CIRCULATION OF F R O M T H E N O R T H E A S T T E M P E R A T U R E PACIFIC O C E A N A N D SALINITY DATA. By Richard James Matear B. Eng. (Geophysics) University of Saskatchewan A THESIS SUBMITTED T H E IN PARTIAL REQUIREMENTS MASTERS FOR OF FULFILLMENT T H E DEGREE OF SCIENCE in T H E FACULTY OF GRADUATE STUDIES GEOPHYSICS We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA August 1989 © Richard James Matear, 1989 INFERRED OF In presenting this degree at the thesis in University of partial fulfilment of of department this or thesis for by his or requirements British Columbia, I agree that the freely available for reference and study. I further copying the representatives. an advanced Library shall make it agree that permission for extensive scholarly purposes may be her for It is granted by the understood that head of copying my or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract T h e temperature, salinity, and pressure ( S T P ) data were coDected during two cruises, one i n earl}' October and the other in early December of 1987, as part of the Ocean Storms experiment. These hydrographic data were analyzed to determine circulation of the northeast Pacific Ocean and to calculate the factors influencing the heat and salt content of the upper ocean. From the depth profiles of the temperature and salinity data, the water mass in the Ocean Storms area was classified as being the Eastern SubArctic Pacific Water Mass. Maps of dynamic height for this area revealed that the current pattern was generally smooth but that mesoscale eddies did exist in the flow. Isentropic analysis of the temperature and salinity fields produced a flow pattern that was generally consistent with the. dynamic height maps. However, this analysis revealed additional details in the flow that were not evident i n the dynamic height maps. To extract additional information from the temperature and salinity data an inverse model was developed. This model assumed that the flow was geostrophic and that the vertical velocity satisfied a linear 5-plane vorticity equation. This inverse model calculated the vertical and horizontal velocities at a reference level of 1000 dbars, and the horizontal and vertical mixing terms by conserving mass, salt, and heat. These conservation constraints were applied to large- boxes defined by four hydrographic stations and two pressure surfaces. The circulation determined using the model showed well-defined flow features. Comparison of the absolute geostrophic flow with the seven-day averaged current meter observations showed much similarity, despite complications from an incident storm. Correlation between the geostrophic flow at the surface and the total flow field inferred from ii drifter data was also high. A n estimate of the ageostrophic flow suggested that acceleration and nonlinear terms played an important role in affecting the flow field during the first cruise. The observed change in salt content of the upper 150 m of the ocean was .004 ppt for the sixty days between the two cruises. The net transport of salt into the study area by the calculated flow field was -.016 ppt. Therefore, to balance the salt budget would require E - P = 9 cm. The upper ocean lost 92 W/m 2 of heat during the sixty days between the two cruises. As the vertical and horizontal transport of heat acounted for 40 W/m 2 loss of heat, the remainder of heat lost, 52 W/m . 2 boundary processes. m was attributed to air-sea Table of Contents Abstract ii List of Tables vii List of Figures x Acknowledgement xv 1 Introduction 1 2 Hydrographic Data 5 2.1 Data Collection 5 2.2 D a t a Calibration 11 2.3 Data Processing 11 2.4 Removal of Internal Waves 12 3 Hydrography of the Ocean Storms Area 15 3.1 Temperature and Salinity Structure 15 3.1.1 Vertical Structure 15 3.1.2 Temperature-Salinity Diagram 19 3.2 The Horizontal Flow Field 21 3.3 Dynamic Height 22 3.3.1 29 3.4 Horizontal Distribution of Temperature and Salinity Summary 31 iv 4 5 6 7 Developing and Solving the Inverse Problem 44 4.1 Determining the Ocean Circulation: The Physical Problem 44 4.2 The Inverse P r o b l e m 47 4.2.1 Formulation 47 4.2.2 Ocean Storms G r i d 50 4.3 Including A Priori Information 52 4.4 Solving the Inverse P r o b l e m 54 Evaluating the Model and Solution 62 5.1 The Importance of Temperature, Salt and Mass Conditions on the Solution 63 5.2 The Effect of the Layering of the Ocean on the Solution 69 5.3 The Sensitivity of the Solution to the Choice of Reference Level 71 5.4 The Importance of the Different Unknown Parameters on the Solution . . 73 5.5 Sensitivity of the Solution to Errors i n the Data 74 5.6 Reference Solution 78 Results of the Model 80 6.1 Vertical Velocity . 80 6.2 Vertical M i x i n g Coefficients 81 6.3 Horizontal Velocities 81 6.4 Direct Current Measurements 89 6.5 M i x e d Layer Drifters 99 Factors Influencing Heat and Salt Content in the Ocean Storms Area 104 7.1 The Observed Change i n Salt and Heat Content 104 7.2 Transport of Salt and Heat 107 7.2.1 108 Salt Transport v 7.2.2 7.3 8 Heat Transport 109 Exchange Across the Air-Sea Boundary 7.3.1 Salt Budget 7.3.2 Heat Budget 110 I l l 112 Discussion and Conclusion 114 Bibliography 119 vi List of Tables 4.1 The levels used to vertically define the boxes used in the model 51 5.2 Correlation and regression coefficients between up-weighted solutions and the reference solution which weighted the three conditions equally ( T = l , S= l, M = l ) 5.3 64 The normalized residual errors i n the three conditions for the null solution, the reference solution, and the solution obtained by alternating the upweighting of the three conditions. The solution for temperature, salt or mass is the solution obtained when this condition is up-weighted by a factor of 5 5.4 65 Correlation and regression coefficients between solutions with various weights for the conservation of mass condition. The weighting of the temperature and salt conditions are held fixed at one 5.5 T h e normalized residual errors i n the conservation of temperature, 67 salt, and mass conditions for various weights for the conservation of mass condition. T h e N u l l column refers to errors i n these three conditions if the solution is identically zero 5.6 68 T h e correlation and regression coefficients between the reference solution, a 12-layer model with solutions obtained from models using different numbers of layers 70 vn 5.7 The correlation and regression coefficients between the reference solution and solutions obtained by doubling the thickness of the layers considered in the model. For the 6-layer model the ocean is divided according to the following pressure surfaces, 150, 250, 400, 600, 800, 1000, 1500 dbar. The 5-layer model divides the ocean according to the following surfaces, 150, 250, 500, 800, 1000, 1500 dbar. Only the reference vertical and horizontal velocities are used i n determining the regression and correlation coefficients. 71 5.8 The correlation and regression coefficient between solutions using different reference levels and the reference solution that used 1000 dbar. The correlation and regression coefficients were calculated using the velocities from 1000 dbar and all of the mixing coefficients 5.9 The correlation and regression coefficient for horizontal reference velocities between solutions obtained with a reduce set of unknown parameters. 5.10 72 75 The correlation and regression coefficients between the reference solution and solutions obtained using data with random errors added. The random errors with standard deviation a are add to the original data as follows: a) salinity, 5.11 ppt and b) temperature, t& °C 75 The correlation and regression coefficients between the reference solution and solutions obtained using data with coherent temperature and salinity errors added to the data 6.12 76 A comparison of the geostrophic velocities calculated from the model to the current meter observations 91 Vlll 7.13 Change i n salt content between cruise one and cruise two. The value given for the change i n salt content is the change in salinity observed i n that layer. T h e mean and standard deviation are obtained using the stations occupied during both cruises 106 7.14 Change i n heat content between cruise one and cruise two. T h e value given for the change in heat content is the change in heat observed i n that layer. T h e mean and standard deviation are obtained using the stations occupied during both cruises 7.15 107 T h e calculated transport of salt using circulation parameters determined by the inverse model. The vertical mixing term is only important at depths of 250 m to 800 m. The errors in these transports are determined from the uncertainty in the circulation parameters and the uncertainty i n the measurement of salinity. A positive transport means that the Ocean Storms area experiences a gain in salinity 7.16 108 The calculated fluxes of heat from using the velocities obtained from the inverse model. The vertical mixing term is only important 500 m to 1200 m. at depths of T h e errors i n these transports are determined from the uncertainty i n the circulation parameters and the uncertainty in the measurement of temperature 110 IX List of Figures 2.1 Location of the Ocean Storms area i n the northeast Pacific Ocean 6 2.2 The planned hydrographic grid for the Ocean Storms area 7 2.3 The S T P stations occupied during the first cruise (8702) i n the Ocean Storms grid and surrounding area. T h e arrows indicate the path travelled by the ship during this cruise 2.4 9 The S T P stations occupied during the second cruise (8704) i n the Ocean Storms grid and surrounding area. T h e arrows indicate the path travelled by the ship during the cruise 2.5 10 Potential density versus pressure plots. T h e upper panel shows the presence of the internal wave i n the data. T h e lower panel shows the filtered version with this internal wave removed 3.6 13 Major surface currents of the North Pacific Ocean. T h e Ocean Storms area is centred at 47.5° N and 139.0° W, just southeast of Ocean Station P 3.7 16 Salinity and temperature profiles from the Ocean Storms area (OS10, 48° N and 139.3° W). In these plots the solid line refers to data from cruise one and the dash line represents data from cruise two 3.8 17 Temperature-Salinity (T-S) diagrams for the first cruise (8702) and for the second cruise (8704) for data obtained from the Ocean Storms area. In these plots the solid line represents the average T-S curve for that cruise. x 20 3.9 Spatial structure of the T-S diagram for cruise one (8702). T h e solid line represents the T-S curve from OS25. These plots show the presence of two different water masses (A) and (B) i n the halocline 21 3.10 Cruise One: Dynamic Height i n dynamic meters relative to 1000 dbar at: a) the surface, b) 150 dbar, c) 500 dbar. In these maps the arrows indicate the direction of the geostrophic flow and the dots denote the location of the S T P stations obtained during this cruise 23 3.11 Cruise Two: Dynamic Height i n dynamic meters relative to 1000 dbar at: a) the surface, b) 150 dbar, c) 500 dbar. In these maps the arrows indicate the direction of the geostrophic flow and the dots denote the location of the S T P stations obtained during this cruise 3.12 Temperature and Salinity distribution for cruise one on the a 26 t = 25.0 surface for: a) the temperature distribution i n deg C, and b) the salinity distribution i n ppt. The arrows i n these maps indicate the interpretative current pattern 3.13 Temperature 32 and Salinity distribution for cruise one on the a t = 26.2 surface for: a) the temperature distribution i n deg C, and b) the salinity distribution i n ppt. The arrows i n these maps indicate the interpretative current pattern 3.14 Temperature 34 and Salinity distribution for cruise one on the a t = 27.0 surface for: a) the temperature distribution i n deg C, and b) the salinity distribution in ppt. The arrows i n these maps indicate the interpretative current pattern 36 xi 3.15 Temperature and Salinity distribution for cruise two on the a t = 25.5 surface for: a) the temperature distribution in deg C, and b) the salinity distribution in ppt. The arrows i n these maps indicate the interpretative current pattern 3.16 38 Temperature and Salinity distribution for cruise two on the a t = 26.2 surface for: a) the temperature distribution in deg C, and b) the salinity distribution in ppt. The arrows i n these maps indicate the interpretative current pattern 40 3.17 Temperature and Salinity distribution for cruise two on the a t = 27.0 surface for: a) the temperature distribution i n deg C, and b) the salinity distribution in ppt. T h e arrows i n these maps indicate the interpretative current pattern 42 4.18 T h e boxes used for the inverse model of the Ocean Storms area. For cruise one, all the boxes are defined 4.19 48 A plot of the normalized magnitude of the singular values versus index number p 58 4.20 T h e reduction in error versus the number of singular values of matrix B that are used in determining the solution. T h e number of singular values used (n), indicates that the first n singular values of matrix B 4.21 are used. 59 T h e trade-off curve for the inverse model. This curve relates the resolution of the model to the size of the variance of the unknowns. The numbers over the points on this curve indicate the cutoff for the singular values. . 61 5.22 The high frequency fluctuations of the temperature measurements obtained from a mooring located i n the centre of the grid xii 77 6.23 The vertical velocity determined from the inverse model. The velocities are i n fim/s and a positive value means the flow is upward. The top map shows the vertical velocity at the base of the mixed layer (50 m). The lower map is the vertical velocity at the base of the halocline (150 m). . 82 . 83 6.24 The vertical mixing coefficients,^, determined from the inverse model. 6.25 The horizontal flow field for selected depth surfaces of the upper zone (0200 m). The velocity is shown at: a) the surface, b) 15 m, c) 100 m, and d) 200 m. . . 85 6.26 The horizontal flow field i n the intermediate zone (200- 900 m) at selected depths. The velocity is shown at depths: a) 500 m, b) 600 m, c) 700 m, and d) 900 m 87 6.27 The horizontal flow field i n the deep zone (900-1500 m) at selected depths/ The velocity at depths of a) 1000 m and b) 1500 m 90 6.28 The velocities from the current meter observations for a seven day average plotted every three days. In this figure the stick points i n the direction of flow, with north being up. The numbers on the left side of the plot are the corresponding depths i n meters of the current meter. On the right side of the plot, the velocities calculated from the inverse model at the location of the mooring are displayed 92 6.29 The velocities from the current meter observations for a fourteen day average plotted every seven days. In this figure the stick points i n the direction of flow, with north being up. The numbers on the left side of the plot are the corresponding depths i n meters of the current meter. On the right side of the plot, the velocities calculated from the inverse model at the location of the mooring are displayed 93 Xlll 6.30 The velocities from the current meter observations for a 21 day average plotted every 10 days. In this figure the stick points i n the direction of flow, with north being up. The numbers on the left side of the plot are the corresponding depths i n meters of the current meter. On the right side of the plot, the velocities calculated from the inverse model at the location of the mooring are displayed 6.31 94 The v-component of the velocity measured at 3000 m on the OSU moor- ing(upper panel) and the v-component computed from the geostrophic relation and data from the bottom pressure sensor array 6.32 95 The horizontal velocity profiles for a) the geostrophic flow, b) the ageostrophic flow, and c) the total flow 6.33 The tracks of the drifter data: 98 The location of the Ocean Storms grid is defined by the dashed line on this map. The oscillation of the drifter tracks is due to inertial oscillations 6.34 The total flow field determined from the drifter data. The location of the Ocean Storms grid is shown by the enclosed region on this map. velocity map 6.35 100 This is obtained from the drifter data from day 275 to 295. . . . 101 The ageostrophic flow field determined from subtracting the geostrophic velocity determined from the inverse model at 15 m from the total flow estimated from the drifter data 103 xiv Acknowledgement I would like to thank my supervisors Dr. W. Hsieh and Dr. G. M c B e a n for their inspiration and guidance during the work on this thesis. I would also like to thank Dr. W. Large for the enthusiatic suggestions he provided during my research. Dr. S. Tabata generously supplied the S T P data as well as invaluable information on the analysis of these data. The current meter observations were provided by Dr. C. Paulson and Dr. M. Levine of Oregon State University. T h e drifter data were supplied by Dr. W. Large of University of British Columbia. I would also like to acknowledge the moral support provided by my wife, Rhonda who had the unenviable task of proof-reading the many drafts of my thesis. Chapter 1 Introduction One of the major goals of physical oceanography is to obtain a description of the large time-scale movement of water i n the ocean. Determination of this general circulation is an important step toward understanding global climate and the distribution of chemical properties i n the ocean [1]. T h e main body of knowledge about the general circulation of the ocean has been drawn from observations of the temperature and salinity distributions i n the ocean. T h e justification behind using temperature and salinity to infer ocean circulation are: 1) these quantities are relatively easy to measure and 2) i n contrast to direct velocity measurements, the signals i n the temperature and salinity fields are far less contaminated by energetic smaller-scale motions induced by eddies and waves [2]. The inference of flow characteristics from temperature and salinity data has evolved along two distinct paths: 1) the water mass analysis techniques (eg. isentropic analysis [3]), and 2) dynamic methods (eg. Beta Spiral method[4]). In the water mass analysis techniques, water masses with a characteristic range of temperature, salinity, and other properties are identified. B y identifying the location of the source of these water masses, a qualitative description of the flow can be inferred by assuming the path of the flow must be from the source to the point of observation [3]. This technique allows one to infer the flow field directly from the observed temperature and salinity distribution, but no quantitative description of the flow can be made. In dynamic methods, the constraint governing the large-scale flow i n the ocean is 1 Chapter 1. Introduction 2 geostrophy, the balance between the pressure gradient and the Coriolis forces = --(sjp) Q kxfu (1-1) The k is the unit directional vector i n the z direction (up). T h e pressure gradient can be related to the density stratification v i a the hydrostatic approximation. From the temperature and salinity information one can calculate the density stratification and determine the flow relative to a prescribed reference level, z , using the thermal wind T equations u - f QJ 0 Q dz y (1.2) Jz r The classical approach to solving this problem is to subjectively determine a depth of no motion and set the reference level z r equal to this depth. Dietrich [5] reviewed five different proposals for ascertaining this depth of no motion and concluded that none of the methods considered were very reliable. In recent years a more mathematical approach has been taken to determine the general circulation of the ocean by dynamic methods. T w o methods that have been developed which pose the problem mathematically as an inverse problem are the Beta Spiral method [4] and the box model method [1]. The Beta Spiral method was originally developed by Schott and Stommel [4] to determine the velocity at the reference level. In this method, an additional dynamic constraint, a linear /3-plane vorticity equation, was used to constrain the velocity at the reference level. This method allowed the determination of the 3-dimensional velocity field. Olbers et al [6] applied this method to a set of grid points i n the North Atlantic Ocean. In their formulation, the conservation of mass, potential density, and potential veronicity (a linear functional of temperature and salinity defined to be orthogonal to the potential Chapter 1. Introduction 3 density i n the T-S diagram [6]) were included to further constrain the solution. These constraints were applied at each grid point for several depths through a finite difference equation. Olbers et al solved the problem locally at each grid point independently of other grid points. Each grid point therefore required its own inverse solution. B y considering enough vertical depths at each grid point, one derives an overdetermined system of equations and finds the least squares solution to the problem. In the box model method originally presented by Wunsch [1], an inverse problem was formulated for determining the general circulation of the North Atlantic by considering the conservation of mass i n several large boxes. These boxes were defined horizontally by large regions of the North Atlantic Ocean bounded by hydrographic stations, and verti- cally by several isothermal layers. A solution to the problem was obtained by determining velocities at the prescribed reference level that best conserved the mass flowing into each of the boxes considered. In applying this method to large boxes containing many hydrographic stations, the inverse problem that was generated was highly underdetermined (ie. an infinite number of solutions existed). In order to obtain only one solution to the problem, Wunsch required the solution to conserve the mass and minimize the energy ofthe unknown reference velocities (smallest model solution). In later papers by Wunsch [7,8], additional constraints on the solution were imposed by conserving salt and heat. W i t h the addition of more constraints, additional unknowns were also included for determining the vertical velocity and horizontal and vertical mixing coefficients. B y using only three conservation constraints the problem still remained underdetermined [7]. In view of this situation, Wunsch [8] chose to investigate the range of possible solutions consistent with the set of constraints rather than to try to determine one unique solution. From the methods presented above, I have developed a box model method that makes use of the linear /3-plane vorticity equation. This method is used i n this thesis to investigate the circulation i n the northeast Pacific Ocean from data obtained as part of the Chapter 1. Introduction 4 Ocean Storms Experiment. The organization of this thesis is as follows. In Chapter 2, I present a brief account of the collection, calibration, and data processing of the hydrographic data collected i n the Ocean Storms Experiment. In Chapter 3, I describe the general oceanic features of the Ocean Storms area through the use of the observed temperature and salinity information and maps of dynamic height. In Chapter 4, I develop the physics behind the model employed and then proceed to show how the inverse problem was formulated and solved. In Chapter 5, the model is investigated to evaluate the uncertainty of the calculated unknowns. the model. Chapter 6 presents the circulation parameters calculated from T h e calculated flow field is compared to current meter observations and Lagrangian Drifters. From these measurements, the time-scale of geostrophic flow is estimated and the ageostrophic flow is calculated and discussed. In Chapter 7, the salt and heat transport due to horizontal and vertical velocities a n d mixing terms is calculated for the upper ocean. These values are compared to the observed changes i n salt and heat content. From the differences between these two measurements, the salt and heat budgets are balanced to estimate the effect of the air-sea boundary processes. Chapter 8 concludes this thesis with a discussion and summary of the limitations of the model, and interpretation of the ocean circulation i n the Ocean Storms area. Chapter Hydrographic 2 Data The hydrographic data made available for this thesis were acquired as part of the Ocean Storms Experiment. This experiment was a joint Canadian-American effort designed to study the interaction of the atmosphere and ocean during severe storms. T h e main goals of this experiment were to: 1) measure the surface fluxes of heat and momentum during severe storms. 2) compute a 3-dimensional heat budget of the oceanic mixed layer and seasonal thermocline. 3) test the oceanic response to severe storms against appropriate physical models. This project was carried out in the open ocean 1000 k m off the coast of Vancouver Island, centred at 47.5° N and 139.0° W (Fig. 2.1). In the Ocean Storms area, it was planned that the hydrographic data would be collected on a 6 x 5 grid with grid spacings of 60 k m (Fig. 2.2). W i t h this grid spacing it was expected that mesoscale features of the ocean would be resolved [9]. 2.1 Data Collection The hydrographic data for this thesis was acquired during two cruises i n the fall of 1987. T h i s is a summary of information presented i n the data report for these two cruises [10]. D u r i n g both cruises, the same equipment was used to obtain the hydrographic data. The salinity/temperature profiles ( S T P ) were collected with a Guildline 8705 Digital STP probe /87102 Deck Control unit. T h e transmission of the data to the surface 5 Figure 2.1 r Lb'cation of the Ocean Storms area in.the.-"northeast Pacific Ocean. Chapter 2. Hydrographic Data s 0 ^ Q- /2 ^ ^ ^ d ) /fry CD' _ e 0 o^—e—£>^-e—e -6) cb (F .6 75 &^ *£ © 23" -0 &L F i g u r e 2.2: T h e p l a n n e d h y d r o g r a p h i c grid for the Ocean S t o r m s area. Chapter 2. Hydrographic Data 8 was performed by a Sea Tech Transmissometer attached to the S T P probe. In addition, hydrographic casts were made with Niskin sampling bottles, each fitted with two reversing thermometers. For samples > 200 m, a single reversing thermometer was used. The salinities from these bottle samples were determined i n duplicate with a Guildline 8400 Autosal salinometer at the Institute of Ocean Sciences (IOS). T h e ship was also equipped with a sea water loop system which provided temperature and salinity readings from a depth of 3 m every 2 minutes. T h e salinities observed from this unit were considered to accurately represent the salinity of the upper mixed layer [11]. For the first cruise (8702), carried out from Sept. 22 to Oct. 16, good weather allowed the completion of all of the planned survey. D u r i n g this cruise, 117 S T P stations were occupied and 8 hydrographic casts were acquired. A l l 34 S T P stations i n the Ocean Storms area were occupied (Fig. 2.3). One hydrographic cast was obtained in the Ocean Storms area at station OS14. In the closely spaced grid of the Ocean Storms area, all the S T P stations were acquired to a depth of 1500 dbar, except for stations OS31 to OS34 which went to a depth of 500 dbar. T h e second cruise (8704), carried out from Nov.25 to Dec. 5, was plagued by persistent gales and storms with winds reaching speeds of 75 knots. Due to the rough conditions at sea, only a fraction of the planned S T P stations were occupied. During this cruise, 32 S T P stations were occupied and 3 hydrographic casts made. Only 17 S T P stations were acquired i n the Ocean Storms grid (Fig. 2.4). The failure of the second cruise to cover all the desired locations i n the Ocean Storms grid limited the usefulness of using this data i n the inverse model. T h e data from this cruise will be used only to calculate the change i n heat and salt content of the ocean. Chapter 2. o rviJ Hydrographic I <g-| I ! 145.0 I ! -143-.0 9 Data. I ! — L | 141.0 I I 139.0 LONGITUDE WEST | I 1 137.0 F i g u r e 2.3: T h e S T P stations o c c u p i e d d u r i n g the first cruise (8702) i n the Ocean grid a n d s u r r o u n d i n g area. this cruise. 1 135.0 Storms T h e arrows i n d i c a t e the p a t h t r a v e l l e d by the ship d u r i n g Cha.pt er 2. Hydrographic F i g u r e 2.4: T h e S T P stations o c c u p i e d d u r i n g t h e second cruise (8704) i n the O c e a n Data S t o r m s g r i d a n d s u r r o u n d i n g area. d u r i n g t h e cruise. 10 T h e arrows i n d i c a t e the p a t h t r a v e l l e d b y the ship Chapter 2.2 2. Data Hydrographic Data 11 Calibration The S T P data were calibrated by comparing these values to the measurements obtained from the hydrographic casts, special calibration samples (sampling bottles placed a few metres above the S T P sensor usually at 1500 dbar), and the sea water loop system. The data report [10] provides more details on the calibration than the summary given below. For the first cruise, the S T P temperature readings were almost identical to the readings obtained from the hydrographic cast and the calibration samples. S T P temp - T hydro/calib. = +0.003°C (standard deviation of 0.05°C) Therefore, no correction was applied to the S T P temperatures. Comparing the STP salinities for the first cruise to calibration readings indicated that a slight instrumental drift occurred during this cruise. The correction for the S T P salinities was accomplished as follows: 1) P01 - P14 subtract 0.016 ppt from S T P salinity 2) P15 - R06 subtract 0.047 ppt from S T P salinity (Ocean Storms grid) 3) R07 - E N D subtract 0.081 ppt from S T P salinity For the second cruise it appeared that the observed S T P temperature and salinity values exhibited a temperature dependence. A regression curve using S T P temperature was used to reduce the observed S T P temperature and salinity values to the "correct" temperature and salinity values [10]. 2.3 Data Processing To convert the S T P conductivity measurements to salinity, the practical salinity units were used [12]. The temperature and salinity values were interpolated to a constant pressure interval of 1 dbar. To smooth any random errors present i n the temperature and salinity values, a running average filter was applied to the data. The following filter Chapter 2. Hydrographic Data 12 lengths were applied to the data: 1) 10 dbar filter length for the interval 0 - 500 dbar 2) 20 dbar filter length for depths greater than 500 dbar From the filtered values of temperature and salinity, potential density and dynamic height relative to 1000 dbar were determined [12]. 2.4 Removal of Internal Waves It is evident i n the plot of potential density as a function of pressure for several stations in the Ocean Storms area (Fig. 2.5), that oscillations occurred in the pycnocline. These oscillations i n part are caused by internal waves [13]. From a study of the temporal variability at Ocean Station P (Station M P 2 6 i n Fig. 2.1), it was determined that most of the variability i n the internal wave is i n the semidiurnal frequency band [13]. Marginally resolvable spectral peaks also occurred within the diurnal and near inertial band (16 to 24 hrs). In order to properly calculate the change i n heat and salt content at any one station it is necessary to remove the temporal fluctuations of temperature caused by internal waves. The reduction of the effects of these internal waves was accomplished by calculating an average potential density function of pressure by averaging several adjacent stations collected over a time period of 12 to 18 hours. The original density function was then shifted i n depth i n order to match the average density curve (Fig. 2.5). The corresponding values of temperature and salinity were not altered but their position i n depth moved up or down i n accordance with the match to the average density function. The temperature and salinity values were then interpolated back to the original pressure sampling. A tapered cut-off was used so that no changes were made to the data for depths having a density > 26.8. Below this point, the effect of the internal waves on the salt or heat Chapter 2. Hydrographic Data. 13 Figure 2.5: Potential density versus pressure plots. T h e upper panel shows the presence of the internal wave i n the data. The lower panel shows the filtered version with this internal wave removed. Chapter 2. Hydrographic Data 14 content of the ocean was not significant because the change i n temperature and salinity with depth was small. This technique of removing the internal tides from the data lacks a sound theoretical basis, but faced with the data available it was deemed to be the best approach. This filtering method will also reduce baroclinic eddies present i n the data. Chapter 3 Hydrography of the Ocean Storms Area The major surface currents of the North Pacific Ocean are shown i n Figure 3.6 [14]. Due to influences of the wind system over the North Pacific Ocean, the surface water tends to flow eastward across the ocean. The Ocean Storms area lies i n the broad boundary between the West W i n d Drift and the Sub-Arctic current [14]. T h e expected surface geostrophic current through the Ocean Storms region should be of the order of 5 km/day ( 5.8 cm/sec) i n an easterly or northeasterly direction [14]. This flow may be complicated by eddies with spatial scales of 50 - 100 k m [15]. 3.1 Temperature and Salinity Structure 3.1.1 Vertical Structure The water mass i n the vicinity of the Ocean Storms area is the Eastern Sub-Arctic Pacific Water Mass [9]. T h e structure of this water mass is characterized by three zones: the upper "seasonal" zone (0-100 m), a principal halocline (100-200 m), and a lower zone (> 200 m) [9]. A n example of the vertical temperature and salinity structure i n the Ocean Storms area is displayed i n Figure 3.7. These plots clearly show the three characteristic zones. The upper seasonal zone is defined by a layer of relatively constant salinity. For the first cruise this zone can be divided into two layers, a well mixed upper layer of constant temperature and salinity, and a lower layer possessing a prominent seasonal thermocline 15 I4-0°E \(oO°E- 180 U lfaO u) 140 W Figure 3.6: Major surface currents of the North Pacific Ocean. T h e Ocean Storms area is centred at 47.5° N and 139.0° W, just southeast of Ocean Station P . Chapter 3. Hydrography of the Ocean Storms STATION SALINITY VS DEPTH Salinity (ppt) 1 Area OS 10 TEMPERATURE VS DEPTH Temperoture ( C) r F i g u r e 3.7: S a l i n i t y a n d t e m p e r a t u r e profiles f r o m t h e O c e a n S t o r m s area ( O S 1 0 , 4 8 ° a n d 139.3° W ) . In these p l o t s t h e solid l i n e refers, to d a t a f r o m cruise one a n d the das line represents d a t a f r o m cruise t w o . Chapter 3. Hydrography of the Ocean Storms Area 18 and a small seasonal halocline. B y the second cruise, this upper zone has undergone a couple of dramatic changes. One, the mixed layer has deepened by 50 m to encompass the first 100 m. Two, the mixed layer has experienced an increase i n salinity of 0.2 ppt and a decrease i n temperature of 2.7°C . These changes to the mixed layer are offset by changes of the opposite sign to the seasonal thermocline and seasonal halocline. T h e changes that occurred to the upper ocean between the two cruises are caused principally by two effects, by a loss of heat to the atmosphere, and by the vertical mixing of the cooler and saltier water of the halocline and seasonal thermocline with the warmer and fresher surface waters [16]. The role of horizontal transport i n altering the salinity and temperature of the upper zone is less well known but is necessary to balance the salt and heat budgets [14]. The second zone, the principal halocline, occurs where the salinity increases rapidly w i t h depth. This zone is clearly present i n both cruises and it represents a transition zone between the upper and lower zones. The temperature i n this zone generally decreases with depth but at some of the Ocean Storms stations, a temperature inversion occurs. This situation is still stable because of the dominating effect that salinity has i n determining the density of the water column. T h e presence of this marked halocline prevents winter convective overturn from being able to mix water deeper than the halocline [9]. Only w i t h the occurrence of intense evaporation at the surface can the halocline be eroded to allow mixing below the halocline [9]. This erosion of the halocline is not evident i n the data from the two cruises. In the lower zone the temperature decreases gradually with depth, reaching 2.8°C at a depth of a 1000 m and 1.5°C i n the abyssal layers (4000 m)[17]. T h e salinity also increases gradually with depth, reaching 34.4 ppt at 1000 m and 34.7 ppt i n the abyssal layers [17]. The differences i n salinity and temperature between the two cruises are small for this zone. This implies that the steady state assumption should be valid i n this zone Chapter 3. Hydrography of the Ocean Storms Area 19 for the time interval between the two cruises. 3.1.2 Temperature-Salinity Diagram The presence of three distinct zones i n the depth structure of the water is illustrated i n the temperature-salinity (Fig. (T-S) diagram by the three main segments of the T-S diagram 3.8). These three zones can be divided according to potential density (cr ): t 1) the upper zone cr < 25.8 t 2) the principal halocline 25.8 < <r < 26.6 t 3) the lower zone a > 26.6 t In addition to the depth structure of the T-S diagram, the first cruise exhibited a distinct spatial structure to the water mass present i n the halocline (Fig. 3.9). O n the south boundary of the Ocean Storms grid, the water i n the halocline is warmer than the water on the west boundary of the grid. A t station OS25, there is a transition between the two water masses. A t this station, the cooler water mass (A) is found at the top of the halocline and the warmer water mass (B) is found at the bottom of the halocline. This transitional T-S diagram is also evident at stations OS20, OS15, OS11, and OS06. This spatial structure of temperature and salinity will become more evident i n the next section. Chapter 3. Hydrography of the Ocean F i g u r e 3.8: . T e m p e r a t u r e - S a l i n i t y Storms Area ( T - S ) d i a g r a m s for the first cruise (8702) a n d for t h e second cruise (8704) for d a t a o b t a i n e d f r o m the Ocean S t o r m s area. solid l i n e represents 20 t h e average T - S c u r v e for t h a t cruise. I n these plots t h e Chapter 3. Hydrography of the T — S Diogrom CRUISE „</ Ocean Storms 21 Area T - S Diogrom 8702 OCEAN STORMS SOUTH BOUNDARY STATIONS: OS28 OS27 OS26 OS25 32.0 33.0 S<iliruty F i g u r e 3.9: 34.0 (ppt 35.0 ) 32.0 33.0 Solinity 34.0 35.0 (ppt.) S p a t i a l s t r u c t u r e of the T - S d i a g r a m for cruise one (8702). T h e solid line represents the T - S curve f r o m O S 2 5 . T h e s e plots show the presence of two different water masses ( A ) a n d ( B ) i n the h a l o c l i n e . 3.2 The Horizontal Flow Field T o p r o v i d e a q u a l i t a t i v e p i c t u r e of the flow field a n d define mesoscale features i n the s t u d y area, maps of d y n a m i c h e i g h t are u s e d . T h e flow field f r o m these m a p s w i l l then be related to the o b s e r v e d d i s t r i b u t i o n of t e m p e r a t u r e a n d s a l i n i t y o n p o t e n t i a l density surfaces, Chapter 3. Hydrography of the Ocean Storms Area 22 and to the flow inferred from these distributions using isentropic analysis. To simplify the discussion of the flow field, the ocean will be divided into the three zones defined by the vertical structure of the temperature and salinity profiles, the upper "seasonal" zone, the principal halocline and the lower zone. 3.3 Dynamic The Height maps of dynamic height presented i n this section display the dynamic height i n dynamic meters relative to a reference level of 1000 dbar. In these maps, the contour lines represent streamlines of the geostrophic flow. For the first cruise, the dynamic height maps for the three zones are similar (Fig. 3.10). These maps reveal three distinctive features: 1) A prominent eddy near OS02 (48.6° N and 141.0° W ) is present i n all zones, adjacent 2) A less well-defined eddy appears i n the south, near station OS27 (46.4° N and 140.1° W ) . Insufficient data exists to properly define this feature. 