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Rossby adjustment over canyons Chen, Xiaoyang 1996

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ROSSBY ADJUSTMENT OVER CANYONS by XIAOYANG CHEN B.Sc, The Ocean University of Qingdao, 1984 M.Sc, The Science and Technology Institute of Beijing, 1989 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Oceanography) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA February 1996 ©Xiaoyang Chen, 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 Q t&VAA>>c\ The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract The influence of submarine canyons on currents is studied using the Rossby adjust-mentmethod for an inviscid fluid on an /-plane. Two geometric models are used: (1) a flat bottom, uniform width, vertical edged and infinitely long canyon which cuts through a flat infinite ocean (F canyon model), and (2) a sloping bottom, uniform wddth, vertical edged and finite long canyon which extends from the shelf break into a semi-infinitely long strait (S canyon model). The waves which exist around an F canyon are composed of both superinertial waves (Poincare waves) and subinertial waves ("canyon waves"). The canyon waves are more important than Poincare waves in the determination of the steady state. The disper-sion relation of the canyon waves is obtained. The canyon waves are dispersive and propa-gate in both directions along the canyon. While both the geostrophic and the transient solutions of Rossby adjustment over the F and the S canyons are studied, the emphasis of this thesis is to study the geostrophic state. For a homogeneous fluid in the northern hemisphere: (1) when a shelf flow ap-proaches an F canyon, a net transport along the canyon is generated to the left (looking downstream) of the approaching shelf flow; (2) when a shelf flow approaches an S canyon, if the flow is left-bound (with the coast to its left looking downstream), a net in-canyon flux is generated, while if the flow is right-bound (with the coast to its right looking downstream), a net out-canyon flux is generated. For a two-layer stratified fluid (regardless of the shear) in the northern hemisphere: (3) over an F canyon in which the interface is above the shelf, the lower layer adjusts over the canyon in a similar way as in a homogene-ous fluid but over the length scale of the baroclinic Rossby radius, while the upper layer adjusts in a way to accommodate the changes at the interface; (4) over an S canyon in which the interface is below the shelf, a left/right-bound shelf flow leads to an in/out-canyon flow throughout the water column. The time scale for adjustment is \lfiot both a homogeneous and a stratified fluid. A parameter, ar e {0, 1}, which is denoted as the Canyon Number, is found to control the geostrophic state over a canyon. The Canyon Number, which is calculated from the geometric parameters of the shelf-canyon system, determines the interactive strength of one canyon edge on the circulation induced by the other edge. The parameter cr can be used to unify theories over canyons, steps, ridges and straits. The research has demonstrated that a canyon can have an important influence on cross-shelf circulation and has gone beyond earlier work in explaining the details of flow patterns around canyons. -iii-Table of Contents Abstract ii table of Contents iv List of Figures viii Acknowledgment xiii Chapter 1 General Introduction 1 1.1 Currents around Canyons 9 1.2 Topographic Waves and Properties of Double Kelvin Waves 11 1.3 Rossby Adjustment and the Choice of Initial Conditions 13 1.3.1 Rossby Adjustment 13 1.3.2 Initial Conditions 17 1.4 Derivation of the Equations 20 1.5 Geometric Models 23 1.5.1 Flat Bottom Canyon (F Canyon) 23 1.5.2 Sloping Bottom Canyon with a Coast and Shelf Break (S Canyon) 23 1.6 Objectives 27 Chapter! Geostrophic Solutions over a Flat Bottom Canyon in a Homogeneous Fluid 30 2.1 Introduction 30 2.2 Governing Equations, Qualitative Analysis and a Simple Case 32 2.2.1 Derivation of Equations 32 2.2.2 Steady State 36 2.2.3 Rossby Adjustment over a Flat Bottom Ocean 37 2.3 Waves over an F Canyon 39 2.3.1 Superinertial and Subinertial Waves 39 -iv-2.3.2 Properties of Canyon Waves 48 2.3.3 Structure of Waves over an F Canyon 50 2.4 Geostrophic State over an F Canyon 53 2.4.1 Long Time Asymptotic Solution 53 2.4.2 Analytic Far Field Solution for a Specified Initial Condition 62 2.4.3 Full Analytic Solution 69 2.4.4 Numerical Integration of the Steady State Equation 78 2.5 Discussion 81 2.5.1 Flux in an F Canyon 81 2.5.2 Extrapolation to Other Topography 82 2.5.3 Dilemma of Terminology 86 2.6 Summary of Rossby Adjustment over an F Canyon 87 Chapter 3 The Geostrophic State over a Sloping Bottom Canyon with a Coast and a Shelf Break in a Homogeneous Fluid 89 3.1 Introduction 89 3.2 The Geostrophic State over an S Canyon 90 3.2.1 Analytic Solutions at Depth Changes and Boundaries 92 3.2.2 Numerical Integration of the Steady State Equation 105 3.3 Discussion 112 3.3.1 Reversal of the Shelf Flow 112 3.3.2 Validity of the Solution 112 3.3.3 Flux in an S Canyon 114 3.3.4 Barotropic Circulation around Juan de Fuca Canyon 115 3.3.5 Extrapolation to Other Topography 119 3.4 Summary of Rossby Adjustment over an S Canyon 122 Chapter 4 Numerical Transient Solutions In a Homogenous Fluid 124 4.1 Introduction 124 4.2 Description of the Numerical Code 126 - v -4.2.1 The Numerical Scheme 126 4.2.2 The Parameters and the Domain Used 131 4.2.3 Stability and Accuracy of the Code 132 4.3 Transient Flow Patterns Observed over an F Canyon 134 4.3.1 Description of the Flow 134 4.3.2 Effects of Changing the Forcing Type 141 4.3.3 Discussion 146 4.4 Transient Flow Patterns Observed over an S Canyon 152 4.4.1 Description of the Flow . 1 5 3 4.4.2 Discussion 160 4.5 Description of the Nonlinear Flow 170 4.6 Summary of the Numerical Simulations for a Homogeneous Fluid 191 Chapter 5 Numerical Transient Solutions in a Stratified Fluid 193 5.1 Introduction 193 5.1. J Outline 193 5.1.2 Theoretical Results of Rossby Adjustment in a Stratified Fluid 194 5.2 The Numerical Code, Parameters and Domain Used 198 5.2.1 The Numerical Code 198 5.2.2 The Parameters and Domain Used 199 5.3 Transient Flow Pattern Observed over an F Canyon 203 5.3.1 Description of the Flow 203 5.3.2 Discussion 215 5.4 Transient Flow Patterns Observed over an S Canyon 217 5.4.1 Description of the Flow 219 5.4.2 Discussion 238 5.5 Circulation over Juan de Fuca Canyon 242 5.6 Summary of the Numerical Simulations for a Stratified Fluid 246 Chapter 6 Conclusions, Discussion and Suggestions 248 -vi-6.1 Conclusions 248 6.2 Discussion and Suggestions 252 6.2.1 Validity of the Results 252 6.2.2 Friction 252 6.2.3 Further Theoretical Analysis 253 6.2.4 The Sources of Nutrient-Rich Water in the La Perouse Bank Region 254 Bibliography 260 Appendix A Solving the System of Linear Partial Differential Equations 263 Appendix B The Full Geostrophic Solution over an F Canyon 268 Appendix C Rossby Adjustment over a Single-Step Parallel to a Coast 273 Appendix D Rossby Adjustment over a Single-Step Parallel to a Coast with Two Constraints 278 -vii-List of Figures Figure 1.1 Canyons off the northeastern United States. [After Fig. 79, Shepard et al. (1979)] Figure 1.2 Contour chart of the Congo Submarine Canyon. [After Fig. 123, Shepard et al. (1979)] Figure 1.3 The La Perouse Bank region and Juan de Fuca Canyon. [After Fig. 1, Freeland & Denman (1982)] Figure 1.4 Fishing statistics for the northern hemisphere. [After Fig. 2, Ware & Thomson (1992)] Figure 1.5 The North Sea region and the South Norwegian Trough. [After Fig. 9, Shepard (1931)] Figure 1.6 Geometry of an F canyon. The bottom of the canyon is flat; the edges are vertical; the width is uniform; the length of the canyon is infinite. The shelf is flat and infinitely wide and long. Figure 1.7 Geometry of an S canyon. The bottom of the canyon consists of four segments: deep mouth part, mouth slope, main body and head slope; the edges are vertical; the width is uniform. The shelf break, coast and strait walls are all vertical. The length of the strait is infinite. The shelf is infinitely long and flat. Figure 2.1 Dispersion relation for canyon waves [the nondimensional form, (2.3.11)] for an F canyon with (a) B = 0.5, y2 = 2, 3, 4 and 5, and (b) y2 = 4, B = 0.01, 0.05, 0.25, 1 and 10. Only the part for the positive wave number is shown. The curves are symmetric about the frequency axis. The horizontal axis rep-resents the wavenumber in the along-canyon direction, k, and the vertical axis rep-resents the frequency, co. Figure 2.2 Contours of surface elevation, n, which are also the streamlines, for an F canyon model in the steady state. Far field solutions apply outside the dashed square. Thick lines represent the position of the canyon edges. Solid lines represent positive 77, while dotted lines represent negative 77. Arrows represent the direction of flow in the northern hemisphere. The position of the initial surface discontinuity is along the liney = 0. The length scale is R2. V2 .= 2, (a) B= 2 and (b) B= 30. Figure 2.3 Shapes of G2(x,<ff) (for the examples of B = 2 and x - 0.2 and x = 0.5) and E(y - 1 ) (for the examples of y - 0.5 and y - -1.0) vs £ Figure 3.1 Top view of an S canyon. The shaded regions represent the canyon bot-tom slopes. The dotted line represents the position of the initial surface discontinu-ity. The width of the canyon as well as the strait is 2L. The depth in the inner strait, on the shelf, over the middle canyon and over the deep canyon (as well as in the deep ocean) are h0, hh h2 and h3, respectively. Surface elevation in the geostrophic state at all depth changes is indicated. See text for other notations. 91 Figure 3.2 Side view of a coastal region where the vertical shelf break is infinitely long and parallel to the infinitely long vertical coast. The line of the initial surface elevation is at_y = 0; the shelf break is aty = dSB', the coast is at_y = dc. 94 Figure 3.3 Close-up views of an S canyon near (a) the mouth of the strait, (b) the canyon head slope, and (c) the canyon mouth slope. Surface elevation in the geostrophic state at all depth changes is indicated. 100 Figure 3.4 Geostrophic state around an S canyon (see text for configuration), (a) Contours of surface elevation, 77. (b) A close-up view around the canyon, (c) Veloc-ity field around the canyon. Thick lines represent the positions of the canyon edges, shelf break, coast, strait walls and boundaries of canyon bottom slopes. Solid lines represent positive 77, while dotted lines represent negative 77. The length scale is R\. The range of 77 contoured is from -770 to 770 and the contour interval is O.I4770 where 770 is half the height of the initial surface discontinuity which was taken as 0.2m in this example. 109 Figure 3.5 Velocity field around the canyon whose configuration was given in Fig-ure 3.4. Thick lines represent the positions of the canyon edges, shelf break, coast, strait walls and boundaries of canyon bottom slopes. 113 Figure 3.6 The domain for the S canyon. Configuration is given in the text. Distance is in units of the Rossby radius on the shelf Thick lines represent the positions of the canyon edges, shelf break, coast, strait walls and boundaries of canyon bottom slopes. Dashed square is the region of most interest. 