TIDAL RESIDUAL CIRCULATION OVER A N SEAMOUNT: AXISYMMETRIC SEAMOUNT GEOMETRY EFFECTS By Jennifer A . Shore B . Math. (Applied Mathematics) University of Waterloo, 1991 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES OCEANOGRAPHY We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA O c t o b e r 1996 © Jennifer A . Shore, 1996 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of c3CeAr/o G£P)PI+H' The University of British Columbia Vancouver, Canada Date DE-6 (2/88) ? / /4L Abstract A numerical tidal model is used to investigate the effect of seamount geometry on tidally rectified residual flow and the resultant particle behaviour. A weakly nonlinear analytical solution of barotropic oscillatory flow over a tall, axisymmetric, isolated seamount is derived and compared to a numerical tidal model solution. The comparison is formulated for a flat topped seamount with parabolic sides and with a standing wave boundary condition that makes the problem analytically tractable and reduces it to a series of ordinary differential equations. Based on the model comparisons, it is concluded that the numerical model gives reasonable solutions for tall topographies. Comparisons to a Gaussian seamount and Kelvin wave forcing illustrate the generality of the results. Residual velocities axe found to be proportional to the aspect ratio of the topography showing that for cases where the deep water depth does not change, wider seamounts will tend to have weaker residuals. Numerical experiments are conducted to investigate the effect of changes in seamount base width, slope and height on the tidal residual field and particle behaviour. Consistent with the analytic solution, results show that as the seamount widens (with its percentage depth remaining unchanged) the magnitude of the maximum residual velocity decreases. Unlike previous results of other researchers, it was found that an increase in slope does not necessarily generate a strengthening of the maximum residual. Results from the particle tracking studies show, in part, that trapping is more sensitive to changes in distances to the shelfbreak than to changes in the base width. Furthermore, retention times are enhanced, more by increases in the flank slope than by increases in base width, or seamount height. ii Table of Contents Abstract ii List of Tables List of Figures vi vii List of Parameters xi Acknowledgements xiii 1 Introduction 1 1.1 Origin of seamounts 1 1.2 Recirculating flows 2 1.3 Motivation 6 2 The numerical model 11 2.1 The Finite Element Method 11 2.2 The numerical model grid 17 2.3 Energy scattering in Finite Element grids 20 3 Analytical and Numerical Parabolic Seamount Comparison 3.1 27 The analytical model 27 3.1.1 Formulation 27 3.1.1.1 Governing equations 27 3.1.1.2 Boundary conditions 30 3.1.1.3 Form of the solution 31 iii 3.1.1.4 3.1.2 3.2 4 32 Analytical solutions 37 3.1.2.1 Zeroth order solutions 37 3.1.2.2 First order solutions with bottom friction 40 3.1.2.3 First order solutions without bottom friction 43 Comparison of model results 45 3.2.1 Zeroth order results comparison 47 3.2.1.1 Zeroth order components without bottom friction . . . . 47 3.2.1.2 Zeroth order components with bottom friction 51 3.2.2 3.3 The weakly nonlinear equations First order results comparison 51 3.2.2.1 First order harmonic components 51 3.2.2.2 First order residual components 54 Summary and discussion 59 Residual Velocity Comparisons 61 4.1 Comparison to Cobb Seamount 62 4.2 Kelvin wave to standing wave boundary condition comparison 67 4.3 Effect of seamount geometry 68 4.3.1 Set I: varying seamount base width 70 4.3.2 Set II: varying flank slope 75 4.3.3 Set III: varying seamount height 83 4.4 Summary 87 5 Tracking of inert and biological particles 90 5.1 Inert particle tracking in two-dimensions 5.2 Creating a 3D velocity field using a 2D velocity field Ill 5.3 Inert particle tracking in three-dimensions 116 iv 91 5.4 Tracking of swimming particles in three-dimensions 134 5.5 Summary 138 6 Conclusions 145 6.1 Analytical and numerical model results 145 6.2 Model validity 147 6.3 Residual flow relationships to seamount geometry 148 6.4 Particle tracking 149 6.5 Epilogue 150 A The Standing Wave Velocity Field 161 B The Degeneracy in the Weakly Nonlinear Expansion 163 C Calculating Parameters at Cobb Seamount 168 D Calculating Energy Density for the Boundary Conditions 171 v List of Tables 1.1 Selected studies of topographically affected flow 1.2 Summary of flow characteristics 10 2.1 Total number of elements and nodes 23 2.2 Maximum and minimum values for residual velocities 24 2.3 Maximum values for real part of residual velocities 25 3.1 Parameter values 46 4.1 Parameter value comparison: plane parabolic, Gaussian and Cobb 67 4.2 9 Some recent studies which explore parameter relationships for flow over isolated topography 71 4.3 Energy integrand values for Set I: varying 74 4.4 Energy integrand values for Set III: varying 8 88 4.5 88 Summary table of physical parameters for Sets I, II, and III 5.1 Mean Stokes and Lagrangian flow 93 5.2 Percentages of trapped drogues for Set I and II 100 5.3 Percentages of trapped drogues for Set III. 106 5.4 Maximum residual velocity estimates over a seamount 114 5.5 Percentages of trapped drogues for an initial release depth of 50 m. . . . 117 5.6 Migration time dependence 136 vi List of Figures 2.1 Comparison of a Finite Difference grid versus a Finite Element grid. . . . 13 2.2 Original domain of Foreman et al. (1992) 18 2.3 Finite element grid with discontinuous node spacing 19 2.4 Example one-dimensional node spacing function 20 2.5 Node positions of Grid I over a 50 km by 50 km square region 22 2.6 Grid spacing functions for first and third test cases 23 2.7 Finite element grid with linear node spacing 26 3.1 Topographical features 28 3.2 Plane parabolic topography used in the semi-analytic solution 38 3.3 Analytical linear sea height solution and enhancement 41 3.4 Top view of seamount 46 3.5 Linear sea height solutions without bottom friction effects 48 3.6 Linear Cartesian velocity components without bottom friction 49 3.7 Calculated seamount flank slopes and the relative error 50 3.8 Linear Cartesian velocity components with bottom friction effects 52 3.9 Harmonic Sea height solutions 53 3.10 Harmonic Cartesian velocity solutions 54 3.11 Residual Sea height solutions with K = 0.003 55 3.12 Residual Cartesian u velocity solutions with K = 0.003 56 3.13 Residual Cartesian u velocity solutions 59 4.1 Comparison of averaged Cobb Seamount bathymetry to a Gaussian profile. 62 vii 4.2 Linear sea height solutions: Gaussian versus plane parabolic 63 4.3 Linear Cartesian u velocity solutions: Gaussian versus plane parabolic. 4.4 Residual u velocity solutions: Gaussian versus plane parabolic 4.5 Residual u velocity solutions: Kelvin versus standing wave boundary con- . 64 65 dition 69 4.6 Seamount Profiles for residual velocity comparison for Set I: Varying r^. . 70 4.7 Residual velocity comparisons for Set I: Varying 73 4.8 Seamount Profiles for residual velocity comparison for Set II: Varying Slope. 75 4.9 Residual velocity comparison for Set II: Varying Slope 76 4.10 Seamount Profiles with decreasing slope 78 4.11 Residual velocity comparison for seamounts of decreasing slope 79 4.12 Advection term comparisons to residual velocity profiles 81 4.13 Residual velocity comparison for Set II: Varying Slope - superinertial forcing. 83 4.14 Seamount Profiles for residual velocity comparison for Set III: Varying 8. 84 4.15 Residual velocity comparisons for Set III: Varying 8 85 4.16 A log-log plot of the maximum residual velocities versus 8 86 4.17 Superinertial and subinertial residual velocity profiles 87 5.1 Particle paths for Seamount profile C: 30 day period 95 5.2 Particle paths for Seamount profiles B and C: 93.15 day period 96 5.3 Particle paths for Seamount profiles D and E: 93.15 day period 97 5.4 Particle paths for Seamount profiles G and H: 93.15 day period 98 5.5 Particle paths for Seamount profiles A and F: 93.15 day period 99 5.6 Definition of the residence time T b of a water parcel 102 5.7 Scaled residence time versus distance to the shelfbreak 103 5.8 Scaled residence time versus slope 104 a viii 5.9 Scaled residence time of particles across the background flow 105 5.10 Scaled residence time versus changes in fractional seamount height. . . . 107 5.11 Contour plot of retention time for seamount Profile A 109 5.12 Particles released at different times in the tidal phase 110 5.13 Horizontal vector flow fields over an axisymmetric seamount 118 5.14 Qualitative sketch of the vertical residual flow structure over a tall seamount. 119 5.15 Vertical residual flow field from Haidvogel et al. (1993) 120 5.16 Residual vertical flow field construct 121 5.17 Qualitative sketch of superposed 2D and vertical residual flow fields. . . . 122 5.18 Particle pathlines for different downwelling magnitudes - Horizontal View 126 5.19 Three-dimensional particle pathlines for different downwelling magnitudes. 127 5.20 Vertical particle excursions in depth over time (total time is 93.15 days) . 128 5.21 Vertical particle positions versus radial distance 129 5.22 Final vertical positions after 30 days of particles released from initial depths of 0 (O), 50 (+) and 400 m (•) 130 5.23 Pathline of a particle released near the seamount base 131 5.24 Pathline of a particle released at the seamount peak 132 5.25 Pathline of a particle released near the seamount shelfbreak 133 5.26 Pathlines of swimming and inert particles released near the seamount summit 140 5.27 Two week long pathlines with and without tidal oscillations 141 5.28 Two week long pathlines of swimming and inert particles released near the seamount summit 142 5.29 Two week long pathlines of swimming and inert particles released away from the seamount summit 143 ix 5.30 Two week long pathlines of swimming and inert particles released over the A.l seamount flank 144 Clockwise rotating velocity vector 162 x List of Parameters Parameter x,y,z r, 8, z Description Page Cartesian co-ordinates or particle position 45 Cylindrical-polar co-ordinates 28 t Time 29 rj Surface height 28 u Velocity, {u, v,w) h Fluid depth 28 / The Coriolis parameter 28 g Gravitational acceleration 28 K Bottom friction coefficient 28 rd Minimum radius to deep water 29 rb Boundary condition radius, hj. The deep water depth 29 77b Sea height at r& 29 UQ Characteristic velocity 29 T Wave period 29 <jj Wave frequency 29 a Temporal Rossby number 29 e Rossby number 29 Rr Rossby radius 29 a Scaled Rossby radius 29 xi 28 > 29 01 A clockwise phase 32 02 A counter-clockwise phase 32 Linear clockwise phase propagating terms 33 ',"0. Vo) Linear counter-clockwise phase propagating terms —* 33 Sea height in the absence of topography 39 Local vorticity 44 Fractional seamount height 66 A height function defining topography 72 Shelfbreak 80 Initial particle position, (x ,yo, z ) 91 Stokes flow 92 u Mean Stokes flow 92 U~L Lagrangian residual 92 u Eulerian residual 92 Tb Residence time 102 T Advective time scale 102 D Seamount base diameter 102 Mean background flow magnitude 102 Tidal excursion 104 Keulegan-Carpenter number 104 Burger number 115 a us s r a S 0 Xll 0 Acknowledgements First and foremost I would like to thank my supervisor, Susan Allen, for the support and encouragement she gave me during this endeavour. I am especially grateful for the extra time and effort she made when the project experienced technical difficulties. I am also thankful for the insightful recommendations and comments from the members of my committee - Mike Foreman, William Hsieh, A l Lewis, and Brian Seymour - all of whom managed to attend my many committee meetings even when their schedules were overtaxed. I would like to acknowledge the efforts of the administrative staff - Chris Mewis, Carol Leven and Olive Lau - who took pity on me when I was encumbered in red tape. I also received extensive help and instruction from the system gurus Denis Laplante and Joseph Tarn. I am grateful to NSERC who supported me throughout my undergraduate and graduate studies. I am indebted to Scott Tinis who was Han Solo to my Luke Skywalker and to Sally Tinis who graciously accepted this. I would like to thank Roger Pieters and Hugh MacLean for providing interesting conversation at work, and also the regular crew of the annex, past and present, who all contributed to my life there. I would like to thank Mike McAllister, Nancy Day and Scott Flinn who gave me extra-curricular lessons on loyalty, friendship and wisdom. I am also grateful for the mischievous distractions thoughtfully provided by Ken Thomson. I am thankful for all the people who made the other part of my life, especially the sporting activities, enjoyable. There are others without whom I would not have gone on to a graduate program and they also deserve my heartfelt thanks: Mans Haider, Burke Pond and John Stockie. And finally, thanks go to my family who bought me food when I couldn't afford to buy my own and never rented out my room at home. Xlll Chapter 1 Introduction 1.1 Origin of seamounts Characteristically, seamounts are submerged extinct volcanos. They are bathymetric features of the ocean floor and can rise up to heights of 3000 m and more 1990; GARRISON, (THURMAN, 1996). In general, a topographic feature is classified as a seamount if its local relief is greater than 1000 m and it has a slope of more than 5 degrees. Typical slopes are not more than 25 degrees (KING, 1975; SEIBOLD and BERGER, 1993). Seamounts tend to have somewhat sharp tops; relatively older seamounts have flatter tops due to wave action and are often referred to as guyots. It is estimated that there are over 20 000 seamounts world-wide with the greatest percentage (over 50 %) occurring in the Pacific Ocean (KING, 1975; SEIBOLD and BERGER, 1993). Volcanic activity in the ocean is usually associated with zones of sea floor spreading. A volcano can form as part of the underwater ridge associated with the central active spreading area. In time, as the volcano grows in size, seafloor spreading will move it away from the central region where it may submerge to become a seamount. It can take up to 10 million years for a seamount to be created and a further 40 million years of weathering before it is classified as a guyot (THURMAN, 1990; GARRISON, 1996). There are other mechanisms by which seamounts arise. An active volcano can form in an oceanic plate over a hot spot (a stationary source of heat in the upper mantle) which, as the plate continues to move, can extinguish and subside under the water, creating 1 Chapter 1. Introduction 2 a seamount. The Emperor Seamount Chain in the North Pacific was created in this manner. Another less common mechanism occurs at subduction zones where an oceanic plate subducts beneath a less dense continental plate. If the continental plate has a frontal margin below sea level, volcanic activity can form an island arc in this margin, part of which may, in time, extinguish leaving a chain of seamounts G A R R I S O N , 1996). (THURMAN, 1990; Further information on the geomorphology of seamounts and other features of the ocean floor can be found in introductory texts on marine geology (e.g. KING, 1975; 1.2 SEIBOLD and BERGER, 1993). R e c i r c u l a t i n g flows In general, anticyclonic (clockwise in the Northern Hemisphere) time mean flows are expected to be found within the seawater above seamounts and there are a few dynamically different mechanisms that can generate this flow type. For example, to conserve potential vorticity in a steady inflow, a column of water will gain negative local vorticity as it is advected up over the seamount. This is referred to as Taylor column formation and results in anticyclonic currents. Another mechanism is the resonant excitation of topographic Rossby waves by subinertial (wave frequency is less than the inertial frequency at that latitude) tidal flow over a seamount. Topographic waves travel with the shallower depth on the right in the Northern Hemisphere; hence, these waves are trapped to the seamount and will propagate clockwise around it. The resultant Stokes drift (the small mean flow due to wave nonlinearity) of these waves is also anticyclonic. Another mechanism, tidally rectified flow, is not limited to the subinertial regime as is the last case. Nonlinear tide - tide interaction transfers energy to other tidal frequencies (these other frequencies will be harmonics of the tidal frequency and include the zero, or mean, frequency) and, on average, advection of vorticity by the tidal stream leaves a negative residual vorticity Chapter 1. Introduction 3 on the top of the seamount. Again, this negative residual vorticity leads to anticyclonic mean flows. There have been many studies focussing on the different aspects of these mechanisms. In 1923, Taylor was able to initiate Taylor columns in a laboratory setting, building on the theory of stationary fluid columns over topography developed separately by himself and Proudman ( P R O U D M A N , 1916; T A Y L O R , 1917; T A Y L O R , 1923). Many studies on Taylor columns have followed, with the addition of other dynamical effects such as stratification ( H O G G , 1973), the /3-plane effect ( M C C A R T N E Y , 1975) and sheared flows (JOHNSON, 1983). These studies were also generalized to study the initialization of these columns (some examples are H U P P E R T and B R Y A N , 1976; S M I T H , 1992; C H A P M A N and HAIDVOGEL, 1992; T H O M P S O N and F L I E R L , 1993). The focus of Chapman and Haid- vogel (1992) was not only on the Taylor column initialization problem but also on the effect of stratification on the Taylor column and subsequent particle trapping. In 1970, Longuet-Higgins presented a linear theory that suggested subinertial, anticyclonic topographic Rossby waves could be resonantly excited about an axisymmetric island by an oscillating current. Brink (1989) examined the effects of stratification on these waves and found that strong stratification forces the resonant waves to be bottom trapped to the seamount. Anomalous enhancement of the subinertial tidal component was reported in velocity measurements taken at Rockall Bank (northwest of the Irish coast) and the Yermak Plateau (in the Arctic Ocean) ( H U T H N A N C E , 1974; H U N K I N S , 1986). In an attempt to explain these observations, Chapman (1989) examined this tidal resonance mechanism more closely. He found that an exact match of the forcing frequency to the resonant wave mode frequency was not required for significant enhancement. Further work has since been done expanding the analytical theory behind this generating mechanism ( B R I N K , 1990; S H E N , 1992). Quasi-geostrophic (small topography, small velocity) effects have been looked at both analytically and in the laboratory Chapter 1. Introduction (VERRON, 1986; BOYER 4 ET AL, 1991) as well. Haidvogel et al. (1993) present results of a three-dimensional study of oscillatory flow over tall topography. They examine resonant generation of subinertial trapped waves, map out the effect of stratification and frequency dependence on wave enhancement and focus on the residual flow field dynamics. The dynamics of topographic rectification of tidal currents have been established by other researchers and it has been shown that rectification, which involves the nonlinear advection of vorticity by tidal currents, generates residual or time independent currents whose existence may form a basis for Lagrangian eddy formation (e.g. Z I M M E R M A N , 1978; HUTHNANCE, 1981; YOUNG, 1982; FOREMAN ET AL, 1992). Wright and Loder (1985, 1988) extended the theory into the vertical regime and determined solution responses to time and space variations in the bottom friction formulation. Boyer and Zhang (1990) examine particle behaviour in the lab for a range of inflow magnitudes in the case of an oscillatory flow past isolated topography and determine three different characteristic flow behaviour regimes including situations where eddies were found above the seamount (fully attached flow) and trapped behind it (attached lee side eddies). These laboratory studies have been expanded upon to examine these flow regimes in more detail ET AL, 1991; ZHANG and BOYER, (BOYER 1993). Other studies have looked at tidally rectified flow over realistic topographies. Numerical simulations of tidal rectification have been been developed for both the Gulf of Maine and Georges Bank regions. This east coast area has been the focus of an extensive number of studies; Lynch and Naimie (1993) present a chronicle of that development. Building on this body of work, they then used a finite element barotropic model of the entire Gulf of Maine-Georges Bank region to present 3D M 2 tidal and residual velocities. Subsequently, Naimie et al., (1994) examined the seasonal variations in the residual signal on Georges Bank, while Chen et al., (1995) explored stratification effects. Specifically, Chen et al., (1995) found that stratification could intensify tidal rectification and mixing. Chapter 1. Introduction 5 A number of numerical studies have also looked at simple tall topographies. For example, Garreau and Maze (1992) examine the mechanism Unking maximum tidal residuals to the shelfbreak. Chen and Beardsley (1995) study the effect of stratification, and seamount height and slope on the residual velocity structure and tidal mixing in a moderately nonlinear regime. Goldner and Chapman (1996) look at particle retention in a stratified ocean for a mean flow combined with weak subinertial tidal forcing. They found that retention was enhanced near the bottom by the bottom trapped subinertial waves. Table 1.1 shows how a number of these studies can be classified by the Rossby number (R ), percentage of the water column that the seamount occupies (p) and by the wave 0 frequencies used. These parameters can be used to distinguish between characteristic flow types; separating the strongly nonlinear, moderately nonlinear and weakly nonlinear regimes which, in turn, characterize expected Taylor column formation from steady flow and eddy generation from oscillatory flow. Values used to distinguish between flow regimes were chosen based on comments made in Boyer et al. (1991), Chapman and Haidvogel (1992) and Lynch and Naimie (1993). Table 1.2 is a summary of results drawn from the studies shown in Table 1.1. For steady inflows over tall topographies, Taylor columns are expected to form. However, as the strength of the inflow increases, the Taylor column will decrease in width until, for strong nonlinearities, it is swept downstream by the inflow. Lee waves also increase in strength with nonlinearity and have been shown to decrease the potential trapping of particles. In a tidal or oscillatory flow, tidal rectification is expected. At subinertial frequencies, resonant trapped waves are expected to dominate and enhance the subinertial response over the seamount. At superinertial frequencies, rectification is dominant and increases with increasing nonlinearity. Laboratory studies have shown that eddy behaviour moves from trapped lee eddies to eddy shedding as the regime changes from weakly to moderately nonlinear. 6 Chapter 1. Introduction Note that there is a history of quasi-geostrophic studies (R < 0.1, p < 10 %) which 0 has not been included here so that the emphasis remains on finite height studies. One could infer that because subinertial enhancement has been shown to exist for both the weakly nonlinear and strongly nonlinear regimes, that it will also be present in the moderately nonlinear regime. However, none of the studies in Table 1.1 examine the problem in that particular regime and, therefore, that conclusion is not included in Table 1.2. Field experiments have also provided valuable data useful for validating the physical models. Measurements of physical data have been collected at locations such as Cobb Seamount, off the Washington State coast, ( D O W E R et al, 1992; F R E E L A N D , 1994) and Fieberling Guyot in the eastern North Pacific (e.g. E R I K S E N , 1991; B R I N K , 1995; K U N Z E and T O O L E , 1996). Researchers have also collected biological information over Minamikasuga Seamount in the northwest Pacific ( G E N I N and B O E H L E R T , 1985) and over a shelf region in the Southern Ocean ( A T K I N S O N et al, 1992). 1.3 Motivation In general, this study falls within the larger scope of investigating Lagrangian eddy formation over a tall, isolated seamount. Seamounts have been observed to be regions of high productivity ( K E A T I N G et al, 1987) and the existence of a Lagrangian eddy in the flow field can increase local particle residence times. Therefore, these eddies may have implications towards plankton dispersal and survival questions, as well as other questions involving pollutant and nutrient transport. Specifically, the goal of this project is to determine, using a finite element tidal model, how mean (tidal residual)flowsand particle trapping are affected by the geometry of an axisymmetric seamount. The flow regime falls within the classes Weakly Nonlinear R < 0.1, p > 70 %, One Frequency and a Chapter 1. Introduction 7 Combined Frequencies listed in Table 1.2. There are a number of papers in Table 1.1 that fall within this classification and they can be further separated into groups. A few of the papers specifically focus on the subinertial regime and the subinertial resonant response. As has been previously mentioned, the laboratory experiments of Boyer and Zhang (1990) delineated different eddy regimes (fully attached flow, lee side eddies, and eddy shedding) within this subinertial regime. Further, in a stratified system, Goldner and Chapman (1996) found that particle retention was enhanced near the bottom by the bottom trapped subinertial waves. Two companion papers focussed on the effect of subinertial flow around a tall isolated seamount. The first (BECKMANN and HAIDVOGEL, 1993) formulated the problem and investigated the accuracy of a sigma-coordinate, semispectral model for axisymmetric tall topographies. The second ( H A I D V O G E L et al., 1993) focussed on the effect of stratification and frequency dependence on wave enhancement. Using frequencies less than and greater than the inertial frequency, Boyer et al. (1991) examined particle paths over a cosine-squared seamount for various flow regimes. Particle behaviour was seen to include tidal oscillation loops whose size was critically dependent on a normalized tidal excursion parameter. A few of the papers do focus more on the superinertial response. Recall from Table 1.2 that tidal rectification is expected to dominate a superinertial flow regime as opposed to the enhanced subinertial response at lower frequencies. Garreau and Maze (1992) analytically examine the mechanism linking maximum tidal residuals to the shelfbreak. Their model employs a co-ordinate transformation which replaces the traditional harmonic analysis and correctly reproduces the Mi - S2 tidal interaction. Zhang and Boyer (1993) examined both superinertial and subinertial flow response for small Rossby number (less than 0.04) flow over a cosine-squared seamount. Their experiments determined that particle behaviour in the superinertial regime (for their range of Rossby numbers) could be classified as following tidal oscillation loops and effort was made to describe Chapter 1. Introduction 8 the flow details of this regime. In 1992, Shen derived linear analytic solutions to flow over a bump and compared them to a nonlinear model. The numerical model was then used to examine weakly (R = 0.1) and strongly (R = 0.5) nonlinear flow over a bump, 0 D examining tidal and mean flows and vorticity generation. Lagrangian particle tracking was also examined in these flow regimes and mixing action due to the bump was found to be highly complex with no simple parametrization evident. This study begins with an analytical extension of Shen's linear work into the weakly nonlinear regime. The analytical solutions to flow over finite-height topography are first compared to a numerical solution. The derivation of the analytic equations and the subsequent solutions can be used for comparison to other weakly-nonlinear flow solutions over finite height topography problems. The numerical model is then used to examine flow over topography for which no analytic solutions exist. The goal of this project is to determine the effects of seamount geometry on residual flow and particle behaviour, and tests are geared to give a more complete picture than has been previously reported. This work is presented as a series of chapters beginning with a brief outline of the finite element method and a description of the numerical tidal model (Chapter 2). Subsequently, the derivation of an analytical solution used to validate the numerical tidal model is presented in Chapter 3, as are results of the comparison of the analytical and numerical models. Following that, the relationship between seamount geometry and tidal residual flow is examined. The relevant parameters which define seamount geometry and the flow relationships to those parameters are presented in Chapter 4. A series of particle tracking results in two and three dimensions are presented in Chapter 5 and a summary of conclusions and discussion of the results is presented in Chapter 6. Chapter 1. Introduction Weakly Nonlinear R < 0.1 10 % < p < 70 % p > 70 % Chapman and Haidvogel, 1992 Smith, 1992 Thompson and Flierl, 1993 Thompson and Flierl, 1993 Beckmann and Haidvogel, 1993 Boyer et al., 1991 Boyer and Zhang, 1990 Garreau and Maze, 1992 Goldner and Chapman, 1996 Haidvogel et al., 1993 Shen,1992 Zhang and Boyer, 1993 Garreau and Maze, 1992 Goldner and Chapman, 1996 Naimie et a l , 1994 0 Steady One Frequency Combined Frequencies Moderately Nonlinear 0.1 < R < 0.2 10 % < p < 70 % p > 70 % Chapman and Haidvogel, 1992 Thompson and Flierl, 1993 Chen and Beardsley, 1995 Boyer et al., 1991 Boyer and Zhang, 1990 Chen and Beardsley, 1995 0 Steady One Frequency Combined Frequencies FuUy Nonlinear R > 0.2 10 % < p < 70 % p > 70 % Chapman and Haidvogel, 1992 Chapman and Haidvogel, 1992 Thompson and Flierl, 1993 Foreman et al., 1992 Shen, 1992 0 Steady One Frequency Combined Frequencies Foreman et al., 1992 Table 1.1: Selected studies of topographically affected flow. Chapter 1. Introduction Steady Weakly Nonlinear 10 % < p < 70 % If p > 30, a Taylor column is expected to form with some associated eddy shedding from flow initialization. One Frequency 10 R < 0.1 0 p > 70 % Taylor column formation. Tidal rectification dominates for superinertial