3) In the centre of the Ocean Storms grid, the geostrophic flow is smoother with the flow generally i n a northeasterly direction. As one goes to deeper levels this flow tends to rotate to the south; at a depth of 1500 dbar the flow is southerly. At this depth the flow i n the centre of the Ocean Storms grid becomes more complicated with smaller eddy features i n the centre of the grid. The flow structure evident by the dynamic height maps for the second cruise (Fig. 3.11) revealed several changes from that of the first cruise. First, the eddy that was present near station OS02 (48.6° N and 141.0° W ) still exists but appears to have moved north, out of the Ocean Storms area. B y the second cruise, the disturbance i n the flow at station OS04 (48.6° N and 139.3° W ) has developed into a weak cyclonic eddy. Due Chapter 3. Hydrography of the Ocean Storms Area. 23 to the l a c k of d a t a i n the lower p o r t i o n of the O c e a n Storms grid for cruise t w o , it is d i f f i c u l t to infer the general flow p a t t e r n i n the O c e a n Storms area. It does a p p e a r t h a t a. n o r t h e a s t e r l y flow c o u l d be consistent w i t h the observed d a t a . o 145.0 143.0 141.0 139.0 137.0 135.0 LONGITUDE WEST F i g u r e 3.10: C r u i s e O n e : D y n a m i c H e i g h t in d y n a m i c meters relative to 1000 d b a r at: a) the surface, b) 150 d b a r , c) 500 d b a r . In these m a p s the arrows i n d i c a t e the d i r e c t i o n of the g e o s t r o p h i c flow a n d the dots denote the l o c a t i o n of the S T P stations d u r i n g this cruise. obtained Chapter 3. Hydrography of the Ocean Storms Area 24 Chapter 3. Hydrography of the Ocean Storms Area Chapter 3. Hydrography of the Ocean Storms Area 26 o 145.0 143.0 141.0 139.0 137.0 -135.0 LONGITUDE WEST Figure a) the of the during 3.11: Cruise Two: Dynamic Height in dynamic meters relative to 1000 dbar at: surface, b) 150 dbar, c) 500 dbar. In these maps the arrows indicate the direction geostrophic flow and the dots denote the location of the S T P stations obtained this cruise. Chapter 3. Hydrography of the Ocean Storms Area. 28 Chapter 3.3.1 3. Hydrography o f the Ocean Storms Area 29 Horizontal Distribution of Temperature and Salinity The horizontal distribution of temperature and salinity contain information on the flow field. In this section, the flow field will be inferred from the temperature and salinity field using isentropic analysis. This flow field will be related to the geostrophic flow evident in the maps of dynamic height. Isentropic analysis is based on the idea that water preferentially moves along isentropic surfaces [3]. A qualitative description of the flow can be inferred directly from the distribution of the temperature and salinity on these isentropic surfaces. T h e bending and stretching of the contour lines of these properties are the result of the spatial features i n the flow field. For example, the presence of an anomalously large flow in a confined area will stretch the contours of temperature and salinity i n this area i n the direction of the flow. In oceanography, these isentropic surfaces can be defined by surfaces of constant potential density (cr ) [3]. t The distribution of temperature and salinity are considered for the three zones which are defined i n the temperature and salinity depth profiles at depths comparable to the pressure surfaces used to display the dynamic height. T h e upper zone is represented by the a = 25.0 surface for cruise one and cr = 25.5 surface for cruise two, the principal t t halocline by the a = 26.2 surface, and the lower zone by the a = 27.0 surface. To better t t define these distributions, S T P stations outside the Ocean Storms grid are also included. O n these maps, the arrows indicate the inferred direction of flow. For the first cruise the upper "seasonal" zone is shown i n Figure 3.12. These maps indicate that a tongue of warm salty water is moving through the southern part of the Ocean Storms grid in a northeasterly direction. This is similar to the dynamic height map for this zone which displays a strong northeasterly flow i n this area (Fig. 3.10a). T h e geostrophic flow indicated by the dynamic height map would tend to stretch the contours Chapter 3. Hydrography of the Ocean Storms Area 30 of temperature and salinity i n the direction of this flow. However, the flow inferred from isentropic analysis is more easterly than the geostrophic flow. This indicates the presence of an additional component to the flow. Also present i n the temperature and salinity fields i n this zone is an oceanic front i n the centre of the grid separating the warm salty water i n the south from the colder fresher water i n the north. Such a feature is not evident in the dynamic height map. Also, the current pattern evident from the temperature and salinity fields gives more details about the flow field than what was evident i n the map of dynamic height. The situation i n the halocline for this cruise is very similar (Fig. 3.13). A tongue of warm salty water still appears to be moving through the southern half of the grid in a northeasterly direction. This flow is slightly more northerly than that which was observed i n the upper zone. This feature is consistent with the geostrophic flow inferred from the dynamic height map (Fig. 3.10b). A ring of warm salty water is present i n this zone i n the northwestern corner of the grid. This ring defines the prominent eddy evident i n the dynamic height map. T h e oceanic front still exists i n the centre of the grid separating the warm salty water from the colder fresher water. This front observed in the T-S diagrams (Fig. 3.9) cuts through the Ocean Storms grid i n a northeasterly fashion. In the lower zone, the temperature and salinity fields no longer display the presence of the oceanic front (Fig. 3.14). Instead, i n the centre of the grid the temperature and salinity fields show no significant flow features this is consistent with the map of dynamic height which indicate that the flow i n this area is weak (Fig. 3.10c). The temperature and salinity fields do show that a tongue of cold fresh water is moving southeasterly through the eastern half of the grid and that a tongue of warm salty is moving north in the southeastern corner of the grid. T h e former feature is not clearly evident i n the map of dynamic height but the latter flow feature is. T h e prominent eddy present i n the map Chapter 3. Hydrography of the Ocean Storms Area 31 of dynamic height is well-defined in the temperature and salinity fields by a ring of warm salty water near OS02 (48.6° N and 141.0° W). For the second cruise, the pattern has changed from what was observed in cruise one (Figs. 3.15). In the upper zone the contour lines are meridional and lack any prominent features. In the halocline the pattern is similar, but there is more evidence of a tongue of cold fresh water moving south through the northern part of the grid (Figs. 3.16). In the lower zone this tongue of water still exists but is less prominent (Fig. 3.17). For the second cruise, flow in all zones appear to be north-south which is similar only to the lower zone of cruise one. T h e analysis of the second cruise is subjective because of the lack of data i n the south sections of the grid. This flow is generally comparable to the dynamic height maps but again the lack of data makes the comparison very subjective. 3.4 Summary It is evident i n the closely spaced grid of stations i n the Ocean Storms area that several mesoscale features existed i n this relatively quiet area of the open ocean. It was interesting to observe that in previous maps of dynamic height at the surface, an eddy did occur near the eddy observed i n this data at OS02 (48.6° N and 141.0° W ) [17]. T h e existence of eddies in this area with spatial scales of 50 -100 k m have been previously observed by Tabata along line P [14]. The circulation pattern deduced from the isentropic analysis is generally consistent with the dynamic height maps. Therefore, the results of the isentropic analysis are considered as supporting any conclusion reached from considering the flow is geostrophic. However, isentropic analysis does reveal more details i n the current pattern than what is evident i n the maps of dynamic height. This suggests that additional information about the flow field is available from the temperature and salinity data than just using this Chapter 3. Hydrography of the Ocean Storms 32 Area Figure 3.12: Temperature and Salinity distribution for cruise one on the a = 25.0 surface for: a) the temperature distribution i n deg C, and b) the salinity distribution in ppt. The arrows in these maps indicate the interpretative current pattern. t Chapter 3. Hydrography of the Ocean Storms Area. 33 Chapter 3. Hydrography of the Ocean Storms Area 34 F i g u r e 3.13: T e m p e r a t u r e a n d S a l i n i t y d i s t r i b u t i o n for cruise one o n t h e cr = 26.2 surface t for: a) t h e t e m p e r a t u r e d i s t r i b u t i o n i n deg C , a n d b) the s a l i n i t y d i s t r i b u t i o n i n p p t . T h e arrows i n these m a p s i n d i c a t e the i n t e r p r e t a t i v e current p a t t e r n . Chapter 3 Hydrography of the Ocean Storms Area (b) o LONGITUDE WEST 3 Chapter 3. 145.0 Hydrography -143.0 of the Ocean Storms 141.0 Area. 36 139.0 137.0 135.0 LONGITUDE WEST Figure 3.14: Temperature and Salinity distribution for cruise one on the a = 27.0 surface for: a) the temperature distribution i n deg C, and b) the salinity distribution in ppt. The arrows in these maps indicate the interpretative current pattern. t p Chapter 3. Hydrography of the Ocean Siorms 38 Area F i g u r e 3.15: T e m p e r a t u r e a n d S a l i n i t y d i s t r i b u t i o n for cruise t w o on t h e a = 25.5 surface t for: a) t h e t e m p e r a t u r e d i s t r i b u t i o n i n deg C , a n d b) the s a l i n i t y d i s t r i b u t i o n i n p p t . T h e arrows i n these m a p s i n d i c a t e t h e i n t e r p r e t a t i v e current p a t t e r n . Chapter (b) 3. Hydrography of the Ocean Storms Area 39 Chapter 3. Hydrography of the Ocean Storms Area. 40 o LONGITUDE WEST Figure 3.16: Temperature and Salinity distribution for cruise two on the a = 26.2 surface for: a) the temperature distribution i n deg C, and b) the salinity distribution in ppt. The arrows in these maps indicate the interpretative current pattern. t Q CO Chapter 3. -145.0 Hydrography 143.0 of the Ocean Storms T~ 141.0 •—r 42 Area 1 139.0 r 135.0 137.0 LONGITUDE WEST Figure 3.17: Temperature and Salinity distribution for cruise two on the u = 27.0 surface for: a) the temperature distribution in deg C, and b) the salinity distribution i n ppt. The arrows in these maps indicate the interpretative current pattern. t Chapter 3. Hydrography of the Ocean Storms Area 43 Chapter 4 Developing and Solving the Inverse Problem 4.1 Determining the Ocean Circulation: The Physical Problem The method to be presented in this thesis for obtaining ocean circulation from temperature and salinity information is based on the following approximations to the equations of motion. In cartesian coordinates, the equations of motion with the hydrostatic approximation and conservation of mass are u + uu t x + V U (4.4) - fv = ~~Vx y v + uv + vv + fu - - -p 1 . . (4.5) Pz = -gg (4.6) t x u x y y = 0 + v + w y z (4.7) The Coriolis parameter is defined as / = (/ +fly).B y assuming the flow is linear and c steady state, and using the Boussinesq approximation, (4.4) and (4.5) can be reduced to the geostrophic equations fv = So fu P = --p x (4.8) (4.9) y Differentiating (4.8) and (4.9) with respect to z and using (4.6) leads to the thermal wind equations fv z = -~9Q X Qo 44 (4.10) Chapter 4. Developing and Solving the Inverse fu z Problem = —gg 45 (4.11) y T h e thermal wind equations relate the vertical shear of the horizontal velocities to the horizontal gradient of the i n situ density, g. If (4.10) and (4.11) are now integrated vertically from a reference depth, z , the horizontal velocity is given by r v = - / g dz + v = v + v x 0 T (4-12) 0 Qof Jz T u = H - / g dz + u = v + v y QoJ 0 r (4-13) 0 Jzt T h e horizontal velocities are made up of two terms. T h e first'term, the relative velocity (u , v ), depends only on horizontal gradients i n the density field. T h e second term is r r the constant of integration, (u , v ), and it refers to the velocity at the reference level z . 0 0 r B y measuring the temperature and salinity distributions at different depths, one can obtain the density field using the equation of state (g(S,T, P)). Knowing the density field, the relative velocities given i n (4.12) and (4.13) can be evaluated. By differentiating (4.8) with respect to y and (4.9) with respect to x, and adding them together, one obtains a linear /3-plane vorticity conservation equation (3v=fw (4.14) z Integrating this equation from a reference level z one obtains r /3 r z w = — / vdz + w J Jz (4-15) 0 r T h e variable w arises from the constant of integration and it refers to the vertical velocity 0 at the reference level z . Using (4.12), one can rewrite (4.15) as T (3 f — / z w J J Z t v dz r j3 + —v (z - Zr) + w 0 0 (3 = w + — v (z - z ) + w r 0 J T 0 (4-16) J To evaluate these unknown reference velocities, u , v , and w , the equations for 0 0 0 conservation of mass (4.7) and conservation of salt and heat are used. T h e equations for Chapter 4. Developing and Solving the Inverse Problem 46 conserving heat and salt are denned by the following advection diffusion equations uS x + vS uT + vT x y y + wS z + wT z = Sj{k\/S) (4.17) = S7{kS7T) (4.18) These equations assume that the ocean is i n a steady-state condition with no sources or sinks. Since this assumption is clearly violated at the surface of the ocean, the depth at which these equations are valid must be considered i n order to properly apply them to the data available. The conservation of heat is equivalent to the conservation of temperature multiplied by the heat capacity qC. Over the range of temperatures, salinities, and pressures i n the ocean, this term can be considered constant and thus eliminated from (4.18). This allows the conservation of heat equation to be replaced by the conservation of temperature equation. The variable k i n (4.17) and (4.18) is referred to as the mixing coefficient. This variable parametrizes mixing due to eddies or other features not resolved by the model. In a stratified ocean, mixing along isopycnals is much stronger than cross isopycnal mixing [6]. In the Ocean Storms area the slope of the isopycnal surfaces are almost horizontal. This enables one to replace the isopycnal and diapycnal mixing coefficients by vertical and horizontal mixing coefficients, k and kh respectively. This simplification z should not introduce large errors because these mixing terms do not play a crucial role of transporting heat and salt i n the Ocean Storms area. Because of the small area covered by the data (250 km by 200 km), the mixing coefficients are only considered to be functions of depth. The horizontal variations of these mixing coefficients were ignored. I now proceed to show how these conservation equations are used to formulate an inverse problem to describe the flow in the ocean. Chapter 4.2 4. Developing T h e Inverse 4.2.1 and Solving the Inverse Problem 47 Problem Formulation To develop a set of equations from the conservation equations, first consider one box in Figure 4.18. This box is bounded horizontally by four hydrographic stations and vertically by two depth surfaces, zi and z . T h e velocity and the mixing coefficients on 2 each face of the box must be determined. For each of these faces, a value for the relative velocity can be determined using the bounding hydrographic stations (equations (4.12), (4.13) a n d (4.16) ). O n l y the velocities directed perpendicular to the box faces can be determined. In this scheme no information is available to constrain the flow parallel to the faces of the box. The conservation of mass constraint for this box can be obtained by integrating the conservation of mass equation around the region enclosed by the box. V • u dx dy dz = 0 Ay Azu\% (4-19) + Ax Az v |£ + A z Ay w \ = 0 (4.20) T B The overbar denotes the average velocity on each face of the box, with the subscripts E, W, N, S, T, and B referring respectively to the east, west, north, south, top, and bottom faces of the box. B y writing the velocities i n terms of known relative velocities and unknown reference velocities, one obtains the following equation Ay Az u -Ay 0 1^ + Ax Az v \g + Ax Ay w D 0 \ = T B Az u | J - Ax Az v \% - Ax Ay w r r r \ T B (4.21) Chapter 4. Developing and Solving the Inverse e 0- Problem 48 © - ^ 4. o e — e o e o 10 8 o e * — w o - .6- 13 12 c > — — it, £3 e : (8 n e 6) 14 — — = — 15 e — o 2o l? e Figure 4.18: T h e boxes used for the inverse model of the Ocean Storms area. For cruise one, all the boxes are defined. Chapter 4. Developing and Solving the Inverse Problem 49 To obtain an equation for the conservation of salt, S, one starts by combining the advection diffusion equation and the conservation of mass equation 0*S n {uS) x + {vS) + {wS) = V t f ( W t f S ) + o - ( k - a - ) y ( -22) 4 z OZ OZ A n equation is obtained by integrating this equation over an enclosed box A y Az uS \% + Ax Az vS \% + Ax Ay wS Ax Ay K ^- \ + Ay Az k ^ 3 T B oz ox \% = \™ + Ax Az k ^h \» oy (4.23) The overbar again denotes the average value on each face of the box. From this, an equation can be generated by rewriting the velocity term in its two parts, the reference velocity and the relative velocity A y Az u S \ J + Ax Az v S \% + A z A y 0 0 Ax Ay k — oz z -Ay I - Ay Az k —ox h £ - Ax w S\lQ Az k -— \ oy h N S = Az u S \ J - Ax Az v S \% - Ax Ay w S \ T r T T B (4.24) This equation relates the unknown transport of S due to the reference velocities and mixing terms to the known transport of S due to the relative velocities. A n identical equation can be written for the temperature T. B y applying this idea to many boxes, a set of equations is obtained. This set of equations can be written as the matrix equation Ax = h. T h e vector x represents the unknown variables u , v„, w , k , k^; the matrix A represents the average properties on 0 0 z the faces of the boxes; and the vector b represents the mass, heat, and salt transport due to the relative velocities. This averaging of the variables on each face of the box results in a reduction i n the resolution of the model but also reduces the susceptibility of the method to errors i n the data. Chapter 4.2.2 4. Developing Ocean Storms and Solving the Inverse Problem 50 Grid In order to use the B o x model, one has to select a set of layers over which one can apply the conservation constraints. Traditionally i n the large regional studies [1,18,7], the vertical layering was based on isopycnal surfaces. T h e use of this layering stems from early work of water mass analysis that states that the flow should occur along isopycnal surfaces [3]. In a small regional study such as the Ocean Storms experiment, the slope of the isopycnal surfaces is nearly horizontal over the study area. Therefore it is considered appropriate and convenient to divide the ocean vertically according to pressure surfaces, instead of isopycnal surfaces. These surfaces of constant pressure are almost identical to surfaces of constant depth (tO.lm). T h e layering of the Ocean Storms area is shown in Table 4.1. By dividing the ocean according to these 17 different levels, one reduces the ocean to 16 layers. T h e reduced thickness of the layers i n the upper ocean is necessary to properly define the features such as the mixed layer, seasonal thermocline, and halocline. The boxes denned i n the Ocean Storms grid are shown i n Figure 4.18. For every layer considered in the model there are 20 boxes; with a m a x i m u m of 16 layers this allows for a maximum of 320 boxes. W i t h three constraints to be applied to each box, the maximum number of equations available to constrain the problem is 960. T h e maximum number of unknowns to the problem is determined by considering that for any layer one must determine the reference velocity on every face of the box. This requires that 24 u 's, 0 25 Vo's, and 20 w s be determined. In addition, since the mixing terms are considered J 0 to be only functions of depth, k^ was determined for each layer (16 maximum) and k z was determined at each level except at the surface (16 maximum). T h e total maximum number of unknowns to the problem is 101. Chapter 4. Developing and Solving the Inverse Level Pressure (dbar) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 0.0 40.0 60.0 100.0 150.0 200.0 250.0 300.0 400.0 500.0 600.0 700.0 800.0 900.0 1000.0 1200.0 1500.0 Problem Depth (m) 0.00 39.8 59.7 99.5 149.1 198.8 248.4 298.0 397.1 496.2 595.2 694.2 793.1 892.0 990.8 1188.3 1484.2 Table 4.1: The levels used to vertically define the boxes used in the model. 