117 Figure 3.7 The results of the integration of the steady state equation, (2.2.8), in the dashed square shown in Figure 3.6. Given are contours of surface elevation, 77, (also the streamlines) in the geostrophic state as the shelf break current flows over Juan de Fuca Canyon. Thick lines represent the positions of the canyon edges, shelf break, coast, strait walls and boundaries of canyon bottom slopes. Arrows represent the direction of flow in the northern hemisphere. The range of 77 contoured is from O.5770 to O.98770 and the contour interval is 0.01770 where 770 is half the height of the initial surface discontinuity. 118 •ix-Figure 3.8 Contours of surface elevation around a stepped bottom canyon. Configu-ration is given in the text. Thick lines represent the positions of the canyon edges, shelf break, coast, strait walls, the canyon bottom step and the canyon head. Solid lines represent positive 77, while dotted lines represent negative 77. The length scale is Ri. 121 Figure 4.1 A sketch of surface change rate(<7 = drj/dt) vs time for "forcing type /". 129 Figure 4.2 The results of the simulation for an F canyon for forcing type 1. The configuration of the canyon is given in the text. Distance is in units of the Rossby radius over the canyon. The contours of surface elevation, 77, are shown after (a) 1 day, (b) 2 days, (c) 3 days and (d) 4 days. Thick lines: positions of the canyon edges; solid lines: positive 77; dotted lines: negative 77; range of 77: -0.4m ~ 0.4m; contour interval: 0.06m. The velocity field at day 4 in the central region [dashed square in (d)] is shown in (e). The three dimensional view of 77 is shown in (f) after 3 days. 135 Figure 4.3 The results of the simulation after 3 days of forcing (type 2) for an F canyon. The configuration is same as that in Figure 4.1. The contours of surface ele-vation, 77, are shown in (a) the whole domain and (b) the central region [dashed square in (a)]. Range of 77: -0.03m ~ 0.03m; contour interval: 0.006m. The three dimensional view of the surface elevation is shown in (c) after 3 days. In (a) and (b), thick lines: positions of the canyon edges; solid lines: positive 77; dotted lines: nega-tive 77. 143 Figure 4.4 A cross-section across the canyon of the numerical solution for the F canyon whose configuration is the same as that in Figure 4.1 except that the width of the canyon is la. The horizontal axis represents the distance in the across-canyon direction with its median marking the central axis of the canyon. The vertical axis represents the relative value at y = 5a of (a) the surface elevation and (b) the v-component of velocity. Thick line: the analytic long time asymptotic solution; thin lines: the numerical solutions after 1 day, 2, 3, 4 and 5 days. 147 Figure 4.5 The results of the simulation for an S canyon. The configuration of the canyon is given in the text. Distance is in units of the Rossby radius in the deep ocean. The contours of surface elevation, 77, are shown after (a) 2 days, (b) 3 days, (c) 4 days and (d) 5 days. Thick lines: the positions of the canyon edges, boundaries of canyon bottom slopes, shelf break, coast and strait walls; solid lines: positive 77; dotted lines: negative 77; range of 77: -0.4m ~ 0.4m; contour interval: 0.06m. The close up view of 77 and the velocity field in the central region [dashed square in (d)] are shown in (e) and (f), respectively. 154 -x-Figure 4.6 The results of the simulation for an S canyon. All specifications are the same as those in Figure 4.4 except that the forcing is reversed and thus the flow is in the opposite direction. The contours of the surface elevation, rj, are shown after (a) 2 days, (b) 3 days, (c) 4 days and (d) 5 days. The close up view of rj and the velocity field in the central region [dashed square in (d)] are shown in (e) and (f), respec-tively. Range of n: -0.4m ~ 0.4m; contour interval: 0.06m. 164 Figure 4.7 The results of Rossby adjustment over an F canyon for 8=2. Distance is in units of the Rossby radius over the canyon. The velocity fields are shown after (a) 7 days for the whole domain, and the close up view of the velocity fields in the cen-tral region [dashed square in (a)] are shown after (b) 7 days, (c) 11 days, (d) 15 days and (e) 19 days. The contours of the surface elevation, n, are plotted after 19 days in (f) where the range of n\ -4.0m ~ 4.0m; contour interval: 0.4m. Thick lines: the positions of the canyon edges; solid lines: positive n; dotted lines: negative rj. 171 Figure 4.8 The results of Rossby adjustment over an F canyon for 6=2. Near the canyon, the contours of vorticity are shown after (a) 7 days and (b) 19 days, and the contours of potential vorticity are shown after (c) 7 days and (d) 19 days. Thick lines: positions of the canyon edges; solid lines: positive values; dotted lines: nega-tive values. 178 Figure 4.9 The results of Rossby adjustment over an F canyon for 8 = 0.02. Dis-tance is in units of the Rossby radius over the canyon. The velocity field is shown after (a) 7 days, and the close up view of the velocity field in the central region [dashed square in (a)] are shown after (b) 7 days and (c) 19 days. The contours of the surface elevation, n, are plotted after (d) 19 days where range of 77: -4.0m ~ 4.0m; contour interval: 0.4m. Thick lines: positions of the canyon edges; solid lines: positive 77; dotted lines: negative 77. 183 Figure 4.10 The results of Rossby adjustment over an F canyon for 8 = 0.02. Near the canyon, the contours of vorticity are shown after (a) 7 days and (b) 19 days. Thick lines: positions of the canyon edges; solid lines: positive value; dotted lines: negative value. 187 Figure 5.1 The results of Rossby adjustment for a two-layer fluid over an F canyon. The configuration of the canyon is given in text. Distance is in units of the baroclinic Rossby radius over the canyon. The contours of the surface elevation and the inter-face elevation are given after (a) and (b) 0.5 day, (c) and (d) 1 day, (e) and (f) 1.5 day. Thick lines: positions of the canyon edges; solid lines: positive values; dotted lines: negative values. The velocity fields at day 1.5 in the central region [dashed square in (a) ~ (f)] are shown in (g) for the upper layer and (h) for the lower layer. The three dimensional views of (i) the surface and (j) the interface at day 1.5 are also given. 204 Figure 5.2 The domain for the simulations of Rossby adjustment over an S canyon. Thick lines represent the positions of the canyon edges, shelf break, coast, strait walls and boundaries of canyon bottom slopes. The length scale is the baroclinic Rossby radius over the canyon. The results of the experiments will be plotted within the region of the dashed square. 218 Figure 5.3 Velocity fields for Rossby adjustment over an S canyon in experiment one. (a) and (b) are for the upper and lower flows, respectively, after 1 day; (c) and (d) are for the upper and lower flows, respectively, after 2 days. 221 Figure 5.4 Velocity fields for Rossby adjustment over an S canyon in experiment two. (a) and (b) are for the upper and lower flows, respectively, after 1 day; (c) and (d) are for the upper and lower flows, respectively, after 2 days. 225 Figure 5.5 Velocity fields for Rossby adjustment over an S canyon in experiment three, (a) and (b) are for the upper and lower flows, respectively, after 1 day; (c) and (d) are for the upper and lower flows, respectively, after 2 days. 229 Figure 5.6 Velocity fields for Rossby adjustment over an S canyon in experiment four, (a) and (b) are for the upper and lower flows, respectively, after 1 day; (c) and (d) are for the upper and lower flows, respectively, after 2 days. 233 Figure 5.7 Velocity fields for Rossby adjustment over an S canyon which has the same configuration as the experiment one but the forcing magnitude is only 10~2 of that for the experiment one. (a) is for the upper flow and (b) is for the lower flow after 2 days. 239 Figure 5.8 General summer circulation pattern in La Perouse Bank region. Open ar-rows show upper layer currents, and solid arrows indicate the sub-surface undercur-rent. [After Fig. 1, Mackas & Sefton (1982)] 244 Figure 6.1 Distribution of trawl hauls during the Pacific hake fishery, 1987-1989 in La Perouse Bank region, [after Fig. 15, Saunders & McFarlane (1990)] 255 Figure 6.2 Contours of the acoustical estimates of euphausiid biomass during a June cruise in La Perouse Bank region, [after Fig. 1, Simard & Mackas (1989)] 256 Figure 6.3 General summer circulation pattern in La Perouse Bank region. Unfilled arrows denote an area of confused flow, [after Fig. la, Ware & Thomson (1991)] 259 -xii-Acknowledgment I was motivated to conduct this research project by my supervisor, Dr. S. E . Allen. Thanks to Dr. Allen for providing me with the numerical code which I modified for this research. I am also grateful for her encouragement, guidance and her meticulous reading of the drafts of this thesis. This research was supported by a subvention of the Department of Fisheries and Ocean and the Natural Sciences and Engineering Research Council and by a research grant of the Natural Sciences and Engineering Research Council. I am grateful to the De-partment of Oceanography, UBC for the partial financial support through my time at UBC. I would like to thank Dr. P. H. LeBlond, Dr. W. W. Hsieh and Dr. R. E . Thomson for their encouragement and many useful pieces of advice. I acknowledge the suggestions from anonymous referees on the submitted paper which contains part of this thesis. Many thanks to my wife, Danya Chu, and my parents for their generous support and love. -xiii-Chapter 1 General Introduction 1. General Introduction Why is the topographic problem worth studying? A good answer was given by Killworth (1989): "One of the first features of the ocean, which strikes its students, is the rich variety of bottom topography possessed by even the simplest ocean basin. The lay person often expresses surprise that so little is known about the interactions between flow and topography, and that so many theories of ocean circulation ignore variations in depth. The reason is obvious: the inter-actions between stratified flow and topography are complicated even in ideal-ized situations. We thus lack the kind of 'thumbnail sketch' of simple topographic effects that we possess for large-scale gyre circulation and many other problems." Consequently, there is no doubt that further knowledge of the interactions between flow and topography is significant for a better understanding the behavior of the ocean. I hope that this thesis can make some contribution to the efforts to improve our knowledge about the topographic problem in ocean science. Submarine canyons are important topographic features of the coastal regions of the world's oceans. For instance, there are more than twenty submarine canyons or valleys -1-Chapter 1 General Introduction along the northeastern coast of the United States (see Figure 1.1). Most canyons are con-fined to the shelf break, but some canyons, valleys and troughs do cut through the whole shelf and extend to the coast, e.g., the Hudson Shelf Valley and the Northeast Channel shown in Figure 1.1, and, furthermore, some canyons extend into estuaries, e.g., the Congo Canyon (see Figure 1.2) and the Juan de Fuca Canyon (see Figure 1.3. Strictly speaking, the Juan de Fuca Strait to which the Juan de Fuca Canyon extends is not a classical estu-ary). Does a canyon have impacts on local circulation, and hence fisheries, etc.? Let us examine the topography of some commercial fishing areas. The average commercial fishing yields from major commercial fishing zones in the northern hemisphere are shown in Figure 1.4. A common topographic feature of all these major commercial fishing areas is their association with major submarine canyons or troughs. For example, Georges Bank and Gulf of Maine are associated with the Northeast Channel (see Figure 1.1), the La Perouse Bank region encloses the Juan de Fuca Canyon (see Figure 1.3), and the North Sea covers the South Norwegian Trough (see Figure 1.5), etc.. Is it a fortuitous association of major commercial fishing areas with submarine can-yons? Investigations have shown that a canyon does have effects on local circulation. Many observations have demonstrated that submarine canyons serve as active conduits for the exchange of inshore and outer shelf water. Under certain circumstances this kind of Figure 1.1 Canyons off the northeastern United States. [After Fig. 79, Shepard et al. (1979)] 3* 6* CONGO SUBMARINE CANYON C M r o u " m £ " m 5 0 W M M S Figure 1.2 Contour chart of the Congo Submarine Canyon. [After Fig. 123, Shepardetal. (1979)] • 49/y 4 8 W 125" 124°»K -5-SE Alaska Nova Scotia North Sea La Perouse Georges Bank G. of Maine 7 7, 7/. 