flow, otherwise expect wave enhancement due to subinertial trapping. Attached lee side eddies are also possible. Weakly nonlinear regime allows the superposition of the above effects. Combined Frequencies Moderately Nonlinear 0.1 < R < 0.2 10 % < p < 70 % p > 70 % Possible small Taylor column Possible Taylor column formation. formation. Associated eddy Lee waves can destroy trapping. shedding. Spatial structure of the residual Tidal rectification evident. flow field does not change Eddy shedding also expected. as the seamount height changes. Q Steady One Frequency Combined Frequencies Steady One Frequency Combined Frequencies Fully Nonlinear 10 % < p < 70 % No real possibility for Taylor column formation. Eddy shedding dominates initialization such that there is no trapping. Eulerian residual acts counter to the Stokes drift, decreasing the Lagrangian residual (reduces trapping). Summation of average tidal contributions to Stokes drift is generally good to first order. R >0.2 o p > 70 % Possible Taylor column formation. Lee waves dominate and destroy trapping. Tidal rectification possible as is subinertial enhancement. Rectification is amplified by strong nonlinearity. Table 1.2: Summary of flow characteristics. Chapter 2 The numerical model 2.1 The Finite Element Method The Finite Element (FE) method is one of a number of methods that developed out of the need for approximate solutions to boundary value problems (BVPs). In general, the mathematical problems that scientists are now required to solve are too complex to allow closed form analytic solutions. With the advent of computers, this difficulty has been addressed through the extensive use of approximate numerical solutions. The F E method has a history which can be traced back and forth through the disciplines of applied mathematics, engineering, physics and computer science. Essentially, what began as an idea to solve BVPs by discretizing the solution domain into a set of triangular spatial elements, found its way into the aeronautical engineering field - used to find loads for shell-type structures with internal spars and struts - and was put to use by physicists studying elastic-continuum problems (HUEBNER and THORNTON, 1982). As the method became more prominent, mathematicians began fleshing out the groundwork of the theory, looking at error estimation and convergence rates. There have been numerous texts and papers written on the subject. Huebner and Thornton (1982) give a somewhat detailed history of the development of the F E method and the text serves as a good introduction to the topic. Becker et al. (1981) detail the computational implementation aspects of the technique. As well, there are introductions and reviews specific to oceanography (e.g. PlNDER and GRAY, 1977; LYNCH, 1983; NAVON, 1988; WESTERINK 11 Chapter 2. The numerical model 12 and GRAY, 1991). The F E method is only one of several numerical analysis methods and is most often compared to the finite difference (FD) method. Traditionally, the FD method involves overlaying the solution domain with a rectangular grid of points allowing derivatives in the differential equation(s) to be approximated with difference quotients at those points. This method gives a pointwise approximation to the solution because the discrete representation of the equation is applied at a set of points. Furthermore, it is assumed that accuracy of the solution can be increased by increasing the number of points in the solution domain. On the other hand, the F E method divides the solution domain up into a number of small spatial elements. This method gives a piecewise approximation to the solution. That is, the final solution is assembled from many small parts of the solution which are solved over these small elements. Accuracy is assumed to increase as the element size decreases. In fact, the F E method solves a weak statement of the problem over these elements which allows small pointwise errors (unlike the FD method) and, instead, focusses on reducing global errors (PlNDER and GRAY, 1977; HUEBNER and THORNTON, 1982). There are advantages and disadvantages to using the F E method instead of the FD method. One advantage is that while the FD method is limited to rectangular point grids, the F E method may make use of simpler shapes (such as triangles) which allows better representation of more complex boundary shapes. Consider the shape of a quarter circle. Fig. 2.1 shows how the two approximation methods may be applied to this shape (the solid line is the boundary of the quarter circle). Note that the F E method (Fig. 2.1b) does a much better job of mimicking the domain shape because it is made up of interconnecting triangular elements (the dashed lines). Points where these triangles connect are generally referred to as nodes (the O's) and in this example there are 3 nodes per element, one on each corner. In this example, the FD method uses a square grid domain with 83 points Chapter 2. The numerical model a) 13 30 25^ 20 15 0—-0—-0—-0—-0— 10 5 —0-—<j>—6>-—-0——-0—-<|>—-4> 0 -5 -10 -30 -25 -20 -15 -10 5 10 -30 -25 -20 -15 -10 -5 Figure 2.1: Comparison of a Finite Difference grid versus a Finite Element grid for a quarter circle domain. Solid line represents the boundary of the quarter circle. Dashed lines in b) represent the element boundaries. O's represent a) grid points and b) nodes. and the F E method uses 68 triangular elements (and 43 nodes). Note also that triangles within the domain are not necessarily similar or of the same size (they are not congruent). It is easy to imagine, based on this flexibility, that the F E method would be useful for coastal oceanography models where the coastline boundaries are very complex. On the other hand, there are also disadvantages to using the F E method. Accuracy errors have been shown to occur for non-regular element shapes (e.g. for non-equilateral triangles) (FOREMAN, 1984) and to occur when the element size does not change smoothly in space (VlCHNEVETSKY and TURNER, 1991). A brief summary of the F E method is now presented. Consider an arbitrary problem on the domain D with boundary T which has a solution u. Let L be the differential operator and M the boundary condition operator. We can then mathematically state the problem as L(u) = f on D (2.1) Chapter 2. The numerical model 14 M(u) =g on T (2.2) where / and g are non-zero functions for a nonhomogeneous problem. Assume a finite series for an approximate form of the solution, u, such as m n e=l z=l where m is the number of elements in the domain, n is the number of nodes per element and c6,- are basis functions judiciously chosen such that the a; represent the value of u at the nodes. The accuracy and nature of the solution will depend on the number of elements used and choice of the basis functions. Since u is an approximate solution, it does not exactly solve the stated problem (2.1) and (2.2) and, therefore, domain and boundary residuals result R = L{u) - / / 0 d R = M(u) -g^O. b (2.4) (2.5) The F E method attempts to globally minimize these residuals in an average sense over the domain and this minimization can be achieved through a method of weighted residuals. The result is a weak statement of the problem: / (L{u) - f)WdD + / (M(u) - g)WdY -)• 0 JD JT (2.6) where W and W are arbitrary weighting functions and are added to generalize the problem (HUEBNER and THORNTON, 1982). The minimization is in an average sense because it is integrated over the domain. Galerkin's Method defines the weighting functions equivalent to the basis functions, </>,-. In fact, (2.6) isn't an exact representation of the F E method as it applies to (2.1) and (2.2). Some of the finer details of the F E method have been omitted in an attempt to give a general overview of the method. Having said Chapter 2. The numerical model 15 that, however, it is worthwhile to present the basis of the formulation of (2.6) because it can now be tied into the variational method by which, traditionally, the F E method has been proven valid. Equation (2.6) is a representation of a variational problem. The calculus of variations is an area of mathematics where problems involving the determination of maxima and minima of functionals are examined. Functionals are functions whose independent variables are functions. For example, a functional J may have an independent variable y which itself is a function of x. J could be written in a general form, similar to (2.6) without the weighting functions as J[y} = f F[x,y{x)^{x),..]dx Ja (2.7) OX where the goal would be to minimize J[y] over the domain (GELFAND and FOMIN, 1963). Traditionally, most of the proofs showing the validity of the F E method use the calculus of variations (HUEBNER and THORNTON, 1982; BECKER et al., 1981). The original problems to which variational principles were applied required the problems to be formulated in the classical variational sense. For example, an incompressible, inviscid fluid experiencing irrotational flow has a kinetic energy which is a minimum. Therefore, this problem can be formulated within a classical variational framework. Solutions can be found by minimizing the kinetic energy formulation using calculus of variations techniques. However, the development of (2.6) described above derives a similar (to (2.7)) variational formulation of the problem but begins with the differential equations (2.1) and (2.2) and does not require the problem to have a variational formulation. This is a useful alternative approach because it allows more flexibility in the types of problems to which the F E method can be applied (ie. those that have no classical variational principle) (HUEBNER and THORNTON, 1982). For the fluid problem studied in this thesis, it should be pointed out that there is Chapter 2. The numerical model 16 a certain amount of manipulation of the mathematical problem before a suitable differential operator (L in (2.1)) is arrived at. It is first assumed that this problem can be approached using the shallow water equations. Since the system being investigated is homogeneous (constant density and therefore incompressible) and assumed hydrostatic (vertical accelerations are assumed small compared to gravitational acceleration), using the shallow water equations is justifiable (details found in A har- P E D L O S K Y , 1979). monic analysis is first applied to the shallow water equations. That is, time dependence of the sea height and velocity variables is assumed to be of the form J2Pj{ >y) ~* x e Ujt which results in a system of elliptic equations. A major milestone in the history of the F E method as it applies to shallow water fluid problems, was the manipulation of these equations into the "wave equation" form shown by Lynch and Gray (1979). In this case, the equation takes a form similar to a Helmholtz equation: k rj + V77 = <f> 2 which replaces the continuity equation in the formulation (LlN and T E R S , 1995). (2.8) 2 S E G E L , 1988; WAL- Their formulation suppresses errors for short wavelength waves (until then, a common problem) without sacrificing accuracy at long wavelengths MYERS and WEAVER, 1995; ( F O R E M A N , 1984; WALTERS, 1995). The differential operator L is taken from this wave equation formulation. In this case, L is a nonlinear equation (due to advection and bottom friction terms in <j> of (2.8)) and, therefore, to solve this equation, the numerical method first approximates the final solution by getting a linear solution (using only the linear terms of L) and then iterates using nonlinear forcing (the rest of the terms in L) to converge to the nonlinear solution. The wave equation formulation decouples the sea height and velocity terms and, therefore, each iteration solves first for the free surface and then for the horizontal velocities. The boundary conditions are tidal amplitudes and phases specified at the Chapter 2. The numerical model 17 boundaries. 2.2 The numerical model grid In 1990, Foreman and Walters presented a finite element tidal model (hereafter referred to as the F W model) which, for our purposes, is used to study tidally rectified mean flows around a seamount. This section briefly describes the numerical grid used to calculate the F E numerical solution in Chapter 3. The FW model is a diagnostic (no time-stepping) model, based on a harmonic decomposition of the shallow water equations as discussed above and which therefore determines solutions at given frequencies (WALTERS, 1986). The original model used eight tidal constituents and two residual components. Residual components are treated as tidal constituents of zero frequency and the two can be differentiated in the numerical model by the dynamics governing their creation (e.g. buoyancy forces or rectification). The grid consists of triangular elements with linear basis functions and was created with Henry's (1988) grid generating package. The original grid covered the southwest coast of Vancouver Island from Estevan Point to just south of Cape Flattery and out past the 2000m depth contour, (Fig. 2.2). In this region there is no tall isolated axisymmetric topographical feature. Initial and boundary conditions are specified as surface height and phase values at each boundary node for each tidal constituent. Unfortunately, this type of boundary condition can lead to problems in that if the initial specified amplitudes and phases are not consistent with numerical propagation in the model domain, then reflected waves can be generated at the boundary (FOREMAN and WALTERS, 1990). For the original grid, these values had been supplied by a combination of field data and model predicted values. Chapter 2. The numerical model *VjJX \ VANCOUVER A> ISLAND 18 V" V -> Figure 2.2: Figure taken from Foreman et al. (1992). Geography and depth contours (in meters) to which the original model was applied. The thick dashed line represents the ocean boundaries of the model. For this study, a new grid was created with 3161 nodes and 6120 elements situated over a 2000 by 2000 square kilometer domain (Fig. 2.3 - similar to Fig. 2.1b, however it is shown without the nodes). Note that the grid is built with a pattern of concentric circles. In this way, the nodes are placed along topographical contours such that one side of each element is along a contour. It is then possible to exploit the feature of easy refinement of the grid in an area of rapidly changing topography to keep the change in depth across the other two sides of the triangular element to a minimum. The grid is Chapter 2. The numerical model 19 slightly asymmetrical to reduce a possible bias in the numerical wave solution (F. Henry, pers. comm.). Again, the grid was created with Henry's grid generating package. The seamount covers less than 6 % of the total domain (situated in the middle) and noise from boundary effects is minimal. Details of the model will be set to match the analytical model described in Chapter 3. I I— I— I— [— In — — I— I— I— I— I— |-r— i— i— I— I— i— i— i— i— I— [— i I I I I— I I I— I— |— I— I— i— iI IIII|III j I •••' -1000 ' ' -500 0 1 500 1000 Figure 2.3: Finite Element grid generated for this study. Boundary length scales are in km. Seamount is centered at the origin. A n increased resolution near the boundary is used to examine boundary noise effects. Chapter 2. The numerical model 2.3 20 Energy scattering in Finite Element grids Tidal interaction with topography results in rectified mean flows whose magnitudes are expected to be less than that of the tidal flow (ZIMMERMAN, 1978). In the same region, nonlinear interaction of the tidal wave with itself produces a (time independent) Stokes flow whose magnitude is expected to be at most the same order as the tidal residual (FOREMAN et al., 1992). In a numerical model, these residuals may be masked by energy reflection (scattering of the numerical wave solution) which is known to occur at discontinuities in the grid spacing (VlCHNEVETSKY and TURNER, 1991). In this section, Vichnevetsky and Turner's (1991) theory is applied to the F E grids used in the latter part of this thesis. Three different grids are tested (in conjunction with the F W finite element numerical tidal model) and maximum residual velocities are used to examine spurious noise effects. From this investigation, the grid that is found to generate the least amount of noise is used in the subsequent models for the seamount comparisons discussed in Chapters 4 and 5. A one-dimensional grid may be generated by defining the grid (or node) spacing, h, as a function of position, h = h(x). For example, h — x + 1 would result in the following node distribution: 1 < i i © © 1 1 1 o 01234567 I 11 o 11 " 1 X l Figure 2.4: Example one-dimensional node-spacing function. The shaded circles represent the nodes. Chapter 2. The numerical model 21 Vichnevetsky (1991) summarized some of the known facts about how and where scattering occurs in model grids. First, scattering is known to occur in piecewise uniform meshes at the discontinuities where grid spacing jumps from one section to the next. Further considering a continuous one-dimensional grid spacing function, h(x), Vichnevetsky shows that if the rate of change of this function with respect to position, dh/dx, is not continuous, then reflection will occur at those discontinuities. The F W numerical model is used to examine relative differences in the amount of energy scattering by examining residual velocities for three different grid test cases. Maximum values of velocity are taken as the measure by which it is possible to calculate these relative differences. Note that tests will be conducted with no topography, therefore, tidal rectification is not expected to occur and mean flows are interpreted to be a combination of spurious noise and the Stokes flow generated by the nonlinear tidal interaction. The Stokes flow in this case is expected to be very small, <9(10 cm/s) (LONGUET-HlGGINS, _1 1969), so it is assumed that a decrease in maximum mean velocities represents a decrease in energy scattering and an improvement in the numerical accuracy of the solution. The numerical model is used to solve for the superinertial oscillatory barotropic flow over a flat bottomed (2000 m depth), square domain (1000 km X 1000 km) which has a boundary condition that prescribes a Kelvin wave travelling through the domain. Three different grids are tested with this model. Each grid consists of triangular elements with linear basis functions and is created with Henry's (1988) grid generating program. The first grid was created as a series of concentric annular regions with each region consisting of a number of equally spaced rings. Nodes are then equally spaced around each ring (see Fig. 2.5 - the entire domain is not shown). It is possible to write the grid spacing function in this case as a function of r (in cylindrical co-ordinates). It appears as a series of flat (uniform) piecewise continuous parts separated by jump discontinuities (see Fig. 2.6) The second grid was generated by Chapter 2. The numerical model 22 25 20 o 0 0 0 0 15 0 0 0 © 0 0 o 0 © 0 0 > 6 o O c (0 0 ° A o , 0 A © © V. 0 °A o o o o o © © 0 0 - > 0o v v o 0 o o 1 o o © o © -10 o o © o 0 0 © 0 0 0 © © 0 © ^°A0A0» 60 0.A A©" o A © ©0 0© 0 0 0 © ©oooo© 0 © © 00©©©© 0 0 © ©©©©©© ©ooooo 0 © © ©©©* 0 © © 0©< *0 o © V © o o o > o o o o © © 0 0 o^o^o%°v/o%^<>©0©©00©0000 © 0 •0 o o o -5 » 0 0 0 0 0 0 - 0 o 0 0 © A o © V© °A © V A A° < >© © 5 0 *AAAA ^ A 10 0 0 0 o 0 0 © © 0 < © © • 0 0 c o © o o o © -15 A © A 0 ©o0 © A, ?0. 0©'©o ©© ©© o o o©©O0v©<»O?oi'o * o© 0 © o © 0 © © o 0 © o 0 0 0 0 © o o A 0 o © © © o > 0 0 ' © 0 0 * . o _ l _ _l_ _1_ _J_ v 0 c 0 -20 v -25 -25 -20 -15 -10 -5 0 5 Distance (km) 10 15 20 25 Figure 2.5: Position of nodes for Grid I within a 50 km by 50 km square region centred in the domain. Note the distinct annular regions and that node spacing changes from region to region. the same method but with fewer rings within the 0 - 9 km annular region, hence its grid spacing function is only slightly different than that of the first test case and is therefore not shown. A linear grid spacing as a function of radial position was chosen to create the third grid. Here, the function is infinitely differentiable except at the origin and the maximum magnitude of the spacing is similar to that of the original grid within the 0 - 3 0 km region (Fig. 2.6). The mesh resolution at the center of Grids I and III is 0.5 km for both cases. For reference, the total number of elements and nodes for each Grid are listed in Table 2.1 Chapter 2. The numerical model 120 23 Grid I: Discontinuous funtion Grid III: Linear function — — 100 80 [ c 60 V 40 [• 20 [• 0 50 0 100 150 200 250 300 350 400 450 Position (km) 500 Figure 2.6: Grid spacing functions for first and third test cases. The second test case is different from the first between 0 - 9 km only. Mesh resolution at the center of both grids is 0.5 km. where Grid I is the original grid of concentric annular regions, Grid II is similar to Grid I, but with less nodes in the 0 - 9 km region and Grid III is created with a linear variation in grid spacing in the radial direction. Maximum and minimum values for residual velocities are summarized in Table 2.2. Grid Number Total number of Elements Total number of nodes I II III 5380 4058 3800 2791 2130 1935 Table 2.1: Total number of elements and nodes for each test grid. From Table 2.2 it is evident from test cases II and III that results for maximum velocities, which are calculated over the entire domain, occur at the boundary (since the Chapter 2. The numerical model 24 Grid Number Domain (km x km) Position of Max I II III 1000 1000 1000 (7.43,-8.1) (-400,500) (-205.8,500) 6.0 2.41 1.80 0.0063 0.0059 0.0034 I II III 30 30 30 (7.43,-8.1) (1.4,-2.65) (7.33,-0.61) 6.0 1.26 0.95 0.0151 0.0206 0.0117 2 2 2 i*,y) 2 2 2 Max/Min (cm/s) Table 2.2: Maximum and minimum values for residual velocities. grid domain is -500 < y < 500 and -500 < x < 500). This may imply that boundary effects are overwhelming the effects of scattering which generally occur at the centre of the grid. A review paper of trends in mesh generation (BAKER, 1989) states that if a mesh expands in all directions, then a wave solution may be trapped within this region. An examination of velocity vector plots indicates that the majority of the energy in the solutions of Grids I, II and III does occur in the centre. Table 2.2 also summarizes maximum values calculated over a smaller area of 30 km x 30 km centred in the domain which excludes boundary condition effects. Note that Fig. 2.6 indicates that the element sizes for Grid I are smaller than those of Grid III within a 30 km radius. Recall from the previous discussion of the F E method in § 2.1, that smaller element sizes are expected to increase the solution accuracy. Therefore, one would expect better resolution of the residuals in this region using Grid I. This is apparently not the case, as is seen in the results tabulated in Table 2.2. Similar arguments can be applied to the results of Grid II where the number of elements were decreased relative to Grid I (hence their sizes increased) within the 0 - 9 km radius. Thus, it appears that energy reflection within Grid I may be contaminating the residual flow field. Chapter 2. The numerical model 25 Table 2.3 summarizes the maximum values of real residual velocities and the percentage decrease in these velocities with respect to Grid I. For comparison purposes, the first test case (Grid I) is taken to be a baseline noise level. The linearly stretched grid (Grid III) reduces maximum values within the grid by approximately 84 %. Grid Number Maximum (cm/s) Percentage decrease I II III 6.0 1.26 0.95 0.0 79 84.3 Table 2.3: Maximum values for real part of residual velocities and percentage decrease in maximum values with respect to Grid I. Thus, when using this particular finite element model to study residual velocities, it would appear that better solution accuracy can be achieved by using a grid spacing function which is continuous and has a rate of change with respect to position that is also continuous, consistent with the theory of Vichnevetsky and Turner (1991). Grid III is used for the numerical models in the Chapters 4 and 5 and is shown in Fig. 2.7. Chapter 2. The numerical model Figure 2.7: New finite element grid generated for this study. Boundary length scales in km. Seamount is centered at the origin. Chapter 3 Analytical and Numerical Parabolic Seamount Comparison As has been mentioned in Chapter 2, the FW finite element numerical model was not originally developed to study isolated finite height topography. In a quasi-geostrophic context, topography which is not finite height normally occupies less than 10 % of the maximum water column. The height of the seamounts used in this study do not occupy less than 70 % of the water column. The goal of this chapter is to validate the F W model for finite height topographies by comparison to an analytic solution. In fact, the solution is semi-analytic in that it is comprised of both closed form parts and parts which require the numerical solution of first order ordinary differential equations. The derivation of the analytic model with a weakly nonlinear constraint and the assumed form of the boundary conditions are presented and discussed in this chapter. 3.1 The analytical model 3.1.1 3.1.1.1 Formulation Governing equations The zeroth and first order equations for a weakly nonlinear perturbation of the shallow water equations over axisymmetric topography can be derived in the following manner. The vertically integrated shallow water equations can be written in full cylindrical polar co-ordinate form as: 27 Chapter 3. Analytical and Numerical ParaboHc Seamount Comparison du du (- u dt dr dv dv dt dr Sr? 1 d where 1 du v . dr) K uu 1—v — --fv +gr + — = 0 r d6 r or h + 77 1 dv uv 1 drj K\U\V r dv r r dO h + 77 . . 1d , . 2 L 7 r r / rj(r,6,t) the surface height u{r,e,t) the horizontal velocity (u,v) T n the fluid depth h(r) f the Coriolis parameter 9 gravitational acceleration and K 28 = the bottom friction coefficient. Wright and Loder (1988) studied the effect of different bottom friction formulations on topographic rectification of tidal currents. They concluded that spatial gradients in a nonlinear bottom friction term enhance the mean current more than the same in a linear formulation. A quadratic bottom friction term is used for this study, i) ii) \ Figure 3.1: Topographical features, i) Top view and ii) side view. Note that the shaded region represents the full horizontal extent of the topography. The boundary condition is applied at the radius defined by T V Chapter 3. Analytical and Numerical Parabolic Seamount Comparison 29 We now non-dimensionalize the equations with scalars pertinent to the topographic feature (Refer to Fig. 3.1 for a diagram of a topography with these scalars): r = TDT* where H = HJH* where HD rjbTf* where rjb = 77 = is the minimum radius to deep water is the deep water depth ^(^b) , ?"b is the radius at which the boundary condition is applied and r& > (u,v) t = Uo(u*,v*) where (7o is the characteristic velocity and = Tt* where T is the wave period = 2ir/ui; u> is the wave frequency. A relationship between rjb and UQ can be derived from a geostrophic balance: grjb/RD — fU . 0 Substituting into the equations of motion and dropping the *'s used to denote the dimensionless variables gives cr du , du 1 du v. dn 2 ,KRD\u\u. ] . — — + e « — + - — - - ) - + -L + e{-^ -j-) = 0 2TT ot or R 89 R or HD H a dv , dv 1 dv uv. ldri ,Krd\u\u. „ — ^- + e{u— + -v— + — + u + --L + {^-±l±-) =0 2ir Ot or r 06 r r 06 H H a drt a , d . , . d e. d , , d , „ v . 3.1 n v .„ 3.2) e D .„ „. 2 + Y,Tt where a e = cu/F 11 1+ w [hv]) + -r Tr «« { [r ]+ a? = ' W ) 0 (3 3) the temporal Rossby number = Uo/(FRD) the Rossby number = a T*'*" S/GHD/F = Rr/RD the Rossby radius and the scaled Rossby radius. From [(3.1)-(3.3)], it is apparent that, assuming NRD/HD is (at most) order 1, if the Rossby number becomes small (e <C 1) then the basic flow behaves linearly, as is expected. Assuming e <C 1 restricts flow solutions to a weakly nonlinear regime. Nonlinear effects, such as the mean tidal residual, may be important even if small in relative magnitude. Chapter 3. Analytical and Numerical Parabolic Seamount Comparison 30 Recall in Chapter 1 that, in general, a weakly nonlinear flow regime exists at Rossby numbers less than 0.1. Since the Rossby number is a ratio of advective to Coriolis forces, small Rossby numbers correspond to Coriolis dominated flows of which there are many geophysical examples at mesoscale (on the order of the Rossby radius) length scales and greater. Furthermore, while the weakly nonlinear regime is only a part of the entire class of flow types, using this assumption allows the formulation of semi-analytic solutions which can be used to compare to numerical solutions. Zhang and Boyer (1993) use the temporal Rossby number to classify six different flow regimes (cr e [0,2.5] and e e [0,0.04]) based on laboratory experiments of rotating, stratified, oscillatory flow over a cos seamount. For our study, the effect of a will be 2 examined in more detail in the section discussing the analytical solutions. 3.1.1.2 Boundary conditions A standing plane wave centered over the seamount is used as the boundary condition. This results in the collapse of the analytical model to a series of ordinary differential equations and gives the semi-analytic solution sought for verification. Other, more realistic, boundary conditions require solution of the full partial differential equations. The boundary condition can be written ^(rj,, 8, T) = sin(t5) cos(27r£). While this boundary condition may not represent realistic sea surface conditions, it does result in a velocity whose direction rotates in much the same way as that of a tidal current (i.e. traces out a current ellipse - see Appendix A) and it is this velocity field which governs the time mean residual field through the momentum equations. As will be shown later on (§ 3.1.2.1), there are 3 distinct domains used to describe the seamount. Matching conditions between these sections are that the sea height and its slope are continuous. In addition, note that boundary conditions must be specified such that components which are 6 dependent will vanish at the origin where 8 is indefinite. Chapter 3. Analytical and Numerical Parabolic Seamount Comparison 3.1.1.3 31 Form of the solution The form of the solution is assumed to consist of a series of ordered components based on the following frequency expansion. The zeroth order solutions are assumed to have clockwise and counter-clockwise rotating components and the first order effects are assumed to result from the interaction of the zeroth order solution with itself. That is, the first order solutions are forced by the nonlinear interaction of the zeroth order terms (this is intrinsic to the meaning of weakly nonlinear). For example, consider the zeroth order component's frequencies to be {cos(c/>i), cos(<^ )}, 2 where (f>i — 9-\-wt and c/> = 9—wt are clockwise and counter-clockwise phases. In this case, 2 advection of the zeroth order solution by itself implies that the first order components will have frequencies which result from all possible combinations of the zeroth order frequencies, having time dependencies of the form {1, cos(20), cos(2u;£), cos(2</>i), cos(2c/> )}. 