51 Chapter 4. Developing and Solving the Inverse Problem 52 Although I have listed the entire layering of the ocean, due to the underlying assumptions made i n discussing the physical problem (ie. steady state and neglection of source and sink terms for heat and salinity), the number of layers actually considered i n the model is less. The layers to be included i n the model are determined temperature by analyzing the and salinity depth profiles obtained i n the Ocean Storms area during the two cruises. These profiles show that depths shallower than 150 m clearly violate the assumptions. This reduces the number of unknowns to 94. One other parameter that must be specified for the inversion is the depth of the reference level used i n determining the relative geostrophic flow. Most of the previous work i n this area uses the 1000 dbar level as the choice of reference level [17]. This level is also used for the inverse model. 4.3 Including A Priori Information Before proceeding to describe how to obtain a solution to the inverse problem, there is still additional information that can be included to obtain a more realistic solution. A priori information is included i n the model through the use of weighting matrices and inequality constraints. B y proper weighting of the system of equations, one can include additional information in the inversion. This method of weighting the equations has been used i n previous work on this problem to force the solution to be more physically real [1,19]. T h e weighted problem can be written as QAW B W-jJ= m = Qb + Q£b (4.25) e + Se (4.26) The matrices Q and W are referred to as the data weighting matrix and parameter Chapter 4. Developing and Solving the Inverse Problem 53 weighting matrix respectively. These weighting matrices represent covariance matrices for the errors in the forcing terms, 8b, and for the unknowns, x. Due to the lack of knowledge about the correlation between unknowns, and the affect of errors i n the data on matrix A , diagonal matrices are used [19,1]. The errors present in the forcing terms for the inverse problem are in part due to measurement errors, but are mostly due to unresolved small scale processes like internal gravity waves and baroclinic eddies [7]. W i t h no definite information about the magnitude of these errors, 8b, and with no clear understanding of how errors i n the data affect matrix A, the rows of A are normalized by their length to force all equations to be treated equally by the inversion [19]. A n additional factor, C, is added to allow one to easily change the importance of the different constraints. T h e data weighting matrix is defined as follows M Qu = Ci/(z*hr (- ) 1/2 4 27 The parameter weighting matrix W is used to include information on the expected magnitude of the different model parameters [19]. This is done by denning a diagonal weighting matrix to be T^H^ 1 / 2 (£4r 1 / 4 (4-28) i= l In the two equations defining the weighting matrices, a^- are the elements of the matrix A, N is the number of equations,M is the number of unknowns, and [xj] is the expected magnitude of the ) t h unknown. The expected magnitudes of the unknowns are set to: 1) |it | and |i> | = 10~ m/s 2 0 0 2) \w \ = l(T m/s 3) \k \ = l C T W / s 4) \k \ = 10 m /s 0 z h 8 2 2 W i t h o u t including any information on the magnitude of the different unknowns, the model will produce a physically unrealistic solution [1]. Thus it is desirable to correct Chapter 4. Developing and Solving the Inverse Problem 54 this problem to ensure that the magnitude of the model parameters are consistent with the physical situation. T h e use of the parameter weighting matrix forces all the model parameters to be of the same order of magnitude and thus not bias the solution to determine the unknowns which have a large magnitude [20]. Inequality constraints are included i n the model to force the mixing coefficients to be positive. This is required because physically meaningful solutions only exist for solutions which have positive mixing coefficients [6]. This extension to the inverse problem is called the least squares inversion with inequality constraints (LSI) [21]. In mathematical notation, the solution to the L S I problem is obtained by minimizing i) = ||Bm - e|| (4.29) Hm > 0 (4.30) subject to the inequality constraints The matrix H is defined to only allow positive mixing coefficients. 4.4 Solving the Inverse Problem As formulated, the resulting inverse problem Bra = e is overdetermined. In reality, the problem is formally underdetermined and an infinite number of degrees of freedom exists in the solution [7]. It is only through simplifying the physics of the problem (ie. assuming the flow is steady-state and geostrophic) and parametrizing the continuous velocity fields that the problem is made to be overdetermined. Through this simplification of the problem, one hopes to make more efficient use of the data available. B u t the results of this inverse problem are only of interest if this simplified model captures the essence of the real situation. This is an important point to consider i n inverse modelling because considerable uncertainty in the results may stem primarily from this simplification of the Chapter 4. Developing and Solving the Inverse Problem 55 model. This point will be discussed i n later chapters when discussing and evaluating the results of the model. The solution to the inverse problem is obtained using the L S I algorithm developed by Lawson and Hanson [21]. Essential to this method is the singular value decomposition ( S V D ) of the matrix B . To gain additional insight into the ability of the model to predict a solution, the problem is analyzed without the inequality constraints. To include the inequality constraints i n such an analysis would be too difficult. The advantage of using S V D to analyze the inverse problem is that i t allows one to identify less certain information i n matrix B . This factorization provides an excellent way of representing and ranking the information contained i n the matrix B [20]. This enables one to minimize the effect of the less certain information on the solution [20,22]. Consider solving the inverse problem with /V equations and M unknowns m = N x M e (4.31) M x 1 TV x 1 (4.32) If there are k independent equations among the N simultaneous equations, then the matrix B has rank k and can be factored using S V D as follows [22] B = N x M U = Ui U A N x k k x k U2 ^3 V T k x M Chapter 4. Developing and Solving the Inverse A x 0 V = Problem 0 0 A 0 ••• 2 ••• : 0 '•• 0 0 0 0 Vi v A fc v 2 The matrix U contains k eigenvectors, 56 3 associated with the span of the column vectors of matrix B . T h e matrix V contains k eigenvectors, V{, associated with the span of the row vectors of matrix B [22]. The matrix A is a diagonal matrix of k singular values, A;, arranged in descending order. Once the matrix has been decomposed using S V D , the generalized inverse of the matrix B is [22] B = V A _ 1 _ 1 U T (4.33) and the solution to the inverse problem is [22] m = V A _ 1 U T e (4.34) or, in component form, m i = (4.35) H 3 J This equation shows that the calculated solution is very sensitive to small singular values. The relative importance of the small singular values in the calculated solution means that these values will tend to greatly amplify any errors present in the data [22]. In order to use the generalized inverse (4.33), it is essential that one identify the number of non-zero singular values, p [22]. One must determine if the small singular Chapter 4. Developing and Solving the Inverse Problem 57 values are really non-zero or caused by errors i n the matrix equations and calculations of the SVD. This problem is dealt with by either truncating singular values below a specified cutoff or by adding a small constant to all singular values. B o t h methods were investigated and there were only small differences i n the results. In the results to follow, a sharp cutoff will be used. To determine how these small singular values should be treated, the following will be used: 1) a plot of the magnitude of the singular value versus the index number, 2) a plot of the error reduction versus the number of singular values used, 3) a trade-off curve relating the size of the model variance to the resolution of the model, and 4) the examination of the solutions obtained with different cutoffs. This analysis will be performed on the data from cruise one. Weighting matrices will be applied to the system of equations generated and all equations will be treated equally (ie. C = 1 ). The results of this analysis will be applied to solving the L S I problem. The range i n magnitude of the singular values of matrix B is displayed i n Figure 4.19. This plot shows a gradual decrease i n magnitude of the singular value w i t h index number, p. Only the singular value with index number, p>82, are significantly smaller than the previous values. These singular value are considered to be zero and will not be used in the solution. The plot of the normalized residual error versus the number of singular values used in the solution is displayed i n Figure 4.20. This plot indicates that at least the first 25 singular values of matrix B should be retained i n order to adequately reduce the error in the solution. To really assess how to handle the small singular values, a trade-off curve was constructed that relates the resolution of the model to the variance i n the solution. To Chapter 4. 1 o - Developing 0 and Solving the Inverse Problem 58 - i o 10~ 1 1 L j — 0 : 1 1 1 20 40 60' Index N u m b e r (p) L 80 Figure 4.19: A plot of the normalized magnitude of the singular values versus index number p. Chapter 4. Developing .0.0 0 and Solving the Inverse 40 No. of Singular 20 Problem 59 60 80 values (n) F i g u r e 4.20: T h e r e d u c t i o n i n error versus the n u m b e r of s i n g u l a r values of m a t r i x B t h a t are used i n d e t e r m i n i n g t h e s o l u t i o n . T h e n u m b e r of s i n g u l a r values used ( n ) , indicates t h a t the first n s i n g u l a r values of m a t r i x B are u s e d . Chapter 4. Developing and Solving the Inverse Problem 60 calculate the variance i n the solution, the following definition was used [22] size of solution variance = ||cr (VA~ V )|| 2 (4.36) t 0 The <x is assumed to be constant. The resolution of the solution is determined using 0 the following definition [22] spread The of solution trade-off curve was constructed the model resolution and variance. resolution = || V V * — I|| (4.37) by changing the prescribed cutoff and calculating The value of the cutoff used implied that singular values with normalized magnitude less than this cutoff were omitted in this analysis. The trade-off curve obtained, Figure 4.21, shows that by demanding a model that highly resolves the solution, the spread of the solution resolution will be small, but the variance of that solution will be large. This trade-off curve allows one to assess and determine the o p t i m u m cutoff based on the compromise between a well-resolved solution and a model with a satisfactory variance. Based on the trade-off curve, 10~ 4 is selected as the best choice of the cutoff. It is important to realize that the trade-off curve is strictly based on matrix B , neither the solution to the inverse problem nor the forcing terms enter into this analysis which only assesses the ability of the model to determine a solution. A cutoff of 10~ 4 produces a satisfactory solution to the L S I problem. If this cutoff is decreased, the magnitude of the solution becomes unreasonably large. This cutoff represents the m i n i m u m acceptable cutoff for determining the non-zero singular values of matrix B . It will be shown i n the next chapter that this cutoff level will not produce a solution that is very overly sensitive to the expected errors i n the data. B y using this cutoff the rank of the matrix B is reduced to 52. As the problem is mixed-determined the condition of minimizing the energy of the solution will be implicitly used to further constrain the solution. Chapter 4. Developing and Solving 20 Spread the Inverse Problem 40 of M o d e l 61 60 8 0 :-, Resolution Figure 4.21: The trade-off curve for the inverse model. This curve relates the resolution of the model to the size of the variance of the unknowns. T h e numbers over the points on this curve indicate the cutoff for the singular values. -- - . Chapter E v a l u a t i n g the 5 M o d e l and Solution In this chapter I evaluate the model and solution i n an attempt to better understand the model and estimate the uncertainty i n the solution. I first investigate the importance of the three conservation conditions - temperature, salt and mass - i n determining the solution and assess whether modifying the weighting of these conditions is necessary. Second, I examine the layering used i n the model. T h i r d , I vary the reference level to see how this affects the solution. Fourth, I evaluate how important the different unknowns are i n the solution and estimate how well resolved they are. Fifth, I investigate how sensitive the solution is to errors i n the data. The initial solutions are based on a 12-layer model w i t h the reference level at 1000 dbar. To compare the solutions obtained i n the following sections, correlations and regression coefficients are used. To calculate these coefficients, all the unknowns parameters are used (total of 94), and they are weighted so that they are of the same order of magnitude. The error reduction i n the different solutions are also used i n comparing the different solutions. The normalized residual errors i n the three conservation conditions (1 = either T, S, or M ), are determined as follows In this equation, N refers to the number of constraint equations for that condition, B{ is the set of rows of matrix B describing these equations, and e; is the corresponding forcing term. The total of these three errors gives the total reduction i n error from the 62 Chapter 5. Evaluating the Model and Solution 63 initial condition where all unknown parameters are zero. Throughout this chapter the null solution will be used to denote the solution where all the unknown parameters are zero. 5.1 The Importance of Temperature, Salt and Mass Conditions on the So- lution The solution obtained to the inverse problem B m = e is dependent on the forcing terms e. If this vector is identically zero then the solution vector rh will also be zero. It is the non-zero forcing terms that force the solution to the problem to be non-zero. These forcing terms are due to three effects: 1) the relative vertical velocity which affects the conservation of mass i n each of the boxes, 2) the transport of heat due to the relative vertical and horizontal velocities, and 3) the transport of salt due to the relative vertical and horizontal velocities. The following analysis will illustrate the incompatibilities between the different conservation conditions (ie. mass, salt, and heat). It will also show which conditions provide the most constraint on the solution. To assess the importance of these three conditions, they are alternately up-weighted by a factor of five. The solution obtained is then compared to the reference solution which weighted the three conditions equally ( S = l , M = l , T=l). The correlation and regression coefficients between the up-weighted solutions and reference solution are shown i n Table 5.2. The biggest value of the correlation coefficient exists for the up-weighted salt condition. This implies that the salt condition the dominant role i n forcing the reference solution. The much smaller values of the correlation coefficient for the up-weighted temperature and mass conditions that this information is of considerably plays indicates less importance i n forcing the reference solution. Chapter 5. Evaluating the Model and Solution 64 Temp. Weighting Salt Weighting Mass Weighting Correlation coefficient Regression coefficient 5 1 1 1 5 1 1 1 5 0.357 0.789 0.165 1.7976 1.8169 0.0075 Table 5.2: Correlation and regression coefficients between up-weighted solutions a n d the reference solution which weighted the three conditions equally ( T = l , S = l , M = l ) . The value of the regression coefficient for the up-weighted salt and temperature conditions (% 1.8) indicates that the magnitude of these solutions are almost twice as large as the magnitude of the reference solution. However, the regression coefficient for the up-weighted mass condition is considerably smaller than one. This indicates that the magnitude of the solution obtained from this weighting is considerably smaller than the magnitude of the reference solution. This points out the importance of the mass conditions i n the reference solution. T h e mass condition is not important i n forcing the solution, as exhibited by the small correlation coefficients. T h e role of the mass condition is to damp the magnitude of the solution in order to force the solution to conserve mass. The normalized residual error for up-weighting the three conditions reveals additional insight into the problem. These errors (Table 5.3), show that the errors i n the three conditions differ greatly. T h e error i n temperature in the null solution is extremely small. This small error in the temperature condition is attributed to the fact that the geostrophic flow is almost parallel to the isotherms. This close connection between the dynamic height and temperature field demonstrates the importance of the temperature information i n determining the relative geostrophic velocities. Most of the relevant information i n the temperature field is used to determine the relative velocities. This prevents this condition Chapter 5. Evaluating the Model and Solution 65 Solution Temperature error Salinity error Mass error Total error Null Reference Temp. Salt Mass 0.005 0.012 0.376 0.190 0.211 0.142 0.210 0.619 0.297 0.297 0.290 0.295 1.000 0.499 0.509 0.731 0.511 0.001 . 