'A i i A 21 2 3 4 5 Catch (tonnes/kmVyr) Figure 1.4 Fishing statistics for the northern hemisphere. [After Fig. 2, Ware & Thomson (1992)] Figure 1.5 The North Sea region and the South Norwegian Trough. [After Fig. 9, Shepard (1931)] -7-Chapter 1 General Introduction horizontal water exchange may affect the local fishery and transportation of sediments or pollutants. The central point of this thesis is to provide a general model to investigate the hori-zontal water exchange induced by a canyon as well as its impacts on the local circulation. Chapter 1 General Introduction 1.1 Currents around Canyons Many observational studies around canyons in the past focused on tides or internal waves and the effects of these relatively high-frequency currents on sediment distribution and resuspension in canyons. The existence of longer time scale (and time mean) currents in and around submarine canyons has drawn attention only in recent years. Although some were designed for different purposes, several studies, e.g., Han et al. (1980), Mayer et al. (1982), Freeland & Denman (1982), Freeland et al. (1984), Hickey et al. (1986) and Hunkins (1988), support the existence of some coupling between shelf and canyon circula-tion. Such coupling exists in both shallow shelf-canyon systems, e.g., Hudson Shelf Valley [Mayer et al. (1982)], and deep ones, e.g., Juan de Fuca Canyon [Freeland & Denman (1982)]. Observations in the Juan de Fuca Canyon during the winter of 1970 showed that relatively large currents and large excursions exist both into and out of the canyon [Cannon (1972)]. The unidirectional out-canyon flow at both 53m and 3m above the bottom is as-sociated with the predominately southerly winds which cause the northwestward shelf cur-rent along the Vancouver Island southern coast. The major flow reversals observed in the canyon appear related to wind reversals (and hence reversals of the shelf current). Cannon (1972) found that not all the observations could be explained using Ekman transport the-ory. Freeland and Denman (1982) conducted an oceanographic experiment in the La Pe-1.1 Currents around Canyons -9-Chapter 1 General Introduction rouse Bank region from 1979 to 1981. Although, unfortunately, direct observations of cur-rents in the Juan de Fuca Canyon were not made in the experiment, the results provide a good picture of the flow field as well as distributions of sigma-r and dissolved oxygen around the canyon. A persistent cyclonic eddy was observed on the shelf at the northern side of the Juan de Fuca Canyon during summer months when the shelf break current is southeastward. Many observational studies suggest that a canyon can affect the circulation around it. 1.1 Currents around Canyons -10-Chapter 1 General Introduction 1.2 Topographic Waves and Properties of Double Kelvin Waves The topographic waves discussed in this thesis are class II waves (explained below). The restoring force of these waves requires a potential vorticity gradient. Class II waves include shelf waves (coastal trapped waves), topographic Rossby waves and double Kelvin waves. Shelf waves are trapped over the continental slope and shelf against a coast. A good review of shelf waves can be found in LeBlond and Mysak (1977). Topographic Rossby waves exist above a smooth topographic depth change which has horizontal length scale much greater than the wavelength of the waves. A discussion of topographic waves can be found in LeBlond and Mysak (1978). The above two kinds of class II waves are not the main topographic waves studied here, so they will not be discussed further. The domi-nant topographic waves in all the cases presented hereafter are double Kelvin waves. In a thin layer of hydrostatic fluid over a flat bottom, where the rotation, / , is uni-form in the domain, only superinertial frequency waves, such as Poincare waves, can exist. A change in depth permits the existence of subinertial waves which are called double Kel-vin waves. The dispersion equation and flow field of double Kelvin waves were first studied by Rhines (1967) and the results published in Rhines (1969a & 1969b). Using the rigid-lid ap-proximation, Rhines analyzed homogeneous flow over a step and a slope on a /?-plane and two-layer flow over a step on an /-plane. The dispersion relation and wave shape for the case of a free surface were derived by Longuet-Higgins (1968a) over a step and by 1.2 Topographic Waves and Properties of Double Kelvin Waves -11-Chapter 1 General Introduction Longuet-Higgins (1968b) over a slope in a homogenous fluid. The forced double Kelvin waves produced over a step by variable horizontal wind stress was investigated by Mysak (1969) in a homogeneous fluid and by Willmott (1984) in a two-layer stratified fluid. The properties of double Kelvin waves in the presence of a free surface will be rele-vant to most of the cases discussed in this thesis. These properties may be summarized as [see Longuet-Higgins (1968a)] (i) the waves are trapped along a discontinuity in depth; (ii) the wave amplitudes attenuate exponentially to either side of the discontinuity in depth; (iii) the waves propagate in the direction keeping shallow water at their right in the northern hemisphere; (iv) the waves are subinertial. Also, two results demonstrated by Longuet-Higgins (1968a) can be noted: first, superinertial waves cannot be captured by discontinui-ties in depth, although they may be partially reflected and partially transmitted, and second, if an initial surface elevation exits along a line perpendicular to the discontinuities in depth, say, a canyon, double Kelvin waves must be generated. The most frequently cited property of double Kelvin waves in this thesis is (iii) in the previous paragraph. This property is due to the conservation of potential vorticity dur-ing the motion of a particle, i.e., the stretching or compressing of planetary vortex lines as columns of fluid move down or up the ocean bottom changes. A full description of this mechanism can be found in Gill (1982, pg. 410-411). 1.2 Topographic Waves and Properties of Double Kelvin Waves -12-Chapter 1 General Introduction 1.3 Rossby Adjustment and the Choice of Initial Conditions 1.3.1 Rossby Adjustment The ocean and atmosphere tend to be close to a state of geostrophic equilibrium [see Philips (1963) for a review], therefore, to study the geostrophic (i.e., steady) state is usually the first and crucial step for understanding their behavior. Rossby adjustment [Rossby (1938)] is an effective approach to find the geostrophic solution while avoiding the complicated, transient, initial value problem. The emphasis of this thesis is the use of Rossby adjustment to study the geostrophic state over topography. The problem of the distribution of mass and resulting pressure in the ocean and at-mosphere in the course of adjustment to equilibrium was first considered by Rossby (1938). He studied a problem in which momentum was supposed to be put into the ocean to give a nonequilibrium velocity distribution. He found that the process of adjustment produces inertial waves with small group velocities which stay in the region of the jet and cause oscillations of the jet width. He also noticed that the final state has kinetic energy and that the fluid is undisturbed outside a narrow region around the initial jet. The scale of the narrow region is given by the Rossby radius of deformation. The time scale of the adjust-ment process is fl, i.e., the inertial time scale. For an initial condition consisting of a still fluid with a surface discontinuity — which is different from that considered by Rossby (1938), the Rossby adjustment in a 1.3 Rossby Adjustment and the Choice of Initial Conditions -13-Chapter 1 General Introduction channel is discussed by Gill (1976), and that in an open ocean is summarized in Gill (1982). The initial surface discontinuity produces superinertial gravity waves ~ Poincare waves. For short waves, the phase and group velocities are in the same direction and approxi-mately equal in magnitude. For long waves, the phase and group velocities tend to be zero. Note that the final, steady state differs from the initial condition only in a narrow (Rossby radius wide) region around the original change in surface elevation. Farther from the initial disturbance the propagating Poincare waves carry energy but no surface height changes. It was also found that 2/3 of the available potential energy is radiated by the Poincare waves while only 1/3 remains as kinetic energy of the mean flow. The problem of Rossby adjustment over topography has been considered in several previous papers. Using a four-layer general circulation numerical model, Hsieh & Gill (1984) studied Rossby adjustment in a channel in which a stepped bottom runs along one wall of the chan-nel. The flow over the topography produces subinertial shelf waves which are trapped by the step. The group velocities of the long waves are in the direction with the wall on the right in the northern hemisphere, whereas the group velocities of the short waves are in the opposite direction. Using linear analysis, numerical simulation and laboratory experiments, Gill et al. (1986) studied, in a homogeneous fluid, the Rossby adjustment over a step with and with-out an adjacent vertical wall. They demonstrated that a topographic step acts as a complete 1.3 Rossby Adjustment and the Choice of Initial Conditions -14-Chapter 1 General Introduction barrier to the approaching geostrophic flow (except for the case in which double Kelvin waves propagate towards the adjacent wall) and that a jet approaching deeper water is de-flected to the right. The geostrophic solution was found and demonstrated to be consistent with the numerical time-dependent solution and the laboratory experiments. Extending the work of Gill et al. (1986) and using an almost identical approach, Allen (1988) studied the Rossby adjustment over a slope in both a homogeneous fluid and a stratified fluid (a two-layer model). The results in a homogenous fluid show similar fea-tures to those over a step (limit of a slope). In the geostrophic state, the fluid over a slope is stagnant (except for the singular region near the wall adjacent to the slope when double Kelvin waves propagate towards the wall), whereas, beyond the slope, the results are ex-actly the same as those beyond a step. However, the stratification introduces new and more complex responses to the adjustment over a slope. In the steady state, the lower layer of the two-layer stratified fluid is stagnant over the slope. If the background flow is principally barotfopic, the incoming jets turn to flow along the slope keeping the shallow water on the right. On the contrary, if the background flow has a baroclinic character, the incoming jets split and parts travel each way along the slope. The geostrophic adjustment over an infinitely long, rectangular, flat bottom canyon was considered in a homogenous fluid [Klinck (1988)] and in a stratified fluid [Klinck (1989)]. In a homogenous fluid, it was found that flow in the canyon responds quickly, in about 0.1 inertial period, to an overlying current and that the time-dependent solution con-sists of standing waves over the canyon and radiating waves on either side of the canyon. 