2 Note that the first two time dependent forms {l,cos(2#)} will represent residual terms because they are independent of time, and that the last three frequencies , the "harmonic" frequencies, oscillate at twice the frequency of the zeroth order frequencies and are sometimes referred to as the overtide when modelling tidal flows. The assumed form of the solutions are based on this frequency expansion. Consider an expansion for sea height written in polar co-ordinates (3.4). Note that there are two extra order e terms (7716 and 7717) which account for bottom friction effects on the zeroth order terms and, further, that second order terms are also given which will be referred to later on in this formulation. The zeroth order clockwise and counter-clockwise components are superscripted with a -f and a — sign respectively. rj{r,e,t) = + + ij+(r)sm(0i)+f/o (r)sin(0 ) 2 e fonW + 7712(7") cos(20) + 7713(7-) cos(47r£) + r/i (r) cos(2</>i) 4 7715(7") cos(2</>2) + 77i (r) cos(</>i) + 7717(7-) cos(c/>2)] 6 Chapter 3. Analytical and Numerical Parabolic Seamount Comparison + e [ {r)+r, (r,e,t)} + 0(e ) 2 where <> /i (3.4) 3 V21 32 22 (f) = (9 + 2itt) a clockwise phase, 2 = [0 — 2irt) a counter-clockwise phase and e, the Rossby number, is assumed small. This regular expansion in e is written such that the zeroth order solutions will be solutions to the linear equations derived by Shen (1992) and are therefore sometimes referred to as the linear solutions. Substituting the assumed form of the solution into (3.1)-(3.3) results in systems of equations for the zeroth and first order terms. The weakly nonlinear expansion including the expansions for sea height and velocity are given in the next section, as are the differential equations. 3.1.1.4 T h e weakly nonlinear equations Consider regular expansions for sea height and velocity written in polar co-ordinates: rj{r,e,t) = i7+(r)sin(0i) + »7d(» ) (^2) , Bin + e[j7n(r) + 7712(r) cos(20) + r] (r) COS(4TT£) + rf {r) cos(2c/>i) + W ") 13 lA ( 02) + Vie{r) cos(0i) + 77i(r) cos(c/>)] COS 2 7 7 2 + e [r, (r) + ri (r,6,t)} + 0(e ) 2 (3.5) s 21 22 u(r,9,t) = -u+(r)cos(0i) - u~(r) cos(c/>) 2 + e[«n(r) + u {r) sin(20) + u (r) sin(47ri) + u (r) sin(2c/>i) 12 13 14 + ui (r) sin(2c/>) + u (r) sin(0i) + u (r) sin(c/>)] 5 2 16 17 + e [u {r) + u {r,e,t)] + 0{e ) 2 3 21 22 2 (3.6) Chapter 3. 33 Analytical and Numerical Parabolic Seamount Comparison V(R,9,T) = -y+(r)sin(0i) + ^ ( r ) sin(</>) + E[VU{R) + V (R) 2 15 cos(20) + + V {R) L2 V1 3 (r) cos(47ri) cos(20 ) + vi6(»") cos(0i) + 2 V (R) 17 + VU(R) cos(2c/>i) cos(c/>)] 2 + e ' M ^ + ^ r . f l . O l + O^ ) (3-7) 3 again <j)\ — (6 + 2TT£) a clockwise phase propagation, <^> 2 = — 27r£) a counter-clockwise phase propagation and e, the Rossby number, is assumed small. As discussed in the previous section, the order e components have frequencies which result from all possible combinations of the zeroth order frequencies (0i and c/>). Sub2 stitution of (3.5)-(3.7) into the governing shallow water equations (3.1)-(3.3) results in systems of equations which are comprised of three differential equations for each frequency component. The zeroth order terms satisfy the following system of first order ordinary differential equations: 1 ±- ±<TUQ ± , ^0 —- Vg H = n0 DR ±AV±-V± + -TI± R = 0 (3.8) d ±<"7o - -[-rirhut) T hv±\ = 0 AR where (UQ , VQ , T]Q ) are the counter-clockwise phase propagating terms and (UQ , v£, 770") are the clockwise phase propagating terms. The solution for (3.8) is discussed in § 3.1.2.1. Assuming that \VQ\ ~ \UQ\ and \UQ | 3> \UQ\, it is possible to show that |u| = UQ (1 + 7cos(20)) for 7 < 1. Physically, these assumptions presume that the flow is, for the most part, going over the seamount and not around. The first two scaled equations of (3.8) support the first assumption and both assumptions are borne out by the solutions. Using the relation for \u\, the first order systems of differential equations can be written as: Chapter 3. Analytical and Numerical Parabohc Seamount Comparison The first order residual components: vn + -3— = Un - gu dr 0 d(rhun) =0 dr . dn V12 12 H ;— = gi2 dr 2 7/12 r uu = — (rhu ) dr 2hv = \pu - 12 , J12 12 a z The first order harmonic components: 2crui3 - vi H — = gi3 3 dr -2av + 1 3 u = 13 2cru 14 + 3 d. a 2 -2o-77i3 fi = P13 — [-r[rhu )\ 13 - v +— —=g 1 4 14 dr 2 -2(TVi4 + Ui 7/14 = 4 r -2cT77i -2o-u 4 1B 0?. d . . Jl4 . . + — — ( r f t u i ) - 2/u>i J = p i - vis + —r- = 9i5 4 4 4 dr 2 2aV i5 + Ui 7/15 = 5 /l5 r ex —[^-{rhum) r dr 2 2o-T/i5 + - 2hvi*\ = p i5 Chapter 3. Analytical and Numerical Parabolic Seamount Comparison 35 First order components due to interaction of zeroth order terms with bottom friction: . drj nr 16 -crv + 16 u d 77i 16 6 = 16 -<ru 17 (3.14) nn d + —[^-{rhuie) r dr - v - hv \ = 0 16 drj nr + - 3 — = JIT dr titi ir 17 2 -TT-U+V+ r -ar] + , _ oo d u u d 1 av + 17 K u 2 u 17 r d fO 1 - K \ (3.15) v nfi d —[-r{rhu ) 0-7717 + + d r] = -T7T o o 17 a T - hv \ = 0 17 r dr 17 with 511 = - K ^ - = -jK 5i2 1. ' 1 12 +u -^) - ^ y + + , dVg « -^r) 0 -dVQ + U 1 / P12 = 7 + ^ c ^ " " ™° 7 1 ,tili +^ „ ) +^ ( K ) +K) ) 2 -^ W o +» o O J. _ . tt 2 ^ ° ~dV ° "dr ^ 2 7 ^ ° = « 0 + U _ 4 ° ^^ ~ u rr w _^0\ 0 + = -^W-^r Pis = ^ ^ 514 r l o r 1 / 4. _ U V U ~o u o -fo-J - ^( o o u v ~ lo o) rr dr — dr (3.17) v _ , + V U I . . 1 , U l v l 0 + 1 + 0 /« ™\ ^ ' + -«o o) v r" (3-21) (3.23) 2r 2r (3-19) - 4-\ V v r°° V (3-22) u •-«o-r- + . /l4 u _ v w + /13 j (3-16) ~oo lo Z) ~ 2(i7o o + »/o o )] + 913 = ~2^ °~dr~ ° ~dV' ~ 2r~^ ° ° ° °' , 1, +dvo -dv£ 1 _ _ U l s - 2 0 2 (Vi) 2r °' K (3.24) Chapter 3. Analytical and Numerical Parabolic Seamount Comparison P14 = Y^i^tO-H^] 1 _du = / l 5 = Pl5 = 2 °~dr~ U (3.25) 1 _ _ 0 9 1 5 36 + 1 , 2r~ ° ° ~ 27 "° V U ( 1 _dvo 1 _ _ 1 ^ ^ r " " ^ ^ ~27 4 0 ^ [ ^ H o «0 ) - 2 ) ( 3 ' 2 6 ) 2 K ) (3 WV] - 2?) ( - ) 3 2 8 Furthermore, an important second order result is: = -/21 - 7 7 ^ 0 i i u i (3.29) v 2 where / 2 1 = 2 " + -(liievd + ^ o [ r 1 6 ^ + U u v l 7 ^ - ^ ° - ^ - wt v u ~ n o)\ v u 0 (3.30) Examination of the first order equations (3.9) - (3.15) shows the form of these to be the first order variables arranged in linear relationships similar to the structure of (3.8) but forced on the right side by the nonlinear interaction of the first order variables (3.16) - (3.28). In general, the weakly nonlinear formulation organizes the equations into the form where the linear terms are in balance with the nonlinear terms of the next lower order. Of the zeroth and first order terms, only the "16" and "17" components directly feel bottom friction effects. Advection terms are given by the / ; and gi (i is the associated frequency label) ; pi arise from the conservation of mass terms. The result, un = 0, is a direct consequence of the residual component equations. Chapter 3. 3.1.2 3.1.2.1 37 Analytical and Numerical Parabolic Seamount Comparison Analytical solutions Zeroth order solutions The zeroth order differential equations can be combined to give the following equations for the clockwise and counter-clockwise linear velocities: A 1 ± , lo dr — i±„-£) u. (3.31) i7 _ ± 1 ( ±^ (3.32) M)dr + where 775^ are governed by the ordinary differential equations: r 2 #Vo . r , dh r drj <r - 1 2 0 2 r dh ± The boundary conditions discussed in § 3.1.1.2 apply here as r]Q(rb) = | and r) E C2 0 over the domain. This boundary condition is derived by assuming that the standing wave boundary condition is fully solved by the linear part of the weakly nonlinear expansion for sea height at r^, i.e. 77(7-6, 8, t) = sin(0) cos(27ii) = rj£(r ) s i n ^ ) + 77^(r ) sin(c/>)b 2 b (3.34) Since 0 i = (8 + 2irt) and <j> = (8 — 2nt) this implies that rj^rb) = | . Further, the 2 matching conditions are assumed to be that the sea height and its slope are continuous across the domain. This is written as 770 € C . 2 Equation (3.33) is similar to those described by Hunkins (1986) and Shen (1992) and has a closed form solution for the cases h(r) oc a constant and h(r) oc r and thus a plane 2 parabolic topography defined by 0.1 Mr) = r 2 1.0 0 < r < .3156 .3156 1.0 <r<1.0 <r<2.36 Chapter 3. Analytical and Numerical Parabolic Seamount Comparison 38 will yield closed form analytical solutions. This topography was derived from similar linear studies (LONGUET-HlGGINS, 1970; HUTHNANCE, 1974; HUNKINS, 1986; CHAPMAN, 1989; SHEN, 1992) and is shown in Fig. 3.2. Radial Distance (km) 1 50.7 25.3 0 76 101.4 126.8 2000 Plane Parabolic Topography Gaussian Topography 0.8 1600 E cd 0.6 1200 •§> a> 0.4 800 0.2 400 .c *-» c o CD CO CO CO c o u w c 3 Q 00 CO c E CD 0.5 1 1.5 2 Dimensionless radial distance 2.5 Figure 3.2: Plane parabolic topography used in the semi-analytic solution. For comparison purposes, a Gaussian topography is also shown. The Gaussian topography is discussed in Chapter 3. Dimensional values for seamount height and radial distance have been added to expedite comparison to subsequent figures. The plane parabolic topography produces a zeroth order solution for 77, for the superinertial case, given by Ji(^ir)(c sin(c/. ) + c osin(c6 )) 1 = < [c r 2 w 1 1 0 2 + c r « ] sin(c^i) + [ c r + c r ] sin(ci> ) w 3 4 0.3156 w 5 2 [(ce-Mftr) + °rY {far)) sin(> ) + (c Jx{/3 r) + c ^ f t r ) ) sin(<£ ) x a 8 2 < r < 0.3156 2 1.0 < r < 1.0 < r < 2.36 . (3.35) Chapter 3. Analytical and Numerical Parabolic Seamount Comparison where Ji?* - Pi = 1)/Q.l a 01 = ^1,2 v V -- 1 a 2 -1.0 ± 1 = -<T -+ a CT /*3,4 39 2 -1.0 ± ^2 + _ + £7 2 2 1 -<r a 2 2 and Ji(Br) and Yi(/3r) are first order Bessel functions of the first and second kind respectively, which are oscillatory and decay geometrically (i.e. oc 1/y/r) over the domain (ABRAMOWITZ and STEGUN, 1972). The coefficients [c ,...,c ] are real and can be de1 10 termined by the boundary conditions. Fig. 3.3a shows the solution for sea height using these boundary conditions, at time t = | T and with a superinertial frequency equivalent to the M tidal frequency (i.e. u = 1.405 x 10~ rad/s). 4 2 The linear solution for 77 in the absence of topography, say 77/, would take the same form as it assumes over the flat top. That is, 77/ = Ji(/5 r)(ci sin(0i) + ciosin(c/> )). 2 2 (3.36) If cr ^ 1 or a ^> 1, then |/3 | <C 1 and J\{B r) ~ B r/2 which implies that 2 2 2 Br 2 In this case, the elevation increases (or decreases) linearly towards the boundary (consistent with Shen (1992)) and oscillates as the sum of two waves counter rotating around the boundary. Incorporating the ideas of both Chapman (1989) and Brink (1990), one could then define the linear sea height enhancement due to the topography as (77 — 77/)/|T7/| and similarly for the linear velocities. Fig. 3.3b shows the linear sea height enhancement using this calculation. Chapter 3. Analytical and Numerical Parabohc Seamount Comparison 40 In the case of subinertial frequencies (i.e. a < 1), f3 are complex and hence the Bessel 12 function in (3.36) becomes a Modified Bessel function (ABRAMOWITZ and STEGUN, 1972). If tx < 1 or a 3> 1, then the Modified Bessel function is still linearly dependent on r and the form of the sea height is similar to the above case. , Consider the topographical effects over the sloping region. If <x is in a bounded domain, say a E [0.7,1.4], then both super and subinertial tidal frequencies are represented. For this frequency domain (again assuming a ^> 1), fi is positive real so that the r^ term 3 3 is exponentially growing and fj,4 is negative real so. that the r decaying. However, fi 2 XI M4 term is exponentially are real only in the superinertial case. In the subinertial case they are complex and hence their terms become oscillatory in space. Since seamount trapped waves occur only at subinertial frequencies (LoNGUET-HlGGINS, 1970) it is expected that there would be differences between the super and subinertial analytical solutions. However, the differences between the final superinertial and subinertial elevations are slight because only the "near field solutions" forced by the standing wave boundary condition applied at a radius encircling the topography is considered. Haidvogel et al. (1993) explore subinertial enhancement around a tall isolated Gaussian seamount for a large range of Burger numbers and subinertial frequencies. They also examine the resulting horizontal and vertical residual circulation. 3.1.2.2 First order solutions with bottom friction The first order solutions consist of the harmonic frequency components (2</>i, 2c/>2 and 47r£), the first order bottom friction terms and the time independent or residual components. As has been previously mentioned, substitution of the assumed form of the solutions into (3.1)-(3.3) results in systems of equations which are comprised of three differential equations for each frequency component (§ 3.1.1.4). The quadratic formulation of the bottom friction in (3.1)-(3.3) results in first order components which oscillate Chapter 3. Analytical and Numerical Parabohc Seamount Comparison 41 D i m e n s i o n l e s s S e a Height 0.655 0.328 -1.11e-16 -0.328 -0.655 "53 « on -100 X Distance (km) 100 50 100 Y Distance (km) E n h a n c e m e n t d u e to S e a m o u n t 1.31 0.656 0 -0.656 -1.31 W) g X Distance (km) -100 100 50 100 -100 Y Distance (km) Figure 3.3: Figures shown are a) the analytical linear sea height solution for (3.35) and b) the sea height enhancement given that the elevation increases linearly towards the boundary for the flat case. Chapter 3. Analytical and Numerical Parabolic Seamount Comparison 42 at the frequency of cf>\ and (f> . These components are governed by equations which have 2 the same structure as the zeroth order equations, but are forced on the right side by the bottom friction terms (3.8)-(3.9). A fifth order Runge-Kutta integration package is used to calculate the first order solutions. Within the superinertial range discussed by Zhang and Boyer (1993), they observe a "weak rectified anticyclonic circulation in the immediate vicinity of the topography" consistent with tidally rectified flow. To locate this type of feature in the analytical solution, the residual velocity solutions must be considered. In the case of the 9 independent component of the residual solution (i.e. for the 7711 term in (3.4) and similarly for the velocity), two of the three equations imply that tin = 0. Therefore, the system is degenerate because it results in only one equation for two unknowns (see Appendix B). However, with tin = 0, if vu is not identically zero, then a gyre will form part of the overall response. It is possible to derive an equation for v from the second order velocity equations. n Using the same numbering convention for the second order velocity components as appears in (3.4) for sea height, it can be determined that for the radial, 9 independent residual component, u , no degeneracy occurs as it does for the T i n component. That is, 21 making use of bottom friction effects, it is possible to obtain a fully determined system for the residual terms. The continuity equation (3.3) gives an integrable equation for (rhu ) 21 in terms of the first order 0 i and c/> terms and the first momentum equation 2 (3.2) gives a relation between u and Vn, hence it is possible to derive an expression for 2i the first order azimuthal, 9 independent residual component: v u = + Kr (3.37) d where F 2 I is a forcing function generated by the nonlinear interaction of the linear velocity components with the first order velocity field. Having previously scaled all the variables, Chapter 3. Analytical and Numerical ParaboHc Seamount Comparison 43 the constants can be factored to the front of the right side. Thus, it can be seen that VU is proportional to hd/rd which is an aspect ratio, a parameter often used to classify the physical characteristics of axisymmetric seamounts (CHAPMAN and HAIDVOGEL, 1992). Equation (3.37) implies that when the deep water depth does not change, wider seamounts will tend to have weaker residuals. Unfortunately, a similar dependence on seamount height, which is intrinsic to the definition of h(r), does not precipitate from the analytical derivation. Nevertheless, it would appear that the aspect ratio is a factor in determining the strength of the time independent residual flows. A similar equation can be derived for the 8 dependent residual component, v\2, using the same method. 3.1.2.3 First order solutions without bottom friction Without making use of the second order bottom friction terms, the governing system of equations for the residual components is degenerate. In this case, for both types of residual components (characterized by their dependence on 0), the set of equations for each is underdetermined over part or all of the domain (this mechanism and the following weakly nonlinear expansion are detailed in Appendix B). In the case of the 8 dependent residual component (i.e. for the 7712, Ui and V12 2 terms) the equations are degenerate only over the constant depth domains much like the steady state geostrophic flow equations are degenerate. In comparison, the other residual components are degenerate over the whole domain. This implies that another equation (i.e. another physical constraint other than bottom friction) will be required for these regions to establish a well-posed system. There is another method of fully determining the system and that is to use conservation of potential vorticity as a governing equation. The use of conservation of potential vorticity is not new as applied to time varying flow over topography (e.g. LONGUETHlGGINS, 1970; HUPPERT and BRYAN, 1976), but is of interest in this study when Chapter 3. Analytical and Numerical Parabolic Seamount Comparison 44 considering the relative strength of the flow due to superinertial forcing compared to the flow when bottom friction is included. This comparison will be briefly discussed in the section on model comparisons of the residual velocities. Applying a weakly nonlinear expansion to conservation of potential vorticity results in additional equations which can be used, in the case of these residual components, to fully determine the system for the domains where h(r) is constant. For example, writing the dimensionless forms of local vorticity and sea height as t = 6 + eh + 0{e ) 2 V = Vo + e*7i + 0(e ) 2 (3.38) (3.39) and substituting these into the equation for conservation of potential vorticity (which may be derived from the shallow water equations given in § 3.1.1.1 as was shown by Ertel in 1942 ( P E D L O S K Y , 1979)) D Dt T) + h = 0 (3.40) yields 6 - ^ =0 (3.41) where linear potential vorticity has been conserved, i.e. the linear solutions satisfy £o — rjo/(a h) = 0, and h and / are assumed constant. From (3.41) it is possible to derive the 2 necessary equations to fully determine the residual components over the constant depth domains. A similar procedure can be followed for the domain where h(r) is not constant. Timeaveraging the equations generated by substituting (3.38) and (3.39) into (3.40) will result in an equation comparable to (3.41) for the 6 independent component of the residual. Chapter 3. Analytical and Numerical Parabolic Seamount Comparison 3.2 45 Comparison of model results The harmonic analysis of the shallow water equations in the finite element numerical approach allows examination of distinct frequencies and/or their combined effects. This is well suited to separate comparison of the zeroth and first order solutions. Consider the interaction of a wave (whose frequency is equal to that of the M tide) 2 with itself, including advective and bottom friction effects. Following the discussion on frequency expansions in § 3.1.1.3, a weakly nonlinear expansion could be written out to second order with the following structure: Total Wave Effect = linear M wave 2 + e (residual + M 4 component + M terms generated by nonlinear bottom friction) 2 + 0(e>). Thus, examination of the M frequency component consists of comparing the numerical 2 solution to the analytical linear M solution plus the first order M solution from bot2 2 tom friction effects. Analysis of the M frequency component consists of comparing the 4 the M numerical solution to the sum of the results for the 2c/>i, 2(f> and 4?ri compo4 2 nents. Similarly, the residual comparison requires summing the two residual frequency components: 1 and 26. Note that for all figures, unless otherwise stated, the horizontal axis is along the line x = 0, y > 0 from an origin placed at the center of the seamount (see Fig. 3.4), time is zero and, for simplicity, velocity comparisons are made with Cartesian velocities (as opposed to the original cylindrical polar co-ordinate forms). To compare the semi-analytic and numerical solutions, it was necessary to determine the values of the dimensionless parameters. In this, the superinertial case, the numerical model used predefined values of rj. = 50.7 km, hj — 2000 m, T = 12.42 hrs and / = 1.031 x 1 0 Table 3.1. -4 1/s. These values along with other solution parameters are summarized in Chapter 3. Analytical and Numerical Parabolic Seamount Comparison 46 120 km Figure 3.4: Top view of seamount. Dashed line indicates radius along which values are presented in the following plots. The radius is arbitrarily chosen to run due north away from the center marked with a 0, hence the arrow pointing east indicates the direction originating from the dashed line which is consistent with that of anticyclonic flow (in the Northern Hemisphere). Parameter ra hd T f e Value 50.7 km 2000 m 12.42 hrs 1.031 x 10" 1/s 0.074 4 Parameter U Vb a 0 Rr a Value 38.01 cm/s 20.27 cm 1.36 1357.90 km 26.78 Table 3.1: Parameter values used to compare the semi-analytic and numerical solutions. The small value for e implies that the system is weakly nonlinear; a is greater than 1 because the M frequency is superinertial. Further, the effective Rossby radius is much 2 larger than the bank radius which would indicate that the dynamics roughly satisfy a rigid lid approximation but this approximation was not used in either the numerical or analytical formulation. Chapter 3. Analytical and Numerical Parabolic Seamount Comparison 3.2.1 3.2.1.1 47 Zeroth order results comparison Zeroth order components without bottom friction The first order terms generated by bottom friction are much smaller in magnitude than the zeroth order terms of the same frequency. Even though (as discussed above) it is more realistic to compare the numerical solution to the linear analytic solution plus the first order bottom friction terms, it may be argued that an initial comparison with no bottom friction present may lead to a better understanding of the models. Figs 3.5 and 3.6 present the comparisons of sea height and Cartesian velocity results between the analytical and numerical models. Fig. 3.5 indicates that the sea height solutions differ by less than 1 cm over a distance of 120 km. Note that, at this phase, the standing wave (the boundary condition) amplitude is negative on the northern most part of the boundary at 120 km which implies that sea height will also be negative in this figure. From the discussion in § 3.1.2.1, it is known that the linear sea height solution to this problem in the absence of topography is an elevation that (at this phase and along this radius) decreases from 0 linearly towards the boundary. Thus, it can be seen that, along this radius, the modification due to the seamount takes the form of a depression in the elevation centered near the point where the topography changes from the flat top to the sloped side. A similar enhancement in elevation occurs in a direction 180° to this radius (i.e. along the line x = 0, y < 0) and it appears as an increase in the elevation instead of a depression (consistent with the linear solutions shown in Fig. 3.3). The velocity profile in Fig. 3.6 shows an intensification of the flow across the top of the seamount (which, at this time, is to the west). It can be seen from this figure that the velocities show reasonable agreement. Note that the v velocity solution from both models is zero along this radius. The most significant difference occurs at the point where the topography changes from plane to parabolic shape, i.e. where a discontinuity Chapter 3. Analytical and Numerical Parabohc Seamount Comparison 48 Analytical Solution F E M Solution E o -10 g> CD .C CO CD -15 CO -20 -25 0 20 40 60 80 100 D i s t a n c e f r o m s e a m o u n t c e n t e r (km) 120 Figure 3.5: Linear sea height solutions without bottom friction effects. Analytical and Finite Element model solution. Sea height is plotted along the dashed line in Fig. 3.4 which runs outwards due north from the origin of the seamount. The standing wave boundary condition is at its maximum negative value at 120 km. in the topographical slope occurs. The relative error is less than 1 % away from the discontinuity, but rises to almost 4 % at the discontinuity. This result leads to a closer look at the analytical model. It can be seen from Fig. 3.5 that the linear sea height and its slope are continuous and this can also be derived mathematically from the linear equations. With these assumptions, it is possible to derive a relation from the zeroth order continuity equation that relates a discontinuity in the slope of the linear velocity to the discontinuity in the topographical slope. Using the notation Aa; to represent a jump discontinuity in x gives ^du _ _'H_^dh dr h dr 0 (3.42) Further, using this relation, a jump discontinuity in any of the first order variables can also be written in terms of the jump in the topographical slope. Therefore, if there is any error in the numerical calculation of this slope it will also appear in the first order Chapter 3. Analytical and Numerical Parabolic Seamount Comparison 49 20 -120 ' 0 20 40 60 80 100 D i s t a n c e f r o m s e a m o u n t c e n t e r (km) ^ 120 Figure 3.6: Linear Cartesian velocity components without bottom friction. Analytical versus Finite Element model solution. Velocity is plotted along the dashed line in Fig. 3.4 which runs outwards due north from the origin of the seamount and hence this velocity is to the west (the V component for both models is approximately zero). results; however, this does not mean that all of the error in the first order results originate from the calculation of this slope. Fig. 3.7 compares the analytical slope and numerical slope in m/km and also shows the relative error. It can be seen that the maximum relative error is just over 10% at about 33 and 50 km. However, as can be seen from (3.42) this error is multiplied by the factor UQ /h and so the maximum absolute effect is moved from the point where the seamount reaches its maximum depth to the 16 km radius where the top of the seamount changes to the parabolic shape. The major differences between analytic and numerical solutions will occur at this radius. Since this weakly nonlinear approach is an extension of the linear work of Shen (1992), the zeroth order components were devised to be solutions to Shen's linear equations. Further, the governing linear equation (3.33) is also similar to that described by Hunkins (1986). Consequently, these zeroth order solutions reproduce Chapter 3. Analytical and Numerical Parabolic Seamount Comparison 50 80 Analytical S l o p e F E M Slope 70 60 E 50 E g CD Q. 40 cn 20 _o 30 10 0 -10 20 40 60 80 100 120 40 Relative Error 35 30 25 in CD > 20 15 CD DC 10 5 0 0 20 40 60 80 100 D i s t a n c e f r o m s e a m o u n t c e n t e r (km) 120 Figure 3.7: Comparison of analytical and numerical slope over plane parabolic topography and the relative error in the slope. Chapter 3. Analytical and Numerical Parabolic Seamount Comparison 51 the same velocity and sea height structures found in those two studies. They are also similar in shape to the linear solutions of the forced subinertial response described by Chapman (1989) even though those solutions are forced by a subinertial Kelvin wave. The enhancement found by Chapman is greater than that observed here as would be expected due to the subinertial resonant response included in his. study. 3.2.1.2 Zeroth order components with bottom friction To be able to compute solutions which are comparable to the numerical results for the barotropic bottom friction problem, the regular expansions must include first order terms which oscillate at the zeroth order frequencies (e.g. 7716 and rjir in (3.4)). In this case, the numerical M solution is comparable to the analytical linear M 2 2 solution plus the first order M effects from bottom friction. Addition of bottom friction 2 has very little effect on the sea height profile, however an effect can be readily seen in the velocity solutions. These results are shown in Fig. 3.8. As is expected, the u velocity has only been altered slightly (its maximum magnitude has been decreased), but the v velocity has changed from 0 cm/s to 20 cm/s across the top of the seamount. Therefore, the major effect of bottom friction on this frequency is to change the direction of the velocity across the top of the seamount from west to approximately west northwest. 