The elements N, P and Si can be used to estimate the NPP rate or a particular contribution to the NPP rate. In the N budget (Eq. 2.79), if N represents only upwelled N (Fig. 3.8), the estimated NPP rate based on N will give an estimate of the new NPP rate. On the other hand, P is used in both new and regenerated primary production. Thus, the estimated NPP rate based on P will give an estimate of the (total) NPP rate. The Si removal is associated with the diatom fraction [Miller 2004, Sarmiento and Gruber 2006] of the total NPP rate because diatoms can use both upwelled and regenerated forms of N. The Si removal is due to the growth of siliceous phytoplankton, mainly diatoms. Before estimating the NPP rate, it is necessary to determine the relationship between the sink rate terms of the four sampled elements (P, N, Si and O2 ) and the unknown element (organic C). It is important to note that the ratios C:O2 :Si are more variable than the C:N:P ratios, and hence are treated differently. Using the notation introduced earlier the NCP rate is defined by: NCP = NPP − Rh  (5.8)  NCP = GPP − Ra − Rh  (5.9)  or  where Rh is the heterotrophic respiration. O2 is not only respired by phytoplankton (Ra ) but also by zooplankton, heterotrophic bacteria and other organisms (Rh ) that are not part of the NPP rate (and not included in the box model). Only if zooplankton and bacterial respiration is negligible compared to phytoplankton respiration, will the C:O2 ratio reflect the NPP rate and be equal to the empirical mole ratio associated with the phytoplankton composition. Before analyzing the relationships between the sink rates of the sampled elements, 145  Chapter 5. Nutrients Uptake and Primary Productivity it is necessary to choose one of them as a common element. C has not been measured, although it is the currency of NPP rate. I decided to use P as the common element for the analysis. Since the C:Si and C:O2 are complex and very variable in time, the choice of a common element for comparison can only be made between N and P. In upwelling coastal regions, new NPP is usually associated with nitrate because recycling of N, as ammonium, is inhibited by light and usually occurs outside the euphotic zone [Sarmiento and Gruber 2006]. Contribution of N2 fixation by cyanobacteria (e.g., Synechococcus and Prochlorococcus) to the surface N cycle is also unlikely in upwelling coastal regions [Miller 2004]. But the cycle of N can be complex and the observations collected for this study are limited to the observed concentration of nitrate and nitrite. On the other hand, the cycle of P is simpler being mainly controlled by external sources and used by both new and regenerated NPP [Sarmiento and Gruber 2006]. The observed mole ratios for C:N:P:Si have been shown to have a global average over the ocean and coastal areas. Average C:P:N:Si ratios are 106:1:16:16 [Brzezinski 1985, Redfield et al. 1963], while the C:O2 had different values according to different authors : C:O2 = 106:138 Redfield et al. [1963] and 106:150 Anderson [1995], Fraga et al. [1998]) as this depends on (among other things) the amount of O in nitrogen − − species (NH+ 4 , NO2 and NO3 ). In this study the C:O2 ratio when considering only  autotrophic respiration is chosen to be 106:150. Following the argument of Fraga et al. [1998], a ratio of 106:150 is preferable to a ratio of 106:138 because the primary source + of nitrogen, in the SoG box model, is nitrate (NO− 3 ), not ammonium (NH4 ), and  organic C can be used to build complex molecules like lipids, not just carbohydrates as assumed by Redfield et al. [1963] in the open ocean. The C:P molar ratio will be  146  Chapter 5. Nutrients Uptake and Primary Productivity assumed to be equal to 106:1. C : P = 106 : 1  (5.10)  If the productivity estimates based on P and N are very similar, then NPP will mostly be new NPP. In this case, there will be little regenerated productivity, so that NPP will be primarily new NPP. The extent to which the C:P uptake ratio approximation and the equivalence between new NPP and NPP rates, and the N and P sink rate terms hold will be discussed later in this chapter and chapter 6. Figs 5.5–5.7 show the surface N sink rate, surface Si sink rate and surface O2 source rate against the surface P sink rate. Each single-survey estimate is plotted with the corresponding error bars based on bootstrap replicate statistics (defined in Chapter 2) and each season is specified by a different marker. The average sink term and the corresponding error range is also indicated for each figure (vertical and horizontal dashed lines). All terms have been already converted into mol m−2 d−1 using the surface area given by the SoG hypsography (Fig. 3.2). Fig. 5.5 shows the surface N sink rate plotted against the surface P sink rate term. The surface N sink rate and the surface P sink rate are strongly correlated. Most of the points are slightly below the line that represents the empirical Redfield ratio N:P=16:1. A least-squares fit method yields a linear slope of (14.6±0.8):1. Most of the points are aligned along a straight line with lowest values of P and N uptake in winter, the highest values in spring, and intermediate values in fall and summer. A smaller ratio than 16:1 might be expected since the surface N sink rate (nitrate and nitrite) is associated with new NPP rate and the surface P sink rate with total NPP rate. The net biological uptake of P is on average (5.8×10−4 ) ± (6×10−5 ) mol P m−2 d−1 with a range of 0 to 16 ×10−4 mol P m−2 d−1 . Multiplying this P average uptake, (5.8×10−4 ) ± (6×10−5 ) mol P m−2 d−1 , by the average N:P ratio, 14.6:1, 147  Chapter 5. Nutrients Uptake and Primary Productivity leads to an average N uptake of (8.5×10−3 ) ± (9×10−4 ) mol N m−2 d−1 . In Fig. 5.5, the negative values are generally not significantly different from 0. Fig. 5.6 shows the surface Si sink rate plotted against the P sink rate. Most of the points are above the Si:P=16:1 line and below the Si:P=32:1 line. The 32:1 line was arbitrarily chosen to provide an upper bound of the largest Si:P ratios (average ratio, spring and winter ratios). The points in fall and summer are the closest to the 16:1 line with the highest ratios of Si and P uptake in fall (19.3±4:1 in fall, 14.4±3.8:1 in summer). Five fits constrained by the least-squares method yield linear approximations of slope (24.7±2.2):1 over the year, (28.1±3.8):1 in spring, (14.4±3.8):1 in summer, (19.3±4):1 in fall, and (28.7±14.4):1 in winter. Thus, high Si:P occur in winter, spring and overall of the year. The net biological uptake from the Si sink term was about (1.3×10−2) ± (2×10−3 ) mol Si m−2 d−1 , with a range from 0 to 6×10−2 mol m−2 d−1 . The negative values were not significantly different from 0. Fig. 5.7 shows the surface dissolved O2 source rate plotted against the P sink rate. The figure suggests that the O2 :P ratio is highly variable, not only from season to season, but also from survey to survey during the same season. Most of the points are scattered away from the Redfield ratio O2 :P=150:1 line. The highest values of O2 production and P sink rates are in spring and the lowest values tend to be in winter. Four fits constrained by the least-squares method yield linear approximations of slope (128±21):1 over the year, (143±29):1 in spring, (78±25):1 in summer, and (110±55):1 in fall. Annual average, spring and summer average ratios have similar uncertainties. The winter values of the O2 net production and P net uptake rates were small and noisy and led to winter uptake ratios with a large range. The best estimate of the winter average ratio is (70±358):1. This average ratio is indistinguishable from zero within the uncertainties. The net biological productivity inferred from the O2 148  Chapter 5. Nutrients Uptake and Primary Productivity source term is on average (8.3×10−2 ) ± (9×10−3 ) mol O2 m−2 d−1 , with a range from 0 to 23×10−2 mol O2 m−2 d−1 . Most of the negative values were generally not significantly different from zero. The analysis of the average ratios suggests a good agreement for N:P between the empirical Redfield ratio of 16:1 and the estimated value of (14.6±0.8):1 throughout the year. It also suggests a very similar variability for the surface P and N sink rates. In contrast, the Si:P and the O2 :P average ratios agreed on the order of magnitude but with less fidelity: Si:P =(24.7±2.2):1 instead of 16:1, O2 :P over the year (128±21):1 instead of 150:1. In the case of Si:P, the uncertainty is about 10% of the average. The spring and winter average ratios are markedly higher than 16:1. The overall annual average, spring and winter averages are greater than expected (between +50% and +100% on average). In the case of O2 , the overall year average, spring and fall averages are smaller than, but still close to, the 150:1 ratio. The spring ratio, 143:1, is the closest ratio to the Redfield ratio. All the O2 :P ratios are lower than 150:1 suggesting that other processes are using up the photosynthesized O2 . The decrease of the O2 :P ratio from spring to summer suggest an increasing heterotrophic respiration, in particular grazers respiration. Further discussion of these Si and O2 results will be deferred until section 5.3.  5.2.4  Estimates of Net Primary Productivity  The previous analysis based on P (being the common element for comparison) provides an estimate of P:N:Si molar uptake ratios of 1:(14.6±0.8):(24.7±2.2). The N:P estimate is very close to 16:1, the Redfield ratio. This suggests that the C:P:N molar ratios can reasonably be approximated by 106:1:(14.6±0.8), found in section 5.2.3. In Tables 5.1 and 5.2, the annual uptake averages of N and P are 543±93 molN s−1  149  Chapter 5. Nutrients Uptake and Primary Productivity  40 spring 35  summer fall winter  30  Average N (−−) and errorbars (...) N:P=(14.6±0.8):1  20  Nitrate 10  −3  mol N m d  −2 −1  25  16:1 line  15  10  5 Average P (−−) and errorbars (...) 0  0  5 10 −4 −2 −1 Phosphate 10 mol P m d  15  20  Figure 5.5: Surface Nitrate and Phosphate Uptake Rates. The slope of the dotted line is the overall N:P ratio 14.6:1, the slope of the dashed line, the Redfield ratio 16:1. 150  Chapter 5. Nutrients Uptake and Primary Productivity  Si:P=(24.7±2.2):1  spring summer fall 5  winter  spring (28.1±2.3):1 summer (14.4±3.8):1 fall (19.3±4):1  winter spring 24.7:1 line  32:1 line  winter (28.7±14.6):1  Silicic Acid 10−2 mol Si m−2d−1  4  fall Average Si (−−) and errorbars (...)  3  summer  2 16:1 line 1  0 Average P (−−) and errorbars (...) −1 0  5 10 −4 −2 −1 Phosphate 10 mol P m d  15  20  Figure 5.6: Surface Silicic Acid and Phosphate Uptake Rates. The overall molar ratio is 24.7:1 (slope of dash-dotted line) and the seasonal ratios (slope of dotted lines) are 28.1:1 (spring), 14.4:1 (summer), 19.3:1 (fall) and 28.7:1 (winter). The empirical ratio, 16:1 [Brzezinski 1985], and an arbitrary upper bound, 32:1, are the slopes of the dashed lines.  151  Chapter 5. Nutrients Uptake and Primary Productivity  150:1 line 25  128:1 line spring  spring summer fall  fall  winter 20  summer  Average O2 (−−)  winter  and errorbars (...)  2  Dissolved O 10−2 mol O m−2d−1  15  10 O :P=(128±21):1  2  2  spring (143±29):1 summer (78±25):1 fall (110±55):1  5  0  winter (70±358):1 −5  Average P (−−) and errorbars (...)  −10 0  5 10 Phosphate 10−4 mol P m−2d−1  15  20  Figure 5.7: Surface Dissolved O2 Release and Phosphate Uptake Rates. The slope of the dash-dotted line is the overall molar ratio 128:1. The slope of the dotted lines are the seasonal ratios 143:1 (spring), 78:1 (summer), 110:1 (fall), and 70:1 (winter). 152 The Redfield ratio 150:1 is the slope of the dashed line.  