1.3 Rossby Adjustment and the Choice of Initial Conditions -15-Chapter 1 General Introduction The limit of a narrow canyon was considered (A canyon is narrow if its width is less than half of the smaller of the radius of deformation or the width scale of the overlying current). In a stratified fluid, the vertical structure of the fluid was handled with two-level and three-level models. The four important length scales in the adjustment process were compared. They are the internal and external radii of deformation, the initial width of the shelf current, and the width of the canyon. For each vertical mode, the shorter of the radius of deforma-tion for that mode and the width of the coastal current determines the decay distance of the canyon influence; the width of the canyon determines the strength of the canyon's effect on the overlying coastal current. It was shown that even in the case of a flow over a narrow canyon, the isopycnals at the top of the canyon are distorted and there is also some residual circulation on the shelf forced by the presence of the canyon. Rossby adjustment over topography proceeds through propagation of superinertial waves and subinertial waves trapped by the depth discontinuities. The trapped waves are double Kelvin waves for a single-step bottom and modified double Kelvin waves for com-plicated topography. The importance of the superinertial and subinertial waves is different for the two phases of the adjustment process. Superinertial waves dominate in the short time scale phase, whereas, trapped subinertial waves prevail in the long time scale phase. Because the main concern of this thesis is the long time scale effects of canyons on the cir-culation around them, the trapped waves (subinertial waves) will be discussed more fre-quently. The theme of the thesis is to use Rossby adjustment to analyze the geostrophic cir-1.3 Rossby Adjustment and the Choice of Initial Conditions -16-Chapter 1 General Introduction culation around different types of canyons. 1.3.2 Initial Conditions The axes are chosen with the x axis perpendicular to the canyon and the y axis along the central axis of the canyon. Some limitations are imposed on the initial conditions that will be used for the Rossby adjustment problem in this thesis. First, the initial state is nongeostrophic. This limitation is obvious. If the initial state were geostrophic everywhere, there would be no need for the potential vorticity field to adjust, and hence the topographic effect on the adjustment process could not be studied. Second, if there is an initial flow, it must be in the x direction; if the fluid is initially at rest, the x-component of the derivative of the initial surface/interface elevation must be zero, i.e., the initial surface/interface elevation must be a function of only one variable, y. A current can be decomposed into an x-component (in the across-canyon direction) and a y-component (in the along-canyon direction). However, the effect of the canyon on the y-component of the flow is much weaker than the x-component of the flow. The second limitation guarantees a geostrophic flow beyond the region affected by the canyon in the across-canyon direction. So, the strongest effect of a canyon to an approaching geostrophic flow will be studied. It also guarantees the existence of trapped waves at depth changes. 1.3 Rossby Adjustment and the Choice of Initial Conditions -17-Chapter 1 General Introduction Third, the ^ -derivative of the initial surface/interface elevation is non-zero only in a finite region, e.g., along a line perpendicular to the canyon, and is zero anywhere beyond this region. The purpose of imposing this limitation is to create a geostrophic jet. There-fore, the infinitely wide, oscillating initial current considered by Klinck (1988 and 1989) is excluded. A classical initial condition used in Rossby adjustment [Gill (1982), Gill et al. (1986), Allen (1988)] is one with a zero initial velocity, (w, v), but with a surface elevation, 77, discontinuity along a line (at.y = 0 in our coordinates), i.e., 77, = -77 0 sgn(j/), (1 3.1a) Wj =Vj = 0, (1.3.1b) where variables with subscript "7" are those at the initial time (t = 0); 770 is the magnitude of the surface displacement and f 1 y>o, sgntj/H (1-3.2) [-1 y<0. The classical initial condition (1.3.1) satisfies the limitations that we imposed. First, it is nongeostrophic. Second, the fluid is initially at rest, and there is a surface displace-ment, but the x-derivative of the surface elevation is zero. Third, the ^-component of the derivative of the initial surface elevation is non-zero only at_y = 0. Longuet-Ffiggins (1968a) noted that the initial condition (1.3.1) will generate 1.3 Rossby Adjustment and the Choice of Initial Conditions -18-Chapter 1 General Introduction trapped waves around discontinuities in depth. For this reason, as well as for its simplicity, the initial condition (1.3.1) will be the most commonly used for analytical studies in this thesis. 1.3 Rossby Adjustment and the Choice of Initial Conditions -19-Chapter 1 General Introduction 1.4 Derivation of the Equations The fluid considered is incompressible, hydrostatic and on an /-plane. When stratifi-cation is considered, the fluid consists of a series of layers, each of uniform density. The pressure within each layer is uniform with depth, and the flow is independent of depth in each layer. The flow will be treated as inviscid which is a reasonable approximation in the oceanographic context. Shallow water motion, i.e., motion with the horizontal scale large compared with the depth, is also assumed (long wave approximation). Therefore, the gov-erning equations are the shallow water equations. The nonlinear shallow water equations for a homogeneous fluid are du ck du r dn a + u & + v # - f i + g & = 0 - ( 1 4 1 a ) & , &n ~ . s ^ + ^ [ ( A + i 7 ) « ] + | ; [ ( A + i 7 ) v ] = 0, (1.4.1c) where 77 is the surface elevation; / is the Coriolis frequency; g is the acceleration due to gravity; (u, v) is the horizontal velocity and h is the undisturbed depth of the fluid. The nonlinear shallow water equations for a two-layer stratified fluid are 1.4 Derivation of the Equations -20-Chapter 1 General Introduction fa du du r &n d% ^ , „ . — + u—+v—--jv+g'-^-+g-r- = 0, (1.4.2a) dv ck dv , dn dE „ a+"a+v^+^+^'f+*#=0' . <1A2b) f-+|[(*+>/)»l+| ;[(A+'/H = 0> (1.4.2c) j r = 0 , (i.4.2d) dV dV dV dt , ^t*+|[(ff+#-,X/]+|[(ff + f - # ] = 0. (1.4.20 where 77 is the interface elevation; £ is the surface elevation;/and g are defined in (1.4.1); Ap g1 is the acceleration due to the reduced gravity given by —g where Ap is the difference P in density between the two layers and p is the average density; (u, v) and (U, V) are the horizontal velocities in the lower and upper layers, respectively; h and H are the undis-turbed depths of the lower and upper layers of the fluid, respectively. A derivation of the shallow water equations can be found in Gill (1982, chapter 4, 5 and 7). Because the full nonlinear shallow water equations are difficult to solve analytically, 1.4 Derivation of the Equations -21-Chapter 1 General Introduction only linear forms of the equations will be considered for the analytic research in Chapter 2 and 3 . The results from linear theory are a good approximation to the real situation, as long as a "small flow" assumption is valid. The effects of the nonlinear advection terms in the primary equations will be discussed in the numerical studies presented in chapter 4. 1.4 Derivation of the Equations -22-Chapter 1 General Introduction 1.5 Geometric Models Two topographic models will be studied. 1. 5.1 Flat Bottom Canyon (F Canyon) The first model is called the "Flat Bottom Canyon": an infinitely long canyon cuts through a flat, infinite ocean with depth hi. The canyon has vertical edges, a uniform width, 2L, and a flat bottom with depth h2. This geometric model is illustrated in Figure 1.6. It will be referred to as an F canyon in this thesis for simplicity. A limitation is imposed on the F canyon, i.e., hi can never equal h2. It is under-standable because there is no canyon if hi = h2. All the analytical results obtained in this thesis are not intended to apply in the limit of a flat bottom ocean. An F canyon will also be occasionally referred to as the primary model. 1.5.2 Sloping Bottom Canyon with a Coast and Shelf Break (S Canyon) The second model is called the "Sloping Bottom Canyon with a Coast and Shelf Break": a canyon cuts through a flat and infinitely long shelf with depth hi. The shelf break and the coast are both vertical. The canyon has vertical edges and a uniform width, 2L. The mouth of the canyon is at the shelf break and the head of the canyon is within an infinitely long, flat bottom strait with depth ho and with uniform width, 2L. The bottom of the can-yon consists of four segments: a flat, deep, mouth part with the same depth as the deep 1.5 Geometric Models -23-Canyon Srir?i llt ^Tu^ °im F C a n y ° n - T h e b o t t o m o f t h * ^nyon is flat- the edges are -24-Chapter 1 General Introduction ocean, h3, a flat, shallower main part with depth h2, a (mouth) slope joining the deep mouth part and the shallower main part and a (head) slope joining the main part with the bottom of the strait. This geometric model is illustrated in Figure 1.7. It will be referred to as an S canyon for simplicity. The canyon is thought of as being finite length with the understand-ing that the canyon is distinguished from the strait by the "canyon head slope". This type of canyon represents submarine canyons which extend from the continen-tal shelf break into an estuary, e.g., the Congo Canyon (see Figure 1.2), or, into a strait, e.g., the Juan de Fuca Canyon (see Figure 1.3). This type of canyon is expected to have an influence on the coastal circulation. The S canyon is a suitable geometric model to simulate the topography of this type of canyon. An S canyon will also be occasionally referred to as an advanced model. 