3.2.2 3.2.2.1 First order results comparison First order harmonic components Figs 3.9 and 3.10 compare the sea height and Cartesian u velocity for the first order harmonic frequency response. A direct result of the assumptions of the weakly nonlinear formulation is that this component is independent of bottom friction (refer to equations in § 3.1.1.4). Chapter 3. Analytical and Numerical Parabohc Seamount Comparison to E u> CD o CL 20 0 -20 -40 E o O -60 o _p -80 CD > 52 U Analytical V Analytical U FEM V FEM -100 Solution Solution Solution Solution 1 on 0 20 40 60 80 100 D i s t a n c e f r o m s e a m o u n t c e n t e r (km) 120 Figure 3.8: Linear Cartesian velocity components with bottom friction effects. Analytical versus Finite Element model solution with bottom friction coefficient K = 0.003. Velocity is plotted along the dashed line in Fig. 3.4 which runs outwards due north from the origin of the seamount and hence this velocity is to the northwest. Examination of Fig. 3.9 shows reasonable agreement between the semi-analytic and numerical solutions for sea height, but that the numerical model does not give the same sharp peaks as the analytical solution. As will be discussed later on, the sign of the velocity solution is affected by the value of these slopes. Consider Fig. 3.10. The v velocity solutions from both models are zero. However, the u velocities are different signs and significantly different magnitudes. These differences result from the structure of the differential equations governing this component (these equations are discussed in detail in § 3.1.1.4). On the seamount flank, the second order harmonic velocity is governed in large part by the advection of the linear velocity by itself. This means that, in this region, the magnitude of the u velocity is determined principally by the derivative of the zeroth order velocity in as much as an increase in the derivative of the zeroth order velocity results in an increase in the magnitude of the u velocity. Chapter 3. Analytical and Numerical Parabolic Seamount Comparison 53 Distance from seamount center (km) Figure 3.9: Harmonic Sea height solutions. Examining the zeroth order velocities in Fig. 3.8 at a radius of 16 km, it is apparent that the derivatives for the analytical case will be much higher than those predicted by the numerical case because the curve is much sharper there. If the derivatives in the analytical case are forced to be of the same magnitude as those in the numerical program (which is assumed to be the more realistic solution) then the maximum u velocities are within 1.5 cm/s of each other. The sign of the u velocity is determined, in part, by the slope of the harmonic sea height (shown in Fig. 3.9) over the flat seamount summit. In the constant depth region, the velocity and sea height are the largest ordered terms. Fig. 3.9 shows that the analytical and numerical results are within 4 mm of each other in this domain. However, at a radius of approximately 14 km, the sea height slope in the numerical solution is approaching zero whereas the slope of the analytical solution continues to increase. This relative difference between the slopes predicted by the models results in the different signs for the two u velocities. Chapter 3. Analytical and Numerical ParaboHc Seamount Comparison 54 25 r 20 r E w c o c o Q_ E o O >% 15 r 10 i5 \ o o o < > -5 \ CD -10 t 20 40 60 80 100 D i s t a n c e f r o m s e a m o u n t c e n t e r (km) 120 Figure 3.10: Harmonic Cartesian velocity solutions. The v velocity solutions from both models are approximately zero along this radius. These sharp peaks in velocity and sea height are related to the discontinuities in the topographical slope through (3.42). Normally, a seamount in the ocean would not be expected to have such sharp corners and hence the numerical solution would represent a more reasonable result. A consequence of this difference in signs is that the numerical model predicts a reversal in velocity across the top of the seamount that does not show up in the analytical solution. As expected the harmonic sea height and velocity magnitudes are much smaller than those of the linear components. 3.2.2.2 First order residual components To make a comparison of residual currents where bottom friction effects are included, the solution derived from (3.37) with re set to 0.003, is compared to the numerical solution (Figs 3.11 and 3.12). Recall that the horizontal axis is taken to be along the dashed line in Fig. 3.4. Hence, the residual velocity profile in Fig. 3.12, which is a plot of the u velocity in Cartesian co-ordinates, is consistent with that of an anticyclonic flow around Chapter 3. Analytical and Numerical Parabolic Seamount Comparison 5 5 the top of the seamount surrounding a circular area of little or no flow. Fig. 3.11 may be interpreted as representing a slice from the center out to 120 km through the residual sea height. In this case, the residual sea height is seen to form a small dome (maximum height about 6 cm) over the seamount, concurrent with the anticyclonic residual flow. 6.5 i * * i '* *' i D i s t a n c e f r o m s e a m o u n t c e n t e r (km) 120 Figure 3.11: Residual Sea height solutions with K — 0.003. This is the time independent component, hence these solutions represent a persistent hill of fluid, approximately 6 cm high, centered over the seamount. Considering the terms in (3.37), the only variables with a possible dependence on K are u i and /21 (since the linear solution, u^, is independent of bottom friction). It is 2 clear, then, that if (« i + /21) is linearly dependent on K, then Vu will be independent 2 of the bottom friction. Numerical tests show that this is indeed the case and the result is consistent with the work of Huthnance (1981). Therefore for this particular analytical formulation, the presence of bottom friction is necessary for significant residual velocities, however its magnitude is irrelevant. There is an intrinsic difference in the two models' approaches to parametrizing the bottom friction. First, the analytical formulation cannot be used in the numerical model. Chapter 3. Analytical and Numerical Parabolic Seamount Comparison 140 r Analytical Solution F E M Solution (cm/ 120 100 - w c 80 - o o a. E o 60 - O 40 - o o 20 - > 0 - >* 56 CD -20 0 20 40 60 80 100 D i s t a n c e f r o m s e a m o u n t c e n t e r (km) 120 Figure 3.12: Residual Cartesian u velocity solutions with n = 0.003. Velocity is plotted along the dashed line in Fig. 3.4 which runs outwards due north from the origin of the seamount and hence this velocity is to the east, consistent with anticyclonic flow. The analytic model is constrained in its formulation of the bottom friction term by the structure of the ordered components. Representing the nonlinear bottom friction term as a product of terms follows directly from the frequency expansion upon which the form of the solution is based (see § 3.1.1.3). This structure leads to the degeneracy of the residual component equations. This formulation for the bottom friction cannot be duplicated in the numerical model without encountering the same degeneracy problem. The harmonic decomposition of the governing equations allows the numerical model to solve for each harmonic component seperately and sequentially. If the numerical model uses the same structure for the bottom friction term as the analytic model, it would solve the linear zeroth order component first but then it cannot converge to a solution for the first order residual component because at that point the equations are underdetermined. It cannot solve for the second order residual terms before it solves for the first order residual terms. Second, the numerical formulation cannot be used in the analytical model. The Chapter 3. Analytical and Numerical ParaboHc Seamount Comparison 57 calculation of velocities in the numerical model's bottom friction term incorporates an iterative numerical procedure which is not used in the analytic method. As a result, the two models cannot be written such that they directly compare. By altering the bottom friction parameter, it was determined that for this case (of a weakly nonlinear regime), increasing the bottom friction results in smaller residual velocities in the numerical model. In this way, bottom friction is seen as a loss term in the numerical model and hence higher frictional values in the numerical model would result in lower residual velocities compared to the analytical model which has no such restraints due to its formulation. Thus the analytical solution is a possible upper bound prediction for the residual velocities. Fig. 3.12 reflects the differences between the numerical and analytical bottom friction formulations in that the maximum numerical velocity is 3.5 times smaller than that of the analytical solution. The residual velocity profiles can be described as flowing anticyclonically in a ring centered about the seamount, with maximum velocities at the seamount rim or shelfbreak. Velocities decay approximately exponentially outwards from the rim. This same structure is seen in other studies of tidally rectified flow (e.g. GARREAU and MAZE, 1992; HAIDVOGEL et al., 1993; CHEN and BEARDSLEY, 1995). Garreau and Maze (1992) report maximum residual velocities of about 7 cm/s for tidal (M ) velocities of about 52 2 cm/s on the shelf. These residual velocities are much less than the 40 cm/s predicted by the numerical model which also had oscillatory flows of over 100 cm/s on the summit. Chen and Beardsley (1995) report maximum residuals of 2.4 cm/s for their tidal (M ) 2 flows of about 80 cm/s over their bank. The subinertial (K~i) results of Haidvogel et al. (1993) show maximum residuals of about 3 cm/s for tidal flows on the order of 10 cm/s over their Gaussian seamount. These results may indicate that the residual velocities forced by the standing wave are higher than might be expected from Kelvin wave or tidal forcing, however, the differences found between these models are large and conclusions are Chapter 3. Analytical and Numerical Parabolic Seamount Comparison 58 difficult to draw. More discussion about these magnitudes with respect to observational data is given in Chapter 5. Note that the magnitudes of the semi-analytic residuals are comparable to those of the linear solution. This suggests that there is a better formulation for the weakly nonlinear solution that has both the M and residual components as zeroth order terms. 2 Inspection of such an expansion shows that the advection terms governing the residual circulation are qualitatively the same in shape but that they are approximately 3.5 times smaller. Moreover, there will be additional bottom friction effects which will further reduce the magnitude of the residuals. Analysis of this formulation would not be significantly different than the original. It is interesting to compare the analytical potential vorticity conserving solution to the bottom friction solution derived from (3.37). Fig. 3.13 compares the Cartesian u velocity for the first order residual for these two cases. Note that the Cartesian u velocity from the potential vorticity conserving solution was scaled up by a factor of 10. From this figure, it is apparent that the residual velocities from the potential vorticity conserving case are much smaller than those generated with bottom friction. These results suggest that the inclusion of bottom friction in the dynamics is very important for the formation of gyres. For, as can be seen from Fig. 3.13, if the dynamics are governed by conservation of potential vorticity, then the maximum value of the residual velocity field is two orders of magnitude smaller than (approximately 100 times less than) the velocity field in the presence of bottom friction. This observation applies only to the residual flow field from the tidalfy rectified analytical solution, because these two mechanisms were used to fully determine the first order semi-analytic equations. It does not apply to Taylor column dynamics where the mechanism of conservation of potential vorticity is applied to all orders. 140 r 120 \ (cm/ Chapter 3. Analytical and Numerical ParaboHc Seamount Comparison 100 \ w c 80 \ 60 f 40 \ o 20 f > 0 r CD o CL E o O & O CD 59 P o t e n t i a l Vorticity S o l u t i o n B o t t o m Friction S o l u t i o n -20 20 40 60 80 100 D i s t a n c e f r o m s e a m o u n t c e n t e r (km) 120 Figure 3.13: Residual Cartesian u velocity solutions. Comparison of the analytical solution for conservation of potential vorticity to the semi-analytic solution with bottom friction (K = 0.003 in (3.37)). The potential vorticity solution has been scaled up by a factor of 10. 3.3 S u m m a r y a n d discussion A weakly nonlinear approach to examine barotropic oscillatory flow over a tall, axisymmetric, isolated seamount gives systems of first order ordinary differential equations for each component. The systems that govern the time independent (residual) frequencies are underdetermined, hence, two separate approaches - i) the addition of bottom friction and ii) the conservation of potential vorticity - can be taken to formulate solutions. These solutions may be used to test numerical models which solve equivalent mathematical systems. Specifically, residual velocities from tidally rectified flow, while small in magnitude, have been known to affect overall circulation around finite height topographies. A semi-analytic formulation for the case with bottom friction shows these velocities to be proportional to the aspect ratio of the seamount. This linear dependence implies Chapter 3. Analytical and Numerical Parabolic Seamount Comparison 60 that for cases where the deep water depth does not change, wider seamounts will tend to have weaker residuals. Hence, this ratio is a factor in determining the strength of time independent residual flows. Further, the relationship shows the residual velocities to be independent of bottom friction in a manner such that while the presence of bottom friction is required to determine significant residual velocities in the analytical case, its magnitude is irrelevant. This implies that the result may be an upper limit for these velocities. Numerical tests showed that the finite element model predicted weaker residuals for higher bottom friction coefficients. An analytical solution may also be derived through a weakly nonlinear expansion of the equation expressing conservation of potential vorticity. This approach predicts much smaller residual velocities than those generated with bottom friction. The analytical formulation indicates that a discontinuity in the topographical slope can result in differences between the numerical and analytical model solutions. The maximum effect is expected to be at the 16 km radius where the top of the seamount changes to parabolic shape. Furthermore, unrealistically sharp peaks in the analytical results affect the first order solutions. Based on the model comparisons, it is concluded that the numerical model gives reasonable solutions for tall topographies. The accuracy of the numerical model is constrained by the amount of discretization of the topography, especially when the topography is non-smooth. Chapter 4 R e s i d u a l V e l o c i t y Comparisons Results from the previous chapter indicate that, based on comparisons to a semi-analytic solution, the numerical model is capable of producing reasonable solutions for oscillatory flow over tall isolated topography. One of the advantages of using a numerical model is that it can be used to find solutions for topographies for which there are no analytic closed form solutions: the F W model was developed for and used to calculate current velocities over the real bathymetry of the Canadian west coast. Comparisons of velocity results between seamounts whose physical characteristics are described by the parameters used to non-dimensionalize the governing equations in Chapter 3 (e.g. and HD) are presented here to explore the relationships between the tidal residual and seamount geometry. For example, residual velocity comparisons may be made between flow solutions of Gaussian shaped seamounts of different heights. These comparisons will be explored in § 4.3 . This chapter is comprised of a number of different comparisons. Bathymetry data for Cobb Seamount is compared to a Gaussian shape, and velocity and sea height parameters for Cobb are related both to a Gaussian seamount and the plane parabolic seamount of Chapter 3. In addition, the resultant flow field from the standing wave boundary condition of Chapter 3 is compared to that forced by a Kelvin wave. Residual velocity comparisons from different seamounts are discussed in the last section of this chapter. 61 Chapter 4. Residual Velocity Comparisons 4.1 62 C o m p a r i s o n to C o b b Seamount Will the resultant flows from a mathematically simple Gaussian seamount be relevant to the real world? To address this question, this shape has been compared to the bathymetry of Cobb Seamount which is situated off the coast of Washington. An azimuthally averaged profile of Cobb Seamount was provided by D. Codiga (University of Washington) using data from a hydrosweep survey by C. Eriksen (University of Washington) (Fig. 4.1). From this figure, one can see that the seamount has a somewhat flatter top than that of a Gaussian but that the slopes are fairly similar and thus the assumption that a Gaussian shape can compare reasonably well to the real bathymetry of some seamounts appears valid. Radial Distance from Seamount Center (km) Figure 4.1: Comparison of averaged Cobb Seamount bathymetry (data provided by C. Eriksen and D. Codiga at the University of Washington) and a Gaussian profile. Raw bathymetry data was collected using a SeaBeam Hydrosweep system on the October 1991 R / V Thompson cruise in the region 131° 05' W to 130° 30' W, 46° 36' N to 46° 56' N . To further investigate how our solutions compare to Cobb Seamount, plane parabolic solutions from Chapter 3 are related to flow solutions for a Gaussian seamount and Chapter 4. Residual Velocity Comparisons 63 resultant parameter values for these cases along with those of Cobb Seamount are summarized in Table 4.1. The numerical model was applied to the Gaussian seamount shown in Fig. 3.2 and results are shown compared to those of the plane parabolic seamount in Figs 4.2, 4.3 and 4.4 for the same standing wave boundary conditions previously used. The plane parabolic seamount tends to pile more water up on the side where the standing wave amplitude is positive - as can be seen as having less water on the side with the maximum negative standing wave amplitude shown in Fig. 4.2 (the maximum difference in sea height is ~ 1.5 cm). This difference is due to the greater volume that the parabolic seamount occupies and is directly related to the higher velocities it supports away from the center of the seamount. 0 20 40 60 80 100 Distance from seamount center (km) 120 Figure 4.2: Linear sea height solutions. Gaussian topography versus plane parabolic topography. Sea height is plotted along the dashed line in Fig. 3.4 which runs outwards due north from the origin of the seamount. The standing wave boundary condition is at its maximum negative value at 120 km and the resulting plane parabolic solution shows a lower sea height than the Gaussian along this radius. Fig. 4.3 shows the comparison of linear velocity results. Note that the Gaussian seamount results in a higher peak velocity across the center of the seamount (112 cm/s Chapter 4. Residual Velocity Comparisons 64 -10 -20 E w ca> o Q. E o O O O CD > -30 / / / / -40 -50 r -60 - -70 - -80 -90 -110 / / j ' j / . / / / / / - 1 / -100 -• -. \ Plane Parabolic Solution G a u s s i a n Solution \ •i 7 • -120 0 ' ' ' 20 40 60 80 100 D i s t a n c e f r o m s e a m o u n t c e n t e r (km) 120 Figure 4.3: Linear Cartesian u velocity solutions. Gaussian topography versus plane parabolic topography. Velocity is plotted along the dashed line in Fig. 3.4 which runs outwards due north from the origin of the seamount and hence this velocity is to the west. The Gaussian velocity field is greater by about 11 cm/s across the summit. versus 101 cm/s) even though the seamounts have the same height. This enhanced response attenuates fairly quickly away from the center of the seamount and therefore the plane parabolic seamount is seen to have more of an effect away from the peak. In the barotropic case, the shape of the velocity profile appears to be directly related to the shape of the seamount or is proportional to l/h(r). Fig. 4.4 compares the Cartesian u residual velocities of the two models. In this case, since the radius along which velocity is plotted runs due north from the summit, then the velocities are directionally to the east, consistent with anticyclonic flow. Both seamounts support an anticyclonic eddy, however the eddy over the Gaussian has a significantly smaller magnitude and does not show a smaller inner cyclonic eddy over the top of the seamount as the plane parabolic case does. Moreover, the plane parabolic seamount has a residual field which appears to have a larger horizontal extent in that its maximum is Chapter 4. Residual Velocity Comparisons 65 farther away from the seamount center. If a large horizontal extent of the recirculating flow is a factor in aiding trapping, then the Gaussian seamount may be expected to trap a smaller number of particles. 40 ^ "co" E w c CD c o CL E o O >> ' "o _o CD > Plane Parabolic Solution G a u s s i a n Solution 35 30 25 20 15 10 5 0 -5 0 20 40 60 80 100 D i s t a n c e f r o m s e a m o u n t c e n t e r (km) 120 Figure 4.4: Residual u velocity solutions., Gaussian topography versus plane parabolic topography. Velocity is plotted along the dashed line in Fig. 3.4 which runs outwards due north from the origin of the seamount and hence this velocity is to the east. Table 4.1 gives a summary of the relevant parameters which define the flow regimes for the plane parabolic seamount used in the analytic formulation, the Gaussian seamount and Cobb Seamount. For the analytic and numerical cases, the standing wave boundary condition is taken to be oscillating at the M frequency and parameter values for Cobb 2 Seamount, which are taken from Freeland (1994) and Mofjeld et al. (1995), are calculated from M tidal values (see Appendix C). Therefore, cr = 1.36 for all three cases (since 2 a- = a;// is the scaled wave frequency). Considering the values for e, the Rossby number, and a, the scaled Rossby number, it is apparent that the semi-analytic and numerical results are reasonable approximations to the Cobb Seamount values. That is, the Rossby number for these cases is much smaller than 1 and the scaled Rossby radius is much Chapter 4. Residual Velocity Comparisons 66 bigger than 1 meaning that the flow regimes will be weakly nonlinear and roughly satisfy a rigid lid system (§ 3.1.2.1, § 3.2). Table 4.1 also gives values for the variable 8, the fractional seamount height, which represents the fraction of the water column that the seamount occupies and can be taken to be the seamount height scaled by the deep water depth. Note that while 8 = 0.9 for the parabolic and Gaussian cases, the pinnacle of Cobb has an additional smaller peak and therefore 8 is approximately between 0.8 and 1.0 (FREELAND, 1994). The residual velocity field for the Gaussian case compares somewhat favourably with the current meter data presented by Freeland (1994). Results from the eight current meter moorings used for that study appear to correlate with the Gaussian model results both for the direction of the recirculation and the position of the maximum currents (which appear to be at the seamount rim). Furthermore, the maximum residual velocities are given by Freeland as approximately 12 cm/s. This scales nicely with the Gaussian results, in that the background tidal flow used for the Gaussian study is 25 cm/s or 1.8 times that given for Cobb (refer to values of Uo in Table 4.1) and the maximum residual velocity for the Gaussian seamount in Fig. 4.4 is approximately 25 cm/s or 2 times that cited by Freeland. Of course, one does not expect linear scaling of these results nor is the residual flow at Cobb Seamount due solely to rectification (but also due to the presence of a Taylor Cap). Nonetheless, results are the correct order of magnitude. In summary, Cobb Seamount bathymetry compares reasonably well to a Gaussian shape, the flow regimes for all three cases are similar, and the Gaussian and plane parabolic solutions appear reasonably consistent with each other, and therefore it is concluded that a Gaussian shape can give realistic solutions. Chapter 4. Residual Velocity Comparisons 67 Parameter Plane Parabolic Gaussian Cobb Seamount cr 1.36 0.9 50.7 km 2000 m 38.01 cm/s 1357.9 km 26.78 0.074 1.36 0.9 77 km 2000 m 25 cm/s 1357.9 km 26.78 0.031 1.36 0.8 - 1.0 20 km 3000 m 13.7 cm/s 1608.5 km 80.43 0.068 8 r hd U d 0 Rr a e Table 4.1: Comparison of parameter values for 3 cases: the plane parabolic seamount in Chapter 3, the Gaussian discussed in section 4.1 and Cobb Seamount off the coast of Washington. Parameter values for Cobb were taken from Freeland (1994) and Mofjeld et al. (1995). Data supplied by D. Codiga and C. Eriksen from the University of Washington. Note that / = 1.066 X 1 0 s for Cobb Seamount. Definitions of these parameters were originally given in § 3.1.1.1. - 4 4.2 - 1 Kelvin wave to standing wave boundary condition comparison The standing wave boundary condition discussed in § 3.1.1.2 is used with the semianalytic solution of Chapter 3 because it is mathematically convenient. The boundary condition allows the problem to collapse to a system of ordinary differential equations. Further, the standing wave has an associated velocity field which traces out a current ellipse pattern. However, this type of wave is not generally expected to be found out in the open ocean, it applies only to specific physical situations such as resonance in estuaries, harbours and channels. Moreover, a standing wave has no net movement of particles, since there is no mass flux through the nodal points. External tidal forcing is generally specified by a Kelvin wave boundary condition. As the relevance of the Gaussian seamount is being examined, it is prudent to similarly investigate the standing wave boundary condition and how it compares to external Chapter 4. Residual Velocity Comparisons 68 Kelvin wave forcing. Because the residual velocity field is of interest and important to recirculation, resultant residual velocities from flow forced by a Kelvin wave are compared to that of the original standing wave boundary condition. The Kelvin wave is used to approximate a superinertial M tide with a maximum height of approximately 2 m. 2 Properties of a Kelvin wave are discussed in Appendix C. The Kelvin wave is taken to be travelling in the negative y-direction, i.e. due south, parallel to the dashed line in Fig. 3.4. Resultant residual flow fields are shown in Fig. 4.5. The relative amplitudes of the two boundary conditions are given by rjk = 1.9T] S where rjk is the amplitude of the Kelvin wave at the western grid boundary and rj is the maximum standing wave e amplitude. There is a relative error between the maximum velocities shown in Fig. 4.5 of approximately 3.2 %. It is possible to calculate the energy density for both types of waves used as boundary conditions and it was found that the energy density of the Kelvin wave was approximately 3.5 times that of the standing wave (see Appendix D). It is possible to interpret this result to mean that the spatial residual velocity profiles are reasonably similar but the standing wave is 3.5 times more effective at generating the residual response. It should be noted, however, that the magnitudes of the background tidal flow for the Kelvin wave and the standing wave are equivalent and that this factor may also contribute to the small relative difference between the residual velocities. In any case, the two types of boundary conditions give similar residual flow structures and magnitudes of the type that are expected around finite height seamounts (GARREAU and MAZE, 1992; FREELAND, 1994). 4.3 Effect of seamount geometry The purpose of making seamount comparisons is to further explore the relationship of the residual velocities to the parameters describing the seamounts. Chapter 4. Residual Velocity Comparisons 25 69 r Standing W a v e B C Kelvin W a v e B C 20 E o w c 15 - po C D c E o O >> o Vel o 10 5 0 -5 0 20 40 60 80 100 D i s t a n c e from s e a m o u n t c e n t e r (km) 120 Figure 4.5: Residual u velocity solutions. Kelvin wave boundary condition versus a standing wave boundary condition. The maximum relative error between these two flows is 3.2 %. Table 4.2 outlines some of the more recent studies which have explored this subject. Some studies have produced observations for specific seamounts (e.g. Freeland reported current vector information from stations directly over Cobb seamount) which are very useful for validating numerical models. Other studies have explored parameter ranges for variables such as 8 (e.g. C H E N and B E A R D S L E Y , 1995; C H A P M A N and H A I D V O G E L , 1992) and cr (e.g. H A I D V O G E L et al, 1993; Z H A N G and B O Y E R , 1993; SHEN, 1992) and some have sought out results in the laboratory (e.g. B O Y E R et al, 1991). Smith (1992) generalized to finite height topography from a quasi-linear based analysis. The all encompassing goal of these studies could be taken to be an attempt to describe and classify flow regimes over isolated topography. The aim, therefore, of this next section fits in well with that intent. The F W model is used to examine the effects of seamount geometry on the residual flow field. To accomplish this, seamounts are first classified into groups based on their Chapter 4. Residual Velocity Comparisons 70 geometrical characteristics (for example seamounts of the same height but with differing base widths - shown in Fig. 4.6). Following this, residual flows are calculated for each seamount using the F W model and then conclusions are drawn about relationships between the flow field and the geometry based comparisons between the model results. Three different groups (or sets) of seamounts are presented in this section and the outcome of the numerical model results are discussed. 