0.299 0.006 Table 5.3: T h e normalized residual errors i n the three conditions for the null solution, the reference solution, and the solution obtained by alternating the up-weighting of the three conditions. T h e solution for temperature, salt or mass is the solution obtained when this condition is up-weighted by a factor of 5. from being more effective i n the reference solution. The errors i n the salt and mass condition for the null solution are two orders of magnitude greater than the error in the temperature condition. T h e errors i n both these conditions are significantly reduced i n the reference solution. B y demanding that the salinity condition be highly satisfied, the error i n the temperature condition increases drastically. This demonstrates the incompatibility between these two conditions. T h e situation for the up-weighted temperature condition further clarifies that the temperature information carries similar information as the salt field but it is the salt field that yields additional information to constrain the flow. This incompatibility between these two constraints may arise from the assumptions made i n developing the model a n d the inability of the temperature condition to determine the absolute geostrophic velocity. It is surprising that the error i n the mass condition is so large. It was felt that this condition should be less significant because its forcing ability is probably based on the weakest assumption i n the model, the linear /3-plane vorticity equation. Also, the solution should approximately conserve mass because the forcing terms for this condition, Chapter 5. Evaluating the Model and Solution 66 the relative vertical velocities, are small. A closer examination of the model does reveal the cause of the large error i n this condition. The large error i n this condition is an artifact of the row weighting of the matrix equations. The forcing terms for this condition are magnified because the vertical reference velocity is not present i n the conservation of mass equations. This is valid because this term does not affect the mass i n the boxes. As a result though, when the row weighting is applied to these equations, the calculated row norm is considerably smaller than the other two conditions. This is shown through the use of a simplified conservation of mass equation d — (u )dydz Ox 0 d d + —(v )dxdz r—(w )dxdy oy oz J 0 — 0 d Oz —(w )dxdy dx = 60000m dy = 60000m dz. = T 100m Without the term for the relative vertical velocity, the norm of this equation is approximately y/2 dz, but i f the relative vertical velocity is included the norm increases to approximately dx dy/dz. Since dx and dy are considerably greater than dz, the norm of the equation is increased greatly by the addition of the third term. B y having a much larger norm, the forcing term is reduced because the equation is weighted by the inverse of the norm of the equations. Due to this formulation and weighting of the problem, the mass condition is given additional importance i n the solution. To assess the importance of the weighting of the mass condition on the solution, various weightings of the mass condition are tried. The solutions for these various weights are considered to determine if the weighting of the mass condition should be altered. The correlation and regression coefficients for the various weights are displayed i n Table 5.4. These coefficients reveal that for weights smaller than 0.5, the solutions do not change significantly. By down-weighting the mass condition to 0.5 or less, the magnitude of the Chapter 5. Evaluating the Model and Solution 67 Correlation coefficients 1.0 0.5 1.000 0.664 1.000 Mass Weights 0.25 0.1 0.05 0.01 0.612 0.605 0.934 0.995 1.000 0.611 0.579 0.918 0.990 0.999 1.000 Regression Coefficients Mass Weights 0.5 0.25 0.1 1.0 0.05 0.01 1.0 0.5 0.25 3.49 1.457 1.047 3.51 1.309 1.012 3.52 1.242 0.992 1.000 0.999 1.000 0.994 1.000 1.000 1.0 0.5 0.25 0.10 0.05 0.01 0.10 0.05 0.01 1.000 2.92 1.000 0.653 0.809 1.000 3.07 1.695 1.000 0.613 0.661 0.964 1.000 Table 5.4: Correlation and regression coefficients between solutions with various weights for the conservation of mass condition. T h e weighting of the temperature and salt conditions are held fixed at one. Chapter 5. Evaluating Temp Sal. Mass Total the Model and Solution 68 Weight of conservation of mass constraint 1.0 0.5 0.25 0.1 0.05 Null 0.01 0.012 0.016 0.017 0.020 0.020 0.020 0.005 0.190 0.174 0.173 0.165 0.165 0.165 0.376 0.297 0.306 0.314 0.329 0.330 0.331 0.619 0.499 0.496 0.504 0.514 0.515 0.516 1.000 Table 5.5: The normalized residual errors in the conservation of temperature, salt, and mass conditions for various weights for the conservation of mass condition. The N u l l column refers to errors i n these three conditions if the solution is identically zero. resulting solutions increases by threefold over the reference solution. This emphasizes the importance of the mass condition in damping the reference solution. The effect of changing the weight of the mass condition on the residual errors is shown i n Table 5.5. A l l errors have been referenced back to the weighting of the reference solution ( T = l , S = l , M = l ) for easy comparison. As expected, reducing the weight on the mass condition increases the error i n this condition slightly. Accompanying this increase in error is a slight reduction in the salt error and a slight increase i n the temperature error. The down-weighting of the mass condition appears to allow the salinity information to be even more significant in forcing the solution. The result of varying the conservation of mass equation reveals that down-weighting the mass condition produces a dramatic change i n the solution without significantly affecting the total reduction i n errors. T h e magnitude of the solution increases abruptly as the mass condition is down-weighted. From this analysis i t seems best not to downweight the mass condition. T h e mass condition includes very relevant information to the problem and should clearly be included i n the model. Furthermore, it can be argued that a physically real solution only exists for the situation where the mass i n the interior of the ocean is exactly conserved. This extreme could be followed to further constrain the solution [23] but it is felt that a more moderate approach should be taken because Chapter 5. Evaluating the Model errors i n this condition may and Solution 69 exist arising from assumptions i n the model and errors in the relative velocities. It is clear that by subjectively changing the weighting of these three conditions, the solution to the problem will change significantly. In light of the present situation, it was not considered appropriate to bias one of the conditions more heavily than the others. The unbiased weighting of the equations will be used i n obtaining a solution to the model. The layering of the ocean will now be investigated using this equal weighting of the three conservation 5.2 The conditions. E f f e c t of the L a y e r i n g of the In this section I will first look at how O c e a n on the Solution the solution is affected by the number of layers used in the model. T h e n I will consider how changing the layering of the ocean i n the model will affect the solution. B y changing the number of layers included in the model, one can determine which layers are important i n forcing the model. In Table 5.6, the solutions obtained using various numbers of layers are compared to the reference solution, a 12-layer model. To calculate the correlation and regression coefficients displayed i n this table only the reference velocities are used, this gives a sample size of 69 parameters. The m i n i m u m depth in the table indicates that data obtained from depths shallower than this level will be neglected i n the model. It is evident from this table that the data from the 150 - 200 dbar layer does not greatly affect the reference solution. B y including data shallower than 150 dbar (149.1 m), solution does change significantly. The the magnitude of the solution increases rapidly as the surface is approached. From the large difference in solutions between the 1213-layer models, 150 m is assumed to represent the upper boundary on the model. and Chapter 5. Evaluating the Model and Solution 70 Min. Depth (dbar) Number Layers Correlation coefficient R Regression coefficient slope ( X ) 400.0 300.0 250.0 200.0 150.0 100.0 60.0 40.0 0.0 8 9 10 11 12 13 14 0.723 0.804 0.875 0.991 1.000 0.900 0.894 15 16 0.873 0.443 0.534 0.610 0.690 0.970 1.000 1.107 1.090 1.082 1.300 Table 5.6: The correlation and regression coefficients between the reference solution, a 12-layer model with solutions obtained from models using different numbers of layers. This table clearly displays the importance of the data between 200 and 500 dbar i n forcing the solution. Basing the solutions largely on data from this range of depth is considered valid because for depths greater than 500 dbar the horizontal gradients of the salt and temperature field are small and they approach the accuracy of the measurements. The assumption of the steady-state condition for temperature may be questioned i n the upper layers of this model but with the salt condition dominating the forcing of the solution, this assumption appears to be valid at least for the time interval between the two cruises. To investigate the problem of how to divide the ocean into layers, results were obtained using a simplified layering of the ocean. T h e solutions obtained for models with different layerings of the ocean are shown in Table 5.7. This illustrates that the solution is quite independent of how the ocean is divided into layers. A 12-layer model will be retained in order to increase the resolution of the vertical mixing coefficients. Chapter 5. Evaluating the Model and Solution 71 Number of Layers Correlation coefficient Regression coefficient 6 5 0.9802 0.9627 0.9729 0.957.1 Table 5.7: T h e correlation and regression coefficients between the reference solution and solutions obtained by doubling the thickness of the layers considered i n the model. For the 6-layer model the ocean is divided according to the following pressure surfaces, 150, 250, 400, 600, 800, 1000, 1500 dbar. The 5-layer model divides the ocean according to the following surfaces, 150, 250, 500, 800, 1000, 1500 dbar. Only the reference vertical and horizontal velocities are used in determining the regression and correlation coefficients. 5.3 The Sensitivity of the Solution to the Choice of Reference Level A l l the previous solutions were obtained using the reference level of 1000 dbar. To observe how sensitive the model is to the choice of reference level, solutions can be obtained using different reference levels. These solutions are compared to the reference solution that used 1000 dbar as the reference level. The velocities from these different solutions are compared to the reference solution at the 1000 dbar level. The calculated mixing coefficients are also included in calculating the correlation and regression coefficients (Table 5.8) The high correlation coefficients indicates that the calculated flow patterns are nearly depth independent. However, the regression coefficient indicates that the magnitude of the solutions are dependent on the choice of the reference level. Most of the variance the solutions occurs i n the vertical reference velocities (+20%), and the horizontal mixing coefficients ( + 4 0 % ) . T h e horizontal velocities and the vertical mixing coefficients are nearly independent of depth being determined to t l 5 % and 1 5 % respectively. This indicates that these unknowns are well resolved by the model. This can readily be shown from the matrix equation. The changing of the reference level of the model is equivalent to adding an additional term to the forcing terms. Due to the formulation of the problem, Chapter 5. Evaluating the Model and Solution 72 Reference Level Correlation coefficient Regression coefficient 500 600 700 800 900 1000 1200 1500 0.957 0.977 0.970 0.996 0.999 1.000 0.998 0.989 0.810 0.857 0.947 0.942 0.974 1.000 1.043 1.060 Table 5.8: T h e correlation and regression coefficient between solutions using different reference levels and the reference solution that used 1000 dbar. T h e correlation and regression coefficients were calculated using the velocities from 1000 dbar and all of the mixing coefficients. this additional forcing term is equal to where b' = Baf . T h e vector x is the velocity c c difference between the 1000 dbar level and the new reference level. Thus the new problem is Bx=b x=B- (b 1 x = B - 1 + b' (5.39) + b) (5.40) l (6) +VV*aT c (5.41) Therefore if the matrix V V , the resolution matrix, is equal to the identity matrix, the l solution will be exactly independent of the choice of the reference level. In the model used here the resolution matrix is not exactly equivalent to the identity matrix. This allows the solution to change slightly depending on the choice of reference level. The standard deviation of the unknown parameters about the solution using the 1000 dbar reference level is used to estimate the errors i n the solution. T h e estimated errors i n the Chapter 5. Evaluating the Model and Solution 73 unknown parameters are 1) u 0 = 10.12 cm/s 2) v = 10.10 cm/s 0 3) w = 10.20 /xm/s 4) ^ = 1100m /5 5) k = 10.3 cm /s 0 z 2 2 It is observed that the errors i n the conservation of salt, heat and mass is minimized by assuming a level of no motion at 700 dbar. This suggests that the best choice for the level of no motion exists at about this pressure level. However the error i n the salt condition is still significantly reduced i n the inverse solution from that of assuming a level of no motion at 700 dbar. Furthermore, the result of the inverse solutions do not indicate that a level of no motion should exist at 700 dbar since there is still a well-defined structure to the flow field. T h e model'will continue to use the 1000 dbar level as the reference level. T h e high correlation coefficient indicates that the flow field will not be significantly affected by this choice. 5.4 The Importance of the Different Unknown Parameters on the Solution The result of varying the reference level confirms that not all the unknowns are equally well resolved. T h e resolution matrix (VV*), described i n the previous section, provides information on how well the model can resolve the different unknown parameters. E a c h i th row of the matrix gives the resolution of the i th unknown parameter i n terms of an If the diagonal element of the i average of all the unknown parameters. th row of the matrix is 1, then the resolution of the \ unknown parameter is perfectly resolved, ie. th 1 0 0 % resolution. T h e vertical mixing coefficients are well determined by the model, the resolution of these parameters ranges from 8 1 % to 9 9 % . T h e shallower mixing terms are Chapter 5. Evaluating the Model and Solution 74 better resolved than the deeper mixing terms. T h e horizontal velocities are also quite well resolved. T h e resolution of these parameters ranges from about 7 5 % to 80%. T h e v- component of the horizontal velocities is slightly better resolved than the u-component. The resolution of the vertical reference velocity and horizontal mixing coefficients is considerably smaller. T h e vertical reference velocity is resolved to about 2 5 % and the horizontal mixing coefficients are resolved to about 10%. To explore the importance of the unknown parameters on the solution, different parameters are neglected i n the model to observe how the solution changes. T h e correlation and regression coefficients for the horizontal reference velocities between the reduced model and the reference model are shown i n Table 5.9. It is evident from this table that the vertical mixing coefficients are important factors in determining the reference solution. However, the horizontal mixing terms do not significantly alter the solution. This result, coupled with the poor resolving capability of the model for this variable, led to the decision to neglect the horizontal mixing term from the model. B y neglecting the vertical reference velocity in the model, a significant change in the solution occurs. A l t h o u g h the vertical reference velocities are not well determined, their importance i n the solution is recognized and thus they are retained in the model. It was observed that by neglecting the vertical reference velocities, the value of vertical mixing coefficients decreases. 5.5 Sensitivity of the Solution to Errors in the Data To evaluate how sensitive the model is to errors in the data, random gaussian errors and coherent errors were added to the original temperature and salinity data before obtaining a solution. B y adding these errors to the original data, both the effect of these errors on matrix B and on the forcing terms e can be observed. Chapter 5. Evaluating the Model and Solution 75 Parameter Not Included Correlation coefficient Regression coefficient K 0.368 0.993 0.684 0.164 0.990 0.621 k h Wo Table 5.9: The correlation and regression coefficient for horizontal reference velocities between solutions obtained with a reduce set of unknown parameters. Random Error cr Correlation coefficient Regression coefficient 0.005 0.01 0.05 0.10 1.00 1.00 0.65 0.11 1.00 1.01 0.84 0.17 Table 5.10: The correlation and regression coefficients between the reference solution and solutions obtained using data with random errors added. The random errors w i t h standard deviation cr are add to the original data as follows: a) salinity, t c ppt and b) temperature, t c °C. T h e addition of random errors to the original data simulates the effect of instrument measurement errors on the data. Table 5.10 shows the correlation and regression coefficients between the reference solution and solutions obtained from the data with additional random errors added. This table shows that the solution is quite insensitive to small random errors i n the data. Random errors with standard deviation greater than t0.05 ppt and t0.05 °C need to be added to the original data to significantly change the solution. This value far exceeds the expected errors introduced by limitations i n the instruments. A d d i t i o n a l errors in the data arise from the unresolved spatial and temporal features. Chapter 5. Evaluating the Model and Depth-Coherent Errors ppt c 0.002 0.01 0.05 0.01 0.02 0.10 0.5 0.10 Solution 76 Correlation coefficient Regression coefficient 1.00 0.87 0.63 0.13 0.99 0.94 0.89 0.15 Table 5.11: T h e correlation a n d regression coefficients between the reference solution and solutions obtained using data with coherent temperature and salinity errors added to the data. The temperatures of the ocean obtained from a deep mooring i n the centre of the Ocean Storms area clearly illustrates the high frequency variability that is present i n the ocean and not resolved by the S T P data (Fig. 5.22). T h e fluctuations i n temperature are attributed to isopycnal displacements caused by internal gravity waves. The affects of turbulent mixing are considered to be small. T h e estimated standard deviation of the all the temperature measurements for the upper 1500 m of the ocean is about t 0.05°C The fluctuations i n the salinity measurements is estimated from the corresponding depth fluctuations shown i n this figure a n d the observed gradient of salinity with depth. T h e standard deviation of salinity measurement is about t 0.01 ppt for the upper 1500 m of the ocean. T h e salinity errors will have the opposite signs as the temperature errors because of the assumption that this error is due to displacements of isopycnal surfaces. To simulate these errors i n the data, the data at each station is perturbed by a random error where the error i n salinity is always -1/5 of the temperature error. Table 5.11 shows the correlation and regression coefficients between the reference solution and solutions obtained from the data with additional coherent errors added. The large correlation coefficients i n this table imply that the pattern of the flow determined the model does not change significantly with the addition of this error. T h e Chapter 7.0 -6.8 5. Evaluating the Model and 77 Solution —1—1—1—1—1—1—1—1—1—1—I—1—1—1—1—1—1—1—1—1—1—1—1—1—1—1—1—1—1—I-— H-6.6 t- 6.4 w h 1 6.6 6.4 6.2 6.0 1 1 I^lis m 1 1 1 1 1 1 1 ^ ' 1 1 1 7 5 m TCHAIN l l I L_ 1 l . i i i i i i i i 1 1 1 1 1 1 r~—i 1 1 1 1 1 1 1 1 1 . 