1.5 Geometric Models -25-Land ShelfBreak Deep Ocean Figure 1.7 Geometry of an S canyon. The bottom of the canyon consists of four segments: deep mouth part, mouth slope, main body and head slope; the edges are vertical; the width is constant. The shelf break, coast and strait walls are all vertical. The length of the strait is uifinite. The shelf is infinitely long and flat. -26-Chapter 1 General Introduction 1.6 Objectives Although many observations demonstrate the existence of currents within canyons, the mechanism for canyon currents and their impact on the local circulation have not been satisfactorily explained. This thesis will provide a perspective on the problem. The results obtained are expected to have general validity and to be applicable in many situations. The difficulties in the study of interactions between flow and topography are well known. Even the simplest problem rapidly becomes too complicated for analytical studies. Even for the simplest topography — a step or a slope — the complexity of the problem can be sensed from the works of Gill et al. (1986) and Allen (1988). As shown in Klinck (1988 & 1989), study of the geostrophic adjustment over a canyon makes the problem much more complicated because of the coupled effects of the canyon edges. To prevent the problem becoming analytically untractable, most of studies of the interactions between flow and topography are limited to topography in only one dimension [GUI et al. (1986), Allen (1988), Klinck (1988 & 1989), etc.]. However, the simplification of the topography makes the research results hard to generalize. For example, Klinck (1988 & 1989) used the simplest canyon geometry — an infinitely long, flat bottom canyon. The assumption of an infinitely long canyon excludes the consideration of the effects of either a coast or a continental shelf break; the assumption of the flat bottom canyon highlights the obvious unrealistic nature of the model. All the simplification of the geometry greatly limits the application of the results obtained by Klinck (1988 & 1989). 1.6 Objectives -27-Chapter 1 General Introduction To make the study of Rossby adjustment over a canyon more realistic (at least similar to some kinds of canyons, e.g., the Juan de Fuca Canyon), a complicated configu-ration is used in the S canyon model. The effects of both a coast and a shelf break as well as a sloping canyon bottom are all included in the model. The study of Rossby adjustment around an S canyon is an attempt to attack the problem for topography changing in two dimensions and is expected to be a great improvement in understanding the circulation around these canyons. Though the F canyon has the same geometry as that used by Klinck (1988 & 1989), the Rossby adjustment over the canyon is studied with a different approach. Therefore, the results obtained from the primary canyon model provide a different point of view on the problem. The thesis consists of two parts: the analytical part which includes chapters two and three and the numerical part which is made up of chapters four and 'five. The objective of the analytical study is to use Rossby adjustment to determine the geostrophic circulation around canyons, whose significance has been explained in section 1.3.1. The objectives of the numerical study are to analyze the trends of time-dependent solutions around canyons to confirm the analytical geostrophic solutions and to analyze the effects of nonlinearity. Chapter two provides a detailed analytical study of the geostrophic circulation around an F canyon. The analytical solution is found for the whole domain for a homoge-neous fluid. An important parameter which governs the geostrophic state around canyons 1.6 Objectives -28-Chapter 1 General Introduction is derived. Results obtained from the primary model can be applied to many situations, e.g., other complicated canyons, straits, ridges, etc.. Chapter three directly applies the results obtained from the F canyon model as well as the properties of both double Kelvin waves and Kelvin waves to achieve the analytical, homogeneous, geostrophic solution for the S canyon model. The analysis in this chapter can be taken as an example of how to apply the results obtained from the primary model, and the method can be generalized to different situations. Chapter four presents numerical transient solutions in a homogeneous fluid for both the F canyon and the S canyon. Nonlinear effects are also discussed. Chapter five shows the numerical transient solutions in a stratified fluid for the F canyon and the S canyon. The stratification considered is a simple two-layer model. A summary of all the results, a discussion of their implications and suggestions for future work are given in chapter six. -29-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid 2. G e o s t r o n h i r Solut ions over a F l a t B o t t o m C a n v n n in a H o m o g e n e o u s F l u i d 2.1 Introduction First, this chapter gives an analysis of the waves which exist in the F canyon model, including the properties of canyon waves which are a form of modified double Kelvin waves, and, second, provides detailed analysis of the geostrophic state over an F canyon in a homogeneous fluid. In section 2.2, the governing equations, which will be used in the Rossby adjust-ment problem for a homogenous fluid, will be discussed. Properties of the solutions are analyzed qualitatively. In section 2.3, the waves which exist in a canyon will be discussed. The basic prop-erties of canyon waves and the dispersion relation will be revealed. The long canyon waves provide the foundation to determine a particular solution for the Rossby adjustment prob-lem (section 2.4). In section 2.4 discussion will be focused on the analytical geostrophic solution over an F canyon in a homogeneous fluid using the concept of Rossby adjustment. In particular, section 2.4.1 presents the process to determine a particular solution constructed from the longest canyon wave generated by a general initial condition. Section 2.4.2 shows this so-2.1 Introduction -30-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid lution to be the geostrophic solution in the far field away from the initial disruption for the special initial condition, (1.3.1). This initial condition will be the only one considered in the analytical studies after section 2.4.2. Section 2.4.3 gives the analytical geostrophic solution for the whole domain by using the far field solution as the required boundary conditions. Section 2.4.4 presents the numerical results from direct integration of the geostrophic gov-erning equation. Section 2.4 is the core of the thesis as well as of this chapter. A discussion of the results obtained in previous sections will be presented in section 2.5. Application of these results to Rossby adjustment over other topography, such as steps, ridges and straits, will be briefly discussed. The barotropic geostrophic solution over an F canyon will be summarized in the last section. 2.1 Introduction -31-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid 2.2 The Governing Equations and a Qualitative Analysis 2.2.1 Derivation of the Equations The governing equations are the shallow water equations, (1.4.1), for a homogene-ous fluid. Only linear problems are considered in this chapter. Nonlinear effects will be in-vestigated using a numerical code when discussing the numerical transient solutions in Chapter 4. The linear form of (1.4.1) is ^->+3- = 0 , (2.2.1a) dt dx f+>+,^  = 0, (2.2.1b) ^ • + | ; ( A » ) + - f - ( A v ) = 0. (2.2.1c) ot ax qy Taking the divergence of the momentum equations, i.e., d I dx of (2.2.1a) plus du av d I cy of (2.2.1b), and substituting from (2.2.1c) for the horizontal divergence, ~^ + ~^ > gives -^--ghV2rj +M + V/i) = 0, (2.2.2) where 2.2 The Governing Equations and a Qualitative Analysis -32-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid dv du is the relative vorticity; V7J is the gradient of the topography and u = (u, v). Taking the curl of the momentum equations, i.e., d I dy of (2.2. la) minus 61 dx, of du dv (2.2.1b), and substituting from (2.2.1c) for the horizontal divergence, + gives the vorticity equation d_ a + -u-Vh = 0. (2.2.3) h As defined in section 1.5, the topography is flat, i.e., Vh = 0, everywhere other than such places as the canyon edges, the canyon bottom slopes (for an S canyon) and the shelf break (for an S canyon). Except at these special places, (2.2.2) reduces to d2n a2 and (2.2.3) reduces to ghW2n+M = 0, (2.2.4) h f h f 3 (2.2.5) where variables with the subscript "7" are those at the initial time (t = 0). The right hand side of (2.2.5) is for the initial state and the left hand side is for a subsequent state. There-fore, it is possible to calculate a state from the known initial state without knowing the 2.2 The Governing Equations and a Qualitative Analysis -33-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid transient process. The initial condition in (2.2.5) is for a general situation. For the reasons explained in section 1.3.2, the initial condition is limited to functions of one variable, y, in this thesis. Defining an initial quantity, which is also a function of single variable, y, a(*oO = y £ ( * J ' ) - ' 7 i O O , (2-2.6) and then substituting (2.2.5), (2.2.6) and R = y[gh '\f\ (the barotropic Rossby radius of deformation) into (2.2.4) gives an equation in only one variable, TJ, namely, 1 ftfcyj) _ ^ ^ X t y . t ) + v ( x ^ t ) = _Qj M t ( 2 > r 7 a ) / St which is associated with the initial condition 'fi(x,yfi) = ni(y)- (22.7b) Equations (2.2.7) are the primary governing equations for Rossby adjustment over a flat topography. Solution of the initial value problem (2.2.7) can be found by decomposing it into two problems and then adding their solutions. One problem, namely, R2V\(*,y) - V,(x,y) = a(x,y), (2.2.8) gives a particular solution of (2.2.7) ~ the steady state solution, rjs(x, y). The other one is 2.2 The Governing Equations and a Qualitative Analysis -34-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid the homogeneous form of (2.2.7) with the corresponding initial condition, i.e., 1 02vw(x,y,Q r a2 -R2V2nw(x>y,t) + 71w(x,y,t) = 0, (2.2.9a) 7* (*, yfi) = Vi O) -1 , (x, y), (2.2.9b) which gives the transient wave solution, rjjjc, y, t). Obviously, it is possible to solve the problem (2.2.9) only after the steady state solution, TJS(X, y), has been found. Consequently, solving (2.2.8) is the first step in solving the Rossby adjustment problem, (2.2.7). It is easy to solve (2.2.8) in a flat bottom ocean [an example for the initial condition (1.3.1) can be found in Gill (1982), chapter 7]. For a stepped bottom ocean with a vertical wall parallel to the step, the process to solve (2.2.8) for the initial condition (1.3.1) is given in Appendix C, and the results will be used in the analysis in Chapter 3. Solving (2.2.8) is complicated for a topography including a canyon. In the region far away from the canyon where the motion is not disturbed by the existence of the canyon, the process of adjustment is the same as that over a flat bottom ocean (for an F canyon) or a stepped bottom ocean with a vertical wall parallel to the step (for an S canyon). To find the solution of (2.2.8) near or within the canyon, solutions at the canyon edges as well as at the shelf break and the canyon bottom slopes (for an S canyon) must be found. Because of the complexity of the waves which exist over topography including can-yons, solving the transient problem, (2.2.9), is more difficult than solving (2.2.8). In this 2.2 The Governing Equations and a Qualitative Analysis -35-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid thesis, (2.2.9) will not be solved. Only the general properties of the waves which make up the solution, especially the canyon waves, will be analyzed in section 2.3. The complete solution of the adjustment problem will be illustrated with a numeri-cal simulation for a homogenous fluid in Chapter 4. 