4.3.1 Set I: varying seamount base width This is the first section devoted solely to the comparison of residual velocities from different seamounts which are distinguished by the geometrical parameters. In this section, the seamounts have different base widths, r - The seamounts compared in this section d 2000 I 1 r- 1 1 1 1 Distance (km) Figure 4.6: Seamount Profiles for residual velocity comparison. To investigate whether seamounts of the same height but with a wider base have smaller maximum residual velocities. The dashed line appears at a height of 0.02^ and is defined as the height at which Td is measured. Profiles A, B and C are defined by (4.1) with r set to 0, 12.5 and 25 km respectively. a Chapter 4. Residual Velocity Comparisons 71 Study / Seamount Goldner and Chapman (1996) Gaussian 0.9 0.001 - 0.036 mean + tidal 0.67 - 0.83 0.06 1.36 0.8 - 1.0 0.068 mixed 0.9 0.0008 0.6 - 1.45 0.875 0.003 - 0.03 0.2 - 2.4 0.5 0.1 0 0.1 - 0.9 0.5 - 3 0 Smith (1992) Gaussian 0.875 0.025, 0.1 0 Shen (1992) Parabolic, Gaussian 0.75 0.1, 0.5 0.34, 0.683, 1.36 Boyer et al. (1991) Cos 0.857, 0.875 0.026, 0.051, 0.102 0.5, 1, 2 Chen and Beardsley (1995) Flat topped Cosine Freeland (1994) Cobb Haidvogel et al. (1993) Gaussian Zhang and Boyer (1993) Cos 2 Thompson and Flierl (1993) Cylindrical Chapman and Haidvogel (1992) Gaussian, Cos, Tanh 2 Table 4.2: Some recent studies which explore parameter relationships for flow over isolated topography. Chapter 4. Residual Velocity Comparisons 72 are shown in Fig. 4.6 and are defined by the height function hf(r) = { 1800 0 <r< r ~ ~ a 1800exp(-(^f) ) (4.1) r >r 2 a where r determines the distance from the center to the "shelfbreak" and was chosen a so that rd was 25.0, 37.5 and 50.0 km respectively. A table is presented at the end of this chapter which summarizes the physical parameters of all seamounts used in this and subsequent sections (Table 4.5). For these profiles, the slope along the side of each seamount is the same, but the position of the maximum slope is at different distances from the origin. Furthermore, the height for each seamount is the same (i.e. 8 = 0.9 is a constant for this set of seamounts). In Chapter 3, equation (3.37) was derived for the azimuthal residual component which stated that the residual velocities were proportional to the aspect ratio hd/rd- The conclusion for the plane parabolic seamount was, therefore, that in a situation where the deep water depth does not change, wider seamounts will tend to have weaker residuals. It is possible to use this set of seamounts to examine whether this relationship holds true for Gaussian seamounts. Fig. 4.7 shows residual velocity profiles for all three seamounts. Again, these results are plotted along the same axis that was used in Chapter 3, shown as a dashed line on Fig. 3.4, but are only.plotted out to a distance of 60 km. As can be seen from Fig. 4.7, the magnitude of the maximum residual velocity decreases as the width of the seamount base increases. There is another quantitative way of examining these results. The energy of the residual component within the area r < rb is the sum of the kinetic energy KE = -p / / v dzd6dr (4.2) / zdzdOdr Jo Jo Jo (4.3) 2 2 Jo Jo J-h and the potential energy PE = pg Chapter 4. Residual Velocity Comparisons 73 4 Profile A Profile B -+-Profile C Q 3.5 3 2.5 f ^ o o o ffl > 1.5 1 0.5 0 -0.5 -1 0 10 20 30 Distance (km) 40 50 60 Figure 4.7: Residual velocity comparisons for Set I: Varying rj. The seamounts are all the same height but the seamount base width is 25, 37.5 and 50 km respectively for Profiles A , B and C. In this case, the maximum residuals decrease as the size of the seamount base increases. where p is fluid density, ru is the radius at which the boundary condition is applied, h is depth, 77 is sea height, v is velocity, g is gravitational acceleration and the domain is in cylindrical polar co-ordinates (r, 9,z). Therefore the total energy is equal to E = TXp I [(h + n)v + grj ]dr 2 Jo 2 (4.4) The value of this integral (excluding the factor of Trp out front) has been calculated from results of the three seamount profiles and are shown in Table 4.3. It can be noted that the contribution of the potential energy to the total energy was less than 1% in all cases for the time-mean residuals (unlike the equipartitioned Kelvin wave forcing). It would appear that the energy contained within the radius rj, decreases as the seamount base width increases. That is, a wider seamount transfers slightly less energy from the tide to the residual flow field. Another feature of Fig. 4.7 is the flow reversal inside the anticyclonic circulation Chapter 4. Residual Velocity Comparisons Profile A B C 74 (km) EnergyIntegral 25.0 37.5 50.0 2.72 2.59 2.01 Table 4.3: The value of the integral given in 4.4 for the seamount profiles shown in Fig. 4.6 Energy contained within the radius r\, decreases with increasing seamount base width. which occurs for profiles B and C. Physically, this represents a smaller (in both width and magnitude) cyclonic flow within the anticyclonic flow at the top of the seamount. At the peak of the seamount, advection is relatively small compared to bottom friction. In this region, bottom friction is transferring energy from the order zero to the order one terms. It causes the residual sea height to form a slight peak at the rim of the seamount (see Fig. 3.11). Due to the relative weakness of the advection in this region, the flow near the peak is nearly geostrophic and, therefore, the slight increase in sea height is balanced by a smaller (relative to velocities on the slope) cyclonic circulation. On the slope, the flow is ageostrophic, dominated by advective forces, which results in the anticyclonic tidally rectified residual. The typical tidal excursion near the rim is approximately 2 km, less than 10 % of the topographic length scales for these seamounts. Note that the tidal flow over the seamount is approximately 33 cm/s and the maximum residual is about 4 cm/s. Chen and Beardsley (1995) had maximum values of 80 cm/s and 2.4 cm/s for tidal and residual flows over their flat topped cosine shaped seamount (previously discussed in § 3.2.2.2). While not exactly the same, these flows are of the same order of magnitude. Chapter 4. Residual Velocity Comparisons 4.3.2 75 Set II: varying flank slope The previous set was used to investigate a property derived from the analytic solution of Chapter 3. The set in this section leads to another relationship between the seamount parameters and the residual flow field. These seamount profiles are shown in Fig. 4.8. Profiles A , D and E are respectively defined by the height functions 2000 Profile A Profile D -4-Profile E -a - 20 30 Distance (km) 40 50 60 Figure 4.8: Seamount Profiles for residual velocity comparison. To investigate where the maximum magnitudes of the residual velocities occur. Note that profile A is the same as that which appears in Fig. 4.6 hf^r) = 1800 e - ( r/125 ) (4.5) 2 hf {r) = 1800 e"( / - ) / p 12 5 4 (4.6) 4 2 hf {r) = 1800 e-^/^.sr/ie 3 ( 4 7 ) Note that hfi is the same as the seamount in the previous section where r was set to a 0. From Fig. 4.8 it can be seen that r , h and 8 (the fractional seamount height) are d d constant for this set (r = 25 km, h = 2000 ra., 8 = 0.9). However, the maximum d d Chapter 4. Residual Velocity Comparisons 76 magnitude of the slope is different for each seamount as is the distance of this maximum from the center of the seamount. Fig. 4.9 shows residual velocity profiles for this set and two items are immediately apparent. 0 10 20 30 Distance (km) 40 50 60 Figure 4.9: Residual velocity comparison for Set II: Varying Slope. Maximum residuals appear to be related to the position of the "shelfbreak". 1. At first glance, it would appear that the maximum residuals decrease as the slope increases, similar to the previous set of seamounts with increasing base width, even though for this set r<i is constant. 2. The position of the maximum magnitude of a residual field is coincident with the "shelfbreak". Consider item number 1. Chen and Beardsley (1995) report in their study of tidal rectification over a finite height symmetric bank that, counter to point 1 above, the strength of the residual flow should decrease with decreasing slope. In Fig. 4.8 it can Chapter 4. Residual Velocity Comparisons 77 be seen that the base width is constant for this set and slopes are changed by adopting non-Gaussian shapes. Consider the seamounts in Fig. 4.10. These seamounts are denned by the height functions hf {r) = 1800 e-( ') r/r 2 4 (4.8) where r is a scalar equal to 12.5, 19.2 and 25.6 for Profiles A, M and N with base widths B of 25.0, 37.5 and 50.0 km respectively. That is, 8 and hd are constant (equal to 0.9 and 2000 m respectively) and is changing. The slopes of these seamounts are changed in a manner similar to the Chen and Beardsley (1995) results in that the shapes remain Gaussian but the base width is altered. The residual profiles for this set of seamounts are shown in Fig. 4.11. In this case it can be seen that the maximum residuals decrease with decreasing slope consistent with the Chen and Beardsley (1995) results. Further, as rd increases, the magnitude of the maximum residual decreases as might be expected from the analytical solution. Note, however, that equation (3.37) is for a fixed geometry and does not account for the slope variations seen here. By itself, the result from Fig. 4.11 would add nothing to the body of work already completed on the effects of changes in slope on the residual flow. However, the results from Fig. 4.9 do. Turning this discussion around, the Chen and Beardsley results say that maximum residuals increase with increasing slope, however, the results from Fig. 4.9 indicate that the maximum residuals do not necessarily increase with increasing slope. Next, consider point 2 which remarks that the maximum magnitude of the residual flow is coincident with the "shelfbreak". We have previously used the term "shelfbreak" when examining the seamount profiles in Fig. 4.6. In that case, the term "shelfbreak" was used to refer to the point where the seamount profile changes from its maximum height to its sloping side. This occurs at 0, 12.5 and 25.0 km respectively for Profiles A, B and C. Examining Fig. 4.7, one can see that the peak residual velocities do not occur Chapter 4. Residual Velocity Comparisons 78 2000 Distance (km) Figure 4.10: Seamount Profiles with decreasing slope. These profiles were created in a manner similar to those in the study done by Chen and Beardsley (1995). Note that profile A is the same as that which appears in Fig. 4.6 at those "shelfbreaks" but at a distance farther out. To examine this feature, consider the governing equations of the plane parabolic system of Chapter 3. The analytical frequency expansion discussed in § 3.1.1.3 dictates the form of the higher order equations. For example, the momentum equations for the residual flow are forced on the right side by advection, whereas, for the order e terms which oscillate at the frequency of the zeroth order terms, the flow field is balanced exclusively by bottom friction (see § 3.1.1.4). Consider the 6 independent residual terms ((3.9) in § 3.1.1.4). In general, advection is a much bigger term than sea height slope, hence, the radial balance is normally between the residual flow and advection. Advection for a wave with velocity u, has the form u VM. In cylindrical polar co-ordinates, the maximum of this term can be written 3 1d max \u (—+-—)UM \ M2 2 (4.9) Chapter 4. Residual Velocity Comparisons 79 4 3.5 3 _ 2 5 "at ^ ? •Si. >, I 1.5 > 1 0.5 0 -0.5 0 10 20 30 Distance (km) 40 50 60 Figure 4.11: Residual velocity comparison for seamounts with slopes changing in a manner similar to that found in Chen and Beardsley (1995). where the zeroth order forcing term is taken to be a wave oscillating at the M frequency. 2 The |x| specifies the norm of X. From the discussion in § 3.2.1.1, for the seamount, we know that there is a jump discontinuity in -§^UM 2 PLANE PARABOLIC and, therefore, this term determines where the maximum advection occurs with respect to R. Furthermore, from (3.42) it is known that this discontinuity is related to the discontinuity in the topography. Therefore, for the plane parabolic seamount, the maximum forcing of the residual occurs at the break in the topography (refer to Fig. 3.2). Again, this radius for the plane parabolic seamount could be referred to as the "shelfbreak", however, a Gaussian seamount has no such topographic discontinuity. In § 4.1, examination of the linear velocity solutions (Fig. 4.3) led to the speculation that velocity was proportional to B y conservation of mass, for a system that L/H(R). satisfies a rigid lid condition, it is not unreasonable to assume that H[R)UM 2 = H UF D (4-10) Chapter 4. Residual Velocity Comparisons 80 where v,f is the uniform velocity over the flat bottom far away from the seamount. Therefore UM 2 = TTTh[r) (-) 4 U The same proportionality can be derived from the continuity equation ((3.9) in § 3.1.1.4) assuming that the tidal flow can cross topographic contours {\VQ \ ~ a n d o > 1. Consider again maximizing the advection term forcing the residual velocities (4.9). The point at which the maximum advection occurs becomes, through (4.11), the radius at which is satisfied. This result is similar to Equation (7) of Garreau and Maze (1992). Garreau and Maze derive their result, which is based on a dynamic balance between advection and bottom friction, using an alternative method to harmonic analysis. The validity of (4.12) is supported by these two independent derivations. On the other hand, Wright and Loder (1985) derived a result which assigns the position of the maximum residual to the position of the maximum of l/h(r) dh(r)/dr. This result is due to their choice of characteristic horizontal length scale, which they have set equivalent to this term. Wright and Loder (1988) give essentially the same result as (4.12). Fig. 4.12 shows the residual velocity profiles for the Profile A and Profile D seamounts versus their respective terms defined by (4.12). Note that the maximum of the advection term, given that (4.11) is true, occurs at nearly the same radius where the residual velocities are a maximum and that this relationship holds true for the non-Gaussian seamount, Profile D , as well. Therefore, it may be said that the maximum residuals occur at the "shelfbreak" only if the "shelfbreak" is defined to coincide with, the radius at which (4.12) is satisfied. For the rest of this study, L shall be considered to mean this radius and it is defined to be the shelfbreak. Chapter 4. Residual Velocity Comparisons 81 a) Advection function Residual Velocities for Profile A -0.5 10 20 30 Distance (km) 40 50 60 2.5 Advection function Residual Velocities for Profile D 2 h 1.5 h 0.5 -0.5 h , ; 10 20 30 Distance (km) 40 50 60 Figure 4.12: Comparison of the advection term denned by (4.12) and the residual velocity profile given that u oc 1/h for a) Profile A and b) Profile D . Note that the advection terms have been scaled so that they are comparable in magnitude to their respective velocities. Chapter 4. Residual Velocity Comparisons 82 With this set, one could also briefly examine a subinertial frequency response to confirm that results are consistent with work done by previous researchers. There have been numerous studies done on subinertial frequency enhancement of flow over seamounts and, from these, one expects a large enhancement of the subinertial flow if the forcing frequency is significantly close to a resonant free-wave mode (e.g. Lc-NGUET-HlGGINS, 1970; H U T H N A N C E , 1974; C H A P M A N , 1989; BRINK, 1990). Fig. 4.13 shows residual velocities forced by a Kelvin wave oscillating at the Ki frequency which has the same amplitude as the M boundary condition used in the 2 previous example. Note that comparison of the scale of this figure to that of Fig. 4.9 shows that, as expected, the K\ response is much greater (in this case by a factor of ~ 34) than the M response. Recall that subinertial enhancement is generated by the resonant 2 response of trapped topographic Rossby waves to the tidal forcing. According to Brink (1990), the subinertial enhancement predicted here is larger than would be expected at Fieberling Guyot where the topography is such that the frequency of the first trappedwave mode is approximately 5.7 x 10 -5 s . However, it is possible that the Gaussian _1 topography has higher frequencies at which resonance is expected to occur - this can be somewhat confirmed by following Chapman's (1989) method for calculating the trapped- wave mode frequencies for parabolic seamounts. Chapman found that resonant excitation does not require an exact match between the subinertial forcing frequency and the excited trapped-wave mode and the results shown in Fig. 4.13 for the diurnal frequency are within the order of magnitude that he predicts. Note also, that in Fig. 4.13 the peak residual response is shifted towards the center of the seamount; not at the point of maximum advection. This is due to the fact that the strong tidal and residual response of the subinertial wave is beginning to break the "weakly nonlinear" constraint of the previous analysis. Furthermore, there is no reversal of flow at the top of the seamount for Profiles D and E as there is in the superinertial Chapter 4. Residual Velocity Comparisons 83 case. 120 Profile A Profile D Profile E --E>—• - 1 _oo > -20 0 10 20 30 Distance (km) 40 50 60 Figure 4.13: Residual velocity comparison for Set II: Varying Slope. External forcing is a subinertial Kelvin wave of frequency Ki. The enhanced subinertial response is much greater than the superinertial M response shown in Fig. 4.9. 2 4.3.3 Set I I I : v a r y i n g seamount height Many numerical models for finite height seamounts have focussed on one particular seamount while varying other parameters such as the Rossby number, the ambient background flow or the temporal Rossby number. The number of studies that use variable fractional seamount heights is somewhat less, but there are a few notables. For example, Huppert and Bryan (1976) use a range of fractional seamount heights between 0.019 and 0.2 for their studies of potential vorticity conserving systems. Chapman and Haidvogel (1992) conduct their numerical studies of Taylor column formation for a full range of fractional seamount height: 0.1 < S < 0.9. Chen and Beardsley (1995) comment on the effect of changing S from 0.67 to 0.83 in their numerical studies of tidal rectification over finite height symmetric banks. They found the horizontal extent of the residual flow to Chapter 4. Residual Velocity Comparisons 84 be independent of the bank height. Fig. 4.14 shows the seamounts that will be compared in the next analysis. This set of six seamounts are all Gaussian and have the same values for and hd, however, S ranges from 0.7 to 0.95. Profiles H, G , A , J , K and F are defined by the height function 2000 I -i 1 1 1 1 1 0 10 20 30 Distance (km) 40 50 60 Figure 4.14: Seamount Profiles for residual velocity comparison. To investigate the effect of different seamount heights. Note that profile A is the same as that which appears in Fig. 4.6 hf {r) = 2000 8 - ( ' / - ) 12 5 e 5 2 (4.13) where 5 is respectively defined to be 0.7, 0.8, 0.9, 0.917, 0.933 and 0.95. Again, profile A retains the same definition as given previously. Fig. 4.15 shows residual velocity profiles for these seamounts. As can be seen from this figure, the magnitude of the maximum residual velocity increases with increasing seamount height. However, it is apparent that this relationship is not linear. Fig 4.16 is a log-log plot of the maximum velocities versus the fractional seamount height. Since the line is apparently not linear, it is expected that the maximum residual velocities may Chapter 4. Residual Velocity Comparisons 85 Profile F Profile K Profile J Profile A Profile G Profile H E o CD > 20 30 Distance (km) 40 50 60 Figure 4.15: Residual velocity comparisons for Set III: Varying 8. Peak velocity increases with increasing seamount height. increase exponentially with increasing seamount height. Further inspection using a curve fitting routine reveals that for the four cases with 8 > 0.9, the maximum velocities closely follow the exponential relationship 0.01 e " 8 8 8 2 ±0.03. (4.14) It should be noted that the resultant velocities for seamount profiles G and H (8 — 0.8 and 0.7 respectively) fall within the noise level predicted for this grid in § 2.3 and summarized in Table 2.3.2. Furthermore, the maxima of the velocity profiles are shifted in towards the center, as is expected from a calculation of (4.12) for Gaussian seamounts of varying heights. Chen and Beardsley (1995) also found that maximum residuals increase as the fractional seamount height increases, but did not comment specifically on the strength of the response. Again, briefly consider the subinertial response. Fig. 4.17 compares the superinertial (u) = M ) response of seamount A (8 = 0.9) to the subinertial (a; = K\) response of 2 Chapter 4. Residual Velocity Comparisons 86 100 _o "3 =3 a 3 E 5 E 0.1 0.6 0.7 0.8 0.9 1 5 Figure 4.16: A log-log plot of the maximum residual velocities versus 5, the fractional seamount height. seamount H (5 = 0.7). In this case it can be seen that the responses are similar in magnitude and thus argue that even though the subinertial response is greater than the superinertial response for a given seamount, the latter is still an important dynamical feature. There are banks where the dominant tidal constituent is superinertial, such as Georges Bank off the coast of Maine (NAIMIE ET al., 1994) and further, there are topographical features situated in regions where mixed tides dominate, such as Cobb seamount where the magnitude of the K tide is half that of the M . x 2 It is possible to add to the energy discussion in § 4.3.1. Energy calculations using (4.4) for this set of seamount profiles have been added to the original Table and presented in Table 4.4. In addition to results for Set I, Table 4.4 also indicates that the energy contained within the radius r increases as the height of the seamount increases. b Chapter 4. Residual Velocity Comparisons 4 87 1 •• 1 3.5 Velocity (cm/s) 3 2.5 2 1 Profile A -«— Profile H ~^«- - il \ \ •i it '• \ \ 1 \\ 1.5 1 i + ^ \ \ \ ± \ V 0.5 \ V^v. 0 -0.5 \ \ 0 ' t 10 20 30 Distance (km) 40 50 60 Figure 4.17: Residual velocity comparisons of a superinertial (a; = K\) response over seamount A to the subinertial (a; = M ) response over seamount H . 2 4.4 Summary Bathymetry for Cobb Seamount (situated off the coast of Washington) was compared to a Gaussian curve to address the question of usefulness of a Gaussian seamount profile. In addition, parameter values for the plane parabolic solution of Chapter 3 were compared to a flow solution for a Gaussian seamount and to those of Cobb Seamount. Compared to the plane parabolic seamount, the numerical model results for the Gaussian seamount imply that it supports a significantly smaller anticyclonic eddy. Further, the Gaussian did not show a smaller cyclonic eddy over the top of the seamount as the plane parabolic seamount did. The Gaussian seamount has a residual field which appears to have a smaller horizontal extent. Inspection of the flow regimes compared to that of Cobb Seamount showed reasonable consistency and it was concluded that a Gaussian seamount shape would give reasonably realistic solutions. Residual velocity profiles from the standing wave boundary condition of Chapter 3 Chapter 4. Residual Velocity Comparisons 88 Set I Profile ra A B C (km) 25.0 37.5 50.0 Set III Energy Integral 2.72 2.59 2.01 Profile 8 F K J A G H 0.95 0.933 0.917 0.9 0.8 0.7 Energy Integral 12.40 6.63 4.13 2.72 1.10 0.99 Table 4.4: The value of the integral given in (4.4) for the seamount profiles shown in Figs 4.6 and 4.14. Energy contained within the radius r decreases with increasing seamount base width and increases with increasing seamount height. b were compared to those forced by a Kelvin wave boundary condition. The two boundary conditions give similar residual flow structures and magnitudes (maximum error of 3.2 %) of the type that is expected around tall isolated seamounts. For reference, a summary of physical parameters of all seamounts used in this chapter is presented in Table 4.5. Profile A B C A D E M N L (km) (km) Set I 2.0 25.0 12.9 37.5 25.3 50.0 Set II 2.0 25.0 7.9 25.0 11.9 25.0 2.8 37.5 3.8 50.0 5 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 Profile F K J A G H L TD (km) (km) Set III 1.2 25.0 1.5 25.0 1.6 25.0 2.0 25.0 2.7 25.0 3.5 25.0 8 0.95 0.933 0.917 0.9 0.8 0.7 Table 4.5: Summary table of physical parameters for Sets I, II, and III. Chapter 4. Residual Velocity Comparisons 89 Numerical experiments using seamounts with varying seamount base width showed that the magnitude of the maximum residual velocity decreased as the width of the seamount base increased consistent with the analytical result (3:37). Experiments also showed that there can be a smaller (in both width and magnitude) cyclonic flow within the anticyclonic flow at the top of the seamount. On the slope, the flow is ageostrophic, dominated by advective forces, which results in the anticyclonic residual. Spurred by the residual flow fields of a set of seamounts with varying seamount slope, an analysis of the governing equations showed that the maximum residuals occur at the radius at which the maximum advection also occurs. Further, the maximum advection occurs at the radius at which max i 1 /i(r) d 9h(r). — or is satisfied. It may be said that the maximum residuals occur at the "shelfbreak" only if the "shelfbreak" is defined to coincide with the radius at which this condition is satisfied. In addition, it was found that an increase in slope does not necessarily generate an increase in the magnitude of the residual flow maximum. Numerical experiments using seamounts with varying height showed that the magnitude of the maximum residual velocity increased nonlinearly with increasing seamount height, and that the energy contained within the radius r increased as the height of the b seamount increased. Chapter 5 Tracking of inert and biological particles The long term behaviour of water parcels and particles in a time varying flow has implications for nutrient and contaminate transport problems, plankton dispersal and survival questions and some search and rescue operations. These problems require, in part, an examination of particle trajectories in time. For example, consider that there is an oil spill in Juan de Fuca Strait. One solution would be to have an existing numerical model designed for that region. It should include all the dynamical features, such as wind effects, tides and stratification, and be able to predict exactly what is going to happen in the next 48 hours including the dispersion and final destination of the oil (both through sinking and evaporation). But how many of these models exist? And do they cover all the areas where spills are most likely to occur? While this may be the one of the best solutions, it is not a very practical one. In the interim, it may be beneficial to shore* up the framework surrounding the general principles of particle behaviour. Specifically, this chapter will explore the resultant delaying or trapping of particles in the neighbourhood of a tall, isolated seamount. Trapped particles are in evidence through the appearance of closed loops in the particle paths, a behaviour which may be reinforced by the anticyclonic residual flows discussed in the previous chapter. Recall, however, that the existence of an Eulerian eddy in a non-steady flow does not necessarily lead to a Lagrangian eddy (CHENG and CASULLI, 1982; LODER ET AL, 1988; FOREMAN ET AL, 1992) and, therefore, particle tracking is a tpun intended. 90 Chapter 5. Tracking of inert and biological particles 91 useful diagnostic tool when studying longterm particle behaviour. In this chapter, inert particles (acting as passive tracers) are tracked in two and three dimensions. Two-dimensional tracking results are presented in § 5.1. The method by which a third-dimension is introduced into the barotropic model is discussed § 5.2 and presented in § 5.3. Following that, an investigation into the effect of vertical migration on particle motion is discussed. 