Chapter 5. Nutrients Uptake and Primary Productivity and 38±6 molP s−1 , respectively. The N uptake is associated with external sources of N and new net primary production (NPP) rate, while the P uptake is associated with any sources (recycling and external supply) of N and total NPP rate. Applying the empirical Redfield ratio N:P of 16:1 to the average uptake of P gives an estimate of the total average uptake of N (nitrate, nitrite and ammonium included): 608±96 molN s−1 . Given, the average accuracy of about 90 molN s−1 , there is no marked difference between the two uptake rates. This is consistent with the similarity between the observed Redfield N:P ratio of (14.6±0.8):1 and the empirical Redfield ratio. As we will see in further detail in section 5.3, the uptake of N is mostly supplied by nitrate. As a result, new and total NPP rates are close. Fig. 5.8 shows the plots of the two estimates of the total NPP rate based on surface net biological uptake rates of P and N. The estimated total NPP rates were computed from net biological uptake rates of P and N scaled as terms of fixed organic C by using the average ratios C:P:N of 106:1:(14.6±0.8) and the atomic weight of C (12 g mol−1 ) to convert from mole number to mass. The ratio P:N, 1:(14.6±0.8), enables us to obtain a N-based total NPP rate scaled to estimate the total NPP rate like the P-based total NPP rate. In this fashion, our estimates of the NPP rate based on N and P are both estimates of the total NPP rate. Note that the cumulative uncertainty from the transport estimates, the sink rates and the uptake ratios is likely larger than the uncertainty on the NPP rates anticipated with the bootstrap. Although a similarity between the SoG observed N:P uptake ratio and the empirical Redfield ratio was found, it does not guarantee a similarity between the SoG C:P ratio and the empirical Redfield ratio of 106:1. At this point in the analysis, it is clear that without any data on the organic C net production rate in the SoG, it is speculative to define the SoG C:P ratio as the Redfield ratio 106:1 (Eq. 5.10, section 5.2.3). Previous studies suggest that, in the ocean, the surface C:N:P ratios are 153  Chapter 5. Nutrients Uptake and Primary Productivity sensitive to growth rate, nutrient concentrations and species composition [Sarmiento and Gruber 2006]. For instance, C:N:P observed disappearance ratios in the Southern Ocean exhibit some variability between phytoplanktonic groups. For 1 mole of P used by diatoms, 9–10 moles of N are used to fix between 63–94 moles of C [Arrigo et al. 1999, 2000, Sweeney et al. 2000, Quigg et al. 2003], while Phaeocystis sp., another dominant phytoplankton group, used between 18.6–19 moles of N to fix between 133–147 moles of C. Experimental cultures of diatoms, dinoflagellates and Phaeocistis sp groups also showed variability, 60:10:1, 120:15:1 and 80:10:1, respectively [Ho et al. 2003]. In these studies, although the average ratios are close to the empirical Redfield ratios (e.g. C:N:P= 124:16:1 in Quigg et al. [2003]), they emphasize that the variability of the individual values of the ratios found at sea is large as a result of the variability of the composition of the species sampled. In the case of the SoG, the species composition is seasonal (Chapter 1, section 1.2), and thus the C:N:P ratios could vary according to the composition of the bloom. In Fig. 5.8, the two estimated NPP rates have very similar seasonality. Largest differences occurred during 2003, when N was larger than P by 0.6 and 0.5 gC m−2 d−1 in spring and summer 2003, respectively. But, in 2005, the opposite occurred, the P-based estimate was larger than the N-based estimate. On average, the difference between the P-based estimate and the N-based estimate was positive, about 0.12±0.08 gC m−2 d−1 (1 standard deviation). This difference is small compared to the average NPP rate of the order of magnitude of 1 gC m−2 d−1 . When averaged over the period April 11th 2002– April 11th 2005 the two estimates yielded annual NPP rates of 212±41 (for P) and 205±36 (for N) gC m−2 yr−1 . On average, the difference between the NPP rate (based on P) and the NPP rate (based on N) is zero. The P-based and N-based NPP rate estimates are slightly different due to minor discrepancy in their variability during spring and summer (Fig. 5.8). Note that 154  Chapter 5. Nutrients Uptake and Primary Productivity although NPP peak was highest during the spring bloom (average 157 gC m−2 yr−1 based on P), the 3-month spring average equalled the average over summer. An independent proxy for the organic C biomass, the chlorophyll-a (chl-a) average biomass integrated over 30 m is plotted along with the NPP estimates to show the spring bloom timing, and the consistency between NPP rates and phytoplankton biomass. Fig. 5.8 shows that the NPP rate is roughly proportional to the biomass. The maximum rates occur in spring when chl-a is the highest. In summer and early fall the NPP rate can be high despite a low level of chl-a. It is important to note that, since we are not using a direct measurement of C biomass, but instead the amount of chl-a pigments, the total chl-a concentration can change because the concentration of chl-a per phytoplankton cell might vary even though the number of phytoplankton cells in the water column is kept constant. Since there is no direct estimate of C biomass in the STRATOGEM dateset to determine the chl-a:C biomass ratio, the C biomass has to be assumed to be proportional to the chl-a biomass even though it is likely not to be the case. The chl-a-normalized NPP rate depends not only on the cell C (QC , gC cell−1 ) and chla quotas (Qchl−a, g chl-a cell−1 ), but also on the cell growth rate (µ, d−1 ) which represents the number of cell division per day: QC µ Qchl−a  (5.11)  gC cell−1 d−1 = gC chl−a−1 d−1 chl−a cell−1  (5.12)  chla nNPP = Thus, chla nNPP =  The growth rate is known to depend on temperature, light and nutrients among other things [Eppley 1972, Laws and Bannister 1980, Geider et al. 1998]. The photosynthetic rate (the NPP rate normalized by phytoplankton biomass) 155  Chapter 5. Nutrients Uptake and Primary Productivity can be approximated by the ratio between NPP rate and chl-a biomass. It represents the ability of a phytoplanktonic cell (represented by the same amount of chl-a biomass) to produce C. Fig. 5.9 shows that the largest chl-a-normalized NPP rates occur in late summer (2002 and 2003) and in spring (2004 and 2005). The chl-anormalized NPP rate ranges between -0.8 and 47 gC gChl-a−1 d−1 with an average of 11 gC gChl-a−1 d−1 . Assuming no nutrient limitation, since phytoplankton receive more light during the summer (with a seasonal maximum over June–July) than in spring and fall, phytoplankton would need less chl-a pigments to produce the same amount of C. Thus, the chl-a-normalized NPP rate should increase throughout the period of Februrary-August. In 2003, chl-a-normalized NPP rate was low during spring (around 10 gC gChl-a−1 d−1 ) and rose during summer (up to 24 gC gChl-a−1 d−1 ). Fig. 5.10 shows the chl-a-normalized NPP rate and an estimate of the averaged PAR over the mixing layer depth during 2002–2003. It is based on STRATOGEM monthly-averaged SW radiation and PAR vertical profiles, and estimated albedo, light attenuation coefficient and mixing layer depth by Collins et al. [2009]. The averaged PAR represents the light received by phytoplankton when they are mixed vertically through the water column. The seasonal changes of the primary productivity were significantly correlated with those of the averaged PAR: on average, there was a positive linear correlation coefficient, about 0.6 associated with a p-value of 0.0024 and a confidence interval of [0.24–0.81]. The averaged PAR over the mixing depth best correlates with the chl-a-normalized NPP rate among the variables derived from the solar radiation observations. There was a smaller correlation with PAR estimated at individual depth (at 5, 10 and 15 m, not shown). There was no clear relationship between chl-a-normalized NPP rate and other environmental variables: e.g., 0-m PAR level, average light extinction coefficient, surface SW heat 156  Chapter 5. Nutrients Uptake and Primary Productivity flux, and surface temperature. This is most likely because of the large scatter of the chl-a-normalized NPP rate.  5.2.5  Estimates of Net Community Productivity  In this section, we will use the net biological production rate of O2 as a crude estimate of the NCP rate. By comparing it with the new NPP rate, scaled in g O2 unit using the empirical Redfield ratio O2 :N 150:16 (see section 5.2.3), one can speculate on the timing and the size of the regenerated NPP rate and heterotrophic respiration Rh . Fig. 5.11 shows the estimate of the net biological production rate of O2 , in gO2 based on the molecular weight of 32 g mol−1 and O2 :N:P= 150:16:1. It is not possible to directly use the O2 :C ratio to find a NCP estimate in organic C since the O2 :C ratio is not known and the constant ratio that we assume, O2 :C= 150:106, is only associated with the net biological production rate of phytoplankton not the entire community, i.e. phytoplankton, zooplankton, and bacteria. Note that the previous analysis based on P (in section 5.2.3) provides an average estimate of P:O2 molar uptake ratios of 1:(128±21). Thus, the net biological production in organic C unit, based on our O2 net biological production rate and C:O2 = 106:(128±21), leads to an average estimate close to the NPP rates found in the previous section: 223±53 gC m−2 yr−1 of over 2002-2005. This value is larger than, but still close to, the P-based (212±41 gC m−2 yr−1 ) and N-based (205±36 gC m−2 yr−1 ) estimates of the NPP rates as expected from our analysis of the O2 :C ratio. In the O2 surface budget, by definition, the NCP is the difference between the total NPP rate and the heterotrophic respiration rate (section 5.2.3). NCP = NPP − Rh  (5.13)  157  Chapter 5. Nutrients Uptake and Primary Productivity  4  2.5 N P Chl−a  3.5  2  2.5  d  −1  1.5  gC m  −2  2  1  1.5  × 10−1 g Chl−a m−2 over 30 m  3  1 0.5 0.5  0  0  Apr  Jul  Oct 2003 Apr  Jul  Oct  04  Apr  Jul  Oct  05  Apr  Figure 5.8: NPP Rate Estimates Based on Net Biological Uptake Rates of N and P . The NPP rates are obtained by multiplying the net biological uptake rates with the C:N:P ratio defined in section 5.2.3. The chl-a biomass (gChla m−2 ) is based on the 158 surface-box average and integrated over 30 m.  Chapter 5. Nutrients Uptake and Primary Productivity Since NPP is the total of NPPn and NPPr NCP = NPPn + NPPr − Rh  (5.14)  NCP − NPPn = NPPr − Rh  (5.15)  Rearranging yields:  In theory, the difference between the NCP and the new NPP rates (Eq. 5.15) would then represent the difference between the regenerated NPP rate and Rh (heterotrophic respiration rate). Eq. 5.15 provides a way to estimate what the “excess” of regenerated NPP (relative to Rh ) is when NCP is larger than new NPP, and what the “excess” of heterotrophic respiration (relative to regenerated NPP) is when new NPP is larger than NCP. The NCP rate in Fig. 5.11 (triangled solid line) corresponds to the net biological production rate of O2 from Fig. 5.4. To be used in Eq. 5.15, the new NPP rate has to be scaled in unit of mol O2 by assuming a 150:16 O2 :N ratio (section 5.2.3) when respiration is only autotrophic (Fig. 5.11, squared dashed line). It represents the net production of O2 by autotrophs supplied by external source of N. A positive difference between the NCP and new NPP rates in mol O2 would suggest that regenerated NPP rate is significantly larger than heterotrophic respiration rate, while a negative difference would suggest a larger heterotrophic respiration rate than the regenerated NPP rate. Fig. 5.12 provides the timing of possible excess of regenerated NPP (relative to heterotrophic respiration) and excess of heterotrophic respiration (relative to regenerated NPP). In the case of excess of regenerated NPP, NCP is larger than new NPP during October 2002-March 2003 (maximum 2.1 gO2 m−2 d−1 ), November 2003-April 159  Chapter 5. Nutrients Uptake and Primary Productivity Annual  Spring  Summer  Fall  Winter  -0.1±1.9 0.2±2.2 -1.8±0.8 -0.6±1.2 1.5±1.1 Table 5.5: Estimates of the Excess of Heterotrophic Respiration (<0) based on Eq. 5.15 . Annual and seasonal averages (with 1 standard deviation) of the excess of heterotrophic respiration (<0) rates (gO2 m−2 d−1 ) based on Eq. 5.15 and Fig. 5.12. Average error is ±1 gO2 m−2 d−1 (Fig. 5.7). 2004 (2.5 gO2 m−2 d−1 in 2003, 3.6 gO2 m−2 d−1 in 2004), in August 2004, December 2004 and March 2005. Thus, large excess of regenerated NPP during spring and winter suggests that regenerated NPP is likely larger once a large amount of organic C has been produced by phytoplankton (early spring and late fall). An excess of heterotrophic respiration has to be consistent with an increase of zooplankton and bacterial respiration. New NPP was larger than NCP in summer (maximum excess of 3 gO2 m−2 d−1 in 2002 and 2.5 gO2 m−2 d−1 in 2003, 2004, and 2005), spring 2003 (peak of 6.4 gO2 m−2 d−1 ). The rest of the time the difference can be considered insignificant given the average error of ±1 gO2 m−2 d−1 (Fig. 5.7). Averaging of the difference NCP- new NPP (Table 5.5) suggests that on average only the summer difference (on average negative) is significant. This suggests that heterotrophic respiration is large on average during summer and is likely associated with an increasing biomass of herbivorous zooplankton feeding on phytoplankton.  