1X2 Steady State In the steady state, (2.2.3) reduces to u1-Vh = 0. (2.2.10) So the geostrophic current must be parallel to the topography for which Vh * 0. For the canyon models, this requirement means that no geostrophic flow will cross the canyon edges, the shelf break or the canyon bottom slopes (except at some singular points as will be explained in Chapter 3).; Therefore, in the steady state, the surface height, TJS, over these places must be uniform (except at those singular points). Finding the surface elevation over these places is crucial in order to solve the steady state, (2.2.8), and then the complete problem. For reference in Chapter 3, the calculation of the volume flux, in the steady state, for flat topography is given here. If h is the undisturbed water depth over a flat bottom ocean, and pq is a line orien-tated in the positive y direction from p(x, yp) to q(x, yq), the volume flux across pq in the 2.2 The Governing Equations and a Qualitative Analysis -36-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid steady state is [from (2.2. la)] ^ * ) = £ ^ = - ^ ( 2 2 1 1 ) Similarly, if pq is a line orientated in the +x direction from p(Xp, y) to q(xq, y), the volume flux across pq in the steady state is [from (2.2. lb)] ^,O0 = £ ^ ^ (2.2.12) Consequently, in a region of uniform depth, the volume flux across a line is de-termined by the difference of the surface elevation between the end points. Note that (2.2.11) is the difference of the end point value and the beginning point value whereas (2.2.12) is the opposite. 2.2.5 Rossby Adjustment over a Flat Bottom Ocean To familiarize readers with the concept of Rossby adjustment and prepare for more complicated analysis later, it is necessary to discuss a simple case — Rossby adjustment over a flat bottom open ocean. Rossby adjustment over a flat bottom open ocean has been briefly introduced in section 1.3.1. For an ocean with a constant depth h and initial condition (1.3.1) the solution of the steady state equation (2.2.8) is 2.2 The Governing Equations and a Qualitative Analysis -37-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid = Vo l-expQ>/i?) -l + exp(-y/K) fory < 0, for^ y > 0. (2.2.13a) Substituting (2.2.13a) into (2.2.1b) and (2.2.1a) gives ut = (gfiJJR)e>rt-)x\/R), (2.2.13b) v, = 0. (2.2.13c) Rossby adjustment produces a geostrophic jet which is centered at.y = 0 — along the initial surface discontinuity. In the final steady state, only the region near (several Rossby radii) the initial dis-continuity has been affected by the adjustment. The adjustment is mediated by Poincare waves, which for a flat bottom have a dis-persion relation As will be shown later for Rossby adjustment over an F canyon, the solution far • (say, three barotropic Rossby radii) away from the canyon edges is the same as for Rossby adjustment over flat bottom ocean. a>2=f2+gh(l2+k2). (2.2.14) 2.2 The Governing Equations and a Qualitative Analysis -38-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid 2.3 Waves over an F Canyon 2.3.1 Superinertial and Subinertial Waves Some properties of the waves which exist over an F canyon are investigated in this section. The definition of an F canyon was given in section 1.5.1. The axes are chosen with the x axis perpendicular to the canyon and the.y axis along the centre of the canyon. The wave equation (2.2.9a) has a wave-like solution »7 w(*,^0«exp[;(/,x + *v-fflr)], j = \, 2 (2.3.1) provided lj and k satisfy l i + k 2 = °>2-f2 (2.3.2) in which G) > 0 is the circular frequency; j = 1 is for a point (x, y) on the shelf, and j = 2 is for a point within the canyon; lj and k are the x- and -^component of the wave number, re-spectively. Note that the -^component of the wave number should be the same on the shelf and within the canyon because of the uniform depth of the fluid in.y dimension, while the x-component of the wave number is not uniform because of the depth discontinuities in x di-mension [see demonstration in Longuet-Higgins (1968a) for a single-step]. There are different situations whereby (2.3.2) is satisfied. 2.3 Waves over an F Canyon -39-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid (i). Superinertial waves (to >f) — Poincare Waves / and k are both real Both lj and k can be both real if co >/in (2.3.2) (necessary but not sufficient), i.e., waves can propagate in both a component across-canyon (described by lj) and a component in the along-canyon direction (described by k). If we let kH;=i;+k\ (2.3.3) equation (2.3.2) has the same form as (7.3.4) in Grill (1982). Waves with this dispersion relation (lj and k real) are referred to as Poincare waves [see Gill (1982), pg. 196]. The pa-rameter, kffj = (lj, k), is the horizontal wave number vector. The dispersion relation curve for Poincare waves can be found in Fig. 7.2 in Gill (1982). Note that the dispersion relation for Poincare waves within the canyon is different from that on the shelf, because the short wave speed within the canyon, ^jghj, is faster than that on the shelf, y[gl\. I and k are not both real One of the wave numbers, lj or k (but not both), can be imaginary satisfy (2.3.2) of co >f. As demonstrated by Longuet-Higgins (1968a), a wave incident on a depth disconti-nuity will be partially reflected and partially transmitted. In the direction normal to the depth discontinuity, the transmitted wave number, (say) / 2, could be imaginary, while the 2.3 Waves over an F Canyon -40-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid incident wave number, A, whose value equals to that of the negative reflected wave num-ber, must be real; in the direction along the depth discontinuity, the incident wave, the re-flected wave and the transmitted wave have the same real wave number, k. For the F canyon model, the superinertial waves on the part of the shelf from which the waves are incident on the canyon must be Poincare waves because both h (say) and k are real and the dispersion relation (2.3.2) is valid. However, within the canyon, because a component of the wave number may be imaginary, the superinertial waves may be "evanescent" Poincare waves (LeBlond, private communication). In summary, the superinertial waves in the F canyon model are Poincare waves, or, perhaps, "evanescent" Poincare waves. Further detailed analysis of Poincare waves will be neglected for two reasons. First, trapped waves at a depth discontinuity are of great interest in this study, whereas, superin-ertial waves cannot exist as trapped waves. Second, the study of the long time scale effects of canyons on the circulation around them is the main concern of this thesis, whereas, Poincare waves play little role in the determination of the steady state over topography. Following the analysis of Gill et al. (1986) and Willmott & Grimshaw (1991), Poincare waves will not be analyzed here. (ii). Subinertial waves (<»</) — Canyon Waves The wave numbers, / ; and k, cannot both be real if co </in (2.3.2), i.e., the motion 2.3 Waves over an F Canyon -41-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid must vary exponentially in at least one horizontal direction. As introduced in section 1.2, waves induced by discontinuities in depth can only propagate along the discontinuities. In the canyon problem, subinertial waves must propagate in the y direction while their magni-tudes exponentially attenuate in the x direction, i.e., lj must be imaginary. The solution of (2.2.9a) may be written in the form nw(x,y, t) = E(x) exp[/(*y - <ot)) (2.3.4a) where E(x) is a real exponential function of x and k (real) is the wave number in the along-canyon direction. Substituting (2.3.4a) into the momentum equations (2.2. la) and (2.2.lb) gives vM=i^Mzm4^{ky_ml)] (23 4b) and g[jE'(x)-cokE(x)] r , niA^ vw(x,y,t)= J . 2 _ Q ) 2 ^-exp^-Grfj] (2.3.4c) where E'(x) is the first derivative of E. It can be seen that rjw and vw have the same phase, but uw is out of phase. Substituting (2.3.4a) into (2.2.9a) gives d2E dx r- a / £ = 0, y = l , 2 (2.3.5) 2.3 Waves over an F Canyon -42-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid where «,>,*) = g f } 6 } , j = 1, 2. (2.3.6) Because co </and k is real, af must be greater than zero. In the following discus-sions |/2-<y2+*Vr. will be considered positive and the minus sign will be noted explicitly where needed. Then the bounded solution of (2.3.5) can be found Al exp(ajjc), x<-L, E(x) =' exp(a2x) + B2 exp(-a2x)], -L<x<L, (2.3.8) B3 exp(-a,x), x > L, where A\, A2, B2 and B3 are all nonzero constants (real) to be determined. Substituting (2.3.8) into (2.3.4a) and (2.3.4b) and then using the requirement of continuity of r/w and huw at the canyon edges, x = ±L, gives exp(-alL)Al - exp(-a2L)A2 - exp(a2L)B2 = 0, (2.3.9a) hl{alco-kf)exp(-alL)A1 -h2{a2co-kf)exp(-a2L)A2 + h2(a2co + kf)exp(a2L)B2 = 0, ' (2.3.9b) 2.3 Waves over an F Canyon -43-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid exp(a 2Z)42 + Qxp(-a2L)B2 - exp(-alL)B3 = 0, (2.3.9c) h^ajO) - kf) exp(a 2I)4j -h^a^ + kf) exp(-a2L)B2 + hl(ala> + kf) Gxp{-a^L)B3 = 0, (2.3.9d) which can be written in the matrix form M L A = 0. (2.3.9e) where exp(-a,Z,) -exp(-a2L) -exp(a2Z.) 0 f\{alQ}- A/")exp(-a,Z.) -h1{a1co-kf)&vp{-a1L) hi(a2a> + kf)exp(a2L) 0 0 exp(a2L) exp(-a2L) -exp(-a,Z,) 0 /Zj(a2a>-A/")exp(a2L) -/ij(a2a> + A7")exp(-a2Z,) hl{ala + kf)e,\-p{-alL) (2.3.9f) For a nontrivial solution of (2.3.9), i.e., Ai, A2, B2 and B3 not all zero, the determi-nant, | M | , in (2.3.9e) must be zero. This yields the dispersion relation for the waves, 9ft-V d V K a2 +a, 2 +2 a,a 2cth(2Z,a 2) (2.3.10) where a}(co, k) > 0 (/' = 1, 2) is given by (2.3.7). In nondimensional form, after substitution of (2.3.7), (2.3.10) becomes 2.3 Waves over an F Canyon -44-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid £{r2-i)2=P+r4(i-s*+P) + (2.3.11) ^ 2 + 2 ^ ( l - « 2 + P ) [ r 2 ( l - « 2 ) + P]cth|y9Vl-e?2+P)} where "cth" is the hyperbolic cotangent function; co = coif; k=kR2; Rr = Jgh~l[f\ (Rossby radius on the shelf); ic^  = fgh\ I\f \ (Rossby radius over the canyon); P=2L/R2 (2.3.12) (nondimensional canyon width), and Y=R2IR,=^hJh\ (2.3.13) (square root of the ratio of the depth of canyon to the depth on the shelf). The parameters B and ^ are fundamental to the description of adjustment over a canyon. Waves with the dispersion relation (2.3.10), or (2.3.11), will be referred to as Can-yon waves. The dispersion relation of canyon waves, (2.3.11), is plotted in Figure 2.1 for the examples of (a) p = 0.5, y2 = 2, 3, 4 and 5, and (b) y2 = 4, p = 0.01, 0.05, 0.25, 1 and 10. The curves of the dispersion relation are symmetric around the frequency axis, but Fig-ure 2.1 only shows the curve for positive wavelengths. Because of the importance of canyon waves to these studies, it is necessary to dis-cuss their properties in a separate section. After some properties of canyon waves are un-derstood, the structure of the wave solution will be discussed again at the end of this section. 