5.1 Inert particle t r a c k i n g i n two-dimensions Using the seamount sets discussed in the previous chapter, it is possible to investigate a number of relationships between physical seamount parameters and the potential trapping of particles. For example, the following questions are of interest: • Does trapping decrease as the fractional seamount height increases? • Does trapping decrease as maximum residual velocities increase? • Does trapping increase as the distance to the shelfbreak increases? These questions, and others, are explored in this chapter. Particle position x — (x,y) is a function of the initial position a and time t. Particle paths are calculated using dx — = u{x{d,t),t) (5.1) and a 5th order Runge-Kutta algorithm originally developed by Baptista et al. (1984) which has since been modified for this application. It is known that particles in an oscillating flow can be transported solely by Stokes flow (LONGUET-HlGGINS, 1969), however, for the systems discussed in the previous sections, the Stokes flow is not large enough to move particles away from the region of the seamount in any reasonable amount of time. Following Foreman et al. (1992), if the Chapter 5. Tracking of inert and biological particles 92 oscillating part of a flow field consists of a single tidal component, u (x,y,t), t of period T, then the Stokes flow can be defined to be the time average of that component: 1 rto+T u {x ,y ,to) s 0 = 0 ^ / u (x(s),y(s),s) t 1 Jt ds (5.2) 0 (See Foreman et al. (1992) for a brief history of this particular formulation and an examination of the case of more than one tidal constituent). Equation (5.2) states that the Stokes flow, us, of a parcel of water is dependent on the initial position (a:o,yo) and release time, io, where t falls within the range [0, T]. Therefore, from an initial position, 0 a water parcel may travel to any number of final positions (over one period) depending on the phase of the tidal component at the release time. A collection of these final positions comprises a locus of displacements and, thus, Foreman et al. define a mean Stokes flow: 1 u {x , s 0 y) 0 = -T7 N Yl s{x , yo,U) u 0 (5.3) i=i with ti = iT/N. For a flow field consisting of one oscillating frequency plus a residual flow, the Lagrangian and Eulerian residuals can be defined through the familiar relationship: Lagrangian residual = Eulerian residual + Stokes drift which, using the mean flows, can be mathematically expressed as: UL{x ,yo) 0 = u (x ,y ) r 0 0 + u (x ,yo). s 0 (5.4) Values for the mean Stokes and Lagrangian flow fields were calculated for one particular seamount (Profile B) to illustrate the relative sizes of these fields. Further averaging has been done giving values for three different regions {(0 < r < L), (L < r < rd), (r > rd)} which are summarized in Table 5.1. Note that for the Stokes flow or the Lagrangian flow without a mean background flow (i.e. the residual field is due to nonlinear tidal interactions only and does not include a Chapter 5. Tracking of inert and biological particles 93 Velocity (cm/s) u~s UL UL (nonlinear) (with background flow) 0 <r < L 0.16 0.27 0.27 L <r <r r > ra 0.095 0.42 2.41 0.012 0.26 13.61 d Table 5.1: Values for the mean Stokes and Lagrangian flow (with and without a background mean flow) in three annular regions for Profile B . mean Eulerian flow) average flows are below 0.5 cm/s. That is, with these flows, it would take at least 174 days for a parcel to pass by this seamount (which has a base diameter of 75 km). On the other hand, using the average flow of 13.61 cm/s for the case where there is a mean background flow, a particle may take as little as 6.37 days to pass by. A mean background geostrophic flow was therefore added to the dynamics to create a means for particles to be lost from the system in a reasonable amount of time. However, the inclusion of this background mean flow implies that there are additional Taylor column effects. If the incoming mean flow is strong enough, then trapping by a Taylor column will not occur. Chapman and Haidvogel (1992) present numerical results for different seamount shapes giving critical Rossby numbers above which Taylor columns are not expected to occur. Using these results, one can determine that the critical incoming time mean flow must be greater than approximately 60 cm/s to prevent the Taylor column from forming and for all the particles to be swept downstream. That is, for the parameters set forth for the seamounts in the previous sections, a uniform flow field can set up a Taylor column whose width will decrease with increasing flow strength and at approximately 60 cm/s the Taylor column will collapse and be swept downstream. However, the incoming flow strength used in the following numerical trials is at most 17 cm/s (chosen to approximate mean flows at Cobb Seamount (FREELAND, 1994)). Tests over a range of small mean background flow magnitudes (less than 17 cm/s), showed Chapter 5. Tracking of inert and biological particles 94 very little difference in overall particle behaviour. Further, examination of the flow with and without the mean background flow show that the flow within the radius r is not d significantly impinged upon by the background flow (due to the nature of Taylor columns at small Rossby numbers). Thus, the background flow is not detrimental to potential particle trapping within the radius r although it adds a significant method by which d particles may be lost from the region around the seamount. The dynamics of these systems do not include three-dimensional effects such as vertical motions, baroclinic or stratification effects. And although Komen and Riepma (1981) found that wind effects dominate residual tidal vorticity generation when the wind speed is greater than approximately 5 m/s (BOYER et al., 1991) wind effects are not included. Further, only one tidal frequency has been examined which precludes conclusions about mixed tides. However, while it may be premature to suggest that these particle paths represent reality, they are qualitatively similar to laboratory and numerical results shown in Fig. 4 of Davies (1972), Fig. 7c of Boyer and Zhang (1990) and Fig. 5b of Chapman and Haidvogel (1992). For reference, the qualitative similarity to the Boyer and Zhang (1990) figure implies that the system described herein corresponds to their "fully attached flow" which they find at very low Rossby numbers (that is, there are no concurrent leeside eddies). Fig. 5.1 is a plot of typical pathlines calculated from one of the seamounts previously discussed and indicates that the largest particle movement (of particles which remain in the region) occurs between the shelfbreak and r , on the seamount flank as is d expected from the structure of the residual flow field. Furthermore, it can be seen that particles initially within the radius r are being lost from the seamount. Note that since d this system is barotropic, pathlines are depth-independent. However, for comparison to the 3D studies in § 5.3, one could consider these particles to be at the surface. The next four figures (Figs 5.2, 5.3, 5.4, and 5.5) show particle pathlines for Seamount Profiles B , C, D , E , G, H , A and F . Particles were tracked for 93.15 days and only the Chapter 5. Tracking of inert and biological particles -60 -60 r 1 -40 95 1 -20 0 Distance (km) 1 20 40 60 Figure 5.1: Particle paths for Seamount C (shown in Fig. 4.6) for a 30 day period. Overall velocity field includes background mean flow (flowing northward - See Fig. 3.4), oscillatory flow and residual field. Outer dashed circle drawn at radius rj] inner dashed circle drawn at the shelfbreak L. The o shape denotes initial positions. Particles are shown at the end of every tidal oscillation (in effect, the tidal oscillations are filtered out). pathlines for those particles which remained in the region are shown. A l l the initial positions are shown (with a o) and thus the initial positions from which particles were lost can be easily seen. Inner and outer dashed lines represent the shelfbreak and seamount base respectively. Using the seamount profiles of Set I: Varying and Set II: Varying slope, particle pathlines are examined to explore the question "Does trapping increase as the distance to the shelfbreak increases?". Table 5.2 summarizes statistics derived by initially releasing particles over the top of each seamount in Sets I and II and tracking them for a 93.15 Chapter 5. Tracking of inert and biological particles 96 E CD O cc -10 0 10 Distance (km) 40 E CD O c CO -20 0 Distance (km) 20 60 Figure 5.2: Particle paths for Seamount profiles B and C for a 93.15 day period. Overall velocity field includes background mean flow, oscillatory flow and residual field. Outer dashed circle drawn at radius r^; inner dashed circle drawn at the shelfbreak L. The o shape denotes initial positions. Only the pathlines of the particles still in the region after 93.15 days are shown. Chapter 5. Tracking of inert and biological particles 97 E CD O c CO -40 -30 -40 -30 -10 0 10 Distance (km) E CD O c co -20 -10 0 10 Distance (km) Figure 5.3: Particle paths for Seamount profiles D and E for a 93.15 day period. Overall velocity field includes background mean flow, oscillatory flow and residual field. Outer dashed circle drawn at radius rj; inner dashed circle drawn at the shelfbreak L. The o shape denotes initial positions. Only the pathlines of the particles still in the region after 93.15 days are shown. Chapter 5. Tracking of inert and biological particles 98 CD O c CO *•—• CO b -40 -30 -40 -30 -20 -10 0 10 Distance (km) o c co w Q -*—> -10 0 10 Distance (km) Figure 5.4: Particle paths for Seamount profiles G and Ff for a 93.15 day period. Overall velocity field includes background mean flow, oscillatory flow and residual field. Outer dashed circle drawn at radius r<j; inner dashed circle drawn at the shelfbreak L. The o shape denotes initial positions. Only the pathlines of the particles still in the region after 93.15 days are shown. Chapter 5. Tracking of inert and biological particles 99 E 0 o c ns "5 b -10 0 10 Distance (km) 40 -10 0 10 Distance (km) 40 o c B w b Figure 5.5: Particle paths for Seamount profiles A and F for a 93.15 day period. Overall velocity field includes background mean flow, oscillatory flow and residual field. Outer dashed circle drawn at radius r ] inner dashed circle drawn at the shelfbreak L. The o shape denotes initial positions. Only the pathlines of the particles still in the region after 93.15 days are shown. d Chapter 5. Tracking of inert and biological particles 100 day period. The time mean background flow is approximately 15 cm/s. Note that the number of particles released within the region r < r is always 89, irrespective of the d actual value of TD- Sensitivity tests (conducted by successively increasing the number of initial drogues) showed that these values of the percentage of trapped drogues are accurate to about 2 %. Profile L RD (km) (km) Initial Drogues A B C 2.0 12.9 25.3 A D E 2.0 7.9 11.9 Set I 25.0 37.5 50.0 Set II 25.0 25.0 25.0 M N 2.8 3.8 37.5 50.0 % Trapped Drogues 89 89 89 49 62 65 89 89 89 49 61 63 89 89 56 58 , Table 5.2: Here, L the shelfbreak is taken as that value which maximizes (4.12). Initial Drogues refers to the number of drogues that were released within the radius RD for each seamount. Percentage Trapped Drogues is calculated using the number of drogues not lost to the mean background flow after 93.15 days (i.e. still within at this time). Examination of Table 5.2 suggests two similar results. First, consider results from Set I (A, B , and C). For this set, both the seamount height and slope remain constant while both the shelfbreak, L, and base width, RD, increase in size. This case indicates that as the distance to the shelfbreak or base width increases, the percentage trapping increases. Further, consider Set II where the shelfbreak increases but does not (A, D and E ) . It would appear that trapping increases at a rate similar to Set I. However, when examining the other results of Set II (A, M , and N), it would appear that if L Chapter 5. Tracking of inert and biological particles only changes slightly and 101 increases, trapping still increases but not by as much as the previous two cases. Therefore, it may be said that trapping increases if the distance to the shelfbreak or the base width increases, and increasing the distance to the shelfbreak has a slightly larger effect. Note that in the last two cases, trapping increases whether the slope increases or decreases. That is, slope is not a good parameter to use when trying to estimate trapping. Recall that the position of the shelfbreak determines the horizontal extent of the maximum residual velocities. This has implications for previous results. The horizontal extent and strength of the plane parabolic seamount residual flow was greater than that of the Gaussian (Fig 4.4), therefore, these results indicate that more trapping would be expected for the plane parabolic seamount than for the Gaussian it was compared to. At this point, a distinction can be made between types of longterm particle behaviour. Consider the area contained within the radius r j . A water parcel may enter and subsequently leave this region, taking longer than it would if there were no seamount present. This is a finite enhancement of its residence time and can be called retention. However, it is possible that the water parcel does not leave this area within a certain specified amount of time. For example, the particle at the center of the flow in Fig. 5.1 has not moved very far in one month and one might estimate that after 2 months or more, it would still be found in the area. It would be reasonable then to describe this case as a permanent detainment of the particle to the region and refer to the result as trapping. Both dynamics have important implications for residence times of nutrients and/or pollution. Boyer and Zhang (1990) examine parcel retention times in the lab for a range of Rossby numbers (0.01 to 0.2 inclusive) and inflow magnitudes in the case of oscillatory flow past isolated topography. Following their example, retention times have been graphed against changes in shelfbreak widths. The definition for parcel residence time is taken to be the same as that put forth in Boyer and Zhang (1990) and given in Fig. 5.6. The residence Chapter 5. Tracking of inert and biological particles 102 Figure 5.6: Definition of the residence time T of a water parcel, after Boyer and Zhang (1990). T b is taken to be the time required for a particle to pass from line a to line b. The two parallel lines are separated by a distance equal to the seamount base diameter, ab a D. time, T b, will be scaled by the advective time scale, r = D/U , where D is the seamount a b base diameter and U is the magnitude of the incoming time mean background flow. b Fig. 5.7 shows the scaled residence time of a particle (released from point c - refer to Fig. 5.6) versus changes in the distance to the shelfbreak. It is immediately apparent that, in these cases, the retention time of a particle is enhanced by the presence of the seamount since all scaled residence times are greater than 1. Compare the results of Set I: Varying r to Set II: Varying slope (A, D and d E only). It would seem that the scaled residence time is not significantly affected by an increase in the width of the seamount but is affected by an increase in the slope (resulting in an increase in the residence time by a factor of about 1.25). Results for all five seamounts in Set II are shown in Fig. 5.8 for scaled residence time versus slope. Chapter 5. Tracking of inert and biological particles 2 103 1 i 1.95 1.9 1.85 r — 1 Set II 1.8 XI 1.75 1.7 / 1 1 1.65 1.6 1.55 i Set I I- A / : 4 B o s\ 1 • ' 10 - c ~" • 15 ' 20 25 30 Shelfbreak (km) Figure 5.7: Scaled residence time versus distance to the shelfbreak for Sets I and II. Of course, the time it takes for a particle to get from Une a to line b will depend on the initial point where it crosses line a. Boyer and Zhang (1990) plot the scaled residence time as a function of the scaled cross-flow axis (parallel to lines a and b, and scaled by the seamount base diameter). They found that the maximum retention time is attained by a particle released approximately at the center (i.e. at point c) and that the enhancement falls off fairly symmetrically away from that point. However, numerical experiments by Goldner and Chapman (1996) find that their distribution is skewed away from the center. Particles released to the left of point c in Fig. 5.6, but still within the horizontal extent of the seamount, take longer to reach Une b. Numerical particle tracking results from Profile A exhibit behaviour similar to that of Goldner and Chapman (1996) (Fig. 5.9). In summary, retention times are enhanced by the presence of the seamount and they are further enhanced as the flank slope of that seamount increases. Will potential trapping be adversely affected by an increase in seamount height? Table 5.3 summarizes data coUected from trials run on Set III: Varying S. Chapter 5. Tracking of inert and biological particles 104 2 1.95 I 1.9 1.85 I 1.8 1.75 $ 1.7 1.65 $ 1.6 1.55 1.5 M A V -60 -80 -100 -120 -140 -160 -180 -200 Slope (m\km) -220 Figure 5.8: Scaled residence time versus slope using Set II. Table 5.3 indicates that trapping may increase slightly as the fractional seamount height increases. The result of increasing the fractional seamount height is to enhance the oscillating velocity field and therefore to increase the tidal excursion. With an increased tidal excursion, one might expect trapping to decrease. That is, the farther a particle travels during a tidal period, the more likely it is to experience changes in the strength of the rectified mean flow field, decreasing the likelihood of it remaining within the vicinity of the seamount. Let us consider the tidal excursion in more detail. Tidal excursion is generally taken to be the net distance travelled by a particle over half a flow cycle (HUTHNANCE, 1981; BOYER et al, 1991; FOREMAN et al., 1992; ZHANG and BOYER, 1993). In this case, the tidal excursion, [i, for an oscillating flow Uosm(ut) can be written (5.5) and, therefore, scaling the tidal excursion by the seamount base width gives (5.6) Chapter 5. Tracking of inert and biological particles 8 •i I 10.8 105 i* 7 6 £ H 5 4 3 2 1 ii.1.1.1.iii.1.1.1.in.1.1.1.ii 0 -25 -20 -15 -10 -5 0 5 Distance (km) 10 15 20 25 Figure 5.9: Scaled residence time of particles released from different points along line a. where K is the Keulegan- Carp enter number ( B O Y E R and ZHANG, 1990). One would c propose then, that for the fractional seamount height (and therefore the tidal excursion) to have a detrimental effect on particle trapping, that K 3> 1/2. The physical meaning c of K c 3> 1/2 would presumably say that the tide acts to move the particle off the seamount and away from the trapping effects of the residual field. However, K ^> 1/2 c implies that e ^> l/2cr and for the M tide, cr = 1.36, so that e ^> 0.68 must occur 2 for significantly large tidal excursions. However, e 0.68 implies a fully nonlinear flow regime which breaks the imposed weakly nonlinear constraint. At mid-latitudes, an enhanced tidal response of 15 cm/s over a seamount, with a concurrent Rossby number of 0.75 implies that the seamount base width is 2 km. Similarly, a response of 50 cm/s with e = 0.75 implies is 6.7 km. These base widths are small compared to the types of seamounts that have so far generated interest (e.g. Cobb Seamount, Fieberling Guyot, the Emperor Seamount Chain). Cases where the tidal excursion may act to decrease trapping have not been investigated and this is reflected in Table 5.3. Results are similar Chapter 5. Tracking of inert and biological particles 106 Set III Profile L rd 8 F K J A G H (km) 1.2 1.5 1.6 2.0 2.7 3.5 (km) 25.0 25.0 25.0 25.0 25.0 25.0 0.95 0.933 0.917 0.9 0.8 0.7 Initial Drogues 89 89 89 89 89 89 % Trapped Drogues 53 49 48 49 47 44 Table 5.3: Here L, the shelfbreak, is taken as that value which maximizes (4.12). Initial Drogues refers to the number of drogues that were released within the radius for each seamount. Percentage Trapped Drogues is calculated using the number of drogues not lost to the mean background flow after 93.15 days. even when considering the subinertial K\ tide. In summary, trapping increases as the fractional seamount height increases but not as much as when there is an increase in slope. Conclusions about the effect of significant tidal excursions on particle trapping cannot be drawn here. Aside from particle trapping, one could also look at the effect of increasing fractional seamount height has on particle retention. Using the definition put forth in Fig. 5.6, results for retention times for Set III: Varying 8 are shown in Fig. 5.10. It is apparent from this figure that while an increase in the fractional seamount height does enhance the retention time, this enhancement is only slight (less than 6 % for a 26 % change in Similar to calculating retention times, it is possible to measure the amount of time a particle spends in the region of the seamount, based on its initial position. Particles were released over seamount Profile A and tracked for a period of up to 150 days (197 particles were used). The time at which they were seen to leave the seamount region (r < Vd) was noted and a contour plot of these times is shown in Fig. 5.11. This figure shows Chapter 5. Tracking of inert and biological particles 107 1.6 1.58 ~- 1.56 1.54 1.52 1.5 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Fractional Seamount Height (5) Figure 5.10: Scaled residence time versus changes in fractional seamount height. that the maximum retention happens over the seamount summit, with particles arriving upstream on the left being detained up to 5 days before leaving the region. There is a discrepancy between these results and those presented by Goldner and Chapman (1996). They estimate maximum detention times of approximately 25 days (versus 145 days shown on Fig. 5.11) with maximum detention in a region slightly to the left over the seamount looking downstream of the mean inflow. This difference is most likely due to the higher (by two orders of magnitude) bottom friction values used here which alters the streamlines (VAZIRI and BOYER, 1971) and decreases the strength of the recirculation. Part of the F W model is based on the harmonic model discussed in a F E study by Snyder et al. (1979) which included a close look at the effects of the bottom friction parametrization. Testing the bottom friction parametrization, Snyder et al. obtain a reasonable fit (although, perhaps not totally satisfactory) to data collected from a small region in the Bahamas using a quadratic and a quadratic plus linear formulation. They note that the bottom stress is significantly larger than that implied by other researchers. Chapter 5. Tracking of inert and biological particles 108 Regardless, values for the F W model here are quite similar to the Snyder et al. results that produced an acceptable fit to the principle M and M 4 tidal constituents. It is of 2 interest to make note of a quote from Haidvogel et al. (1993) With so little known concerning small-scale mixing in the ocean, particularly in the neighbourhood of seamounts, it would be useless to try to defend any specific frictional parametrization too strongly . . . Equation (5.1) states that the path of any specific particle is dependent on its initial position a and the time of its initial release and (5.5) shows that calculating the net distance travelled over half of a period by that particle would give its associated tidal excursion. Generalizing, it can be stated that a particular system can have a tidal excursion field associated with it, where tidal excursion varies spatially (similar to the Stokes velocity). Since the final position of the particle is dependent on its initial release time, it is possible that the percentage trapping over a seamount may also be affected by the initial release times of the particles. Fig 5.12 shows the different paths taken by a particle released from the same point at different times in the tidal cycle (equally partitioned over the M tidal period with t = 0 concurrent with a maximum southward 2 flood tide). Particles were tracked for a period of 90 days and released in a region where the tidal excursions are less than 2 km and the mean flow field is affected by both the background geostrophic flow and the tidal residual. It can be seen that for some release times the particle remains over the seamount, whereas in others it is lost to the mean background flow. To examine this property further, forty-nine particles were spatially distributed over the seamount described by Profile A and released at quarter periods throughout the M tidal period. Again the time mean background flow was included to 2 induce loss from the system. Results found overall differences in particle loss to be within 5 %. Chapter 5. Tracking of inert and biological particles 109 30 20 Disi 10 (—r 0 w G> (km) •10 P •20 •30 •30 -20 -10 0 10 20 30 X Distance (km) Figure 5.11: Contour plot of retention time from 197 particles released over seamount Profile A . Particles were tracked up to 150 days and were considered lost if they left the region r < rj and were carried away in the background mean flow. Contours are given at 5, 40, 75, 110 and 145. The solid circle represents the seamount base of Profile A at 25 km. Chapter 5. Tracking of inert and biological particles r i i -30 -20 -10 .gQ . i 0 110 i i 1 10 20 30 Distance (km) Figure 5.12: The particle pathlines of one particle released from the same initial point at different times in the tidal phase. Particles were tracked for a period of 90 days. Pathlines are calculated using seamount Profile A . Chapter 5. Tracking of inert and biological particles 5.2 C r e a t i n g a 3 D velocity field using a 2 D velocity 111 field. Recent observational and numerical results have determined that there may be a weak vertical flow field centered over the seamount summit (FREELAND, 1992; HAIDVOGEL et al., 1993; CHEN and BEARDSLEY, 1995; BRINK, 1995) It is worthwhile to incorporate these results into the model in an attempt to simulate three-dimensional particle motion. With the addition of vertical motion, one can measure the effect of upwelling and downwelling on potential particle retention. Furthermore, this addition precedes the implementation of a vertical migration tendency into the particles. There are existing 3D numerical models that could be used to study the vertical movement of these particles. However, to do this would require switching to a model that has different dynamical and numerical constraints built into it (e.g. sigma-coordinate, primitive equation, semispectral models or three-dimensional, diagnostic, baroclinic finite element models). It may be possible to use the 3D model which is an adaptation of the F W 2D model, however, testing of the vertical velocities of that model is still continuing. The goal here is to provide a first estimate of predicted particle behaviour. Results presented in the next section show that the overall effect of the vertical recirculation on the horizontal particle pathlines is small. Therefore, making use of a more complicated 3D model to study seamount geometry effects on particle behaviour within this flow regime is unlikely to add to this discussion. The creation of an appropriate vertical flow field is presented in this section and the effect of the upwelling and downwelling and of the vertical migration on particle movement will be discussed in the following sections. Two dimensional tracking of inert particles for tidal and residual flow over tall, isolated seamounts was discussed in the first section of this chapter. The purpose of creating a 3D flow field using the 2D field is to extend the particle tracking results into the vertical Chapter 5. Tracking of inert and biological particles 112 regime. We have previously discussed the tidal and time mean (residual) 2D flows which were calculated with the F W numerical model for twelve separate tall isolated seamounts. The tidal flow is generated by an oscillatory flow applied at a distant boundary which is far enough removed such that there are no appreciable boundary effects. A geostrophic time mean background flow is also included. The resulting horizontal velocity field from a standard Gaussian shaped seamount is shown in Fig. 5.13. Note that the usual velocity field (or streamline pattern) for Taylor column formation shows an enhancement to the left of the seamount looking downstream. In the case of Fig. 5.13, bottom friction in the model inhibits this response and the results more closely follow those predicted by Vaziri and Boyer (1971) from their study of bottom friction effects on rotating flow over shallow topographies. To extend these results into the vertical regime, two processes must be accounted for. First, consider a particle in the middle of a water column in a barotropic flow such that as the water column moves, the particle moves with it. If the water column is advected up onto the seamount, it is compressed and since the particle must remain in the middle of the water column, it will intrinsically gain elevation with the horizontal movement. This is the first process that is included in the model. The second process consists of a vertical recirculation centered over the peak of the seamount. Researchers have found vertical flow structure in their three-dimensional studies of oscillatory flow over tall topography. In general, there is downwelling over the top of the seamount which is turned radially outwards as it approaches the bathymetry. Fluid is replaced by a weak upwelling farther away from seamount which returns - towards the summit - in the region above the top of the seamount (FREELAND, 1992; HAIDVOGEL et al., 1993; BRINK, 1995). Fig. 5.14 is a qualitative sketch of those findings and Fig. 5.15 shows the radial and vertical flow structure given in the Haidvogel et al. (1993) paper (hereafter, referred to as the H B C L Chapter 5. Tracking of inert and biological particles 113 results). The magnitude of the downward vertical velocities can either be inferred from numerical models and/or chosen to represent a particular site such as Cobb Seamount or Fieberling Guyot. One constraint is that the magnitude of the vertical circulation must be weak (by at least one magnitude) compared to the horizontal flow to be consistent with a weakly nonlinear regime. Table 5.4 presents maximum velocity estimates from four different studies for the vertical residual recirculation. Freeland (1992) presents observational results from a hydrosweep survey made around Cobb Seamount. The tides at Cobb Seamount are mixed and the dynamics there are also expected to include Taylor column formation. Numbers from the H B C L study are also presented, as are observations from Brink (1995) taken from a survey of Fieberling Guyot. Tides at Fieberling Guyot are mainly diurnal. The H B C L results are for the case of a stratified ocean with subinertial forcing equivalent to the resonant frequency of the first trapped-wave mode over a Gaussian seamount. Numerical results from Chen and Beardsley (1995) for the case of a flat-topped cosine squared seamount shape are also included. Those results are for a weakly stratified ocean at a superinertial ( M ) frequency and indicate that the same 2 structure described in Fig. 5.14 is expected to occur for both the super- and subinertial frequency regimes. Values derived from the vertical flow construct for this study, which is discussed below, are presented for later reference. From Table 5.4 it is apparent that the maximum downwelling velocities can be tens of meters per day. Note that the radially inwards velocity for Cobb appears to be somewhat smaller than that observed by Brink at Fieberling, even though their azimuthal residual and radially outwards velocities are of the same order. At present, it is unclear as to the particular reason for this. Note also that the maximum downwelling predicted by Chen and Beardsley is much smaller than that given by Brink or the H B C L results. This difference is due to the fact that the latter two studies are subinertial studies near resonance and hence see an enhanced response. Chapter 5. Tracking of inert and biological particles Study Freeland (1992) Haidvogel et al. (1993) [HBCL] Brink (1995) Chen and Beardsley (1995) Velocity Construct* Azimuthal Velocity (cm/s) 114 Downwelling Upwelling (m/day) (m/day) 12 Radially Inwards (cm/s) Radially Outwards (cm/s) 0.3 3 3 40 6 0.2 0.3 10 40 - 110 - 3 5 2.4 0.9 4.3 0.1 0.1 3.5 37 12 0.5 0.5 Table 5.4: Maximum residual velocity estimates over a seamount. Observational and numerical results are presented for comparison to the vertical velocity construct presented in this section. t These values are discussed at the end of this section. It is possible to create a function which describes the vertical and radial velocity field. The contours of such a function are shown in Fig. 5.16 (for comparison to Fig. 5.15, the same fraction of the upper part of the Profile A seamount is shown.) A vertical velocity field qualitatively similar to that shown in Fig. 5.15 was constructed and then conservation of mass was imposed to solve for the concurrent radial velocity field. That is, given the vertical velocity field, w(r, z), conservation of mass 1 d(ru) r Or dw oz n .