5.3  Discussion  In the previous sections of this chapter, the NPP rate of the SoG and its variability have been examined through estimates of NPP and NCP rates based on a budget of  160  Chapter 5. Nutrients Uptake and Primary Productivity 50  40  gC gChl−a  −1  d  −1  30  20  10  0  −10 Apr  Jul  Oct 2003 Apr  Jul  Oct  04  Apr  Jul  Oct  05  Apr  Figure 5.9: Estimate of the Chl-a-normalized NPP Rate (gC gChl-a−1 d−1 ) , Using NPP Based on P (gC m−2 d−1 ) and Chl-a (gChl-a m−2 , over 0-30 m) in Fig. 5.8. Shades indicate spring and summer. 161  Chapter 5. Nutrients Uptake and Primary Productivity  Integrated PAR based on modelled 0−m PAR (W m−2) and observed PAR vertical profiles  50  chla−normalized NPP rate (gC gChla−1 d−1)  200  100  0  0 May  Sep  2003  May  Sep  04  May  Sep  05  May  Figure 5.10: Comparison of the Chl-a Normalized NPP Rate (thick line, circles, left y-scale) With Averaged PAR over the Mixing Layer (thin line, squares, right y-scale) . The averaged PAR is based on STRATOGEM PAR vertical profiles, and estimated albedo, light attenuation coefficient and mixing layer depth by Collins et al. [2009]. 162  Chapter 5. Nutrients Uptake and Primary Productivity  10 3.5  3  8  2.5 6  −2  gO2 m  1 2 0.5  × 10−1 g Chl−a m−2 over 30 m  1.5  4  d  −1  2  0  0  −0.5 −2  Net Community PP rate based on O (gO m−2 d−1 and 32 g mol−1O ) 2  2  2  New NPP rate (gO m−2 d−1 using N uptake and O :N=150:16) 2  −1  2  −2  Surface chlorophyll−a integrated over 30 m (g chl−a m )  −4 Apr Jul Oct 2003 Apr Jul Oct  04 Apr Jul Oct  05 Apr  Figure 5.11: Net Community Production Rate (gO2 m−2 d−1 ). The estimates are based on net biological productivities of O2 , new NPP rate (gO2 m−2 d−1 , using N uptake and O2 :N=150:16 Redfield ratio), and integrated chl-a biomass (gchl-a m−2 , right y-scale) is also plotted. Average error is about ±1 g O2 m−2 d−1 (Fig. 5.7)  163  Chapter 5. Nutrients Uptake and Primary Productivity  NPP  n  Depth−integrated Chl−a biomass 10  2  −1  1 2  0.5 0  0 Apr  × 10  2  gO m  1.5  4  −2  6  −2  d  −1  2.5  g Chl−a m  3  8  over 30 m  3.5  Jul  Oct 2003 Apr  Jul  Oct  04  Apr  Jul  Oct  05  Apr  Excess Recycling  4 NCP−NPP  n  0  gO2 m  −2  d  −1  2  −2 −4 Excess Respiration  −6 −8 Apr  Jul  Oct 2003 Apr  Jul  Oct  04  Apr  Jul  Oct  05  Apr  Figure 5.12: Estimates of the Excess of Regenerated NPP Rate (NPPr ) and Heterotrophic Respiration (Rh ). Estimates are based on the difference between NCP and new NPP (scaled with C:O2 =106:150 and molar masses of O and C), in mol O2 m−2 d−1 . Positive values indicate that NPPr >Rh , and negative values that Rh >NPPr . Average error (dotted lines) is about ±1 mol O2 m−2 d−1 (Fig. 5.7). All rates are in gO2 m−2 d−1  164  Chapter 5. Nutrients Uptake and Primary Productivity the sink rate terms of nutrients and production of dissolved O2 . The analysis of the sink rate terms shows that the nutrients near the surface were mainly supplied by the net upwelling with a small seasonal variability while both the biological uptake rate and the advective export were responsible for drawing them down. However, in spring, the budget was unbalanced with a residual rate (the storage rate terms in Tables 5.1 and 5.2) of -20±4 mol P s−1 and -299±59 mol N s−1 indicating that the external sources of P and N were not sufficient in spring to sustain sinks (net biological uptake rate and advective export). Nutrients were drawn (storage terms) from the surface nutrient pool (initially in spring 3.8 ×108 P moles and 4.8×109 N moles over 30 m). This surface nutrient pools provide a sufficiently large storage of nutrients to compensate for the unbalance between the spring sources and sinks but are not completely depleted at the end of spring. The estimates of the spring sink rates (net biological uptake and advective export) were similar within error bars although the net biological uptake rate eventually became the largest of the spring sink rates because of their trends (Fig. 5.1–5.3: biological uptake rate reaching a maximum and advective export decreasing). The estimates of NPP rates were obtained from two assumptions: first, the net biological uptake rates of N and P in the SoG are associated with new NPP and total NPP respectively, and second, the molar ratios C:P and C:N in the SoG are relatively constant, and close to the empirical Redfield ratios. The first assumption is reasonable given my analysis of the terms in the budget equations of P and N, although it still requires validation by other studies. The net biological uptake rate of P and N had very similar seasonality. The small difference between them suggests that overall N is supplied to phytoplankton by external sources of N. NPP (average NPP, 212 gC m−2 yr−1 ) is mainly new production (average new NPP 187 gC m−2 yr−1 ). The overall f-ratio is the ratio of average new NPP over average NPP, f=187/212=0.88. It is 165  Chapter 5. Nutrients Uptake and Primary Productivity larger but close to the maximum observed value in coastal systems [Eppley and Peterson 1979, f=0.8, see new:total production p. 679] and larger than the minimum for systems where the nutrients are mainly supplied by circulation [Legendre et al. 1999, f-ratio>0.7]. In coastal areas, high f-ratio (f>0.8) can be observed when spring blooms are dominated by diatoms [Kristiansen et al. 2001](using estimated uptake rates based on radioactive isotopes of nitrate and ammonium). These SoG NPP rate estimates are consistent with other studies and this suggests that they are reasonable estimates of the primary productivity. The ratio of the net biological uptake rates shows a strongly linear relationship of P:N ratio with a slope coefficient of 1:(14.6±0.8) very close to the Redfield ratio of 1:16 (Fig. 5.5). The small difference between the 2 ratios could be due to N being mainly supplied by external sources of N. On average, it is consistent with averages of surface measurements of C:P:N ratios [Sarmiento and Gruber 2006], although individual values vary within a larger range [Quigg et al. 2003]. For this reason assuming a C:P:N ratio of 106:1:(14.6±0.8) seems a reasonable choice to estimate the average NPP rate, but a more speculative choice to estimate time series of NPP rate. The second assumption made to estimate the NPP rates in C unit, that of constant surface C:P and C:N ratios, is also speculative. There is no simultaneous measurements or estimates of the SoG C net production rate, and SoG P and N uptake rates to suggest such a relationship is consistent in the SoG. At the surface of the ocean, it is traditionally assumed that the C:P and C:N ratios are sensitive to the nutrient concentrations, the growth rate, and the species composition and are more variable than the ratios found deeper [Arrigo et al. 2000, Sweeney et al. 2000, Quigg et al. 2003, Ho et al. 2003]. However, this allows one to compare these estimated NPP rates with those of other studies. 166  Chapter 5. Nutrients Uptake and Primary Productivity The annual average rate of NPP was estimated between 205±36 (new NPP rate based on N) and 212±41 (NPP rate based P) gC m−2 yr−1 which is consistent with recent estimates of NPP rates: 220 gC m−2 yr−1 using the same dataset [Pawlowicz et al. 2007], 120–345 gC m−2 yr−1 [Harrison et al. 1983, on average 280 gC m−2 yr−1 ]). The similarity between these estimates and my NPP rate estimates suggests that my NPP rates are realistic rates. The estimates from Harrison et al. [1983]’s review are based on C14 incubation uptake experiments over several hours. An estimated 280 gC m−2 yr−1 has to be regarded as a weighted average between NPP and GPP rates [Marra and Barber 2004]. Their measurements are also spot measurements with a large variance, while ours are time and space averages with smaller variance. The classification of estimated PP rate into GPP or NPP rate depends on the period of incubation of water samples among other things. A recent study [Williams and Lefevre 2008] suggest that a better knowledge of the internal sinks and sources of 14  C of the phytoplanktonic cells are necessary to be able to differentiate the type of  PP rate. The annual estimate of the NPP rate over the spring-summer period alone was on average 157 gC m−2 yr−1 (based on P) equally contributed by spring and summer biomass. Spring biomass was characterized by higher chl-a level and higher nutrient uptake and drawdown rate (negative storage rate), while summer biomass was characterized by lower chl-a level but still large nutrient uptake rate. In Fig. 5.9, the estimate of chl-a-normalized NPP rate (average 11 gC gChl-a−1 d−1 ) suggests that the summer biomass is more productive that the spring biomass because of greater availability of light [Miller 2004]. The effect of warmer temperatures on diatom growth rate can either follow an exponential law (doubling with a 10◦ C increase) or a slower linear relationship [Montagnes and Franklin 2001]. In either case, the surface temperature in the SoG increases by 2.78◦ C month−1 maximum over summer and between 167  Chapter 5. Nutrients Uptake and Primary Productivity depths of 0–30 m by 0.7◦ C month−1 on average (Fig. 3.4, section 3). This suggests a small effect of temperature on diatom growth on a monthly scale over the water column (maximum 2.78◦ C month−1 ). However there may be a significant effect on the phytoplankton assemblage right at the surface over the spring-summer period (10◦ C and more over 6 months). Using direct measurements of chl-a and C14 uptake rate (not shown) in the euphotic zone during 1988–1991 in the SoG, the chl-a-normalized NPP rate in the SoG provide similar average and range to our estimates [Clifford et al. 1992, average 24.5 gC gChl-a−1 d−1 , and range between 3 and 188.4 gC gChl-a−1 d−1 ]. The range of estimated chl-a-normalized NPP rates found in the present work (from -8 to 47 gC gChl-a−1 d−1 , Fig. 5.9) is smaller than maximum chl-a-normalized NPP rates found in other studies in the SoG [Forbes et al. 1986]. My estimates are time and space averages, while the other studies took spot measurements. Forbes et al. [1986] parametrized the PI curve at locations along the coast of British Columbia, in particular in the SoG. Based on  14  C uptake and PAR measurements at the depth  of chlorophyll-a maximum they estimated the maximum chl-a-normalized NPP rate to be on average 12.3±1 mgC mgChl-a−1 h−1 that is 295±24 when scaled to gC gChl-a−1 d−1 . But averaged over a day and over depth this becomes much smaller. Fig. 5.13 shows the estimated C production rate of autotrophic phytoplankton and diatoms based on species abundance at station 4-1 at 5 m (see section 3.2.1 in chapter 3. During summer blooms, the phytoplankton biomass was a mixed assemblage of phytoplankton species [Harrison et al. 1983], but it usually contained a large amount of diatoms. As expected, on average summer nutrients were drawn down in the N:P Redfield proportion (=15.6, Tables 5.1 and 5.2, -844 molN s−1 and -54 molP s−1 , respectively), and in the theoretical Si:N (=1) and Si:P (=15.6) proportions (Table 5.3, -840 mol Si s−1 ) [Brzezinski 1985]. During the summer bloom, 168  Chapter 5. Nutrients Uptake and Primary Productivity 45 Total Autotrophs Total Diatoms S4−1, 5 m 40  35  gC m−2 average over 30 m based on S4−1, 5 m  30  25  20  15  10  5  0 Apr  Jul  Oct 2003 Apr  Jul  Oct  04  Apr  Jul  Oct  05  Apr  Figure 5.13: Estimates of the Diatom and Total Autotrophic Biomass Over 30 m Based on Cell Counts at Station S4-1 at 5 m. Shaded boxes indicate spring. 