1 1 1 1 1 1 I 1 t 1 1 1 1 It -5.6 S5.4 15 m h- 5.2 5.0 *4.8 1 I • * T i i i r T I 1 t 1 i I • J • I " I i 1 i I i 1 i i i i i i i i i i i i i i 1 1 1 1 1 1 1 1 1 t 1 1 1 1 t 1 1 1 1 ,| | | |, 2 2 4 m TCHAIN 1 1 ! t 1 t 1 ) 1 1 1 T | j 1 5 4.4 1 —i 1 ( 1 i i i i i I t I i I i I i • i i i i i i i i i i 1 1 1 1 i i l i | | | | j T| l | | | | 1 i 1 1 1 1 1 i ) • i • i i i 15 m-k" i r .i i J | 1 1 1 1 " 1 5 0 0 m AANDERAA i iT i j - | 3 7 5 m TCHAIN , m 1 -4.2 i i . t 3 0 0 m TCHAIN 4.6 -- 1 \ 1 1 1 1 i i i i i i i u i - 3.0 + 1 5 o 2.0 »- 1.9 + ioo 1 F i g u r e 5.22: 3 5 7 9 11 h—4—H 13 1 m - 1 0 0 0 m AANDERAA T j[ 1 15 .17 OCT 87 •• 1 1 2 0 0 0 m AANDERAA 1 1—1 19 1 21 1 1 23 T h e h i g h f r e q u e n c y f l u c t u a t i o n s of the t e m p e r a t u r e ffrorr r o m a m o o r i n g l o c a t e d i n the centre of the g r i d [24] m 1 1 1 1 1 I 1 1 1 1 1 1 1 1 1 1 i I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 i 1 1 25 1 1—H 27 1 29 1 -1 31 measurements obtained Chapter 5. Evaluating the Model and Solution 78 regression coefficient implies that the magnitude of the solution is not overly sensitive to the expected magnitude of the error present i n the data. Using the estimated errors determined from the mooring the estimated one standard deviation errors about the reference solution are: 1) u 0 = 10.20 cm/s 2) v = 10.15 cm/s 0 3) wQ — 4) k = 10.7 cm /^ • 10.30 /zm/s 2 z These error estimates do show that the errors introduced by features not resolved by the data do place significant limitations on applying this inverse model to synoptic data. 5.6 Reference Solution T h e results of the model to be presented i n the next chapter are based on a 12-layer model using a reference level of 1000 dbar. This model will not use data shallower than 150 dbar (149.1 m) because this data does not satisfy the assumptions made in developing the model. T h e unknown horizontal and vertical reference velocities and the vertical mixing coefficients will be determined using this model. The horizontal mixing coefficients are omitted because they are poorly determined by the model and unimportant i n conserving salt and heat. The reference solution calculated from this model does achieves a 5 0 % reduction i n error over specifying the 1000 dbar level as the level of no motion (reference solution i n table 5.3). T h e estimated one standard deviation errors i n this solution are as follows 1) Uo = 10.32 cm/s 2) v = 10.25 cm/s 0 3) w = 10.5 fim/s 0 Chapter 4) k 5. z Evaluating the Model and Solution 79 = ±1.0 c m / 5 2 These errors are determined from the sensitivity of the solution to the choice of reference level and to the presence of error i n the temperature and salinity measurements. The results obtained using the reference solution will be presented i n chapter 6. Chapter 6 Results of the Model The results of the inverse model for the data collected during cruise one will be presented in this chapter. This inverse model determined the vertical mixing coefficients for depths > 149.1 m, and the vertical and horizontal velocities at the reference level. T h e determination of the reference velocities does allow one to calculate the flow field at any depth by applying the thermal wind equations and linear /3-plane vorticity equation. The only restriction i n applying these equations is that the vertical velocities at the surface of the ocean must be zero. The results of the model for the mixing coefficients and vertical velocities are difficult to verify because of the lack of independent measurements for these quantities. The hor- izontal flow obtained from the inverse model will be compared to velocity measurements obtained from a current mooring and mixed layer drifters. 6.1 The Vertical Velocity vertical velocity at the base of the mixed layer (50 m) and in the halocline (150 m) are displayed in Figure 6.23. For depths shallower than 50 m the vertical velocity is assumed to linearly decrease to zero at the surface. In these maps a positive value indicates that the vertical velocity is directed upward. Both these maps are similar, with the vertical velocity i n the halocline slightly smaller than the vertical velocity at the base of the mixed layer. Evident i n both these maps are the two downwelling north central and south central part of the Ocean Storms grid. 80 zones in the Separating these two Chapter 6. Results of the Model 81 zones is a weak upwelling area. O n the west portion of the Ocean Storms grid, a zone of intense upwelling occurs. T h e vertical velocity at the base of the mixed layer appears to correlate with the convergent and the divergent zones evident i n the horizontal flows at the surface (Fig. 6.25). T h e average vertical velocity at this depth is 0.96 t 0.22 u.m/s, the error is the two standard deviation error determined from the estimated uncertainty in the vertical reference velocities. Although no direct measurements of this vertical velocity exist, this value is consistent with the vertical velocity of 0.63 fim/s (20 m/yr) upward determined by Tabata [14]. He estimated this value from the freshwater budget, divergence of the E k m a n transport, and convergence of the deep geostrophic transport. 6.2 Vertical Mixing Coefficients The model results for the vertical mixing coefficients are shown i n Figure 6.24. plot of the vertical mixing coefficients shows that k z a maximum of 9.8 cm 2 This gradually increases with depth to / s at 600 dbar. Below this maximum, the values of k gradually z decrease to zero. T h e magnitude of these values is large, but this is attributed to the presence of small scale features i n the data. These features i n the ocean not resolved by the model were evident i n the higher frequency sampling of the ocean (Fig. 5.22). The estimate uncertainty i n the these coefficients is about ±1.0cm /s. The lack of other 2 measurements for comparison makes it difficult to assess the calculated the vertical mixing coefficients. 6.3 Horizontal Velocities In this section, I will present the model results for the horizontal velocities. In the two sections to follow, the flow field determined from the model will be compared to current meter observations and to velocities inferred from Lagrangian drifters. 82 LONGITUDE WEST Figure 6.23: The vertical velocity determined from the inverse model. The velocities are in fim/s and a positive value means the flow is upward. The top map shows the vertical velocity at the base of the mixed layer (50 m). The lower map is the vertical velocity at the base of the halocline (150 m). Chapter 6. Results of the Model 83 V e r t i c a l Mixing T e r m v e r s u s Depth Or Figure 6.24: T h e vertical mixing coefficients, k , determined from the inverse model. z Chapter 6. Results of the Model 84 The horizontal velocities determined from the inverse model represent the "absolute" geostrophic flow field in the Ocean Storms area. As stated i n the previous chapter, the one standard deviation i n the horizontal velocities is approximately ± 0.3 cm/s. This uncertainty present in the reference velocities does not detract greatly from the observed flow pattern. To simplify discussion of the results, I will divide the ocean into three zones, an upper zone (0-200 m), an intermediate zone (200-900 m), and a deep zone (900-1500 m). To display the flow field in these zones I have selected several depth surfaces. The flow i n the upper zone of the ocean, Figure 6.25 exhibited three distinct features. One feature is the presence of an eddy i n the northwest corner of the grid. A second feature is the smooth northeasterly flow displayed i n the centre of the survey area. T h e third feature is the distortion i n the flow in the southeast corner of the grid. As one moves down from the surface to deeper levels within this zone, the flow i n the centre of the grid rotates slightly to the south. The eddy present i n the northwest corner of the grid remains almost unchanged within this zone. The feature i n the southeast corner of the grid does evolve with depth. W h a t appears as a slight distortion i n the flow at the surface develops into a strong countercurrent by a depth of 200 m. A t this depth the boundary between the countercurrent and the northeasterly flow is correlated with oceanic front observed i n the principal halocline i n the maps of the temperature and salinity fields (Fig. 3.13). In the intermediate zone (Fig.6.26), the three features observed i n the upper zone are still evident. The strong eddy found i n the northwest corner of the grid is present throughout this zone but by a depth of 900 m this eddy has severely weakened. T h e smooth flow i n the centre of the grid decreases in magnitude with depth. The countercurrent that developed in the lower half of the upper zone is found down to a depth of 900 m. B y this point, it is this feature that is most prominent in the flow and encompasses almost half of the survey area. Chapter 6. Results of the Model 85 REFERENCE VELOCITY OF 4 c « / s 1<J2.0 141 .0 140.0 LONGITUDE -139.0 WEST Figure 6.25: The horizontal flow field for selected depth surfaces of the upper zone (0-200 m). T h e velocity is shown at: a) the surface, b) 15 m, c) 100 m, and d) 200 m. Chapter 6. Results of the Model 86 Chapter 6. Results of the Model 87 REFERENCE VELOCITY OF 4 c n / s ^ a x o ZD . i i r O ZD" j n 142.0 141.0 ! 140.0 ! LONGITUDE , — i39.0 WEST i r 138.0 137. Figure 6.26: T h e horizontal flow field in the intermediate zone (200- 900 m) at selected depths. The velocity is shown at depths: a) 500 m, b) 600 m, c) 700 m, and d) 900 m. Chapter 6. Results of the Model REFERENCE VELOCITY OF 4 ci/s 142.0 141 .0 T 140.0 1 LONGITUDE T 1 39.0 WEST 133.0 Chapter 6. Results of the Model 89 In the deep zone (Fig.6.27), the flow exhibits much smaller length-scales than what were observed i n the previous two zones. The prominent countercurrent is present i n this zone but the other two features have disappeared. This leaves the northwest half of the grid without a distinct flow field. A t 1000 m the smaller scale features dominate the flow in the northwest. Proceeding to deeper levels, the flow becomes more southerly. The flow pattern at 1500 m appears to reverse from what was observed at the surface. The flow at 1000 m from the inverse model results shows the difference between the model results and the flow displayed by the maps of dynamic height (Chapter 3). The most significant difference between these two flow fields is the presence of the strong southwesterly flow i n the southeastern corner of the grid of the inverse results. From the flow field determine by the model it is clear from that at no point does the flow field display a clear depth of no motion. As mentioned in the previous chapter, the 700 dbar reference level is appears to be the best choice of a level of no motion. calculated flow field from the inverse model at 700 m The clearly is not consistent with a depth of no motion existing at this level. 6.4 Direct Current Measurements To validate the velocities determined from the inverse model and to obtain an estimate of the time scale of geostrophic flow, the model results are compared to the velocity measurements made by current meters moored i n the centre of the Ocean Storms grid, the O S U mooring (at 47° 28.4' N, 139° 16.0' W). The current meter observations from this mooring are provided by M. Levine and C. Paulson of Oregon State University. In the upper part of the mooring, seven Vector Measuring Current Meters ( V M C M ) were distributed from 60 m to 195 m. The lower part of the mooring included five Aanderaa meters distributed from 500 m to 4000 m. Chapter 6. Results of the Model 90 REFERENCE VELOCITY OF 4 c . / s a Q: o "i i i r ~~\ 1 1 i r b or o I 0£ 142.0 141.0 140.0 1 LONGITUDE 1 139.0 WEST 1 I 138.0 ~T 137 Figure 6.27: The horizontal flow field in the deep zone (900-1500 m) at selected depths. The velocity at depths of a) 1000 m and b) 1500 m. Chapter 6. Results Depth (m) 1000 500 195 160 140 120 100 60 of the Model Geostrophic Velocity (cm/s) U V -0.28 0.39 1.70 1.98 2.28 2.70 2.82 2.90 -0.36 0.17 0.43 0.53 0.66 1.00 1.20 1.70 91 Averaging Interval (days) 7 7 7 7 7 7 7 40 Time Interval (yeardays) start end 277 263 255 255 257 246 233 251 284 269 261 260 263 252 244 290 Current Meter Velocity (cm/s) U V -0.24 0.40 1.75 1.99 2.16 1.70 2.82 2.27 -0.39 0.16 0.30 0.6.1 0.61 0.13 0.86 1.67 Table 6.12: A comparison of the geostrophic velocities calculated from the model to the current meter observations. The raw data that I received from this mooring had already been low-passed filtered to remove the inertial oscillations and averaged to one-day intervals. From these velocity measurements I further reduced the variability i n the data by calculating several time averages. T i m e averages of the velocity data were obtained for 7 day, 14 day and 21 day time intervals. T h e results of these times averages are displayed i n Figures 6.28, 6.29, and 6.30. I n these diagrams the stick points in the direction of the flow, with north being up. T h e last stick on the right side of the plots are the corresponding geostrophic velocities determined from the inverse model. T h e time of the first cruise is from 274 to 284 yeardays. The occurrence of the October 4th storm (278 yeardays), is evident i n the shallow current measurements by the large increase i n velocity that occurred around this time. The comparison between the current meter observations and the inverse model are summarized in Table 6.12. T h e calculated geostrophic flow field at the current mooring is comparable to a seven day average of the current meter observations. A t 500 m and 6. Chapter Results of the Model 92 -0.36 | 1 1 1 ^ 1 1 1 1 I | I 1 I I I 235. 244. 254. 264. 274. 284. 294. 304. 314. 324. 334. 344. 354. 364. 374. 384. 394. YERRDRYS 1987 F i g u r e 6.28: T h e velocities f r o m the c u r r e n t meter o b s e r v a t i o n s for a seven day average p l o t t e d every three d a y s . In this figure the stick p o i n t s i n the d i r e c t i o n of f l o w , w i t h n o r t h b e i n g u p . T h e n u m b e r s on the left side of the p l o t are the c o r r e s p o n d i n g d e p t h s i n meters of the c u r r e n t meter. O n the r i g h t side of the p l o t , the velocities c a l c u l a t e d f r o m the inverse m o d e l at the l o c a t i o n of the m o o r i n g are d i s p l a y e d . Chapter 6. Results of the Model 93 -0.3S 2 cm/ S I I I I !*' I 1 i I I I 1 I I I I 235. 244. 254. 264. 2 7 4 . 2 8 4 . 2 9 4 . 304. 314. 324. 334. 344. 354. 364. 374. 384. 394. YERRDRYS 1987 Figure 6.29: The velocities from the current meter observations for a fourteen day average plotted every seven days. In this figure the stick points i n the direction of flow,.with north being up. The numbers on the left side of the plot are the corresponding depths in meters of the current meter. On the right side of the plot, the velocities calculated from the inverse model at the location of the mooring are displayed. 6. Chapter Results of the Model 94 60 100 120 140 160 195 500 1000 -0.36 2 Cm/s I 1 1 1 . I <\ I i I I I I I I I I I 235. 244. 254. 2 6 4 . 274. 284. 294. 304. 314. 324. 334. 344. 354. 364. 374. 384. 394. YEARDAYS 1987 F i g u r e 6.30: T h e velocities f r o m the current meter observations for a 21 day average p l o t t e d every 10 d a y s . In this figure the stick p o i n t s i n the d i r e c t i o n of f l o w , w i t h n o r t h b e i n g u p . T h e n u m b e r s o n the left side of the p l o t are the c o r r e s p o n d i n g depths i n meters of the current m e t e r . O n the right side of the p l o t , the velocities inverse m o d e l at the l o c a t i o n of the m o o r i n g are d i s p l a y e d . calculated from the 6. Chapter 1 Results of the i 1 95 Model i-L i i i ' i i 1 1 2-> Q n o™ oo m 3 -2 3 2 1 0 -1 -2 -3 i" i 1 2+ 3 o 1 1 1 < 1 | 1 1 —i 1 •A n A K . V V V AUG F i g u r e 6.31: 1 SEP 1 OCT 1 NOV — V 1—: D€C JAN , , FEB i 1 UAR APR 1 MAY JUN T h e v - c o m p o n e n t of the v e l o c i t y m e a s u r e d at 3000 m o n the O S U m o o r - i n g ( u p p e r p a n e l ) a n d the v - c o m p o n e n t c o m p u t e d f r o m the g e o s t r o p h i c r e l a t i o n a n d data.f r o m the b o t t o m pressure sensor array. •[24] 1000 m , the agreement b e t w e e n these t w o measurements is excellent. A t a d e p t h of 1000 m , t h e g e o s t r o p h i c v e l o c i t y i n f e r r e d f r o m the d e n s i t y field is c o m p a r a b l e to the average of the c u r r e n t meter o b s e r v a t i o n s o b t a i n e d d u r i n g the t i m e of this cruise. A b o t t o m array of pressure gauges d e p l o y e d i n the O c e a n S t o r m s area also appears to s h o w t h a t the deep geostrophic current measurements i n f e r r e d f r o m this array are c o m p a r a b l e to the current m e t e r o b s e r v a t i o n s for p e r i o d s of a few days to a week ( F i g . 6.31) [24]. A s one moves u p w a r d f r o m 1000 m t o w a r d the surface, the g e o s t r o p h i c v e l o c i t y appears to l a g the c u r r e n t meter m e a s u r e m e n t s . p o s s i b l e effects. T h i s s i t u a t i o n m a y be e x p l a i n e d by two O n e , it takes a f i n i t e a m o u n t of time for the d e n s i t y field t o to the present c u r r e n t s y s t e m . adjust T h i s allows the d e n s i t y field, a n d h e n c e the geostrophic v e l o c i t y , to r e t a i n i n f o r m a t i o n o n the p r e v i o u s c u r r e n t field. T h i s suggests t h a t it is t h e , c u r r e n t s y s t e m t h a t forces the g e o s t r o p h i c .flow i n the ocean. H o w e v e r , this effect does not e x p l a i n w h y the g e o s t r o p h i c flow at the surface lags the c u r r e n t m e t e r m o r e t h a n the deeper c u r r e n t meters. observations T h e second reason is u n r e l a t e d to the first b u t - i s r e l a t e d to the effects of the O c t o b e r 4 t h ( y e a r d a y 278) s t o r m on the c u r r e n t meters. The Chapter 6. Results of the Model 96 energy input into the ocean by this storm at the surface slowly propagated downward. Thus the deeper current meters experienced the effects of the storm at a later time. This implies that at the time of the storm, the flow at the deeper levels was less corrupted by the storm. A t 1000 m the average of the current meter observations during the time of the cruise is comparable to the geostrophic flow from the model. As one moves upward toward the surface, the storm effects arrive earlier i n time therefore one must move further back i n time to observe the geostrophic flow. This effect does not explain why there is a large difference between the current meter observations and geostrophic velocities at a depth of 120 m. A t this depth another effect must be considered to explain this large difference. The lack of an exact agreement between the two velocity measurements can be explained by several reasons. One, direct current measurements do not provide any in- formation on the spatial structure of the flow. B u t it is clear from the maps of the geostrophic flow that significant spatial features existed i n the vicinity of the current mooring. T h e velocities determined from the inverse model assume some spatial averaging of the velocity field. T h e most noticeable spatial structure of the flow i n the vicinity of the current mooring occurs for depths greater than 500 dbar. Two, the existence of a significant ageostrophic component to the flow may exist. This effect is expected to be greatest near the surface where the forcing function, the wind, is largest and is capable of generating a large ageostrophic component. Although a seven day average of the current meter measurements provided a good agreement to the geostrophic flow up to 140 m, i t does appear that the geostrophic flow may represent time-scales longer than seven days. It is clear from the current meter observations that after the storm the flow again becomes consistent with the calculated geostrophic flow. This most clearly holds true for the depths shallower than 500 m. This observation is also evident i n the longer time-averaging of the current meter observations Chapter 6. Results of the Model 97 i n the upper 200 m. Although these longer averages produce a good comparison between the two velocity measurements, significant differences still exist. This suggests that the ageostrophic component has a significant low frequency component. Using a seven day average of the current meter observations obtained during the time of the first cruise, 277 to 284 yeardays, an estimate of the ageostrophic component of the flow is calculated. This estimate of the ageostrophic flow is obtained by subtracting the geostrophic velocities determined by the inverse model from these seven day averages of the current meters. The profile of the geostrophic, ageostrophic, and total flow field are displayed i n Figure 6.32. This figure shows that the ageostrophic component of the flow is greatest i n the upper ocean (60 - 195 m). T*'? ageostrophic flow is m a x i m u m