2.3 Waves over an F Canyon -45-D i s p e r s i o n R e l a t i o n f o r C a n y o n W a v e s 0 2 4 6 8 10 N0NOIMENSI0NRL WRVENUMBER RL0NG CRNY0N DIRECTI0N (a) 12 Figure 2.1 Dispersion relation for canyon waves [the nondimensional form, (2.3.11)] for an F canyon with (a) B =0.5, y2 = 2, 3, 4 and 5, and (b) V = 4, /9 =0.01, 0.05, 0.25, 1 and 10. Only the part for the positive wave number is shown. The curves are symmetric about the frequency axis. The horizontal axis rep-resents the wavenumber in the along-canyon direction, k, and the vertical axis rep-resents the frequency, 65. -46-D i s p e r s i o n R e l a t i o n f o r C a n y o n Waves 0 _ 8 -] ' 1 ' 1 • 1 i i i i i 0 2 4 6 8 10 12 N0NDIMENSI0NRL WflVENUMBER AL0NG CRNY0N DIRECTI0N (b) -47-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid 2.3.2 Properties of Canyon Waves Several conclusions can be reached from the dispersion relation of canyon waves. First, canyon waves are subinertial. Consequently, we have l-ed2 > 0 in (2.3.11). Note that the canyon width, /?, is always positive and the "cth" function in (2.3.11) is al-ways above 1. The upper frequency limit for canyon waves can be found from (2.3.11) The upper frequency limit is determined only by the depth ratio of the canyon sys-tem. The shallower the canyon, the lower the maximum frequency. The waves in the deep-est canyon, i.e., y -> oo, have the widest frequency band, covering the whole subinertial frequency band. Expression (2.3.14) is confirmed by Figure 2.1. For instance, if y2 = 4, the wave frequency must be below ,0.6/independent of the value of /?. In other words, there will be no waves in the band 0.6/to/ Second, canyon waves are dispersive. The phase and group velocities are unidirec-tional but both are functions of wave number. For short waves (k » I/R2), the group ve-locity tends to zero (the phase velocity also approaches zero but at a slower rate than the group velocity). For long waves (k « I/R2 and hence <y «f), the phase and group ve-locities are approximately equal in magnitude. The maximum phase/group speed is ob-tained by allowing k -> 0 (and hence a -> 0) in (2.3.11), that is 0<c3< (2.3.14) 2.3 Waves over an F Canyon -48-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid co \Y2-"i (2.3.15) k C l i]y2+2ycthB +1 where c, = -^/g^ . If the width of the canyon is much larger than the Rossby radius over the canyon, i.e., p » \ (2.3.15) reduces to which is identical to the long double Kelvin wave speed given by (2.18) of Gill et al. (1986). Because in the wide canyon case the effects of the canyon edges are decoupled, either canyon edge acts as a single-step. The canyon waves, therefore, degenerate into two sets of double Kelvin waves which exist one on each canyon edge and propagate along the edge in the direction keeping the shallow water to their right in the northern hemisphere. Long waves are expected to determine the final steady state as shown by Willmott & Grimshaw (1991) and Allen (1995). From Figure 2.1, for a given wavelength, two further conclusions can be obtained. If the canyon width is a constant, the deeper the canyon, the faster the phase and group speeds are [see Figure 2.1(a)]. Therefore, flows in deep canyons adjust to the geostrophic state faster than in shallow canyons. This property of canyon waves will be a guideline when choosing parameters for the numerical simulations in chapters 4 and 5. If the depth ratio of a canyon-shelf system is a constant, the wider the canyon, the faster the phase (2.3.16) 2.3 Waves over an F Canyon -49-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid speed is [see Figure 2.1(b)]. However, the effect of the width of the canyon on group speed is a function of wavelength: the wider the canyon, the faster the group speed at long wavelengths, but the slower the group speed at short wavelengths. Third, canyon waves propagate in either direction along the canyon (the phase and group velocities have both positive and negative signs). This property is due to the sym-metric nature of the dispersion curves. Therefore, if a canyon wave (for a given frequency) propagates along an F canyon in one direction, there must be a counter-part which propa-gates in the opposite direction. As shown above for the wide canyon case, canyon waves can be considered as "modified" double Kelvin waves. One can say that canyon waves are a composition of two sets of double Kelvin waves which are induced by both canyon edges, propagate in opposite directions and are concentrated at different edges. Canyon waves retain the properties of double Kelvin waves. Therefore, canyon waves must propa-gate along a particular canyon edge in the direction keeping the other canyon edge to their left in the northern hemisphere. 2.3.3 Structure of Waves over an F Canyon It has been demonstrated in section 2.3.1 that both superinertial and subinertial waves can exist over an F canyon. These two types of waves can co-exist but possess dif-ferent properties. The complexity of superimposed waves makes it almost impossible to solve the wave equation (2.2.9) analytically. Therefore, only qualitative analysis of the wave solution will be presented in this section. 2.3 Waves over an F Canyon -50-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid Though waves passing a point in the domain consist of both superinertial waves and canyon waves, they are separable in the frequency spectrum. Waves with frequency above / v 2 - l , are the superinertial waves, whereas, waves with frequency below ^ 2 ^ / , where is the depth ratio of canyon to that on the shelf, are canyon waves. Note that there is a frequency r2-i gap, i.e., no waves are found with frequencies between — — - / and/ Y +1 The wave solution of (2.2.9) may be found by integrating (2.3.1) across the whole frequency spectrum, i.e., T]w(x,y,t) = R | | q (superimposed waves)cfcyj f r — / r-= Rj Jj ' 2 + 1 (Canyon Waves)*/o + (Poincare Waves)da f rlzlf = Rj jr1+i E(x) exp[/(£y - at^co + (Poincare Waves>fc» (2.3.17) parti partU where K{p} means the real part ofp; E(x) is an exponential function which may be written in the form (2.3.8). The wave number, k, in part I is given by the dispersion relation for canyon waves, (2.3.10). Consequently, the wave solution of (2.2.9) is composed of a superposition the can-yon waves [part,I, (2.3.17)] and ("evanescent") Poincare waves [part n, (2.3.17)]. The canyon waves can only propagate in the y (along-canyon) dimension, whereas, the 2.3 Waves over an F Canyon -51-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid ("evanescent") Poincare waves may propagate in any horizontal direction. As indicated in section 2.3.1, the superinertial waves [part U, (2.3.17)] do not pro-vide much information about the topography except the local depth, whereas the canyon waves [part I, (2.3.17)] depend strongly on topographic parameters such as ^and B. So, if the full wave solution could be found by some method, it would have two parts: one which is the contribution from the superinertial waves containing only local depth information and the other which is the contribution from the canyon waves containing topographic infor-mation of a larger region. Based on the analysis of solutions of the wave equation (2.2.9), one may speculate that the solution of the steady state governing equation (2.2.8) may also be composed of two parts (if the solution is separable): one contributed by the Poincare waves which confer only the topographic information of the local depth and the other contributed by the can-yon waves which show the topographic information of the overall canyon structure. This speculation will be confirmed by the geostrophic solution obtained in section 2.4.3. 2.3 Waves over an F Canyon -52-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid 2.4 The Geostrophic State over an F Canyon 2.4.1 The Long Time Asymptotic Solution The term far field will refer to regions far enough (say, 5 Rossby radii where the amplitude has dropped to less than 1%) away either from the initial disturbance region or from the closer canyon edge. The term long time will refer to a long enough time (say, 3 inertial periods) after a initial state. Solving the steady state equation, (2.2.8), requires that first the far field solution, as well as solutions at the canyon edges, be found. A solution of the primary equations, (2.2.1), will be found and demonstrated to be the geostrophic far field solution. A solution of the shallow water equations (2.2.1), or, in other words, a solution of the general governing equation of Rossby adjustment, (2.2.7), can have the form n(x,y,t) = 77/0)-E(x) exp[?(*y ~ cot)], (2.4.1) where rjAy) is the initial condition which satisfies the limitation imposed in section 1.3.2; E(x) exp[i(ky - cot)] is the longest canyon wave, i.e., k -> 0 and hence co —> 0 (from the dispersion relation for canyon waves), where co = Cok in which Co is the long canyon waves' phase/group speed given by (2.3.15), and E(x) is a exponential function given by (2.3.8) with a, = l/Rj (j = 1, 2) for long canyon waves. It will be demonstrated that (2.4.1) is the geostrophic solution of (2.2.7) for j > » R. 2.4 Geostrophic State over an F Canyon -53-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid If a point (x, y) is chosen in the region where dn1  dy-Q away from the initial surface discontinuity, the velocity components corresponding to (2.4.1) have forms given by (2.3.4b) and (2.3.4c). Now using (2.4.1), (2.3:4b) and (2.3.4c) to examine the magni-tude of each term in the momentum equations, (2.2.1a) and (2.2.1b) (remember: k - » 0 for the longest canyon wave) yields a get oE'-M (2.4.2a) \jv\*g\E'\, (2.4.2b) (2.4.2c) a> a gc0\E'\ f (2.4.2d) (2.4.2e) g dy = g\E\k, (2.4.2f) where E' is the first derivative of E(x). We assume that / i s finite throughout this studies. Obviously, the magnitude of the first term of momentum equation (2.2.1a), i.e., (2.4.2a), is much smaller than those of both the second and the third terms, i.e., (2.4.2b) and (2.4.2c), so the first term is negligible in 2.4 Geostrophic State over an F Canyon -54-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid (2.2.1a); the magnitudes of all the terms in momentum equation (2.2.1b), i.e., (2.4.2d), (2.4.2e) and (2.4.2f), are small but of the same order, so all the terms must be considered in (2.2.1b). Therefore, if (2.4.1) is a solution of the shallow water equations (2.2.1), the equations will reduce to - > + ^ * 0 , (2.4.3a) ~a+fu+g%=0- ( 2 A 3 b ) For an F canyon where the two canyon edges are at x = ±L and the Rossby radii on the shelf and over the canyon are Ri and R2, respectively, (2.4.1) can be written in another form by using (2.3.8), ^(y,0exp[(x + L)/^], x<-L, J5Cy,0exp[-(x + Z)/i?2] + CO,0exp[(x-Z)/i?2]}, -L<x<L, .DO,0exp[-(x-I)/i?J, x>L, (2.4.4) where A, B, C and D are functions of two variables, y and t. JfA,B, C and D are deter-mined, a solution of the shallow water equations (2.2.1) is also found. One advantage of assuming a solution of the form (2.4.4) is that there is a simple initial condition for A,B,C and D, 2.4 Geostrophic State over an F Canyon -55-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid A(y"fi) = B"&^ (2.4.5) Substituting (2.4.4) into (2.4.3a) gives the velocity in the along-canyon direction, namely, v(x,y,t) = -— exp[(x+ £)/;?,], x<-L, ^-j-JBexp[-(x + Z)/i? 2] + Cexp[(x-i:)/ic 2]}, -L<x<L, (2.4.6) D v--^\-{x-L)IR\, x>L. Then substituting (2.4.4) and (2.4.6) into (2.4.3b) gives the velocity in the across-canyon direction, namely, u(x,y,0 = -/ \"iy-( A \ B. Qxp[(x + L)/R.]