169  Chapter 5. Nutrients Uptake and Primary Productivity the concentrations of nitrate and phosphate were very low near the surface and low throughout the water column (Fig. 5.14) as a result of large uptake of nutrients during summer and, earlier, during the spring bloom. However the silicic acid concentration was either increasing or plateauing during summer blooms (Fig. 5.15), possibly because of the additional riverine supply during the freshet [Harrison et al. 1991]: 596 molSi s−1 on average in summer (Table 5.3), and up to 1000 molSi s−1 (Fig. 5.3). Under low nutrient concentrations, diatoms had to compete with other better adapted phytoplankton groups [Miller 2004]: for instance, silicoflagellates (e.g.,Dictyocha speculum) competed with diatoms for, in particular, silicic acid, while phototrophic ciliates, abundant Myrionecta rubra [R Pawlowicz, A Sastri, S E Allen, D Cassis, O Riche, M Halverson and J F Dower; unpublished data], for nitrate and phosphate only [Crawford and Tore 1997, Lagus et al. 2004]. Myrionecta rubra is a ciliate species known to prey on phytoplankton, to retain them as endosymbionts, and to sometimes retain them as permanent autotrophic organelles [Stoecker et al. 2009]. A recent study suggests that Myrionecta rubra is particularly well adapted to low nutrient conditions because it can swim to reach deep nutrients [Lagus et al. 2004]. Spring bloom biomass was dominated by diatoms (Fig. 5.13) adapted to strive in replenished nutrient conditions [Miller 2004, Sarthou et al. 2005]. Although, on average, the nutrients were taken up in near N:P Redfield proportion (14.6, Tables 5.1 and 5.2, -796 molN s−1 and -57 molP s−1 , respectively), the Si:N (=1.8) and Si:P (=25) ratios (Table 5.3, -1460 molSi s−1 ) were higher than expected [Brzezinski 1985]. The surface concentration of silicic acid was steadily decreasing during the spring bloom (Fig. 5.15). There was an excess drawdown of Si (on average, 1.7 higher than in summer) and it peaked during spring with values 2 to 3 times larger than the spring average (e.g. 11.3 gSi gChl-a−1 d−1 in spring 2003, Fig. 5.16). Seasonality was also 170  Chapter 5. Nutrients Uptake and Primary Productivity individual station value at 0 m  35  stations average at 0 m SoG 0−30 m average  30  N concentrationµM  25  20  15  10  5  0 Apr Jul  Oct 2003 Apr  Jul  Oct  04  Apr  Jul  Oct  05  Apr  Jul  Figure 5.14: SoG Nitrate Average Concentration at 0 m and Over 0–30 m. The plot shows the average of SoG nitrate at 0 m and the top box average (0–30 m) for comparison. The averages depart from each other during spring and summer when near surface nitrate tends to be very low (on average 3µM) compared to surface average concentration (about 11µM). Note that the surface and depth-average phosphate concentrations (not shown) exhibit similar trends.  171  Chapter 5. Nutrients Uptake and Primary Productivity observed when investigating the Si:N and Si:P uptake ratios. In spring (and in fall as well), the Si:N and Si:P uptake ratios (Fig. 5.17) were higher that the Brzezinski’s Si:N (=16:16) and Si:P (=16:1) average ratios for healthy diatom cells. Since the N:P uptake ratio (14.6±0.8):1, remained close to the Redfield ratio (Fig. 5.5), the reason for higher than expected Si:N and Si:P ratios is not a nutrient limitation. There are three ways to explain the excess drawdown of Si with respect to N and P during the spring bloom: abundance of heterotrophic silicoflagellates, abundance of heavily silicified diatoms, and a neglected abiotic and non-advective sink of dissolved Si. Heterotrophic silicoflagellates could have been abundant and taken up Si, but not N and P. Unpublished data from the STRATOGEM program suggests that, indeed, at least one species of heterotrophic silicoflagellates, Ebria partita, was abundant in the SoG during springs, e.g. 2004 and 2005 [R Pawlowicz, A Sastri, S E Allen, D Cassis, O Riche, M Halverson and J F Dower; unpublished data at station S4-1, 5-m depth]. Other silicoflagellates were present in the SoG, but they bloomed after the spring bloom, in summer, and were autotrophic species (e.g. Dyctiocha speculum). Heavily silicified and large diatoms tend to uptake Si faster than N and P. Spring blooms mainly composed of heavily silicified diatoms could explain the higher than expected SoG Si:N and Si:P uptake ratios. For instance, a study of Central Equatorial Pacific phytoplankton community investigated the effects of iron- and Si-addition on diatoms bloom using mesocosm experiments [Marchetti et al. 2010]. They showed that Si:N and Si:P uptake ratios could vary manifold above the expected Si:N (16:16) and Si:P (16:1) ratios [Brzezinski 1985] depending on the concentrations of dissolved iron and Si. Note that the Si concentration in this mesocosm experiment varies within 5–14 µM, lower than in the SoG (Tables 5.6– 5.7). In turn, they show that this could affect the species composition of the diatom assemblage and the cell division rate of diatoms. 172  Chapter 5. Nutrients Uptake and Primary Productivity 70  60  Si 0 m Si 0−30 m N0m N 0−30 m  µM Concentration  50  40  30  20  10  0 Apr Jul Oct 2003 Apr Jul Oct  04  Apr Jul Oct  05  Apr Jul  Figure 5.15: SoG N and Si Average Concentrations at 0 m and Over 0–30 m. The plot show the averages of SoG N and Si concentration at 0 m and their top box averages (0–30 m) for comparison. Only the Si averages at the surface (0 m) and throughout the euphotic zone (0–30 m) remain high during summer (maximum about 40µM in 2002). 173  Chapter 5. Nutrients Uptake and Primary Productivity 5  4.5  3.5  3  Si Net Biological Uptake Rate × 10 mol s  −1  4  3  2.5  2  1.5  1  0.5  0 Apr  Jul  Oct 2003 Apr  Jul  Oct  04  Apr  Jul  Oct  05  Apr  Figure 5.16: Estimate of the Net Biological Uptake Rate of Si Normalized by Chl-a (gSi gChl-a−1 d−1 ). Shaded boxes indicate spring (dark) and summer (light). 174  Chapter 5. Nutrients Uptake and Primary Productivity 3.5  51.1 Si:N 43.8  2.5  36.5  2  29.2  1.5  21.9  1  14.6  0.5  7.3  0 Apr Jul Oct 2003 Apr Jul  Si:P Uptake rate ratio  Si:N Uptake rate ratio  Si:P 3  0 Oct  04  Apr Jul Oct  05  Apr  Figure 5.17: Estimate of the Si:N (lefthand) and Si:P (Righthand) Ratios Based on the Net Biological Uptake Rates of Si, N and P . Shaded boxes indicate spring. The dashed line indicates the average ratios (Si:P=24.7, Si:N=1.7). Some of the fall and winter values are <0. But they are negligible within the uncertainties (see Figs 5.5–5.6). 175  Chapter 5. Nutrients Uptake and Primary Productivity Other studies of iron- and Si-replete conditions for heavily silicified diatom blooms have also been carried out at sea, in the Southern Ocean, with naturally-occurring Sireplete and Si-depleted surface conditions [Coale et al. 2004, Brzezinski et al. 2005]. The experiments in iron-replete conditions are of interest to our study because they shared some similarities with the surface Si and N concentrations, the Si:N uptake rate and the abundance of diatoms in the phytoplankton during the SoG spring and summer blooms. The Si-replete and iron-replete conditions (60–65 µM Si, Table 5.6) correspond to early spring bloom in the SoG when Si level is the highest (16–57 µM Si, Table 5.6). The associated N concentration in the experiment also corresponds to the maximum concentration found in the SoG (26–29 µM and 9–26 µM, respectively, Table 5.6). In the Si-replete conditions, the bloom was dominated by diatoms as it happened during SoG spring blooms. The Si:N uptake rate was (2.1±0.5):1 [Coale et al. 2004] (Table 5.6) and similar to the SoG spring rate of (1.9±0.14):1 (based on spring rates of Si and N uptake, Figs 5.1–5.3). For both SoG estimate and Coale et al. [2004]’s estimate, the Si:N uptake rate was larger than expected. Thus, Si:N uptake ratio of 2 may be normal during spring condition in the SoG. However, these Si-replete and iron-replete conditions are promoting less silicification than the Si-replete and iron-depleted conditions. The Si:N uptake rate decreased from (8.1±1.5):1 to (2.1±0.5):1 when iron was added. In Si-depleted conditions the Si:N uptake rate remained around 1 regardless of iron-addition [Coale et al. 2004]. The concentrations in the Si-depleted conditions were somewhat different (0.5–4 µM, Table 5.7) in Coale et al. [2004]’s experiment from the summer Si-concentrations in the SoG (25–41 µM, Table 5.7), possibly because, on average, the SoG summer supply of Si was larger than its sink (Table 5.3). The associated N concentration in the experiment somewhat corresponds to the maximum SoG concentration (18–22 µM and 6–14 µM, in Coale et al. [2004] and this study respectively, Table 5.7). In 176  Chapter 5. Nutrients Uptake and Primary Productivity addition, in the Si-depleted and iron-replete conditions, the phytoplankton was dominated by non-diatoms in both Coale et al. [2004]’s experiment and the SoG during summer, the Si:N uptake rate was (0.85±0.36):1 [Coale et al. 2004] (Table 5.7) and similar to the SoG summer rate of (0.9±0.28):1 (based on summer rates of Si and N uptake, Fig. 5.1–5.3). Loss of dissolved Si out of the system might also explain the higher than expected Si:N and Si:P uptake ratios in the SoG. If there is an another sink of dissolved Si in Coale et al. [2004] this study Si-replete bloom  spring bloom  Si (µM)  60–65  16–57  N (µM)  26–29  9–26  Si:N uptake ratio (2.1±0.5):1  (1.9±0.2):1  Table 5.6: Comparison of Si-replete and Iron-replete Bloom in Coale et al. [2004] and SoG Spring Bloom in This Study. Values come from Figs 3 E–H and pp 411–412 in Coale et al. [2004] and Figs 3.8,3.9,5.1 and 5.3 of this study. Coale et al. [2004] this study Si-depleted bloom  summer bloom  Si (µM)  0–4  25–41  N (µM)  18–22  6–14  Si:N uptake ratio (0.85±0.36):1  (0.9±0.3):1  Table 5.7: Comparison of Si-depleted and Iron-replete Bloom in Coale et al. [2004] and SoG Summer Bloom in This Study. Values come from Figs 3 E–H and pp 411–412 in Coale et al. [2004] and Figs 3.8,3.9,5.1 and 5.3 of this study.  177  Chapter 5. Nutrients Uptake and Primary Productivity the surface Si budget (Eq. 2.81), it has to be abiotic and non-advective. Since the net biological and advective sinks have already been taken into account in Eq. 2.81 (see section 2.4), the excess drawdown of Si would have to be a chemical uptake. If one considers spring blooms when diatoms dominate phytoplankton biomass, the rate of Si biological uptake has to be close to the rate of N biological uptake. The difference between Si and N rates would be the rate of Si chemical uptake (1460-796=664 mol s−1 ), of the same size as the Si and N biological uptake rates (796 mol s−1 ) since our estimated spring Si:N ratio is close to 2:1 (Fig. 5.6). In fall, diatom blooms also occur with a Si:N ratio still larger than 1:1 but close. To account for the fall difference by adding a chemical Si uptake, the fall chemical uptake would have to be about 211 mol s−1 , that is about a third of the spring chemical uptake (664 mol s−1 ). Then, one could assume that the chemical uptake rate should be very small the rest of the year, in particular in summer when the rates of Si and N biological uptake are very close to each other either we assume or not that the excess drawdown of Si is a chemical sink. Thus, the chemical uptake of Si would have to be seasonal with a maximum in spring as large as the biological uptake of Si. The only chemical sink of dissolved Si that could be considered is the spontaneous precipitation of dissolved Si into silica, particulate Si, in the water column. When Si precipitates, it is not accounted to as dissolved Si anymore and can sink out of the euphotic zone. However, this process is balanced by dissolution of silica back into dissolved Si. Note that precipitation also occurs inside the diatom cells, but this has been accounted for as the biological uptake rate, (φ1 )Si in Eq. 2.81. In the water column, the balance between precipitation and dissolution of Si depends on different factors, the first one is the difference between the observed and saturation concentrations of Si [Sarmiento and Gruber 2006]. In the SoG, observed concentrations of Si range between 16–62 µM (Fig. 5.15) on average, with a maximum of 73 µM measured on a March 2003 water sample, while 178  Chapter 5. Nutrients Uptake and Primary Productivity theoretical saturation concentration ranges from ∼900 µM up to 1500 µM [Sarmiento and Gruber 2006]. There is a factor of one order of magnitude, at least, between the observed and saturation concentrations. This means that spontaneous chemical precipitation is very unlikely to happen in the SoG anytime, in particular during spring. In a recent study in Monterey Bay, the estimated dissolution rate is smaller than the estimated precipitation rate during spring bloom (close to 10%, Brzezinski et al. [2003b]), because rate estimates include the phytoplanktonic production of biogenic silica in the total precipitation rate [Brzezinski et al. 2003b]. This strongly suggests that a large chemical uptake of Si apart from