at 60 m 1 but the magnitude of this component is quite uniform between depths of 60 -195 m. T h e ageostrophic component also undergoes a cyclonic rotation as one moves downward from the surface. This is opposite to the rotation of the E k m a n spiral, but this is acceptable because the E k m a n spiral only exist i n steady winds which is not the case during a storm. The absence of a strong decay i n the ageostrophic component with depth suggest that this flow is not forced by the wind. However, the lack of significant total current flow between 140 m and 195 m implies that the geostrophy is not forcing the flow at these depths. Does a depth of no motion exist at this between these depths? If one assumes a depth of no motion at 150 dbar, the errors i n the conservation of salt and heat increases drastically. Based on this the 150 dbar level is a poor choice for the level of no motion if one desires that temperature and salinity i n the interior of the ocean be conserved. To explain these features i n the ageostrophic flow and total flow, it appears that one must evoke additional terms i n the equations of motion du 2 dz (6.42) 2 8v 2 — + u— dt dx -f- v— + fu = dy q dy o~z~ 2 (6.43) Chapter 6. Results o f tiie Model 98 6C \ / 160 195 - 500 (b) ("CO CO Figure 6.32: The horizontal velocity profiles for a) the geostrophic flow, b) the ageostrophic flow, and c) the total flow. If this flow is not consistent with being forced by the pressure gradient ( V P ) windstress (vd ujdz , 2 2 vd v/dz ), 2 2 a n d the the nonlinear terms and acceleration terms must be important. Any error present in the determination of the velocity of the flow at the reference level will be equivalent to adding an additional constant to all of the ageostrophic measurements. term. The depth structure of the ageostrophic flow will not be affected by this Chapter 6.5 6. Results Mixed Layer of the Model 99 Drifters In the Ocean Storms area, 49 satellite-track T R I S T A R drifters were deployed. These instruments consist of a small surface float containing radio transmitting equipment and a large three-axis symmetric drogue element centred at a depth of 15 m. T h e processed position of all the Ocean Storms drifters are shown i n Figure 6.33. T h e flow evident by this data have similar features to the geostrophic flow determined from the inverse model. The three distinct features evident i n the surface geostrophic flow are evident i n this data. In order to better compare these two velocity measurements, a simple algorithm was developed to calculate the velocity from the drifter positions. The area of the Ocean Storms experiment was divided into the same boxes that were used i n the inverse model. A n average value of the velocity i n each of these boxes was determined by calculating the velocity from the change i n position of the drifter over one inertial period. This time sampling is necessary to remove the inertial oscillations from the data. Using the time interval of one inertial period, each inertial period provides one independent estimate of the flow for each drifter. B y averaging all of these velocities from all of the drifters, the average flow i n each of the boxes defined is determined. T h e resulting flow field estimated from the drifter data over the period of 275 to 295 yeardays is shown i n Figure 6.34. Considering a longer time interval does not alter the observed flow in the Ocean Storms area because by this point most of the drifters are no longer present i n the grid. The calculated flow field from the drifter data displays the three characteristic features that were present in the geostrophic flow. These features are a prominent eddy i n the northwest corner of the grid, a distortion i n the flow i n the southeast corner of the grid, and the generally smooth northeasterly flow through the centre of the grid. T h e fact that these features are evident i n this velocity map implies that these major features i n o Lort&iTUOE Figure 6.33: The tracks of the drifter data. The location of the Ocean Storms grid is defined by the dashed line on this map. inertial oscillations. The oscillation of the drifter tracks is due to o o Chapter 6. Results of the Model 101 Figure: 6.34: T h e total flow field determined from the drifter data. T h e location of the Ocean Storms grid is shown by the enclosed region on this map. This velocity map is obtain*ed from the drifter data from day 275 to 295. Chapter 6. Results of the Model 102 the flow field are controlled by geostrophy. A n estimate of the ageostrophic flow can be obtained by subtracting the geostrophic flow from the total flow determined by the drifter data. The inverse model results at a depth of 15 m are subtracted from the total flow field. This difference in flow fields is termed the ageostrophic flow. T h e ageostrophic flow in the mixed layer is shown i n Figure 6.35. T h e ageostrophic flow shows the presence of a jet i n the north with a strong easterly flow which moves i n a south easterly direction through the east half of the Ocean Storms area. A smaller flow exists to the south of this jet moving in a southerly direction. The magnitude of this flow field is significant but possesses smaller spatial features than the geostrophic flow at 15 m. Of the total flow through the Ocean Storms grid, 6 4 % is due to the geostrophic flow and 3 6 % is derived from the ageostrophic component. If the ageostrophic flow was produced by the windstress one would expect that the flow field would exhibit larger spatial features attributed to the larger spatial features present in the wind field. The small spatial features present i n the ageostrophic flow field suggest that the windstress is not the dominant forcing term at this time. From this it is inferred that the acceleration and nonlinear terms in the momentum equations (6.42) and (6.43) must play an important role in the affecting the flow. This is consistent with the depth profile of the ageostrophic flow in the centre of the grid. This analysis of separating the total flow field into ageostrophic and geostrophic flow is essentially independent of the inverse model results because the flow at the reference level does not significantly change the geostrophic flow at the surface. Chapter 6. Results of the Model 103 Figure 6.35: T h e ageostrophic flow field determined from subtracting the geostrophic velocity determined from the inverse model at 15 m from the total flow estimated .from the drifter data. Chapter 7 Factors Influencing Heat and Salt Content in the Ocean Storms Area The factors that influence the salt and heat content i n the ocean can be described by the following two equations ^ The ot = - „ - . V ( ^ ) + > OZ . ^ ) oz + «?' (7.45) change i n salt and heat content of the ocean is affected by three factors 1) the horizontal a n d vertical transport of salt and heat by the flow field, u; 2) the mixing of salt and heat by the mixing coefficient k ; 3) the exchange of freshwater and heat across z the air-sea boundary, Q" and Q , respectively. l In this chapter I first determine the change in salt and heat content of the ocean i n the Ocean Storms area during the time between the two cruises. Second, I calculate the transport of salt and heat by the flow field and mixing coefficients. T h i r d , by balancing the salt and heat budgets, I estimate the exchange of heat and freshwater across the air-sea boundary. 7.1 The Observed Change in Salt and Heat Content A n estimate of the change i n salt and heat content i n the Ocean Storms area is determined from the differences i n salinity and temperature between the seventeen stations occupied during both cruises. These values of the change in salt and heat content between the two cruises exhibit a great deal of variability over the Ocean Storms grid. 104 To reduce Chapter 7. Factors Influencing Heat and Salt Content in the Ocean Storms Area 105 the variability and increase the accuracy of these values, the mean value for the entire Ocean Storms area will only be considered. T h e variability of the change in heat and salt content estimated from the different stations is evident from the standard deviations about the mean values. A n estimate of the error i n the change i n salt and heat content is determined from the high frequency temperature fluctuations observed at an O S U mooring. The estimated two standard deviation errors i n the salinity and the temperature measurements for the upper 1500 m of the ocean is t 0.02 ppt and t 0.1°C respectively. For each station these error estimates are considered independent of the other stations. The change i n salt content, dS/dt, of a layer of the ocean bounded at the top and bottom by the depths z and z& respectively is t (7.46) AS is the change i n salinity of this layer during the time interval At. The change i n the salt content between these two cruises is shown i n Table 7.13. For the upper 1500 m of the ocean the observed change i n salt content is equivalent to a 0.014 t 0.005 ppt increase i n salinity during the time between the two cruises (60 days). Such a change is w i t h i n the accuracy of the measurements of salinity [17]. Considering only the upper 150 m of the ocean, the observed change i n salt content is equivalent to an increase i n salinity of 0.004 t 0.005 ppt. Previous estimates of the change i n salt content of the upper ocean at Ocean Station P indicate that the upper ocean should experience a significant decrease i n sabnity during the fall season [17]. This discrepancy between these two measurements of the change i n salt content of the upper ocean confirms that the salt content can exhibit significant year to year variability. This variability is influenced greatly by the atmospheric conditions but at the present time the affect the atmosphere has on the ocean is not well determined. This is attributed to the difficulty of trying to Chapter 7. Factors Influencing Heat and Salt Content in the Ocean Storms Area 106 Change i n Salt Content (10- ppt/s) Mean Error Standard Deviation 3.5 0.9 2.1 0.7 0.9 13.7 2.8 0.9 2.8 2.7 0.9 2.4 9 Top Bottom 0 0 0 0 100 150 1000 1500 Table 7.13: Change i n salt content between cruise one and cruise two. T h e value given for the change i n salt content is the change i n salinity observed i n that layer. T h e mean and standard deviation are obtained using the stations occupied during both cruises. directly measure the rate of evaporation and precipitation i n the open ocean. The change i n heat content, d(gCT)/dt, of a layer of the ocean bounded at the top and bottom by the depths z and Zf, respectively is t AT is the change i n temperature of this layer during the time interval At. T h e heat capacity (gC) is assumed independent of depth and equal to 4 x 10 k J / m . 6 T h e observed changes i n heat content of the ocean during the time between the two cruise are shown i n Table 7.14. T h e observed change i n heat content for the upper 1500 m of the ocean is equivalent to a decrease in temperature of this layer of 0.047 1 0.018°C during the time between the two cruises. Such a temperature difference i n temperature is definitely within the accuracy of the temperature sensors [14]. In the upper 150 m, the average loss of heat of this layer during the fall 1987 is equal to 92 t-2 W/m . 2 Large et al [16] calculated that the average loss of heat i n this area during the fall for the upper 120 m of the ocean was approximately 90 W/m . 2 Chapter 7. Factors Influencing Heat Layer (m) Top Bottom 0 0 0 0 0 60 100 150 1000 1500 and Salt Content in the Ocean Storms Area 107 Change i n Heat Content (W/m ) 2 Mean Error -124 -96 -92 -65 -54 1.0 1.4 2.0 14.0 21.0 Standard Deviation 28 32 41 128 120 Table 7.14: Change i n heat content between cruise one and cruise two. T h e value given for the change i n heat content is the change i n heat observed i n that layer. T h e mean and standard deviation are obtained using the stations occupied during both cruises. 7.2 Transport of Salt a n d Heat A n estimate of the transport of salt and heat can be calculated from the ocean circulation parameters, determined by the inverse model and the gradients of the salinity and temperature fields observed during the first cruise. T h e results of the inverse model allows one to determined the transport of salt and heat by the vertical and horizontal velocities and the vertical mixing coefficients. B y using the circulation parameters several assumptions are implied: 1) the horizontal flow determined from this model is geostrophic; 2) the vertical velocity satisfies the linear /3-plane vorticity equation, and above the base of the mixed layer (50 m) the vertical velocity linearly decreases to zero; 3) the vertical mixing coefficient is determined only for depths greater than 150 dbar; 4) for depths greater than 150 dbar, one is uses the residual errors of the inverse problem to determine the transport of salt and heat. The transport of the salt and heat from the flow field and vertical mixing terms are determined for the entire Ocean Storms grid. A n estimate of the error i n these transports are calculated using the errors i n the circulation parameters and the estimated Chapter 7. Factors Layer (m) Top Bottom 0 0 0 0 150 60 150 1000 1500 1500 Influencing Heat and Salt Content in the Ocean SALT Horizontal Transport Vertical Transport ( 1 0 - ppt/s) ( 1 0 - ppt/s) Mean Error Mean Error 9 20 27 13 12 10 9 3 3 3 3 3 2 -30 -17 -3 -0.4 2 3 4 4 4 Storms Area 108 Total Transport ( i o - ' Ppt/s) Mean Error £ 22 -3 -3 9 10 4 4 5 5 5 Table 7.15: The calculated transport of salt using circulation parameters determined by the inverse model. The vertical mixing term is only important at depths of 250 m to 800 m. The errors i n these transports are determined from the uncertainty i n the circulation parameters and the uncertainty i n the measurement of salinity. A positive transport means that the Ocean Storms area experiences a gain i n salinity. uncertainties i n the temperature and salinity measurements. 7.2.1 Salt Transport The change i n the salt content of the ocean due to vertical transport, horizontal transport and total transport are shown i n Table 7.13. In the upper ocean 0 - 150 m), the increase i n salinity due to horizontal transport is offset by the larger decrease i n salinity caused by the vertical transport. The vertical transport of salt is negative even though the vertical velocity is upward because more salt leaves through the top of the layer than enters through the bottom. The calculated transport of salt i n the upper 150 m of the ocean is equivalent to reducing the salinity i n this layer by 0.016 10.02 ppt during the time between the two cruises (60 days). In the deeper layers of the ocean (> 150 m) the horizontal transport of salt continues to increase the salinity. T h e vertical transport of salt works to reduce the salt content i n these layers. T h e vertical mixing of salt by the mixing coefficients is significant only Chapter 7. Factors Influencing Heat and Salt Content in the Ocean Storms Area 109 for depths between 248.4 m and 892.0 m, where it causes the salt content i n ocean to increase. In the 150 m to 1500 m layer of the ocean, the calculated transport of salt would result i n an increase i n salinity of this layer by 0.05 ±0.03 ppt for the time interval between the two cruises. T h i s exceeds the observed change i n salinity of 0.018 ±0.004 ppt determined from the differences i n salinity between the two cruises. This shows that the model does not conserve salt more effectively than the actual interior of the ocean. If the model was more effective i n conserving salt this would suggest that the salinity information is being given too much importance i n determining the inverse solution. T h e fact that these two values are comparable implies that the conservation of salt condition in the inverse model was adequately weighted. 7.2.2 Heat Transport The change i n the heat content of the ocean due to vertical transport, horizontal transport and total transport of heat are shown i n Table 7.16. In the upper ocean (0 - 150 m), the vertical and horizontal transports of heat are nearly equivalent. T h e vertical transport of heat results i n a loss of heat i n this layer of 20 ±6 W/m . 2 This value is 21.5% of the observed heat lost by this layer for the time during the two cruises (Table 7.14). This is consistent with previous estimates of the vertical transport of heat which attribute approximately 2 0 % of the heat lost i n the upper ocean to this effect [17,16]. The horizontal transport of heat is also significant, i t reduces the heat content of the upper ocean by 20 ±10 W/m . 2 For the layers i n the interior of the ocean between the depths of 150 m to 1500 m, the calculated net transport of heat from the model is -1 ±100 W/m . 2 T h e observed heat loss of this layer determined from the differences i n temperature between the two cruises is 38 ± 21 W/m . 2 This shows that the model results does conserve heat more effectively than the real ocean. This suggests that the conservation of heat constraint Chapter 7. Factors Layer Influencing Heat and Salt Content in the Ocean HEAT Horizontal Transport Vertical Transport Storms Area Total Transport Top (m) Bottom Mean Error Mean Error Mean Error 0 0 0 0 150 60 150 1000 1500 1500 -18 -20 -9 -9 11 4 10 60 100 100 -9 -20 -34 -32 -12 1 6 7 8 8 -32 -40 -43 -41 -1 4 12 60 100 100 {W/m ) (W/m ) 2 110 (W/m ) 2 2 Table 7.16: The calculated fluxes of heat from using the velocities obtained from the inverse model. T h e vertical mixing term is only important at depths of 500 m to 1200 m. The errors in these transports are determined from the uncertainty i n the circulation parameters and the uncertainty i n the measurement of temperature. used i n the model could have been downweighted to produce a better agreement with observed heat loss. However, the large error attached to the model value does imply some consistency between this two values thus supporting the weighting of the conservation of heat condition used i n the model. 7.3 Exchange Across the Air-Sea Boundary A n estimate of the exchanges of heat and fresh water across the air-sea boundary are determined by calculating the sources and sink terms that are required to balance the salt and heat budget i n the upper ocean (equations 7.44 and 7.45). In balancing the salt and heat budgets, I will restrict the analysis to the upper 150 m of the ocean where the air-sea boundary process are most pronounced. Below 150 m, there is little evidence of the annual cyclic variations of salinity and temperature that would be attributed to the air-sea boundary process [17]. In this section, I will first balance the salt budget and then I will proceed to balance the heat budget. Chapter 7.3.1 7. Salt Factors Influencing Heat and Salt Content in the Ocean Storms Area 111 Budget The air-sea boundary processes that affect the salt content of the upper ocean are precipitation and evaporation. T h e reduction of salinity i n the open ocean is accomplished by precipitation i n the form of rain, snow, and hail. O n the other hand the increase in salinity is accomplished by the evaporation of fresh water from the sea surface. T h e fresh water exchange across the air-sea boundary due to precipitation and evaporation is the most important factor influencing the salinity of the upper surface water [17]. In the northeast Pacific Ocean, this influence is confined primarily to the water above the halocline [17]. In the time interval between the two cruises, the upper 150 m of the ocean experience an increase i n salinity of 0.004 ±0.005 ppt. T h e estimated vertical and horizontal transport of salt for the upper ocean during the same time interval is -0.016 ±0.02 ppt. Therefore, to balance the salt budget would require that fresh water exchange across the air-sea boundary increase the salinity of the upper ocean by 0.020 ± 0.020 ppt. This can only be accomplished by having evaporation (E) exceed precipitation (P) during the fall of 1987. The change of salinity, AS, due to E - P, during the time interval At can be evaluated from the following formula [17] AS = S - Si % S ^ ~ ^ f E P (7.48) Sf is the final salinity (ppt) Si is the initial sabnity (ppt) z is the depth (cm) of the column considered (0 - 150 m ) E — P is the difference of evaporation and precipitation i n centimetres during the time interval At Chapter 7. Factors Influencing Heat and Salt Content in the Ocean Storms Area 112 In the Ocean Storms area the initial salinity of the upper zone, Si is approximately 33 ppt. Thus, to balance the salt budget would require that E - P equals 9 I 9 cm. 7.3.2 Heat Budget The air-sea boundary processes that influence the heat content of the upper ocean are the combined effects of incident solar radiation, reflected solar radiation, effective back radiation, evaporation (and condensation) and conduction of sensible heat. The heat transfer equation across the air-sea boundary during a time interval At can be written as Q l = Qi - Qr - Qb - Q e - Qh (7.49) Q is the net heat transfer across the air-sea boundary l Qi is the short-wave radiation from the sun and sky, both direct and diffuse. Q is the reflected short-wave radiation from the sea r Qb is the effective back radiation (long wave radiation from the sea surface minus that from the atmosphere) Q is the heat transferred by evaporation (and condensation) e Qh. is the conduction of sensible heat between the atmosphere and the sea. The observed loss of heat i n the upper 150 m of the ocean during the fall of 1987 is 92 t 2 W/m . 