„ „. was imposed to give u(r, z), the radial velocity field. Note that the flow field structure and magnitude are being modelled after the enhanced subinertial response of the H B C L results even though the recirculation is going to be superposed with a superinertial flow. It will be seen that the addition of this field has a very slight affect on the horizontal particle paths even though it would appear that the vertical magnitudes may be too Chapter 5. Tracking of inert and biological particles 115 large. The effect of altering the maximum magnitudes is examined in the next section. Assuming linear or very weakly nonlinear flow fields (i.e. no significant interaction between the tidal and residual flows), it is possible to couple the 2D flow field of Chapter 3 to the vertical field discussed above. To create the complete three-dimensional picture, the 2D and vertical flow fields are superposed along with the intrinsic gain/loss of elevation function. Fig. 5.17 shows the 2D flow (shown on the surface and which is assumed to extend downwards to the bathymetry) combined with the vertical structure shown in Fig. 5.14. Recall that there is an intrinsic gain in elevation with the horizontal movement. This is not represented in either Fig. 5.14 or Fig. 5.17. There are a number of assumptions intrinsic to this work which should be noted. First, the H B C L results presented in Fig. 5.i5 are for a moderately stratified case with Burger number (S = NH/fL) of 1.5. Given the known behaviour of stratified versus barotropic flow (an increase in the amount of the vertical structure), one can compensate for this discrepancy either in the construction of a vertical velocity function or by taking it into account in the analysis of the particle behaviour. Second, the H B C L results do not follow the conservation of mass principal given in (5.7). Those results satisfy the full conservation of mass equation: 1 d(ru) r or ldv r dv dw oz n .„ „. where the recirculating residual, v(r, z), also forms part of this balance. To construct the vertical velocity function, it is assumed that v(r, z) comes solely from rectification and is not involved in this balance. Further, the H B C L solution includes lateral viscosity which supports, in part, the vertical flow structure (HAIDVOGEL et al., 1993), and yet, lateral viscosity is not included in the F W velocity solutions used for the particle tracking study. In the end, however, the validity of the vertical residual field is based on its structure Chapter 5. Tracking of inert and biological particles 116 (compare Figs 5.15 and 5.16) and magnitude (given in Table 5.4). Referring to Table 5.4, one can see that the final velocities are of the same order as those predicted by the H B C L (the value of 3.5 is taken from Fig. 4.7 for Profile A ) . The radial values are somewhat the same as seen by Freeland (1992) at Cobb and Chen and Beardsley (1995) but underpredict those observed by Brink (1995) at Fieberling Guyot. It could be argued that the vertical recirculation velocity field is better denned by the superinertial Chen and Beardsley (1995) results, however, it will be seen in the next section that the overall effect on particle behaviour through the addition of the recirculation is small. Therefore, further refinement of the field appears unwarranted. 5.3 Inert particle t r a c k i n g i n three-dimensions In this section, the vertical velocity field construct (developed in the previous section) is used to predict three-dimensional particle movement. Retention and trapping behaviours similar to those discussed in § 5.1 are examined. Further, vertical excursion distances and changes in the magnitude of the downwelling are also explored. Examination of particle retention times (see Fig. 5.6) were carried out at 50 m, 400 m and 1100 m depths. The 50 m depth was chosen because it is representative of the summertime average depth of the chlorophyll maximum observed at Cobb seamount in 1990 (DOWER et al., 1992) and the 400 m and 1100 m depths were chosen because they lie below the seamount summit. Consider a particle released from any point along line a (in Fig. 5.6) at a depth of 50 m. As it is advected past the region of the seamount by the mean flow, it will likely not encroach upon the shelfbreak region (for if it did, it would most likely be caught and trapped - see Figs. 5.2 through 5.5). Since the vertical velocity field construct has a small horizontal extent, it does not significantly affect this pathline. The conclusion, Chapter 5. Tracking of inert and biological particles 117 then, is that retention times do not change with depth in this barotropic flow field. This behaviour is observed at 50 m, 400 m and 1100 m depths but is not shown because it is not significantly different from that shown in Fig. 5.9. Particle trapping results for an initial release depth of 50 m are presented in Table 5.5. In general, results are not significantly different than those presented for the case with no vertical movement (Tables 5.2 and 5.3). Profile S L (km) (km) A . B C 2.0 12.9 25.3 25.0 37.5 50.0 A D E 2.0 7.9 11.9 25.0 25.0 25.0 M N 2.8 3.8 37.5 50.0 F K J A G H 1.2 1.5 1.6 2.0 2.7 3.5 25.0 25.0 25.0 25.0 25.0 25.0 Set I 0.9 0.9 0.9 Set II 0.9 0.9 0.9 0.9 0.9 Set III 0.95 0.933 0.917 0.9 0.8 0.7 Initial Drogues % Trapped Drogues 89 89 89 49 61 65 89 89 89 49 63 65 89 89 56 56 89 89 89 89 89 89 53 49 48 49 47 45 Table 5.5: Percentages of trapped drogues for an initial release depth of 50 m. Here, L the shelfbreak is taken as that value which maximizes (4.12). Initial Drogues refers to the number of drogues that were released within the radius r for each seamount. Percentage Trapped Drogues is calculated using the number of drogues not lost to the mean background flow after 93.15 days. d In summary, the horizontal pathlines calculated previously are not largely affected Chapter 5. 118 Tracking of inert and biological particles ' ' 1 ' 1 ' ' 1 ' 1 I I , ' , * * V V . . ' i ' • t \ \ V ' 1 ^ \ \ \ \ ' \ ( \ \ ' i i < i > ' / / . • : ' ' I' ' t 1 \ , ' ' » 1 1 1 ' , 1 i ',111 ' ! ! : t i » , \ \ » » * \ \ , \ \ < v ; » • ' • • ' • ' / / ' / - - - - - - . \ * \ , ' ' 120 km i \ " < » » ; ; / / / ' \ • •' i'' ' \ \ \ * V * v * * , » V x * 1 . . / ' , • ,- ' , ' , ' . v > ' ' ', 1 / ' i i ' ; ' ' i i i , > ' ' ' 14km Figure 5.13: Typical flow field showing a) a geostrophic background mean flow (flowing northwards) overlaying an anticyclonic tidal residualfieldcentered on top of the seamount and b) a close up of the anticyclonic flow. Chapter 5. Tracking of inert and biological particles 119 Figure 5.14: Qualitative sketch of the vertical flow structure over a tall seamount. Note that the radial movement is designed to satisfy conservation of mass with the given vertical flux. The dotted line represents the zero level of the sea surface. Chapter 5. Tracking of inert and biological particles Vertical Velocity < 110 km > x Along-Channel Velocity Figure 5.15: Results from (HAIDVOGEL et al, 1993) which show the mean (c) vertical and (a) radial residual velocity profile over the top part of their Gaussian Seamount. Contour intervals are .1 cm/s for the along-channel (taken to represent radial) velocity and 2 m/day for the vertical velocity. Dotted contours are negative. Chapter 5. Tracking of inert and biological particles 121 •a OH Q 10 15 20 25 OH Q 10 15 20 25 R a d i a l Distance (km) Figure 5.16: Residual vertical flow field construct, a) Contour intervals for the vertical field are: -31.9, -27.6, -23.2, -18.8, -14.4, -1, 1, 5, 10. The vertical velocity contours show a downwards (negative contour values) flow over the top of the seamount and an upwards (three positive contour values) flow over the flank between 5 km and 10 km. b) Contour intervals for the radial field are: -.04, -.08, -.12, - .2, 3.72, 2.73, 1.75, 0.5. The radial flow consists of a large area of inwards flow over the top of the seamount (denoted by negative contour values) and a thin area of outwards flow between the inwards flow and the bathymetry (denoted by positive contour values). Chapter 5. Tracking of inert and biological particles 122 Summit Shelfbreak R a d i a l Distance Figure 5.17: Qualitative sketch of the tidal and residual flows generated by the finite element model superposed over the given vertical flow structure. The dotted lines represent the zero level of the sea surface. The dashed lines represent the horizontal 2D flow field (tidal plus residual). Stratification effects are not present. Chapter 5. Tracking of inert and biological particles 123 by the imposed vertical circulation. There are, however, interesting results that arise in the vertical. For example, increasing the magnitude of the downwelling has the effect of moving particles off the summit into the shelfbreak region sooner. Fig. 5.18 shows the pathlines of a particle released from the same initial point and time, under different downwelling magnitudes. The magnitude of the vertical velocity is multiplied by 0.5 (O), 1.5 (+) and 2.0 (•) respectively and tests were run using seamount Profile B (see Fig. 4.6). On Fig. 5.18, particle position is shown every 3.105 days, hence, it is easy to see that the O particle takes much longer than the + particle to reach the shelfbreak region (approximately 21 versus 6 days). Note that the final behaviour for each particle is that it is advected to the shelfbreak region where it remains for the rest of the time period. The significant difference is that the particle which experiences the weaker downwelling takes longer to reach the shelfbreak. That is, the strength of the downwelling over the seamount affects the flushing time for the peak region. This is an expected result since the radial velocity, u(r,z), is defined by (5.7) which implies that an increase in the downwelling magnitude will result in a similar increase in the radial velocity. There is additional information from tests using these three downwelling magnitudes. Fig. 5.19 is a three-dimensional plot of the pathlines. Fig. 5.20 shows the vertical particle excursions over time and Fig. 5.21 shows vertical particle positions with respect to their distance away from the seamount center. In the latter figure, the seamount profile is shown for reference. A n interesting result is that since the O particle (from the weakest recirculation) spends a longer time in the region near the summit, within the shelfbreak, it experiences the weak upwelling part of the vertical recirculation for longer. This is reflected in the slight upward gain in elevation seen in Fig. 5.20 in the period from the 10th to the 20th day. Furthermore, as the particles move around the seamount, excursions of approximately 100 m result (Figs 5.19 and 5.20). At first glance, one might conclude Chapter 5. Tracking of inert and biological particles 124 (incorrectly) that as the particles circle around to the northwest, they sink to a depth of approximately 300 m and as they travel back around to the southeast, they rise to a depth of approximately 200 m. However, careful examination of Fig. 5.18 would also seem to indicate that the recirculation is off center of the seamount. In fact, particle positions are shown every 3.105 days (which removes tidal oscillations) and at the same phase of the tide (which predisposes the center to appear shifted to the northwest). If particles were shown every 3.105 days but at a different phase of the tide, results would be shifted in another direction (this is evident in Fig. 5.27 of the next section). This does not, however, exclude the vertical excursions. As the particle travels anticyclonically in a mean sense around the seamount at the shelfbreak, tidally induced horizontal movement also intrinsically moves the particle vertically on the order of 100 m. The predictions for the magnitudes of the vertical excursions from particles released at different initial depths are also of interest. Fig. 5.22 shows the final positions, after 30 days, of particles released from initial depths of 0 (O), 50 (+) and 400 m ( • ) . This figure shows that vertical excursions range up to approximately 400 m. In a stratified flow, vertical excursions are not expected to be as large (e.g. BOYER et al., 1987; BRINK, 1989; CHAPMAN and HAIDVOGEL, 1992; ZHANG and BOYER, 1993). Figs. 5.23, 5.24 and 5.25 show characteristic particle tracking results for three dif- ferent particles released over seamount Profile A , in three different regimes: outside the shelfbreak region, over the peak, and near the shelfbreak. The particle in Fig. 5.23 was released outside the region of the seamount at a radial distance of 30.1 km and at a depth of 50 m. Examining part a) of this figure, it can be seen that the particle moves into the region above the seamount and is then deflected around to the right (looking downstream of the mean flow). It then moves off the seamount and after 13 days it is almost 40 km away from the seamount center. This particle does not experience the region of vertical recirculation centered over the seamount, and thus its vertical motion, shown in parts Chapter 5. Tracking of inert and biological particles 125 c) and d), is determined fully by the action of the water column as it moves around the seamount. The particle in Fig. 5.24 was released at the summit of seamount Profile A , at a depth of 50 m. Part a) and b) of this figure show the two and three-dimensional plots of this pathline. The particle position is shown every 3.105 days. It can be seen in parts b) and c) of this figure, that the particle starts initially at 50 m and then sinks to a depth of about 300 m as it circles about the seamount. Part d) shows that at distances greater than 3.5 km away from the summit, the depth of the particle is controlled by the intrinsic change in depth which it gains or loses as it moves horizontally across isobaths. The effect of the downwelling is clearly seen in parts c) and d) within the first 15 days and within the 3 km radius. The particle in Fig. 5.25 was released outside the shelfbreak at a radial distance of 7.3 km from the summit and at a depth of 50 m. As can be seen in part a) of this figure, the particle travels around in a circle before being drawn inwards towards the shelfbreak. Note that as it circles around, it moves slightly inwards, resulting in a slight gain in elevation which can be seen in the first 13 day period in part c). At one point in its path, it's motion becomes dominated by the downwelling and is advected downwards to about 300 m from 50 m - see part b) and c). Part d) shows that its depth is controlled both by intrinsic depth changes and the vertical velocity field. In summary, if the particle is outside of the region where the vertical recirculation has no effect, then particle movement is governed solely by the action of the water column as it moves over the topography. However, if the particle is within the region of downwelling centered about the seamount summit, then particles can sink to a depth of about 300 m. Chapter 5. Tracking of inert and biological particles 126 -15 : -20 -15 -10 -5 0 5 X Distance (km) 10 15 20 Figure 5.18: Three different pathlines for a particle released at (-2.3 km, 2.3 km, 150 m) for different downwelling magnifications. The downwelling (contours given in Fig. 5.16) is multiplied by 0.5, 1.5 and 2.0 for particles O, + and • respectively. Particles are tracked every 3.105 days for 93.15 days over seamount Profile B . Chapter 5. Tracking of inert and biological particles 127 Figure 5.19: Three-dimensional particle pathlines for different downwelling magnitudes. Perspective is looking downstream of the mean flow (south to north runs parallel to the y-axis) showing a rise in depth in the east and a sinking in the west. Downwelling is multiplied by 0.5, 1.5 and 2.0 for particles O, + and • respectively. Chapter 5. Tracking of inert and biological particles 128 Figure 5.20: Vertical particle excursions in depth over time (total time is 93.15 days) for different downwelling magnitudes. Downwelling is multiplied by 0.5, 1.5 and 2.0 for particles O, + and • respectively. Chapter 5. Tracking of inert and biological particles 2 4 6 8 10 12 14 Radial Distance (km) 129 16 18 Figure 5.21: Vertical particle positions versus radial distance for different downwelling magnitudes (the seamount profile is also shown for reference as the solid line). Downwelling is multiplied by 0.5, 1.5 and 2.0 for particles O, + and • respectively. Chapter 5. Tracking of inert and biological particles 130 • 111111111111111111 0 • 1111111111 *•O-O-O-Ofl**-- H- ++ ++ +# + 100 200 300 400 500 i• i• • t • I 1 i -40 -30 -20 -10 0 X Distance (km) 10 20 I I I 30 Figure 5.22: Final vertical positions after 30 days of particles released from initial depths of 0 (O), 50 (+) and 400 m (•). Chapter 5. Tracking of inert and biological particles 131 Figure 5.23: Pathline of a particle released near the seamount base of Profile A . Particles are shown every .5175 days (at the end of every M tidal cycle), a) Horizontal pathlines. Inner and outer dashed circles are the seamount shelfbreak and base width respectively, b) Three-dimensional pathlines. c) Vertical particle excursions in depth over time (total time is 13.46 days), d) Vertical particle positions versus radial distance. 2 Chapter 5. Tracking of inert and biological particles -5 132 0 X Distance (km) d) !'" 1 1 '' " 1 • 111 11 1 0r 50 <> 100 • E 150 • —' o. a) Q 200 • 250 • 300 • 350 • 0 10 20 30 40 50 60 70 80 90 100 Time (day) 400 - 2 3 4 Radial Distance (km) Figure 5.24: Pathline of a particle released at the seamount peak of Profile A at (0 km, 0 km, 50 m). Particles are shown every 3.105 days, a) Horizontal pathlines. b) Three-dimensional pathlines. c) Vertical particle excursions in depth over time (total time is 93.15 days), d) Vertical particle positions versus radial distance. Chapter 5. Tracking of inert and biological particles 133 b) -5 10 0 X Distance (km) d) c) 0 0 50 50 < 100 150 200 1 E 250 I 300 \f\ A - Depth ? Depth 100 1 0 "I" o '0 • 0 111 < ^ oo<*>oo O 0 \ 150 200 0 0 250 300 350 350 400 ' 0 ' i i i 10 20 30 40 50 60 70 80 90 100 Time (day) 400 1 1 • •••< 1 1 3 4 5 6 Radial Distance (km) 1 7 Figure 5.25: Pathline of a particle released near the seamount shelfbreak of Profile A at (-2.31 km,-6.92 km, 50 m). Particles are shown every 3.105 days, a) Horizontal pathlines. b) Three-dimensional pathlines. c) Vertical particle excursions in depth over time (total time is 93.15 days), d) Vertical particle positions versus radial distance. Chapter 5. Tracking of inert and biological particles 5.4 134 Tracking of swimming particles in three-dimensions The dispersion and survival of plankton depends, in part, on the characteristics of the plankton's vertical migration patterns. As such, extensive observational studies exist which describe the migration patterns of many coastal and pelagic species. In general, there are many factors which can affect dispersal and survival, not the least of which are the physical motion of the environment, species life cycles, physiology, migration, and the inter-relationships thereof. Swimming behaviour of plankton is further complicated by the fact that plankton can respond to environmental factors such as changes in gravity and pressure (see Day and McEdward (1984) and references within). The focus of this section will be on the vertical migration process. This tendency is incorporated into the particle behaviour and tracked in the flow over seamount Profile A . Results are presented below. It is known that some ocean dwelling species migrate, however, the characteristics of the migration are dependent on the species (fish may migrate faster and farther than smaller Crustacea), the stage of the life cycle (planktonic larvae may migrate while the adult benthic stage does not), gender (some female copepods undertake vertical migration while the males remain in deep water) and the relevant time intervals (some species have a diel cycle while other cycles are seasonal) and M c E D W A R D , 1984). ( B A I N B R I D G E , 1960; B A R Y , The variation in migration patterns is extensive, indicated by the number of contrasting observations collected (e.g. WALKER, 1984; DAY and 1967; D A Y B A I N B R I D G E , 1960; B A R Y , 1967; MCEDWARD, 1984; GENIN and BOEHLERT, 1985; LOCKE and COREY, 1989; ATKINSON et al., 1992). A l l this serves to illustrate that conclusions about the effects of migration on plankton dispersal will be species and site specific. This section will discuss the effect of placing a diel vertical migration tendency onto the previously inert particles. In this regard, Chapter 5. Tracking of inert and biological particles 135 the particles have been modelled after the "negative phototaxis" (descends during the day away from increasing light levels) migration pattern of planktonic Crustacea (BAINBRIDGE, 1960; ATKINSON et al., 1992). In general, the migration is assumed to follow this pattern: quick descent beginning before dawn, slowing as mid-day approaches, maintenance of a day depth, moderate ascent as dusk approaches, accelerating as midnight approaches and while not yet at the evening depth, maintenance of the evening depth until the next diel cycle begins. There is observational evidence consistent with this migration pattern that may be used to infer possible day and evening depths and migration speeds. For example, summer data gathered in Loch Fyne, Scotland shows a species of copepod that has the following pattern. The ascent occurs during the approximate period of 4 p.m. to 10 p.m. and descent from approximately 1 a.m. to 7 a.m. with maintenance of evening and day depths in between. The migration takes place between the surface and about 120 m and average speeds are about 0.6 cm/s (BAINBRIDGE, 1960). Another example of copepod migration comes from the Southern Ocean (ATKINSON et al., 1992). A period of slow ascent precedes a period of rapid ascent, for a total duration of about 8 hours. This is followed by a quiescent period of 3 hours and then a period of rapid descent for 3 hours followed by a 5 hour period of slow descent. A final quiescent period of 5 hours ends the diel cycle. The copepods range as deep as 130 m up to about 15 m for some life cycle stages. Average migration rates are inferred to be about 0.2 cm/s. Examination of fish migrations in Saanich Inlet, British Columbia shows a similar pattern (BARY, 1967). In this case, the ascent and descent times are slightly asimilar in that the former can take up to 3 hours where the latter falls somewhere between 1.5 and 2.5 hours. The vertical migration was found to closely coincide with sunrise and sunset, average speeds were calculated at about 3 cm/s and evening and day depths were around 35 m and 100 m respectively. It can be seen then that the swimming speeds are faster than the copepods Chapter 5. Tracking of inert and biological particles 136 in the previous example and that swimming duration is concurrently shorter. The migration pattern used in this model is based on these examples. Both space and time dependency are incorporated into the vertical migration. Table 5.6 shows the time dependency of the migration. The time dependency closely follows those described by Time duration Behaviour 5 a.m. - 10 a.m. rapid descent 10 a.m. - 1 p.m. slow descent 1 p.m. - 4 p.m. flutter about the day depth 4 p.m. - 7 p.m. slow ascent 7 p.m. - midnight rapid ascent midnight - 5 a.m. flutter about the evening depth Table 5.6: Migration time dependence for swimming particles. Behavioural tendencies are broken up over time periods. the above examples. Both the ascent and descent are broken into slow and rapid stages, similar to the B . C . fish and the Southern Ocean copepods, and the time duration of each stage more closely follows that of the latter as well. The evening and day depths were chosen from an amalgamation of observational data to be 20 m and 80 m respectively. The particles are set to migrate at a constant speed towards the final evening or day depth, but once they range within 5 m of the target depth their velocity will decrease linearly to zero to that depth. Note that for the quiescent time periods (1 p.m. - 4 p.m. and midnight - 5 a.m.) there is no general consensus as to particle behaviour during this time and so the particles axe set to slowly oscillate over a small vertical range during this time period. The benefit of using this particular pattern is that it is general enough to suffice as a reasonable first step in modelling migration. This type of pattern would never be able to encompass all migration patterns due to their high variability from species to species. Chapter 5. Tracking of inert and biological particles 137 For example, some species exhibit positive phototaxis (WALKER, 1984). Fig. 5.26 shows the results of a particle released at a radius of 3.3 km and a depth of 50 m over seamount Profile A with and without the imposed diel migration. The initial time (the beginning of day 0) is set to begin at 5 a.m. in the diel cycle, so that the particle begins to descend immediately. Particle tracking is shown for a period just over one day (shown approximately every 50 minutes) and it can be seen from Fig. 5.26a) that the horizontal difference between the two pathlines is negligible. Note in parts b) and c), however, that the vertical variation can be seen clearly and that the particle with the imposed diel migration covers more distance in the z-direction (between 20 m and 94 m depth). Three different initial positions were chosen for similar studies and are shown in Fig. 5.27. Note that two pathlines are shown for each initial position in this figure. Each pathline runs for a total of 15.525 days. The solid line shows the implied pathline when particles are shown every 2.07 hours, and the dashed line shows the pathline when particles are shown at the end of every tidal oscillation (in effect, the tidal oscillations are filtered out). There are two items of note in this figure. First, the pathlines with no tidal oscillations showing give an overall view of the longterm behaviour of the particle, but it should be kept in mind that a particle may have horizontal excursions that aren't readily apparent. Second, consider the pathlines circling around the seamount summit. Looking at the pathline with the tidal oscillations filtered out, it would appear that the recirculation is centered slightly to the northwest of the peak. However, looking at the dashed pathline, it is apparent that that result is an artifact of releasing the particle at that particular phase of the tide and that the full pathline does appear to be centered over the summit. These particles were again tracked for a period of just over 2 weeks and those results are shown in Fig. 5.28. In this figure, particles are shown at the end of every M tidal cycle and the initial time again corresponds to 5 a.m. in the diel cycle. 2 Chapter 5. Tracking of inert and biological particles 138 Note that the horizontal paths do diverge from each other close to the end of the 2 week period (see the last 5 positions of the swimming (O) and non-swimming (•) paths in part a)). Fig. 5.28c) illustrates the overall effect of the difference a diel migration can have on the final depth a particle may attain. It can be seen that, as time progresses, the nonswimming particle sinks slowly down towards a depth of 300 m, while the impetus on the swimming particle allows it to stay in the upper part of the water column. This result may have implications for survival strategies of zooplankton. For example, if oxygen levels decreased drastically beyond a depth 125 m, the migratory trait would allow it to remain above that depth. Similar studies were carried out in regions farther away from the shelfbreak at radial distances of 7.3 and 11.8 km (refer to Fig. 5.27). Fig. 5.29 shows swimming versus nonswimming paths for a particle released just outside the shelfbreak. Note that again the horizontal paths - shown in part a) - do not significantly diverge within this time period. However, it can be seen in part c) that while the non-swimming particle is slowly being upwelled, the swimming particle clearly shows its diel migration pattern. Results for a particle released farther away from the summit are similar and are shown in Fig. 5.30. In summary, imposed vertical migration patterns can alter the region in which zooplankton are expected to be found but detailed conclusions on plankton dispersal remain species and site specific. 5.5 Summary A number of particle tracking experiments show that the largest particle movement (of particles which remain in the region) occurs between the shelfbreak and rj, on the seamount flank, consistent with the residual velocity field structure. Results indicate that as the distance to the shelfbreak or base width increases, the percentage trapping Chapter 5. Tracking of inert and biological particles 139 also increases and that trapping is more sensitive to changes in the shelfbreak. Trapping was found to increase in cases where the slope increased or decreased. Therefore, slope is not a good parameter to use when trying to estimate trapping. It is possible that trapping may increase slightly as the fractional seamount height increases. However, cases with a large tidal excursion to bank radius ratio (which may act to decrease the trapping and counteract this effect) were not investigated. It is known that the final position of the particle is dependent on its initial release time; results found differences in particle loss to be within 5 %. Under the assumptions outlined, it was possible to extend previous 2D particle tracking work into the vertical regime. This was accomplished by linearly adding a vertical flow field to a horizontal flow field calculated using Foreman and Walters' tidal model. It was seen that the previously calculated 2D (horizontal) pathlines are not largely affected by the imposed vertical circulation. Three-dimensional inert particle tracking results show that the time spent in the region near the seamount, within the radius defined by the shelfbreak, is sensitive to the magnitude of the downwelling due to the formulation of the vertical recirculation. Vertical particle excursions are found to be up to 400 m, which may be expected to be smaller in stratified fluids. Outside of the region where the vertical recirculation has no effect, the particle movement is governed by the action of the water column over the topography. However, within the region of downwelling located at the seamount summit, particles were seen to sink to a depth of about 300 m. Furthermore, diel vertical migration patterns were imposed on particles. In one case, a migratory particle was seen to stay near the surface over a two-week period unlike an inert particle in the same case which sank to a depth of about 300 m. Detailed conclusions on plankton dispersal remain species and site specific. Chapter 5. Tracking of inert and biological particles 140 Figure 5.26: Pathlines of a swimming (O) and non-swimming (+) particle released near the seamount summit of Profile A . Particles are shown approximately every 50 minutes, a) Horizontal pathlines with the initial position marked with a shaded circle and direction of movement shown with arrows, b) Three-dimensional pathlines. c) Vertical particle excursions in depth over time (total time is 1.035 days) and diel migration periods of descent and ascent are marked with a D and A respectively, d) Vertical particle positions versus radial distance. Chapter 5. 