the net biological uptake by diatoms is very unlikely. Finally, there is a last explanation, but also very unlikely, for the higher than expected Si:N and Si:P uptake ratios. Fast remineralization of N and P could occur during spring bloom and result in low nitrate uptake and high and immediate ammonium uptake by diatoms. Since the Si uptake rate is 1460 mol s−1 , this implies that the remineralization rate of N would be the total uptake rate of N (equal to Si uptake rate) minus the uptake rate based on nitrate (our estimate): 1460-796=664 mol s−1 , and similarly, using the Brzezinski Si:P proportion (16:1), a remineralization rate of P, 35 mol s−1 . This would imply that the f-ratio is about 0.5, an unlikely value for remineralization during diatom spring blooms that mainly rely on nitrate not ammonium. In addition, remineralization of particulate organic N and P from diatoms would occur once the organic coating on the diatom frustules [Sarmiento and Gruber 2006, Miller 2004] have been scavenged or broken down to allow the particulate organic N and P to be in contact with the water column. Note that frustules dissolution can be enhanced by increasing salinity and bacterial scavenging [Roubeix et al. 2008] as well as diffusion through the frustule pores [Leterme et al. 2010]. Recent estimates of sinking biogenic Si flux suggest that most of the diatoms from the spring bloom do 179  Chapter 5. Nutrients Uptake and Primary Productivity not remain in the euphotic zone [Johannessen et al. 2005]: e.g., their daily average of sinking biogenic Si particles varies between 3.6×10−3 mol Si m−2 d−1 and 2×10−2 mol Si m−2 d−1 based on yearly on-site trap measurements below 150 m in central SoG. My average Si uptake rate have a similar size (Fig. 5.6). This suggests that, during spring, some of the diatom organic matter, possibly most of it, sinks below the euphotic and, thus, is not remineralized in the euphotic zone. The above discussion strongly suggests that Si is decoupled from the other nutrients (P and N) during spring and fall because another biogeochemical process other than photosynthesis is taking place: e.g., higher rate of silicification, competition of diatoms and silicoflagellates for Si, and higher rate of N and P remineralization. Higher silicification rate is plausible based on previous observations and experiments [Coale et al. 2004, Brzezinski et al. 2005, Marchetti et al. 2010]. Taxonomic analysis of spring bloom water samples at one STRATOGEM location also suggests the presence of heterotrophic silicoflagellates, Ebria tripartita. Since the analysis is based on only one location, although a central station representative of the SoG, one would require an analysis at additional locations to determine the impact of heterotrophic silicoflagellates on spring Si uptake. Finally, the qualitative analysis of the NCP rate estimates (Fig. 5.12) showed that regenerated NPP could be also at work during spring blooms and winter after fall blooms (Table 5.5). However, it did not occur every spring and fall. Thus, on average the regenerate NPP is small (Table 5.5). This is consistent with the quantitative analysis of the NPP rate (based on P uptake) and the new NPP (based on N uptake), and their comparison that suggested that on average the regenerated NPP was low (section 5.2.5). This can be further quantified by using the annual NPP rates based on P and N (section 5.2.4) and assuming the P:N Redfield ratio of 1:16 instead of the observed ratio of 1:14.6 (Fig. 5.5). Any positive difference between P and N would be 180  Chapter 5. Nutrients Uptake and Primary Productivity due to the unaccounted regenerated N. The NPP rate based on P, the total NPP rate, is unchanged, 212 gC m−2 yr−1 (section 5.2.4), and the NPP rate based on N, the new NPP rate, becomes 187 gC m−2 yr−1 . This suggests an annual regenerated NPP rate of 25 gC m−2 yr−1 about 12% of the NPP rate. This yields to an average f-ratio of f=0.88. In comparison, a maximum f-ratio of 0.8 was observed in upwelling regions [Eppley and Peterson 1979] and a minimum f-ratio of 0.7 was suggested in systems where nutrients are mainly supplied by circulation [Legendre et al. 1999]. F-ratios ≥0.9 were observed in a highly productive fjord during diatom spring bloom [Kristiansen et al. 2001]. Thus, the result found here that SoG diatom bloom is mainly fuelled by upwelled nutrients is consistent with previous observations in temperate systems. The analysis of the near surface O2 sink rate terms shows that the O2 was mainly supplied by NPP and net upwelling and mainly removed by advective export. In particular, in spring the system was in quasi-steady state within the uncertainties, while in summer the O2 source was smaller than the sink leaving a residual (storage term in Table 5.4) of about -1901±239 mol O2 . This suggests a possible decrease of the net biological production of O2 by a decrease of phytoplankton biomass (by heterotrophic grazing) and an increase of heterotrophic respiration. The speculative comparison of the NCP and new NPP rates (section 5.2.5) also suggested that on average the summer difference NCP-NPPn was negative. Thus, there was an average summer excess of heterotrophic respiration (relative to regenerated NPP that uses O2 ) in the euphotic zone, about 1.8 gO2 m−2 d−1 (Table 5.5), while the rest of the year the average difference was zero. A comprehensive study of 28 US estuaries, based on high time-resolution measurements of dissolved O2 over two years, looked at the net ecosystem metabolic rate, the difference between the gross production of oxygen by autotrophs and the respiration rate [Caffrey 2003]. They showed that 181  Chapter 5. Nutrients Uptake and Primary Productivity most of these estuaries were heterotrophic, producing less O2 that was respired. In particular, their average annual rates of O2 respiration were based on night respiration rates and varied from about 4 to 19 g O2 m−2 d−1 [Caffrey 2003]. Although my O2 excess heterotrophic respiration (1.8 gO2 m−2 d−1 ) is smaller and the SoG total heterotrophic respiration is unknown, my respiration estimate is consistent with the rates found by Caffrey [2003].  182  Chapter 6 Discussion and Conclusion 6.1  Thesis contributions  In this section, I summarize the contributions of this research to the understanding of the SoG circulation and its primary productivity. First, in chapter 2, a formal mathematical framework and a time-dependent inverse two-box model of the Strait of Georgia (SoG) were defined in order to infer the monthly variability of the SoG circulation. In addition, a careful bootstrap of the input data of the box model [Efron and Tibshirani 1993] is used to estimate and analyze the uncertainty of the estimated transports. Although none of the mathematics used in the two-box model is new, this is one of the few times that this combination of math has been applied to an estuarine system and use to analyze the seasonal variability of the estuarine circulation. A careful analysis of the approximations made in the derivation of the equations of budgets was carried out (2.66–2.71, sections 2.2 and 2.3) and was later applied to estimate and analyze the uncertainty associated with the data and the solutions of the inverse problem (SoG circulation transports, Chapter 4) and the forward problem (sink/source terms in nutrients and O2 budgets, Chapter 5). Note that a resampling strategy was used to estimate the uncertainty in the physical and biological estimates. In chapter 4, transport time series of the SoG estuarine circulation were estimated over three years with a monthly time resolution, for the first time. Uncertainties were 183  Chapter 6. Discussion and Conclusion also carefully estimated. These estimates are based on consistency of the transports with observations of salinity and freshwater (FW) input, temperature and surface heat fluxes. Using these estimated SoG transports, I provide the first observational analysis of the relationship between R (FW input) and surface seaward transport U1 , certainly in the SoG and perhaps for any large estuary. The analysis of the transport time series provides additional novel findings. It suggests that the seasonality of the total upward transport W2 is very small (on average ±11% of 6.2×104 m3 s−1 , the mean W2 ; maximum ±19.3%), even when the seasonality of R is large. Based on 2002–2005 data, the annual seasonality of U1 is weakly linked to R, while the annual seasonality of other transports is possibly independent from R. The transport analysis will be discussed in more detail in section 6.2. In chapter 5, the seasonal and annual average rates of the net primary production (NPP rates) based on SoG nutrient budgets over three years are estimated. Errors are also estimated. Later, in section 6.3, I will compare the estimated NPP rates with previous estimates in the SoG and in other estuaries and find that the estimated NPP rates estimated here are typical of NPP rates in temperate estuaries. The analysis of the nutrient budgets (Chapter 5) showed that, as suggested first by Mackas and Harrison [1997] for N, the estuarine entrainment is the largest supply of P and Si. The comparison of estimated NPP rates based on N and P budgets suggests that the average f-ratio is large (f=0.88, based on annual averages, see section 6.3). These results will be discussed in more detail in section 6.3. Recommendations for future work will be given in section 6.4.  184  Chapter 6. Discussion and Conclusion  6.2  Seasonality and Variability of Water Transports in Estuaries  To my knowledge only a few studies have tried to analyze the seasonal transport variability of estuarine circulation. One useful feature of the SoG system is the extreme seasonality of the freshet. This is a rare feature among large rivers [WWF ( 2006)]. In the SoG, a few recent studies tried to determine the seasonal variability of the circulation by estimating seasonal flushing times [England et al. 1996], or summer/winter water transports [Pawlowicz et al. 2007], or using sophisticated numerical modelling combined with analysis of observed salinity and temperature [Masson and Cummins 2004], and seasonal water mass analysis [Masson 2006]. Although relatively simple estimates of estuarine circulation transports are a staple of the “gray literature”, e.g. Burrard Inlet Environmental Program [1996], published analyses are less common [Petrie and Yeats 1990]. A study of the exchange transport rate in the Long Island Sound, a major urban estuary, was more successful at determining the seasonal variability (winter/summer transport magnitude (1.8–3)×104 m3 s−1 ) of the circulation, but this system is not dominated by the river inflow [Codiga and Aurin 2007]. Although some of the previous studies in the SoG and other estuaries have used a large volume of data, they tended to focus on the average circulation and ignored the time variability. Previous studies also relied on long time series of the transports [Pawlowicz et al. 2007, Austin 2002] and the assumption of steady state [Savenkoff et al. 2001, Pawlowicz 2001]. Austin [2002] used a long multi-year time series (19852001 and 1992-2001) of salinity of the Chesapeake Bay (CB) in a time-dependent box model to infer an average exchange rate between the estuary and the ocean. Despite  185  Chapter 6. Discussion and Conclusion the long multi-year time series used, only an average exchange rate was determined. Interannual variability was only qualitatively analyzed. Determining the seasonality of the water transports is yet to be done in CB [Austin 2002]. Savenkoff et al. [2001] used a quasi-steady state inverse box model of the Gulf of St.Lawrence (GSL) and both physical and biogeochemical budget equations to estimate the water transports in and out of important regions of the GSL. In Savenkoff et al. [2001], eight transports over 4 different depth ranges over July to September were estimated in 32 boxes and required the use of biogeochemical tracers. However, only data from July to September were used. Thus, these transport estimates are only representative of late summer-early fall. Using a 2 box model of the SoG circulation is a useful simplification of the vertical structure of the circulation. But, this approach leads to several limitations due to the number of boxes, the averaged properties in each box, and the shape of the boxes. For instance, the transport in our bottom box combines the