2 The calculated vertical and horizontal transport of heat i n the upper 150 m of the ocean is -40 t 12 W/m . 2 In order to balance the heat budget i n the upper 150 m of the ocean this upper layer would lose 52 +12 W/m 2 of heat. This implies that the flux of heat at the air-sea boundary accounts for about 5 7 % of the observed heat lost by the upper ocean. From the calculated E - P value of 9 cm the m i n i m u m amount of heat lost by evaporation would be 43 W/m . 2 This is comparable to calculated heat lost by the upper ocean. However, i n this comparison one must keep i n mind that there are other Chapter 7. Factors Influencing Heat and Salt Content in the Ocean Storms terms which affect the transfer of heat across the air-sea boundary (equation Area 7.49). 113 Chapter 8 Discussion and Conclusion The work presented i n this thesis was based on the analysis of hydrographic data collected in the northeast Pacific Ocean as part of the Ocean Storms experiment during the fall of 1987. This work will be summarized in four parts: 1) the qualitative description of the hydrography of the ocean during the fall of 1987. 2) the development and application of an inverse model to the hydrographic data. 3) the ocean circulation of the study area determined from the inverse model. 4) the factors that influenced the salt and heat content of the ocean. In the first part of this thesis, chapter 3, the hydrographic data collected revealed that vertical structure of the ocean represented the Eastern Sub-Arctic Pacific Water Mass and could be divided into three zones: 1) the upper "seasonal zone (0 - 100 m); 2) the principal halocline (100 - 200 m); and 3) the lower zone ( > 200 m). Maps of dynamic height of the study area showed that the horizontal flow field was quite smooth but did possess prominent mesoscale eddies i n the flow. Isentropic analysis of the temperature and salinity fields generally supported the flow field inferred from the maps of dynamic height. However, this analysis revealed additional details i n the flow field not evident in the maps of dynamic height. To extract this information from the data an inverse model was developed. In the second part of thesis, chapters 4 and 5, an inverse model was first developed to calculate the absolute geostrophic velocities, the vertical flow field, and the vertical and horizontal mixing coefficients. These parameters were determined by calculating a model 114 Chapter 8. Discussion and Conclusion 115 that best tried to conserve salt, heat, and mass. Unlike finite differencing schemes [6,19], this inverse model calculated the unknown parameters by conserving heat, salinity, and mass in finite volumes. In developing this inverse model, four assumptions were made to simplify the physics of the problem. 1) T h e flow i n the ocean is geostrophic. 2) T h e vertical velocities satisfy a linear /3-plane vorticity equation. 3) A steady state advection-diffusion equation with no sources or sinks can be used in to conserve salt and heat i n the interior of the ocean. 4) T h e mixing terms in the advection-diffusion equation are only considered functions of depth and can be parametrized according to vertical and horizontal mixing coefficients. The effects of these assumptions on calculating ocean circulation were considered by comparing the results of the model to other available information. Second, the inverse model was then applied to the hydrographic data collected. In applying the inverse model to the observed data several problems and limitations arose which affected the results of the inverse model. These limitations are summarized below with a brief explanation of how they affect the inverse problem A x = b. 1) T h e choice of the cutoff level used in determining the singular values that should be included in inverting the matrix A. The cutoff level chosen in part determined how sensitive the solution was to errors in the data. 2) T h e weighting of the conservation of temperature, salinity, and mass equations used i n solving the inverse problem. In evaluating the model, i t was apparent that the solution was very sensitive to the adopted weighting of these three conditions. 3) T h e choice of reference level at which the unknown reference velocities were determined. T h e uncertainty introduced by the choice reference level is intimately related to the ability of the model to resolve the different unknown parameters. Chapter 8. Discussion and Conclusion 4) T h e existence of errors i n the temperature 116 and salinity caused by unresolved tem- poral and spatial features and limitations i n measuring equipment. These errors i n the data affect both the forcing terms b and the matrix A . A n estimate of the uncertainty i n the solution was made from the sensitivity of the solution to the choice of reference level and from the sensitivity of the solutions to errors i n the data. In this estimate of the uncertainty i n the solution, the choice of cutoff level was implicity used because it affects how sensitive the solution is to errors i n the data. T h e weighting of the equations was not used because this would require better understanding of how well the three conservation conditions were satisfied by the approximate equations used i n the model. A t present, this k i n d of information is not well understood i n the literature. T h e errors due to the model assumptions were also neglected; these limitations were assessed i n analyzing the results of the model. T h e estimated one standard deviation errors i n the solution are as follows: 1) u = 10.32 cm/s. 0 2) v = 10.25 cm/s. 0 3) w = 10.5 fim/s. Q 4) k = ±1.0 cm /s. 2 x 5) kh is neglected. The horizontal mixing coefficients kh were omitted from the final solution because these unknowns were poorly determined by the model and did not play a significant role i n conserving salt or heat. In third part of this thesis, chapter 6, the results of the inverse model for the vertical mixing coefficients and horizontal and vertical flow fields were presented and discussed. These results were compared to other measurements made i n the area allowing additional conclusion about the ocean circulation to be made. The vertical mixing coefficients determined by the model were non-zero between the Chapter 8. Discussion and Conclusion 117 depths of 200 m and 1200 m, with a maximum at 600 m. T h e values of these coefficients were large (10 cm /s) but this appears to be consistent with other box model estimates 2 of the vertical mixing coefficients [6]. The vertical velocity field determined from the model gave velocities i n the upper ocean consistent with previous estimates. The average vertical velocity for the upper ocean i n the Ocean Storms area was 1.0 fim/s. These velocities did exhibit prominent spatial features which appeared to be correlated with divergences and convergences i n the horizontal flow at the surface. The geostrophic flow field determined from the model revealed the presence of mesoscale eddies i n the flow. However, the pattern of the flow determined was still generally smooth w i t h well-defined flow features. A t the surface the geostrophic flow produced the major features of the flow field evident from the drifter data. T h e geostrophic velocities calculated from the model were comparable to a seven day average of the current meter observations. The results of the model for the horizontal flow field also appeared to improve on the assumption that a depth of no motion existed at 1000 dbar. In the halocline (100 - 200 m) the flow field from the model showed the existence of the oceanic front that was clearly present i n the temperature and salinity fields but absent from the maps of dynamic height. For depths greater than 1000 m, the horizontal flow field obtained from the model, gave a better defined flow pattern than those obtained with the depth of no motion assumption. Furthermore, the model results were not consistent with the assumption that a level of no motion exists i n the flow field. A depth profile of the ageostrophic flow was determined from the difference between the current meter observations and the geostrophic flow. This profile lacked features that would associate this flow with windstress. The spatial pattern of the ageostrophic flow field in the mixed layer determined from the drifter data and the geostrophic velocities Chapter 8. Discussion and Conclusion 118 also implied that this ageostrophic flow was not forced only by the wind. From these results it was inferred that the acceleration ,and nonlinear terms had an important effect on the flow field during the time of the first cruise. This lack of a connection between the windstress and current meter observations was observed i n the J A S I N experiment [25]. These investigators concluded that at most times the current i n the upper ocean was not wind forced especially when the windstress was small. In the fourth part of this thesis, chapter 7, the factors influencing the heat and salt content of the upper ocean the were calculated. During the 60 days between the two cruises the upper ocean, (0 - 150 m), i n the study area experienced a slight increase i n salinity of 0.004 ± 0.005 ppt. From the model results the calculated net transport of salt into the upper ocean during this same time period was -0.016 t 0.2 ppt. Therefore, to balance the salt budget evaporation h a d to exceed precipitation by 9 ± 9 cm for the 60 days between the two cruises. D u r i n g this same period the upper ocean lost 92 +_ 2 W/m 2 of heat. The horizontal and vertical net transport of heat each account for a 21.5% reduction i n heat content of the upper ocean. T h e remaining 5 7 % of the heat lost by the upper ocean was due to the net loss of heat at the air-sea boundary. In conclusion I would like to emphasize that more work still needs to be done on applying inverse models to oceanographic data. A detail study of the problem is required to assess how much information about the flow field can be extracted from tracer data by inverse models. A natural approach to this problem would be to apply the inverse model to data generated from numerical ocean circulation models. Bibliography [1] C. Wunsch, "The North Atlantic General Circulation west of the 50 W determined by inverse methods." Reviews of Geophysics, vol. 16, pp. 583-620, November 1978. [2] M . E . Fiadeiro and G. Veronis, "On the determination of absolute velocities i n the ocean," Journal of Marine Research, vol. 40, pp. 159-182, Supplement 1982. [3] R. B. Montgomery, "Circulation in the upper layers of southern North Atlantic deduced with use of isentropic analysis," Pap. Phys. Oceanogr. Meteorol., Mass. Inst. Tech. and Woods Hole Oceanogr. Inst., vol. 6, no. 2, p. , 1938. [4] F. Schott and H. Stommel. "Beta Spiral and absolute velocities in different oceans," Deep-Sea Research, vol. 25. pp. 961-1010, A p r i l 1978. [5] G. Dietrich, General 588pp. Oceanography. New York, N.Y.: J. Wiley and Sons, 1963. [6] D. J. Olbers, M. Wenzel. and J. Willebrand, "The inference of North Atlantic circulation patterns from climatolgical hydographic data," Reviews of Geophysics, vol. 23, pp. 313-356, November 1985. [7] C. Wunsch and J. Minster, "Method for box models and ocean circulation tracers: mathematical programing and nonlinear inverse theory." Journal of Geophysical Research, vol. 87, pp. 5647-5662, July 1982. [8] C. W'unsch, "An eclectic Atlantic Ocean circulation model, part 1: the meridional flux of heat," Journal of Physical Oceanograhy. vol. 14, pp. 1712-1733, Nov. 1984. [9] S. Tabata, "The general circulation of the Pacific Ocean and a brief account of the oceanographic structure of the North Pacific Ocean, part i - circulation and volume transports," Atmosphere, vol. 13, no. 4, p. , 1975. [10] S. Tabata, L. A. F. Spearing, R. H. Bigham, B. G. Mmkley, J . love, D. Yelland, J . Linguanti, and P. M. Kimber, "STP/hydrographic observations along line P, Station P, line R and associated lines i n the "Ocean Storms" area: cruise I 22 119 Sept. - 16 Oct. 1987, cruise III 24 Nov. - 9 Dec. 1987.,'' Candian Hydrography and Ocean Science, Data Report of vol. 70, p. , 1988. [11] S. Tabata, "An evaluation of the quality of sea surface temperatures and salinities measured at station P and line P in the northeast Pacific Ocean." Journal of Physical Oceanography, vol. 8, pp. 970-986, 1978. [12] N. P. Fofonoff and R. C. M. Jr., "Alogorithms for computation of fundamental properties of seawater," Unesco Technical Papers in Marine Science, vol. , no. 44. p. , 1983. [13] S. Tabata and R. Thomson, "Barochnic oscillations of the tidal frequency at Ocean Weather Station P," Atmosphere-Ocean, vol. 20, no. 2, pp. 242-257 1982. r [14] S. Tabata, "Variability of oceanographic conditions at Ocean Station 'P i n the : northeast Pacific Ocean," Transactions of the Royal Society of Canada, vol. 3, pp. 367-418, June 1965. [15] S. Tabata and N. P. Fofonoff, "Variability of oceanographic conditions between Ocean Station P and Swiftsure Bank off the pacific coast of Canada.." Journal of Fish Research Board of Canada, vol. 23, no. 6, pp. 825-868. 1966. [16] W. G. Large, J. C. M c W i l l i a m s , and P. P. Niiler, "Upper ocean thermal response to strong autumnal forcing of the northeast Pacific," Journal of Physical Oceanography. vol. 16, p. , Sept. 1986. [17] S. Tabata, "Temporal changes of salinity, temperature and dissolved oxygen content of the water at Station 'P' i n the northeast Pacific, and some of their determining factors," Journal of Fish Research Board of Canada, vol. 18, no. 6, pp. 1073-1124, 1961. [18] C. Wunsch, D. Hu, and B. Grant, "Mass, heat, salt, and nutrient fluxes in the South Pacific Ocean," Journal of Physical Oceanograhy, vol. 13. pp. 725-753, 1983. [19] E. Tziperman and A. Hecht, "Circulation in the eastern Levantine Basin determined by inverse method," Journal of Physical Oceanography, vol. 18, pp. 506-518, March 1988. [20] R. A. Wiggins, "The general linear problem: impbcation of surface waves and free oscillations for earth structure," Reviews 120 of Geophysics and Space Physics, vol. 10, pp. 251-285, Febuary 1972. [21] C. L. Lawson and R. J . Hanson, Solving N.J.: Prentice-Hall, 1974. 1054pp. [22] W. Menke, Geophysical Data Analysis, Least Squares Problem. Discrete Inverse Theory. Englewood Cbff. Orlando Florida: Academic Press Inc., 1984. 260pp. [23] B. Bobn, A. Bjorkstrom, K. Holmen, and B. Moore, "On inverse methods for the chemical and physical oceanographic data: a steady-state analysis of the atlantic ocean.," Tech. Rep. C M 71, Department of Meterology, University of Stockholm. August 1987. [24] E. D'Assaro. "Ocean Storms update," Dec. 1988. Applied Physics Laboratory. University of Washington, Seattle Washington. [25] R. A. Weller and D. Halpern, "The velocity structure of the upper ocean i n the presence of surface forcing and mesoscale oceanic eddies," Philosophical Transactions of the Royal Society of London, vol. 308, no. A, pp. 327-340, 1983. 121
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Circulation of the Northeast Pacific Ocean inferred...
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
Circulation of the Northeast Pacific Ocean inferred from temperature and salinity data Matear, Richard James 1989
pdf
Page Metadata
Item Metadata
Title | Circulation of the Northeast Pacific Ocean inferred from temperature and salinity data |
Creator |
Matear, Richard James |
Publisher | University of British Columbia |
Date Issued | 1989 |
Description | The temperature, salinity, and pressure (STP) data were collected during two cruises, one in early October and the other in early December of 1987, as part of the Ocean Storms experiment. These hydrographic data were analyzed to determine circulation of the northeast Pacific Ocean and to calculate the factors influencing the heat and salt content of the upper ocean. From the depth profiles of the temperature and salinity data, the water mass in the Ocean Storms area was classified as being the Eastern Sub-Arctic Pacific Water Mass. Maps of dynamic height for this area revealed that the current pattern was generally smooth but that mesoscale eddies did exist in the flow. Isentropic analysis of the temperature and salinity fields produced a flow pattern that was generally consistent with the. dynamic height maps. However, this analysis revealed additional details in the flow that were not evident in the dynamic height maps. To extract additional information from the temperature and salinity data an inverse model was developed. This model assumed that the flow was geostrophic and that the vertical velocity satisfied a linear β-plane vorticity equation. This inverse model calculated the vertical and horizontal velocities at a reference level of 1000 dbars, and the horizontal and vertical mixing terms by conserving mass, salt, and heat. These conservation constraints were applied to large- boxes defined by four hydrographic stations and two pressure surfaces. The circulation determined using the model showed well-defined flow features. Comparison of the absolute geostrophic flow with the seven-day averaged current meter observations showed much similarity, despite complications from an incident storm. Correlation between the geostrophic flow at the surface and the total flow field inferred from drifter data was also high. An estimate of the ageostrophic flow suggested that acceleration and nonlinear terms played an important role in affecting the flow field during the first cruise. The observed change in salt content of the upper 150 m of the ocean was .004 ppt for the sixty days between the two cruises. The net transport of salt into the study area by the calculated flow field was -.016 ppt. Therefore, to balance the salt budget would require E - P = 9 cm. The upper ocean lost 92 W/m² of heat during the sixty days between the two cruises. As the vertical and horizontal transport of heat acounted for 40 W/m² loss of heat, the remainder of heat lost, 52 W/m² was attributed to air-sea boundary processes. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-08-21 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0052972 |
URI | http://hdl.handle.net/2429/27594 |
Degree |
Master of Science - MSc |
Program |
Geophysics |
Affiliation |
Science, Faculty of Earth, Ocean and Atmospheric Sciences, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
Download
- Media
- 831-UBC_1989_A6_7 M38.pdf [ 5.41MB ]
- Metadata
- JSON: 831-1.0052972.json
- JSON-LD: 831-1.0052972-ld.json
- RDF/XML (Pretty): 831-1.0052972-rdf.xml
- RDF/JSON: 831-1.0052972-rdf.json
- Turtle: 831-1.0052972-turtle.txt
- N-Triples: 831-1.0052972-rdf-ntriples.txt
- Original Record: 831-1.0052972-source.json
- Full Text
- 831-1.0052972-fulltext.txt
- Citation
- 831-1.0052972.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0052972/manifest