141 Tracking of inert and biological particles -14 -12 -10 -8 -6 -4 -2 0 2 4 X Distance (km) Figure 5.27: Two week long pathlines of three swimming particles with and without tidal oscillations. Solid line pathlines have particles shown every 2.07 hours; dashed lines every 12.42 hours. Chapter 5. Tracking of inert and biological particles a) 142 b) 4 3 E 2 i S CD O 1 •4-» 0 c <fl Vl if/ A a u °50 w b >- 0 / •1 / / / / JI / -2 •3 ' i ' 1 1 - 2 - 1 0 1 X Distance (km) d) E x: a 4-" a a) Q CD Q 6 8 10 Time (day) 12 14 16 0.5 1 1.5 2 2.5 3 3.5 Radial Distance (km) 4 4.5 Figure 5.28: Pathlines of swimming and inert particles released near the seamount summit of Profile A . Particles are shown every 0.5175 days (at the end of every M tidal cycle), a) Horizontal pathlines with the initial position marked with a shaded circle, b) Three-dimensional pathlines. c) Vertical particle excursions in depth over time (total time is 15.525 days), d) Vertical particle positions versus radial distance. 2 Chapter 5. Tracking of inert and biological particles 143 b) -7.5 -7 -6.5 -6 -5.5 -5 -4.5 -4 -3.5 -3 -2.5 -2 X Distance (km) d) o 20 40 a O 0 O + O ° + ++ + *+++++1 + + h O + + t 0 60 a) a 0 80 0 0 100 0 2 4 6 8 10 Time (day) 12 14 16 0 0 7 7.05 7.1 7.15 7.2 Radial Distance (km) 7.25 7.3 Figure 5.29: Pathlines of swimming and inert particles released away from the seamount summit of Profile A . Particles are shown every 0.5175 days (at the end of every M tidal cycle), a) Horizontal pathlines with the initial position marked with a shaded circle, b) Three-dimensional pathlines. c) Vertical particle excursions in depth over time (total time is 15.525 days), d) Vertical particle positions versus radial distance. 2 Chapter 5. Tracking of inert and biological particles a) 144 b) 1.5 Y -12 - 1 1 . 9 -11.8 -11.7 -11.6 X Distance (km) -11.5 d) 0 T7TTiiTT mTTTrrr T T l-TTT1-l.rjlTIT-.TT.. | 10 20 • o>oo 0 0 0 0 0 0 30 • 0 E r a a 40 . 0 80 2 4 6 8 10 12 14 Time (day) 16 o • 0 50 • *rtttfH-++ + + + + + + iV 60 0 o 70 . o 0 ;%oo.° 11.75 • + o' 0 • ill 11.8 1 1 . 8 5 11.9 1 1 . 9 5 1 2 Radial Distance (km) 12.05 Figure 5.30: Pathlines of swimming and inert particles released over the flank of seamount Profile A . Particles are shown every 0.5175 days (at the end of every M tidal cycle), a) Horizontal pathlines with the initial position marked with a shaded circle, b) Three-dimensional pathlines. c) Vertical particle excursions in depth over time (total time is 15.525 days), d) Vertical particle positions versus radial distance. 2 Chapter 6 Conclusions This study falls within the larger scope of investigating Lagrangian eddy formation over an isolated seamount. The goal of this project was to determine, using Foreman and Walters' finite element tidal model, how mean (tidal residual) flows and particle trapping were affected by the geometry of a tall, axisymmetric seamount. The flow regime was weakly nonlinear (Rossby number < 0.1) and was applied to a tall (occupying more than 70 % of the water column), isolated seamount. This study began with an analytical extension of Shen's (1992) linear work into the weakly nonlinear regime. The analytic solutions were used to validate the numerical solutions and then the numerical model was used to examine flow over topography for which no analytic solutions exist. The goal of this project was to determine the effect of seamount geometry on residual flow and particle behaviour, and tests were geared to give a more complete picture, by decoupling height, width and slope effects, than had been previously reported. 6.1 Analytical and numerical model results A n analytical solution was generated from a weakly nonlinear analysis of barotropic oscillatory flow over a tall, axisymmetric, isolated seamount. Solutions were assumed to be of the form of a series of ordered components and the weakly nonlinear approach resulted in systems of three first order ordinary differential equations for each ordered component. In fact, the solution was semi-analytic in that it required the numerical solution of the ordinary differential equations for the higher order components. 145 The Chapter 6. Conclusions 146 analytical solutions may be used to test other numerical models which solve equivalent mathematical systems. The equations that governed the time independent (residual) frequencies were underdetermined, and two separate approaches - i) the addition of bottom friction and ii) the conservation of potential vorticity - were employed to formulate solutions. The formulation for the case with bottom friction showed the residual velocities to be linearly dependent on the aspect ratio of the seamount. This dependence implied that for cases where the deep water depth does not change, wider seamounts will tend to have weaker residuals. Hence, this ratio is a factor in determining the strength of time independent tidally rectified flows. The formulation for the case with bottom friction also showed the residual velocities to be independent of bottom friction. That is, while the presence of bottom friction was necessary to determine residual velocities in the analytical case, its magnitude was irrelevant. This implied that the analytical result may be an upper limit for these velocities. Numerical tests showed that the finite element model predicted weaker residuals for higher bottom friction coefficients. A n analytical solution was also derived using a weakly nonlinear expansion of the conservation of potential vorticity equation. This approach predicted much smaller residual velocities than those generated with bottom friction. The analytical model solutions were compared to the F E numerical model results for a plane parabolic seamount shape. A discontinuity in the topographical slope was linked to differences between the numerical and analytical model solutions. The maximum effect was seen to be at the radius where the flat top of the plane parabolic seamount changed to its sloping side. Unrealistically sharp peaks in the analytical results affected the first order solutions. Chapter 6. Conclusions 147 Based on the model comparisons, it was concluded that the numerical model gives reasonable solutions for tall topographies. The accuracy of the numerical model was constrained by the amount of discretization of the grid in the area of the topography, especially when the topography was non-smooth. Energy scattering can occur in a piece-wise continuous finite element grid and the effects appear in the residual response. A linearly stretched grid reduced maximum values of residuals from the original piece-wise grid of Chapter 3 by approximately 84 %. When using this finite element model to study residual velocities, it was determined that the linear grid would give a more accurate response and was therefore used for the models in Chapters 4 and 5. 6.2 M o d e l validity Before investigating the relationships between the residual velocity and seamount geometry, some investigations were made into the validity of the model bathymetry and numerical boundary condition. Bathymetry for Cobb Seamount (situated off the coast of Washington) was compared to a Gaussian curve to address the question of usefulness of a Gaussian seamount profile and the result was favourable. Residual velocity profiles from the standing wave boundary condition (originally used to simplify the analytic equations) were compared to those forced by a Kelvin wave boundary condition. The two boundary conditions gave similar residual flow structures and magnitudes (maximum error of 3.2 %) of the type expected around tall isolated seamounts. Thus, the unrealistic standing wave boundary condition resulted in a realistic response and provided a useful test for the numerical model over tall topography. Furthermore, parameter values for the analytic plane parabolic solution were compared to those from a numerical flow solution for a Gaussian seamount and to observations Chapter 6. Conclusions 148 taken from Cobb Seamount. Compared to the plane parabolic seamount, the numerical model results for the Gaussian seamount implied that it supported a significantly smaller anticyclonic eddy. Further, the Gaussian did not show a smaller cyclonic eddy over the top of the seamount as the plane parabolic seamount did and the Gaussian seamount had a residual field which appeared to have a smaller horizontal extent. Inspection of the flow regimes compared to that of Cobb Seamount showed reasonable consistency and it was concluded that a Gaussian seamount shape would give reasonably realistic solutions. 6.3 Residual flow relationships to seamount geometry Numerical experiments using seamounts with varying seamount base width showed that the magnitude of the maximum residual velocity decreased as the width of the seamount base increased, consistent with the analytical result. Experiments also showed that there can be a smaller (in both width and magnitude) cyclonic flow within the anticyclonic flow at the top of the seamount. On the seamount flank, the flow was ageostrophic, dominated by advective forces, which resulted in the anticyclonic residual. A n analysis of the governing equations showed that the maximum residuals occurred at the radius at which the maximum advection occurred and that the maximum advection occurs at the radius at which 1 dh{r) h(r) 3 dr was satisfied. It may be said that the maximum residuals occur at the "shelfbreak" only if the "shelfbreak" is defined to be at the radius at which this condition is satisfied. In addition, it was found that an increase in slope does not necessarily generate an increase in the magnitude of the maximum residual flow. Numerical experiments using seamounts with varying height showed that the magnitude of the maximum residual velocity increased nonlinearly with increasing seamount height, Chapter 6. Conclusions 149 and that the energy contained within a prescribed radius increased as the height of the seamount increased. 6.4 Particle tracking A number of particle tracking experiments showed that the largest movement of particles (for particles that remained in the region of the seamount), occurred between the shelfbreak and base width, on the seamount flank, consistent with the residual velocity field structure. Results indicated that as the distance to the shelfbreak or base width increased, percentage trapping also increased. The trapping results were more sensitive to changes in distances to the shelfbreak than to changes in the base width. It was also found that seamount flank slope is not a good parameter to use when examining increases or decreases in trapping. It is possible that trapping may increase slightly as the fractional seamount height increases. However, cases with a large tidal excursion to bank radius ratio (which may act to decrease the trapping and counteract this effect) were not investigated. Retention times (finite detention of particles in the region of the seamount) were enhanced by increases in the seamount flank slope more so than changes in distances to the shelfbreak, base width or fractional seamount height. It is known that the final position of the particle is dependent on its initial release time, however, the effect of changing the initial release times resulted in differences in particle loss of less than 5 %. Under the assumptions outlined, it was possible to extend previous 2D particle tracking work into the vertical regime. This was accomplished by imposing a vertical flow field on to the horizontal flow fields calculated using the F W tidal model for the twelve seamounts studied. It was seen that the previously calculated 2D (horizontal) pathlines Chapter 6. Conclusions 150 were not largely affected by the imposed vertical circulation. Three-dimensional inert particle tracking results showed that the time spent in the region near the seamount, within the radius defined by the shelfbreak, was sensitive to the magnitude of the downwelling due to the formulation of the vertical recirculation. Vertical particle excursions were found to be up to 400 m, which may be expected to be smaller if the background fluid were highly stratified. Outside of the region where the vertical recirculation had no effect, the particle movement was governed by the action of the water column over the topography. However, within the region of downwelling centered over the seamount summit, particles were seen to sink to a depth of about 300 m. It was also found that imposed diel vertical migration patterns could act to maintain the particles closer to the surface, whereas in similar situations, inert particles would be lost. Detailed conclusions on plankton dispersal remain species and site specific. 6.5 Epilogue The goal of this project was to determine the effect of seamount geometry on residual flow and particle behaviour in a weakly nonlinear regime. It was found that a strong residual is not necessarily the best measure for enhanced trapping. Stronger maximum residuals are expected at narrower (and taller) seamounts; however, enhanced trapping is expected at wider (especially at the shelfbreak) seamounts. That is, having a larger horizontal extent for the maximum residual circulation is more effective at enhancing trapping than an increase in the strength of the recirculation. On the other hand, steeper seamounts intensify particle retention. Future work should remove some of the limits on the model which stem from the assumptions laid out in its original definition. First and foremost, the problem was assumed to be barotropic and, therefore, the next step in the procession of this work Chapter 6. Conclusions 151 would be the inclusion of baroclinic effects. Specifically, other researchers have included stratification in their models and it would be appropriate to continue the analysis of the residual flow structure and particle behaviour in this regime, making the conclusions more applicable to real world situations. 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Appendix A The Standing Wave Velocity Field The standing wave boundary condition creates a velocity field which rotates in much the same way as a tidal flow, in that it has an associated current ellipse. This can be shown using the linear dimensionless momentum equations. The linear dimensionless momentum equations can be written in Cartesian co-ordinates as: cr du 27r at a dv 2 ^ dri ox dn " + + a r ° . . „. , - . L < A - 2 > These equations may be rearranged to give: a_, (Pu __ dj _ a d r, 2*> dt> ~ dy lixdtdx a dv _ a d rj dr) 2 T o¥ -~2^d7^ dx~2 2 { + - U 2 [A 6) 2 2 ( j + v + { j Assume that 77 represents a standing wave. In Cartesian co-ordinates, we can write T](x,y,t) = ycos(2irt) and substitute it into the above equations, obtaining: ( ^ ) (£)>^ 2 ^ + ^ = -cos(2 ri) (A.5) + v = *«n{2irt) (A.6) 7 which can be solved for u and v giving: u u = = (^3T) - ( ^ 3 T ) s i c o s n 161 ( 2 7 r i ) ( 2 ^ ) . (-) A 7 (-) A 8 Appendix A. The Standing Wave Velocity Field 162 Consider the superinertial case (cr > 1) at the point (1,1). Fig A . l shows u and v and a sketch of the velocity vector as it rotates over one period. The tip of the vector traces out an ellipse whose semi-major and semi-minor axes depend on the amplitudes of u and v. 0 t _ 1 I Figure A . l : Clockwise rotating velocity vector. At the point x = 1, y = 1 the u and v components will oscillate in time in a clockwise sense. In the upper right figure, the direction of positive u and v vectors are shown. The two left hand figures show how u and v oscillate in time, and the direction of the resulting velocity vector is shown at four different times in the period in the lower right figure. As time progresses, the vector rotates to the right. Appendix B The Degeneracy in the Weakly Nonlinear Expansion For the derivations in this appendix, some equations are used which were previously defined in § 3.1.1.4. Those equations are labelled as such but, for expediency, the definitions of the scaling constants and the variables will not be repeated here. The weakly nonlinear expansion of the shallow water equations resulted in two types of residual components: those independent of, and dependent on, 6. These equations were given previously in § 3.1.1.4 as: +! n fa ( B . l\l ) = gn 0 (B.2) ^(rfcun) = 0 (B.3) + —j^- = 012 (B.4) ?7l2 = /12 (B.5) «ii = - f dr V12 2 ™12 r d 1 — {rhu ) - 2hv = —p\ ar a* 12 12 (B.6) 2 where all the variables were also previously denned in § 3.1.1.4. Consider the first set of residual components: rj , n and v . u The degeneracy arises because (B.2) identically solves (B.3). This leaves an underdetermined system of 1 equation, ( B . l ) , in 2 unknowns, rjn and v . n 163 Appendix B. The Degeneracy in the Weakly Nonlinear Expansion Consider the second set of residual components: 7712, 1*12 and v . i2 164 Substitution of (B.4) and (B.5) into (B.6) gives ^-[2hr, or + rhf ] 12 12 + 2h[-^- + g] or 12 = lp 1 (B.7) 2 G T which expands to become: dh 2-5-1712 or 1 = -JP12 - cr d o-(rhf ) or ia + 2hg (B.8) 12 When h(r) is a non-constant, continously differentiable function of r, (B.8) may be used to solve for the residual sea height. If h(r) is a constant, the left hand side of (B.8) is zero. It takes a bit of algebra, but it is possible to show that when the depth is constant, the right hand side is also identically zero. In this case, the system is degenerate in that equations (B.4) and (B.5) identically solve (B.6). In two different cases, two different physical constraints were used to create well defined systems of equations for the residual components. In one case, the addition of bottom friction is used, in the other, conservation of potential vorticity is enforced. The Weakly Nonlinear Expansion of Conservation of Potential Vorticity The expansion of the equation for conservation of potential vorticity is presented here. The results are discussed in § 3.1.2.3. The linear components of the horizontal velocity, (u,v), were previously given in polar co-ordinates as (see § 3.1.1.4): u 0 = -ii+(r) cos(c/>i) - (r) cos(c/> ) v 0 = -D(j"(r) sin(</>!) + v „ (r) sin(</> ). 2 2 (B.9) (B.10) Note the use of "uo" to denote the sum of the zeroth order components of the radial Appendix B. The Degeneracy in the Weakly Nonlinear Expansion 165 velocity. Similarly, u can be used to represent the sum of the first order components. x This convention is used throughout this appendix for all variables. If we take the linear part of the local vorticity to be £o = k • V x ito I d = = du 0 r dr[ {vo + { r V o ) --d6 ~Q^r) r s i n ] ( ^ i ) io v + + ~g^~) r s i n (^2) - «o s m (</>i) - o sin(c6 )] u 2 then, we can write the components of the linear local vorticity as: & =4 8 + - or r - - • (B.1D r Furthermore, the dimensionless forms of local vorticity and sea height can be written as £ = to + e^ + Oie ) (B.12) V = + em + 0(e ) (B.13) 2 2 where the above relationship, ( B . l l ) , holds and similar relationships can be derived for the higher order components of vorticity. The equation for conservation of potential vorticity may be derived from the shallow water equations given in Chapter 3 and was formalized by Ertel in 1942 (PEDLOSKY, 1979). The equation is given here for a barotropic flow field D_ C + f Di I- - L I . I 0 ( B - U ) where the overbars, ~, are used to express variables which have yet to be nondimensionalized. The equation is expanded and non-dimensionalized in the following manner: Appendix B. The Degeneracy in the Weakly Nonlinear Expansion 166 D_ (B.15) Di j + h D / ( l + U /(fr ) Dl h{l+ /h n/h) D_ /(I + et) h(l + er}/(a h)) 0 Vb 0 d (B.16) J d (B.17) 2 (B.18) which becomes, using (B.12) and (B.13), Dt h [ ^ (1 . + e &+ ^ )(l-.^-^JL 1 r / . ^° \ . J*f it :[£ + 4(6>-:£) fc^" a /i + s £^t f i D* /i L 2 a y + ^ I* ) fa* ah ah 2 2 ] (B.19) i + T^br)] (a h) 2 2 (B.20) to the second order in e. Note that we must also expand the material derivative T8t Dt r (B.21) U d (B.22) which substituting into (B.20) and collecting terms gives: r* 9 (f\ i * a 'dr h' x ' 9 Jit 27T dt h W> a h' 2 Appendix B. The Degeneracy in the Weakly Nonlinear Expansion + ^(6-^)1. 167 (B.23, This is the expanded conservation of potential vorticity equation. Since the terms are arranged by their size (that is, grouped by powers of e), constraints for each e order can easily be seen. The 0(e) terms give a constraint for the zeroth order terms, and so on. For example, if h and / are assumed constant, then conservation of potential vorticity for the zeroth order terms would be: d_ 0. dt (B.24) However, a little bit of algebra shows that the solutions for the linear velocities, given in § 3.1.2.1, identically solve this equation. Hence, no new information is gained for the zeroth order equations in this case. This result, does however, lead to a constraint for the first order terms, giving: 6-^ = 0. (B.25) From (B.25) it is possible to derive the necessary equations to fully determine the residual components over the constant depth domains. Returning to (B.23), it is possible to derive similar constraint equations for regions where the depth is not constant by taking a time average of the equation. Appendix C C a l c u l a t i n g Parameters at C o b b Seamount For comparison purposes, values for the parameters discussed in § 3.1.1.1 and shown in Table 4.1 is determined from observational data taken around Cobb Seamount. Most of the parameters describing the physical geometry of the seamount are taken from Freeland (1994) , however, tidal data are derived from measurements presented in Mofjeld et al. (1995) . Freeland (1994) describes the seamount as rising from a depth of approximately 3000 m to within 24 m of the surface. His description of the height and width of a smaller pinnacle sitting on the summit narrows down the fractional seamount height, 8, to between 0.8 and almost 1.0. It is possible to estimate a seamount base width, rd, from his figure showing the bathymetry of Cobb Seamount and its environs. In addition, Freeland notes the latitude of Cobb Seamount, from which the Coriolis parameter, / , may be calculated. Given / = 2fJsin(A), where f2 = 27r/24 rad/day and A, the latitude, is approximately 47 degrees N, then / = 1.066 x 1 0 - 4 1/s. Freeland also discusses other parameters in detail such as the mean background flow strength and the incident Rossby number. Measurements for the tidal constituents of sea height may be found in Mofjeld et al. (1995) and from these the tidal velocity flow strength can be calculated. The equation for a Kelvin wave is: n^ o e - ' ^ e W ^ - " " ) 168 (C.l) Appendix C. Calculating Parameters at Cobb Seamount where 77 = the surface height which is a function of [x,y,t) rjo = the surface height at the coast R the Rossby radius r = ijj = the tidal frequency and / the Coriolis parameter. = 169 This formula is derived using the assumption that pressure p = pgn (g is gravitational acceleration, p is density). Knowing this, it is possible to use the equations governing conservation of momentum dp 1 dp * = ^ w 5 ) < - " ai ei ! + s ' ] (c • = W^(-^jr-/;r> p(w - f ) 2 dy 2 2) (as) dx to derive equations for the magnitude of the u and v velocity components away from the seamount, assuming that the velocity components have the same dependence on y and t as the sea-height, 9 u ° R {u; U =l g V o r V ui + p 2 jR-Z^p = m -x/R - P) ° 2 r Tloe { -xlR A - 5 ) (n r • ( C A\ C ) K\ Hence, we see that the Re(u ) = 0 as is expected for a Kelvin wave. 0 Given: g = 9.8 m/s 2 h = 3000 m / = 1.066 x l O d - 4 s- 1 Rr = 1605.8 km We need the background M tidal amplitude, r]b. Along the west coast, the M tidal 2 amplitude is 88.0 cm (MOFJELD 2 et al., 1995). Since the amplitude of a Kelvin wave Appendix C. Calculating Parameters at Cobb Seamount 170 exponentially decays away from the coast, we can calculate the background M tidal 2 amplitude given that Cobb Seamount is approximately 500 km away from the coast. Therefore, r] = b => v = 13.7 cm/s 88 e - 5 0 ° / 1 6 0 8 - 5 = 64.5 cm 2 b => U = iJO + vZ = 13.7 cm/s b In Table 4.1, the equivalent analytical parameters are given. Note that the value for Uo in the analytical case is compared to this background tidal flow for Cobb. For comparison, we could calculate the velocity at Cobb where observations show an M 2 tidal amplitude of 81.1 cm v = 17.2 cm/s c U c = \fo + v c (MOFJELD et al., 1995). Substitution into (C.5) 2 = 17.2 cm/s The difference here is apparently from topographical effects reflected in the different sea surface heights over Cobb and away from Cobb. Appendix D Calculating Energy Density for the Boundary Conditions The relative energy densities between the standing wave and Kelvin wave boundary conditions are derived in this section. Formulas for the energy densities of each are brought forth, and then the relative amplitudes of the boundary conditions are used to determine their relative magnitudes. The mean potential and kinetic energy densities of a Poincare wave are given by (LEBLOND and MYSAK, 1979) <PE> = \pgrfr (D.l) (D.2) < K E > l^^J^ = where p is fluid density, g is gravitational acceleration, rjp is the amplitude of the Poincare wave, LO is the wave frequency and / is the inertial frequency. At a latitude of 45 degrees N , and with a semi-diurnal wave frequency, the total energy density of the Poincare wave becomes: < E >= P 1.083 pgrjp. (D.4) The standing wave boundary condition is the sum of two Poincare waves (two plane gravity waves in a rotating frame of reference) and, therefore, the energy density of the standing wave is twice the energy density of a Poincare wave. < E >= s 2 < E >P = 2.17 pgrjp. 171 (D.5) 172 Appendix D. Calculating Energy Density for the Boundary Conditions Further, the amplitude of the standing wave is twice the amplitude of the Poincare wave: (D.6) ris = 2r, P and, therefore, by substitution of (D.6) into (D.5), the mean energy density of the standing wave boundary condition is: <E> = OMpgril S (D.7) In Appendix C, equations governing conservation of momentum of an inviscid fluid are given in the form + 1 u = p(u -f y 1 2 dx 2 , . dp (D.8) dy J dp. g r Assume a Kelvin wave of the form VK = Vo e~ ' e^—*) (D.10) x/R where the amplitude, rjo, is measured along x = 0, Rr is the Rossby radius and k is the wavenumber. Substituting (D.10) into the momentum equations, assuming p = pgr}, and squaring the results gives: .2 u v _ ^ 9 2 -4-5?/.., *™1K Rr - P) 2^2 y \<" T j * " \ f R {u> -f ) 2 A •> 2 - { 2 2 (D.H) 2 2 2+ 2 f 2 2 ) 2 K 2 (D.12) r The mean kinetic energy per unit area can be derived following (GlLL, 1982) <KE>= -pHu +v 2 2 (D.13) and substituting ( D . l l ) and (D.12) in for the velocity terms, giving <KE>= \pgr, e- */ ' 2 2 0 R (D.14) Appendix D. Calculating Energy Density for the Boundary Conditions 173 The energy of a Kelvin wave is equipartitioned, that is, its potential energy is equal to its kinetic energy and, therefore, the total energy density of a Kelvin wave can be written as: <E> = K \pgiile-*'** (D.15) For comparison, the standing wave and Kelvin wave amplitudes, rjs and TJK, axe measured along a line passing over the seamount center, running north-south through the domain (x = 0) and it is found that their relative amplitudes are given by m = 1-9*75 (D.16) and, therefore, ^7 = 3 5 <> D17 defines the relative energy densities of the two boundary conditions. Since these relative amplitudes lead to similar maximum residual velocity magnitudes (within 3.2 %), it would appear that the standing wave is more efficient (by a factor of about 3.5) at generating the residual response.
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Tidal residual circulation over an axisymmetric seamount : seamount geometry effects Shore, Jennifer A. 1996
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Title | Tidal residual circulation over an axisymmetric seamount : seamount geometry effects |
Creator |
Shore, Jennifer A. |
Date Issued | 1996 |
Description | A numerical tidal model is used to investigate the effect of seamount geometry on tidally rectified residual flow and the resultant particle behaviour. A weakly nonlinear analytical solution of barotropic oscillatory flow over a tall, axisymmetric, isolated seamount is derived and compared to a numerical tidal model solution. The comparison is formulated for a flat topped seamount with parabolic sides and with a standing wave boundary condition that makes the problem analytically tractable and reduces it to a series of ordinary differential equations. Based on the model comparisons, it is concluded that the numerical model gives reasonable solutions for tall topographies. Comparisons to a Gaussian seamount and Kelvin wave forcing illustrate the generality of the results. Residual velocities axe found to be proportional to the aspect ratio of the topography showing that for cases where the deep water depth does not change, wider seamounts will tend to have weaker residuals. Numerical experiments are conducted to investigate the effect of changes in seamount base width, slope and height on the tidal residual field and particle behaviour. Consistent with the analytic solution, results show that as the seamount widens (with its percentage depth remaining unchanged) the magnitude of the maximum residual velocity decreases. Unlike previous results of other researchers, it was found that an increase in slope does not necessarily generate a strengthening of the maximum residual. Results from the particle tracking studies show, in part, that trapping is more sensitive to changes in distances to the shelfbreak than to changes in the base width. Furthermore, retention times are enhanced, more by increases in the flank slope than by increases in base width, or seamount height. |
Extent | 7880448 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2009-04-03 |
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. |
IsShownAt | 10.14288/1.0053335 |
URI | http://hdl.handle.net/2429/6800 |
Degree |
Doctor of Philosophy - PhD |
Program |
Oceanography |
Affiliation |
Science, Faculty of Earth, Ocean and Atmospheric Sciences, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 1997-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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