transports of intermediate water and bottom water. Deep water renewals contribute to deep inflow of dense water into the SoG circulation [LeBlond et al. 1991, Masson 2002]. A recent study combined 3 boxes, instead of 2 boxes, with a simple mixing-box approach to determine the transports within 0–50 m, 50–200 m and 200–400 m in the SoG [Pawlowicz et al. 2007]. The transport within 200–400 m is associated with the deep water renewal and it has a summer/winter seasonality ranging from (0–4)×104m3 s−1 . In the 3-box model, the maximum magnitude of the deep water transport can be as large as the average intermediate water transport (4×104 m3 s−1 ). In contrast, in our 2-box model, the transports within 30–50 m, 50–200 m and 200–400 m are combined as a weighted average, the bottom transport, U2 . Given the size of our error bars and the magnitude of seasonality of U2 , the small seasonality of U2 does not provide any useful information about the bottom water renewals. On the other hand, the rela186  Chapter 6. Discussion and Conclusion tively constant magnitude of U2 suggests a very small variability of the combined intermediate and deep water renewal. S1 and S2 represent the box averages of salinity in the top and bottom boxes in both the Knudsen’s relationship (Eq. 6.1) and in our equations (Eqs 2.66–2.71). In both cases, the box averages appear instead of the averages just above or below the separation boundary between the two boxes. This is a simplification of the equations and the calculations of the transports (section 2.2). This is an important problem when the top box represents two or more water masses, for instance in the river plume during the Freshet: the high boundary salinity is approximated by the average of the high boundary salinity and the low surface salinity. Regardless of the tracer considered, the corresponding approximation error on the tracer is proportional to the difference between the box average and the boundary average. In the case of points close and below the separation boundary, the vertical gradient of the tracers tend to be small, while in the case of points close and above the separation boundary the vertical gradient is large (Fig. 3.1). In addition, there are two issues with my representation of the SoG with two domains within 0–30 m and 30–400 m. First, the data only cover the southern part of the SoG, but it was assumed they were still representative of the whole. Other recent sampling surveys over the whole SoG suggests that this is a reasonable assumption [Masson 2006, Masson and Pena 2009]. Comparison of the STRATOGEM temperature, salinity and nitrate box averages with box averages using their data also suggests that this is a reasonable assumption (Figs 6.1–6.3). Fig. 6.1 shows that average surface and bottom temperatures over the Southern SoG and over the whole SoG are very similar. Fig. 6.2 suggests similar salinity seasonality but higher bottom salinity in the STRATOGEM data. Fig. 6.3 suggests similar surface nitrate seasonality but more variable summer concentration during the freshet. Summer deep nitrate 187  Chapter 6. Discussion and Conclusion levels are similar, but no obvious trend can be seen. Secondly, we are averaging the properties of water at different locations. Thus, the averaging of the surface properties ignore the spatial variability of the surface water masses. There are three primary surface masses of water in the SoG that have different properties: the freshwater plume, sinking dense water at the Southern Entrance, and SoG surface mixed water everywhere else (see Chapter 1). This could be further investigated by analyzing seasonal TS properties of these water masses. For instance, it is not clear if the influence of the riverine input of freshwater and silicic acid should be extended to the whole SoG, although the SoG top box averages of the salinity and the silicic acid concentration are strongly influenced by the riverine input (Fig. 3.3a and Fig. 3.9a) [Harrison et al. 1991]. Although we take into account the SoG bathymetry to determine the box averages (section 3.3.2), our estimates of the transports represent the overall inflow and outflow. When comparing the SoG with another system, one can take into account any narrowing or broadening of the cross-section along an estuary to improve the accuracy of the estimated transports. For instance, with the more sophisticated approach of a tapered box model, one could estimate the transports upstream of the mouth in a chosen segment of an estuary and compare them with estimated transports based on current meter measurements. Tapered boxes have been successfully used in estuary box models to provide a better estimation of the estuarine circulation and help compare different systems [Gay and O’Donnell 2009]. The freshwater (FW) input from rivers is the main forcing of estuarine circulation. The FW input (R) in the SoG estuary has a large seasonal change compared to its average value, on average 58% of mean R (Fig. 6.5). The main source of FW is the Fraser River, one of the last large (longer than 1000 km) free-flowing rivers in the world [WWF ( 2006), Figs 3–4 and Appendix 1]. One might then expect that 188  Chapter 6. Discussion and Conclusion  STRATOGEM 0−30 m IOS 0−30 m 14 13 Temperature ( ° C)  12 11 10 9 8 7 6 Apr  Jul  Oct 2003 Apr  Jul  Oct  04  Apr  Jul  Oct  05  Apr  STRATOGEM 30−400 m IOS 30−400 m  10.5  Temperature ( ° C)  10 9.5 9 8.5 8 7.5 Apr  Jul  Oct 2003 Apr  Jul  Oct  04  Apr  Jul  Oct  05  Apr  Figure 6.1: Comparison of STRATOGEM and IOS SoG Temperature Box Averages  189  Chapter 6. Discussion and Conclusion  30 STRATOGEM 0−30 m IOS 0−30 m  29.5 29  Salinity (psu)  28.5 28 27.5 27 26.5 26 25.5 25 24.5 Apr  Jul  Oct 2003 Apr  Jul  Oct  04  Apr  Jul  Oct  05  Apr  31  Salinity (psu)  STRATOGEM 30−400 m IOS 30−400 m  30.5  30  29.5 Apr  Jul  Oct 2003 Apr  Jul  Oct  04  Apr  Jul  Oct  05  Apr  Figure 6.2: Comparison of STRATOGEM and IOS SoG Salinity Box Averages  190  Chapter 6. Discussion and Conclusion  STRATOGEM 0−30 m IOS 0−30 m  35  Nitrite+Nitrate µM  30 25 20 15 10 5 0 Jul  Oct 2003 Apr  Jul  Oct  04  Apr  Jul  Oct  05  Apr  STRATOGEM 30−400 m IOS 30−400 m  35  Nitrite+Nitrate µM  30 25 20 15 10 5 0 Jul  Oct 2003 Apr  Jul  Oct  04  Apr  Jul  Oct  05  Apr  Figure 6.3: Comparison of STRATOGEM and IOS SoG Nitrate Box Averages  191  Chapter 6. Discussion and Conclusion this large seasonal change in FW input will lead to similar large seasonal change in the water transports of the SoG. Knudsen’s relationship (a simplified box model) suggests that in quasi-steady state the circulation magnitude is proportional to R assuming that the salinity field is roughly constant: U1 =  S2 R. S2 − S1  (6.1)  However, in Fig. 4.1 the surface seaward transport U1 exhibits only a small seasonal change compared to its mean over 38 months, on average 14% of U1 , although high outflow can be reached during certain years (for instance 2002, 6×104 m3 s−1 ). This expectation is clearly incorrect. The seasonal change mostly affects the salinity field. In section 4.3.5, it is shown that the relationship U1 =f(R) resulting from the inverse box model (Fig. 4.9) is compatible with theoretical relationships developed by Hetland and Geyer [2004], MacCready and Geyer [2010]. In Fig. 4.9, the theoretical curve and the estimated U1 (R) curve both suggest that sensitivity of the estuarine circulation to changes of freshwater flow is low on a seasonal timescale. Recent advances in the theory of the exchange flow in partially mixed estuaries [MacCready and Geyer 2010] suggest that on a seasonal timescale estuarine circulations are sensitive to the river inflow as Ra (with a ≪ 1). The exponent a tends to be small for long estuaries. Thus, the theory suggests that the sensitivity of circulation to freshwater inflow is high only with very small values of R. The results of chapter 4 suggest that this increased sensitivity could only happen in the SoG with R inflows smaller than 2×103 m3 s−1 . In contrast, all of our R values are in the range 2×103 –104 m3 s−1 . In the SoG, very low values are very unlikely. According to the 1912–2008 historical data (Fig. 6.4), very few values are lower than 2×103 m3 s−1 (only 4 monthly means out of 1162 range within 1.9×103 –2×103 m3 s−1 ). This value is not significantly different from the minimum observed over 2002–2005. That is, the observed winter inflows 192  Chapter 6. Discussion and Conclusion R used in our box model were already typical minimal values (Fig. 6.4). While U1 is sensitive to the seasonal changes of R according to Ra (with a ≪ 1), the analysis of similar relationships between U2 and R, and W2 and R suggests that either the sensitivity of U2 and W2 to R is very small or that there is no sensitivity at all. This can readily be noticed in Fig. 4.1. The seasonality magnitude of U2 and W2 is equal or smaller than the seasonality magnitude of U1 despite the large seasonal changes of the FW forcing. How then does the estuary responds to changes in R? Eq. 6.1 suggests an increase in R is nearly balanced by an increase in stratification near the surface. For instance, Fig. 3.3 shows that S2 changes by less than 0.25 from April to August in 2002 and 2003, while S1 can vary by 3 in 2002 and by 1.5 in 2003, closely following the changes of R during the freshet in 2002 and 2003. In our box model, the estimated variability of the vertical mixing M is large, on average 44% of the mean with maxima during the freshet and winter, but this is partly due to large errors in the estimation (Table 4.1). The monthly sampling period used in our box model is too large to capture the effect of spring-neap tidal effect on M. So, the tidal effect on the vertical mixing over the sampling period cannot be observed. The changes in M must be explained by another mechanism. Vertical mixing can be reduced by an increase in stratification [MacCready and Geyer 2010]. In particular, during the freshet peak, one expects to see a maximal effect of the stratification on the vertical mixing because the stratification increase with R. In Eq. 2.57, the fluxes of correlated variations were neglected. If there is any vertical flux of correlated variations over a monthly time period, neglecting this term increases the error associated with the vertical transports W1 and W2 in Eq. 2.60. Li et al. [1999]’s numerical box model of the SoG also showed a small seasonal variability of the SoG estuarine circulation (see Chapter 4, section 4.3.1) with a 193  Chapter 6. Discussion and Conclusion 8  7  3 −1  Bins are centered and of size 200 m s  % of 1162 FW monthly averages  6  5  4  3  2  1  0 0  0.2  0.4  0.6  0.8  1  1.2  1.4  1.6  1.8  2 4  x 10  Figure 6.4: Histogram of FW Input in the SoG over 1912-2008 parametrization of vertical mixing [Li et al. 1999, Equations (2) and (3)] depending on stratification. The vertical mixing exchange was set to be high at spring tides, low at neap tides and lowest during the freshet. The order of magnitude of the vertical mixing in Li et al. [1999] (pp 10–11, MLi99 = ωg Ag = 104 m3 s−1 ) is similar to mine (on average 1.9×104 m3 s−1 ). In addition, the timing of this lowest vertical mixing at  194  Chapter 6. Discussion and Conclusion the freshet in Li et al. [1999] is similar to the timing of the low values of M during the freshet in our box model. In section 4.3.1 (comparison between our transport U1 with Li et al. [1999]’s) the similarity of the seasonal changes between the two sets of results suggests that reduced vertical mixing [MacCready and Geyer 2010] during the freshet may be diagnosed in our box model. Despite seasonal adjustments to the FW forcing, relatively steady estuarine circulation suggests a small sensitivity of the seasonal changes of estuarine circulation to the FW inflow changes (Fig. 4.9): e.g. maximum increase of 33% above mean U1 (4.6×104 m3 s−1 ) in summer 2002 for a peak of R of about 3 times mean R (5.3×103 m3 s−1 ), maximum decrease of 11% below mean U1 in winter 2003 for the lowest R, about 36% of mean R. The changes of R observed during 2002–2005 are representative of the changes of R recorded between 1912 and 2008. So these limits should hold in general. An important implication of this relatively steady estuarine circulation is that nutrients are upwelled relatively constantly from depth to near surface. This is shown by the analysis of section 5.2.1 where it was found that transports of upwelled nutrients and advected nutrients have a small seasonal variability (Tables 5.1–5.3).  6.3  Estimates and Variability of NPP Rates in Estuaries  Table 6.1 shows the range of PP rates for a large number of temperate estuaries in North America and Europe [Wilson 2002, Heip et al. 1995, Therriault and Levasseur 1985], including the NPP rates from our study. Since there are many different studies, the estimates correspond to a variety of types of measurement of PP rates. However,  195  Chapter 6. Discussion and Conclusion  1.4  1.2  FW discharge (×104 m3 s−1)  1  0.8  0.6  0.4  0.2  0 Jan Feb Mar  Apr May Jun  Jul  Aug Sep Oct Nov Dec  Figure 6.5: Mean Monthly FW Input in the SoG from 97 years of data for Fraser River scaled as described in section 3.2.3. Vertical bars are uncertainty of the means.  196  Chapter 6. Discussion and Conclusion Reference  Annual euphotic  Time Period  Location  PP rate gC m−2 yr−1 This study  net 212  2002-2005  SoG  1977-1995  12 estuaries in Eu-  (spring/summer 157) 7-560  Heip et al. [1995]  rope and North America Therriault and Levasseur  7-470  1974-1984  20 estuaries  [1985] in North America 0-192  Wilson [2002]  1981-2000  5 UK and US estuaries  Harding et al. [2002] Tian et al. [2000]  mean 408 range 282-538  1982-1995  CB  160  1992-1994  GSL model  estimates range 100–212 Savenkoff et al. [2000] Boyer et al. [1993] Pennock  and  Sharp  range 0-500, max in spring  1992-1994  GSL (Fig. 2 in ref.)  mean 465 range 395-493  1985-1988  Neuse River  mean 307 range 190-400  1981-1985  Delaware  90-820  1950s-80s  10 US estuaries  [1986]  (820 Hudson) Testa and Kemp [2008]  net 0-116  Table 1 in ref. 1985-2008  Patuxent River  assuming net uptake rates of P and N Fig. 7 in ref. and Redfield ratios Table 6.1: Ranges of PP Rates in Temperate Estuaries.  197  Chapter 6. Discussion and Conclusion most of them are based on techniques of  14  C uptake in incubation. In general, most  of these estimates would be somewhere between gross and net PP rates [Marra and Barber 2004, Williams and Lefevre 2008]. In some cases, this is known more precisely [Harding et al. 2002]. In Table 6.1, most estimates have numerical values of several hundreds of gC m−2 yr−1 . The average NPP rate of the SoG is 212 gC m−2 yr−1 (based on P uptake rate). The biological analysis shows that the f-ratio could range between 0.5 and 0.88. The former comes from the analysis in discussion section 5.3, while the latter is based on annual NPP rate of 212 gC m−2 yr−1 and annual new PP of 187 gC m−2 yr−1 . However, comparison of P and N uptake rates suggests that the f-ratio leans towards values 0.8, and thus, the total NPP rate overall tends to be greater but close to 212 gC m−2 yr−1 . The SoG total NPP rate overall of 212 gC m−2 yr−1 sets the NPP rate of the SoG as average and typical when compared to other NPP rates in Table 6.1. The most productive estuary in Table 6.1, Hudson estuary [Pennock and Sharp 1986] has a P Macronutrient Range (µM)  N  0.5–1.25 6–14  Si 25–41  Redfield and Brzezinski factors 1  16  Normalized Range  0.4–0.9 1.6–2.6  0.5–1.3  16  Table 6.2: Macronutrient Limitation During Summer in the SoG. The first row shows SoG surface range of the 3 macronutrient (Phosphate (P), nitrate (N), silicic (Si) acid) concentrations during Summer (using Figs 3.7–3.9). Second row shows the Redfield and Brzezinski ratios. Third row shows the ratio between the first and second rows. The limiting nutrient has the lowest value on the third row.  198  Chapter 6. Discussion and Conclusion primary productivity (820 gC m−2 yr−1 ) that is 4 times larger than the primary productivity of the SoG. The primary productivity of Hudson estuary is high because its primary productivity is partly fuelled by anthropogenic nutrient waste inputs [Fisher et al. 1988] and the estimate is based on GPP rate [Howarth et al. 2006, Hudson estuary, 850 gC m−2 yr−1 GPP rate over 1990s in the most productive area of the estuary] (see Eq. 5.6). Since anthropogenic inputs to the SoG are negligable [Mackas and Harrison 1997], during summer nutrient limitation is probably the most important factor in controlling primary productivity in the SoG. Table 6.2 suggests that nitrate limitration is probably the most critical of the 3 macronutrients. Generally in temperate estuaries, a diatom bloom occurs in spring [Heip et al. 1995, Miller 2004] (seasons defined in Glossary p. xiii). In most of the estuaries of Table 6.1, blooms can occur not only during spring but also during summer and fall: e.g., GSL, CB, Delaware, Patuxent River estuaries [Heip et al. 1995, see Table 2]. Diatoms strive when light and nutrients are both plentiful [Sarthou et al. 2005, Aiken et al. 2008]. Thus, in deep estuaries, spring conditions are ideal for diatoms to grow. Surface nutrients have been replenished during winter and light level is increasing. In shallow estuaries, the increase of turbidity, due to sediments or selfshadowing of phytoplankton, can lower light level and delay the large bloom until summer. In shallow estuaries, a large summer bloom can be fueled by an increase of nutrients from rivers or regeneration of spring bloom nutrients [Harding et al. 2002, Testa and Kemp 2008], combined with the increased summer light level. In the SoG, conditions are favourable to diatom spring blooms: diatoms dominate the biomass of autotrophic phytoplankton (Fig. 5.13), and nutrients are plentiful (Figs 3.7–3.9). Fig. 5.8 shows that the largest chla biomass occurs in spring: 0.3 gChl-a m−2 in April 2002, 0.16 gChl-a m−2 in April 2003, 0.37 gChl-a m−2 in March 2004, 0.3 gChla m−2 in March 2005. They are all associated with the largest annual total NPP rate 199  Chapter 6. Discussion and Conclusion (>1.2 gC m−2 d−1 based on P uptake). The NPP rate represents the production of organic carbon by photosynthesizers; a seasonal maximum NPP rate will probably be associated with a high biomass (large number of photosynthesizers) while a low biomass (small number of photosynthesizers) can still produce organic carbon but not necessarily at a high rate [Miller 2004]. Thus, there maybe a correlation between the NPP rate and the phytoplankton biomass [Miller 2004, see Fig. 1.1]. Using our estimate of the NPP rate (Fig. 5.8), the coefficient of correlation between new NPP rates and integrated chla biomass (a proxy for C biomass, assuming C:Chl-a does not very greatly) is 0.65 with a 95% confidence interval ranging between 0.44 and 0.79. Many mathematical models of daily and annual PP rates also assume a proportionality relationship between production and productivity [Boyer et al. 1993, Harding et al. 2002]. In agreement with low winter biomass and in contrast with summer high productivity, winter productivity is low: winter productivity is 0.1 gC m−1 d−1 at most (while summer average is 1 gC m−1 d−1 ), but the winter uncertainty is of the same order of magnitude and the uncertainty suggests that there is no winter productivity on average (using C:P:N=106:14.6:1 and 4th column in Tables 5.1–5.3). The maximum magnitude of the NPP rate can vary from year to year. In other estuaries, studies interested in interannual variability suggest that it is associated with variability in the nutrients supply. In CB, the interannual variability is large and depends on the river input of nutrients: 282–538 gC m−2 yr−1 over 1989-1998 [Harding et al. 2002]. On the other hand, in Patuxent River estuary the interannual variability depends on atmospheric deposition of DIN (dissolved inorganic N) and wet years tend to have higher NPP rates than dry years [Testa and Kemp 2008]. Since phytoplankton blooms depend on light and nutrient availability, the more variable of the two, nutrient availability, is usually thought as the more likely to explain the inter200  Chapter 6. Discussion and Conclusion annual variability of the NPP rate. According to our results, in the SoG there was no large interannual change over 2002–2005. This is not surprising since we found that the net entrainment of nutrients has small variability over seasons and over years. Our estimates of the SoG NPP rate are based on new and total NPP rates. The variability of the regenerated NPP rate might have a non-negligible contribution to the total NPP rate and its interannual variability. However, in the SoG the regenerated NPP rate is not well quantified and the timing of the maximum regenerated NPP rate suggests that regeneration might not be seasonal, but only episodic (Fig. 5.12). For instance, our estimate of the f-ratio varies between 0.5-0.9, although our average estimate suggests a high f-ratio (average f=0.88) based on estimates of the total and new NPP rates (section 5.3). Such high f-ratios have been previously observed in similar systems during diatom blooms using different measurement methods: e.g., high productive fjord (≥0.8 and up to 0.96) [Kristiansen et al. 2001] and in Monterey Bay (average f-ratio=0.83) [Brzezinski et al. 2003b].  6.4  Recommendations for Future Studies  Review and discussion of the results in the previous chapters suggest that several open questions in the analysis of the mass transports and primary productivity still remain. These open questions need to be addressed in future studies. Here these questions are discussed and recommendations for future work are made. The physical analysis of the water transports failed to show the U1 =f(R) relationship at low and high FW inflows (Fig. 4.9), where a match or mismatch with the theoretical U1 =aR1/m (1/m ≪ 1) might be more obvious. Unfortunately, obtaining more data and studying further the estuarine circulation in the SoG will not provide the additional needed information. Indeed, a closer look at the history of Fraser 201  Chapter 6. Discussion and Conclusion River daily discharge (Environment-Canada, see source detail in section 3.2.3) suggests that R <2×103 m3 s−1 only 0.25% of the time over 1912-2005, with a minimum of 1.73×103 m3 s−1 , and R > 2×104 m3 s−1 only 0.13% of the time, with a maximum of 2.64×104 m3 s−1 (Fig. 6.4). Thus, the 2002-2005 data combined with the regular Fraser River cycle has already provided enough information to fully determine the variability of the FW forcing and, through our box model approach, the monthly estuarine circulation. Thus, it would be necessary to study other estuaries to gain further knowledge on the U1 =f(R) relationship. There are three conditions that estuaries have to satisfy if one wants to be able to apply the theoretical U1 =f(R) relationship, U1 =aR1/m [Hetland and Geyer 2004, p. 2689]: 1. Salt budget is dominated by the estuarine circulation. 2. Estuarine circulation is larger than FW flow: mass budget is dominated by estuarine circulation. 3. Estuary domain lies within an approximately rectangular and prismatic channel. Most estuaries are likely to satisfy the first two conditions on a monthly timescale. If one investigates another system, it is recommended to choose an estuary in which ranges of estimated transports would cover several orders of magnitude: e.g. in the case of the SoG, 103 m3 s−1 , 104 m3 s−1 and 105 m3 s−1 would have been more useful than only 104 m3 s−1 . A range with two observable orders of magnitude for FW discharge is unusual, but perhaps not completely unreasonable, since the range is at least one order of magnitude for many large rivers: for instance, the discharge of the Fraser (0.2×104 m3 s−1 –2×104 m3 s−1 ), Seine (0.03×104 m3 s−1 –0.25×104m3 s−1 ), 202  Chapter 6. Discussion and Conclusion and Columbia (0.3×104 m3 s−1 –2×104 m3 s−1 ) rivers [Pawlowicz et al. 2007, Huang et al. 2009, Hughes and Rattray 1980]. In the physical analysis, the next open question concerns the average transports due to deep water renewals (DWR) and their variability in the SoG. The physical analysis (sections 4.2.1 and 6.2) emphasized that a 2-box model had provided little information on the deepwater renewals apart from the average transport magnitude, although this information was present in the data [Pawlowicz et al. 2007]. A 3-box model could provide the information on the seasonal variability of the transports associated with DWR. This would involve a reanalysis of the SoG data with a modified box-model layout. The SoG would be divided into top, deep, and bottom boxes. New transports of water and tracers, and storage rates would be introduced in the budget equations. As for the 2-box model, a critical choice is to decide the separation depths between the boxes. There is no a priori reason to change the separation depth between the top and deep boxes (d) beyond the choices made here in Chapter 4. On the other hand, the separation between the deep and the bottom boxes (D) could be based on the shallowest depth where the SoG water renewals can occur. Mass transports between boxes may (or may not) be relatively insensitive to changes of the separation depth between the deep and the bottom boxes. In the case that one or more transports were to become markedly sensitive to one or both of the separation depths (d and D), a remedy could be to use a discretization of the water column: use a larger number of boxes (>3) limited by the size of the bins of the CTD vertical profiles (section 3.2.1). Such a discretization would have a secondary benefit. In the physical analysis (section 6.2), we discussed the approximation of the boundary average of the tracers by the volume average. In the limit of a large number of boxes, the difference between the volume average and the boundary average would tend to be negligible. 203  Chapter 6. Discussion and Conclusion Another unknown in the physical analysis is the relative size of the transports at the Northern Passage compared to the ones at the Southern Passages. The only estimates of the Northern Passage transports based on observations date back to Godin et al. [1981] (cross-channel transect in Johnstone Strait, or JS). No recent estimates based on observations were available to be compared with my estimates of the SoG transports. All of the analysis is based on observations collected in the central and southern parts of the SoG (Figs 1.1 and 1.2). My estimated transports cannot be clearly associated with either the Southern Passage or the whole SoG. I chose to assume that they were the transports at the Southern Passage based on the assumption, found in the literature, that the transports at the Northern Passage are about 7% of the transports at the Southern Passage. This assumption is based on the comparison of the area of the cross-channel sections at the Southern and Northern Passages, assuming similar current speed (section 1.1). An examination of the estimates from Godin et al. [1981] suggests that the ratio of transport magnitudes between the Northern Passage (average transport about 3×104 m3 s−1 ) and the Southern Passage (average transport 105 m3 s−1 ) is 30%, but they considered transects in the SoJdF (Southern Entrance) and in JS (Northern Entrance, farther from Seymour Narrows) instead of exiting the SoG itself. The SoJdF is separated from the SoG by the HS. Entrainment along the straits results in the amplification of the mass transports in each layer in the downstrait direction. Thus, the layer transports at the Southern Passage cannot be approximated by the SoJdF transports. Similarly, transports at Seymour Narrows cannot be approximated by those in JS. Transport magnitudes and their variability in both Northern and Southern Passages could be estimated using our inverse box modelling approach, but it would require additional data from the Northern end of Vancouver Island. The last unknown in the physical analysis concerns the average magnitude of 204  Chapter 6. Discussion and Conclusion transports and their variability over short timescales (< 1 month). In the physical analysis (section 4.2.1), we noticed large oscillations with 2-month period (when biweekly sampling was possible during March-April 2003 and 2004). The magnitude of these oscillations is very large compared to the average magnitude of the transports over spring, in particular for transport M. One possible explanation for these large oscillations is that they could be due to shorter timescale (< 1 month) processes like spring-neap tides [Li et al. 1999] and episodic wind-driven surface mixing [Collins et al. 2009]. Thus, it would be useful to gather data on a shorter timescale than the monthly timescale used in our study. Since a larger number of cruises would be carried out, a smaller number of stations could be sampled. The biological analysis of the Net Primary Productivity (NPP rate) suggests that the spring and summer blooms have a stable seasonal pattern over a long timescale: in spring NPP rate and biomass both reach their annual peak, in summer NPP rate and biomass levels are lower than spring, but they remain high over a longer interval than in spring. However both spring and summer NPP rates have a roughly equal contribution to the annual NPP rate. How stable is this pattern? Collecting data over a larger timescale (≫ 3 years) would help to answer this question. The NPP rates could be estimated with a forward box model approach (section 2.4) based on the estuarine circulation using the relationship U1 =f(R) found here (Fig. 4.9) with additional observations of the nutrient levels. Recently the Victoria Experimental Network Under the Sea (VENUS) observatory has begun providing near real-time and archived data at different discrete depths within the southern SoG [Dewey et al. 2010]. One of the objectives of the observatory is to provide long-term observations of the SoG physical environment and ecosystem at various timescales. In the near future, some of the VENUS sites may be equipped with automatic vertical profilers to collect various physical and biogeochemical properties at mid-water. Ferries are 205  Chapter 6. Discussion and Conclusion to be fitted with sensors to sample the surface of the SoG [Dewey et al. 2010], as has been already done during STRATOGEM [Halverson and Pawlowicz 2008]. All these data could be used to build long term vertical profiles of the biogeochemical tracers necessary for a long timescale study of the SoG NPP rate and NPP. The biological analysis suggested that there was a difference between spring and summer in NPP rate, biomass (Fig 5.8), taxonomy (Fig. 5.13), Si:N and Si:P ratios of uptake rates (Fig. 5.17). In particular, the Si:N and Si:P ratios were above the average ratios (16:16 and 16:1) expected in coastal waters [Brzezinski 1985, Brzezinski et al. 2003a]. Could these differences be explained by different species of diatoms in the PP: driven in spring by heavily silicified diatoms and in summer by less silicified diatoms? Recent studies of co-enrichment with silicic acid and iron, using central Pacific Equatorial water in mesocosms and seeding of phytoplankton patch in the Southern Ocean, suggested that different concentration levels of dissolved silicic acid could heavily favour or disadvantage silicified diatoms depending on the Si levels and drive the Si:N and Si:P ratios above the 16:16 and 16:1 expected ratios [Coale et al. 2004, Marchetti et al. 2010]. Note that recent estimates of the biogenic silica content of sinking particles in the SoG aphotic zone were consistent with high Si uptake during spring blooms [Johannessen et al. 2005]. It is not clear if the high Si:P and Si:N in the SoG is due to only one cause. First, the Si:P ratio is not as high as expected when there is a high demand of Si by diatoms as suggested by the experiments of co-enrichment with silicic acid and iron. Secondly, the possible spring competition between diatoms and heterotrophic silicoflagellates for Si, suggested above, is based on the taxonomy analysis of only one STRATOGEM station although a central location and the representative of the SoG. At this point, the combined effect of these two Si sinks cannot be ignored. Thus, I recommend a reanalysis of the STRATOGEM 206  Chapter 6. Discussion and Conclusion taxonomy data looking for differences in diatoms composition and silicoflagellates abundance, and a reanalysis of the size-fractioning data of extracted chlorophyll-a. This reanalysis could provide a preliminary study. Secondly, an extensive study similar in scope to Marchetti et al. [2010], but with SoG water could be carried out to determine precise uptake rates, taxonomic and chemical compositions of phytoplankton. In the biological analysis, using the estimates of the new and total NPP rates, we attempted to estimate the f-ratio, the relative importance of external supply of N (assuming negligible atmospheric N2 fixation) relatively to the total primary productivity. Low f-ratios ( 0.5) were found in previous studies of SoG biological oceanography [Harrison et al. 1983] indicating that the internal supply, e.g. regeneration of surface organic N, primarily contributes to the supply of N. However, our biological analysis suggested a higher average f-ratio, e.g. annual average of 0.88 over 3 years (section 5.2.4), with possible events of low f-ratios (section 5.2.5). The results of these studies are not incompatible. High f-ratios (f>0.7) are expected in highly productive ecosystems where the primary supply of nutrients is external [Eppley and Peterson 1979, Legendre et al. 1999]. During coastal diatom blooms, succession of high and low f-ratio events had been previously associated with the succession of blooming diatoms and collapsing diatom blooms due to nutrient depletion [Kristiansen et al. 2001]. Kristiansen et al. [2001] noted a clear transition between PP based on external supply and PP based on nutrient regeneration occurring with the depletion of available nutrients. Thus, we recommend that future studies in the SoG include not only nitrate+nitrite and phosphate measurements, but also ammonium, to directly estimate the regenerated NPP (see Chapter 5). Additional short-term 15  N incubation experiments uptakes have been used to measure the nutrient uptakes  in previous studies [Kristiansen et al. 2001, Elskens et al. 2008] and these isotope 207  Chapter 6. Discussion and Conclusion uptake rates could provide independent estimates of the nutrient uptakes rates. In the biological analysis, we focused on the surface biogeochemical processes as they provide information about spring and summer blooms, the NPP and the new NPP rate. We did not analyze the rest of the SoG water column, below 30 m down to 400 m. Thus, one of the questions left open by our biological analysis is whether there is a net biogeochemical production or consumption of nutrients and dissolved O2 in the aphotic zone of the SoG, and if we can identify the corresponding biogeochemical transformations. If there are deep biogeochemical processes, do they interact with the surface biogeochemical processes? A similar box-model approach has been used in a partially-stratified estuary to address this question [Testa and Kemp 2008]. Unfortunately, the quality of the nutrients data was not very high. The quality of this data could be improved by updating the nutrients protocol and using small batches of fresh nutrient samples for the analysis, instead of large batches of frozen samples. However, this would make the sampling and analysis logistically more complex. I estimated and compared the differences of nutrient concentrations between replicates of fresh and frozen samples analyzed within a week, and frozen samples analyzed within a few months. Nutrient concentrations tend to be more accurate when samples were analyzed within a week, in agreement with Barwell-Clarke and Whitney [1996]. In the biological analysis, I estimated the rate of the primary productivity which is essential, as a food source, to the rest of the SoG ecosystem and adjacent systems. One of the next open issues is now the export production due to SoG phytoplankton and what are its downward (contribution to pelagic and benthic ecosystems) and seaward (possible contribution to adjacent systems) components? Is the SoG estuary a net autotrophic or a net heterotrophic system? To carry out such an analysis of the transport of organic carbon, I recommend 208  Chapter 6. Discussion and Conclusion collecting data on dissolved and particulate organic carbon (DOC and POC) in addition to the nutrients and dissolved O2 in both throughout the water column in the SoG and outside of the SoG (HS, SoJdF, JS). Collecting data on size-fractionated biogenic silica could provide a way to separate the contribution of diatoms from other phytoplankton species and estimate the contribution of the silica pump to the exported production [Brzezinski et al. 2003b]. As introduced in the first chapter, the motivation of this research was to study the SoG ecosystem condition at the lowest level of the foodweb, the primary production, and the related parameters, the seasonal average of the NPP rate and its seasonal variability. 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