\, x<-L, [ ^ - [ ^ - ^ J e x p [ - ( x + Z ) / ^ ] -f C \ exp[ (x -£ )/ .&, ] \, -L<x<L, \"iy \ y M « p [ - ( x - Z ) / ^ ] , x>L, (2.4.7) where variables with subscripts / and y denote derivatives with respect to time and the co-ordinate in the along-canyon direction, respectively. At a discontinuity in depth, say, a canyon edge, it is appropriate to assume that the 2.4 Geostrophic State over an F Canyon -56-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid surface elevation and the normal component of the volume flux are continuous, i.e., atx = ±L: [77)2 = 0 and [hu]\=0. (2.4.8) Substituting (2.4.4) and (2.4.7) into (2.4.8) gives four linear partial differential equations in four unknowns: A,B,C and D A=B + Cexp(-B), (2.4.9a) f J_ ^ T 7 * A y JRXAJ •• hjy^ -By-Cy exp(-/5) -jjjrV-B, + Ct exp(-/5)]}, (2.4.9b) Bexp(-0) + C = D, (24.9c) »[lfy -Dy+ jj^ A) = h)\»ly - By exp(-/3 ) ~ C y ~ [-B, exp(-/3 ) + cj}, (2.4.9d) where /?, as defined by (2.3.12), is the ratio of the canyon width to the Rossby radius over the canyon; hi and h2 are the depths on the shelf and over the canyon, respectively. Combining (2.4.9a) and (2.4.9c), B and C can be expressed in terms of A and D, i.e.. Aexp(B)-D B-^B)-,xp(-BY ( 2 A 1 ° a ) exp(/3)-exp(-/3) 2. 4 Geostrophic State over an F Canyon -57-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid Then substituting (2.4.10) into (2.4.9b) and (2.4.9d), a system of two linear partial differ-ential equations in two variables, A and D, is obtained (for/> 0), namely, ^ ( y . 0 - ^ ^ A ( y , 0 - ( ^ - ^ H 0 ' , 0 = - ( ^ - * i ) ^ (2.4.11a) IK i g sinh/? yig \gj Dt(y,0 + (h2-hl)Dy(y,t) = (h2-hi)TjIy. (2.4.11b) Note that, first, the equations will have similar forms but the signs of the coeffi-cients change i f /< 0; second, if the surface elevation is assumed to be 7±z,(y, 0 at the can-yon edges x = +L, from (2.4.4), we will have (2.4.12) Combining (2.4.10) and (2.4.12), and then substituting into (2.4.4) gives a form in terms of V±L(y, t), namely, V(x,y,t) = TjI+i-(>7-£ - 7/)exp[(x + Z)//?,], cosh(x/ Rz) sinh(x IR^) - 217,) c o s h ( L / R 2 ) Hn, - V-L) S I N H ( L / I U (nL-Tjj)exp[-(x-L)/R}], x<-L, , -L<x<L, x>L. (2.4.13) 2.4 Geostrophic State over an F Canyon -58-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid Therefore, if either the surface elevation at canyon edges, t]±iiy, t), or A and D are known, a solution of the shallow water equations for an F canyon will have been obtained. ) By using a nonstandard method, (2.4.11) is solved (see Appendix A), and the solu-tions (for/> 0) are Kiy + ebO " ^ ~^-Wpiy ~ c0t), (2.4.14a) Diy, t) = r?j (v) -^r^-Wn{y + c0t) Wp(y-c,t). (2.4.14b) Note that if/< 0, A will equal the right hand side of (2.4.14b), and D will equal the right hand side of (2.4.14a). Therefore, flow patterns in the southern hemisphere are exact the reverse of those in the northern hemisphere. Without loss of generality, discussions hereaf-ter will only consider the northern hemisphere. An explanation of solutions (2.4.14) is given below. (i) Co > 0 is the group/phase speed of the longest canyon wave as defined by (2.3.15). (ii) Wn(y + c0t) and Wp(y-cQt) are the information (surface elevation) carried by the longest canyon waves, transmitted in the negative and positive^ directions, respectively. Specifically, Wn(y + c0t) is the information carried by the longest canyon wave that is induced by and intensified on the canyon edge at x = -L, whereas, Wp(y - c0t) is the in-formation carried by the longest canyon wave that is induced by and intensified on the can-2.4 Geostrophic State over an F Canyon -59-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid yon edge at x = L. Because the longest canyon wave travels at the fastest group/phase speed and covers the whole canyon no matter how long the canyon is, Wp(y-c0t) should be the surface elevation at large negative^ over the F canyon, and Wn(y + c0t) should be the surface elevation at large positive^ over the F canyon. The physical meanings of Wn(y + cQt) and Wp(y-c0t) introduced above are impor-tant in later calculations of the geostrophic state around an F canyon and an S canyon. (iii) B and y, defined by (2.3.12) and (2.3.13), are the ratio of the canyon width to the Rossby radius over the canyon and the ratio of the Rossby radius over the canyon to that on the shelf, respectively. These two basic geometric parameters combine into a very im-portant parameter, oiy, B), which is defined as a(y,B) = l--ll/2 /(cosh/3 -1) + sinh/7 (^cosh/5 +1) + sinh/5 (2.4.15) The parameter a will be referred to as the Canyon Number, o s {1, 0} for B e {0, oo}. The wider the canyon (hence, the weaker the effects of one canyon edge on the circulation at the other edge), the smaller the Canyon Number. Therefore, as reflected by the magnitudes of Wn(y + c0t) and Wp(y-c0t) in (2.4.14), the parameter bfy, B) determines the interactive strength of one canyon edge on the circulation induced by the other edge. For the narrowest canyon (J3 « 1) cr —» 1. In this limit, the interaction of one canyon edge on the circulation induced by the other edge is 2.4 Geostrophic State over an F Canyon -60-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid strongest, and the effect of the two canyon edges have same weight, 0.5, on the circulation everywhere. On the contrary, for the widest canyon (B » 1) or -» 0. In this limit, the inter-action of one canyon edge on the circulation induced by the other edge is weakest, and the effect of the canyon edge, say, at x = L on the circulation near the canyon edge at x = -L [described by — W (y'—c0t) in (2.4.14a)] is zero; and the circulation there is solely deter-mined by the effects of the edge at x = -L which achieves the maximum weight, 1. In the latter case (cr-> 0), the effects of the two canyon edges are decoupled, and each edge acts as a single-step. Generally, Wn(y + c0t) has stronger influence than Wp(y-c0t) on the circulation near the canyon edge at x - -L (described by A) and vice versa on the circulation near the canyon edge at x = L (described by D). Now, after substituting (2.4.14) into (2.4.10) and then using (2.4.4), or, after sub-stituting (2.4,14) into (2.4.12) and then using (2.4.13), a solution of the shallow water equations, (2.2.1), for an F canyon is obtained in terms of W„(y + c0t) and Wp(y-c0t). This solution is independent of t if the surface elevation at large ±y over the canyon [i.e., Wn(y + c0t) and Wp(y-c0t)] is constant, and, therefore, this solution must be a geostrophic solution. This nature of the solution will be made more explicit when applying it to a specific initial condition in the next section. 2.4 Geostrophic State over an F Canyon -61-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid 2.4.2 The Analytic Far Field Solution for a Specified Initial Condition The discussion in the previous section was for an general initial condition which satisfies the limitation imposed in section 1.3.2. Although the results are general and, hence, powerful, they are in a sense obscure and hinder intuitive understanding. To illus-trate the significance of the results and show their applicability, a specified initial condition described by (1.3.1) will be used in the analytic studies hereafter. In this section only the far field geostrophic solutions will be considered. These so-lutions are important in that they form the boundary conditions for the geostrophic solution for the whole domain. The far field geostrophic solutions will be discussed in separate regions — those far away from the canyon and those far away from the line of the initial surface discontinuity. (i) Far away from the canyon (\x\ »Rl+L) For a flat bottom, open ocean with depth hu the initial condition (1.3.1), in which fluid is at rest but has a surface discontinuity along the x-axis, will lead to a geostrophic state [see Gill (1982), pg. 191-195] (2.4.16a) (2.4.16b) 2.4 Geostrophic State over an F Canyon -62-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid v = 0, (2.4.16c) For the F canyon model, (2.4.16) describes the geostrophic state on the shelf away from the canyon ( | x | » ic, +Z). As has been discussed in section 2.3, the geostrophic ad-justment around an F canyon is mediated by both canyon waves and superinertial waves which are mainly Poincare waves. However, canyon waves are confined to a narrow region around either canyon edge, i.e., they attenuate rapidly away from either canyon edge. So, far away from the closer canyon edge, the adjustment is solely controlled by the superiner-tial waves and the steady state is just like that for a flat bottom open ocean. In this regions, geostrophic state is independent of x. (ii) Far away from the initial surface discontinuity The information transmitted by the longest canyon wave propagating in the +y di-rection is the initial surface elevation from large negative y, while the information transmit-ted by the longest canyon wave propagating in the -y direction is the initial surface elevation from large positive .y,. Substituting Wn(y + c0t) = -r/0 and Wp(y-c0t) = rjo into (2.4.14) gives >>>0, y<0, (2.4.17a) y>0, y<0. (2.4.17b) 2.4 Geostrophic State over an F Canyon -63-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid Note that (2.4.17) has been obtained by using the physical meaning of Wn(y + c0t) and Wp{y- c0t). This method, which will be used again when studying the S canyon model, is simple and conveys physical meaning. However, (2.4.17) can also be obtained using the purely mathematical method shown below. By using condition (2.4.5), (2.4.14) yields Wn(z) = Wp(z) = Vl(z) (2.4.18) where z is a free variable. So (2.4.14) can be written as My, t) = % GO - (1 - y )»& (y + <tf) ~rjj(y- c0t\ (2.4.19a) D(y, o = 17, OO - f i& (y+cj) - (i - f H (y - co0- (24. i9b) Equations (2.4.19) are valid for any initial condition that satisfies the limitation im-posed in section 1.3.2. Consider now the special initial condition (1.3.1). Ahead of the canyon wave front (Ivl > Cot) There are two cases, i.e., y <-c0t<0: y + c0t<0 and y-c0t <-2c0t <0, 7/0) = ^ o> *li(y + c0t) = Vo a n d Vi(y-c0t) = JJo> 2.4 Geostrophic State over an F Canyon -64-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid or, y>c0t>0: y + cQt > 2c0t > 0 and y-c0t>0, *li(y) = -Vo> Vi(y+ C0O--V0 and Vj(y-c0t) = -rj0. For either case, (2.4.19) gives A(y,t) = D(y,t) = 0, and (2.4.10) gives B(y,t) = C(y,t) = 0. Therefore, from (2.4.4), (2.4.6) and (2.4.7), we have T}(x,y,t) = fyOO and u(x,y,t) = v(x,y,t) = 0, i.e., the state remains unadjusted ahead of the canyon wave front Behind the canyon wave front (Ivl < c<>/) There is only one case, i.e., -c0t <y <c0t: 0<y + c0t <2cQt and -2c0t <y-c0t <0, ii (y) = n0\ !' y > ° ' "i(y+coO = -no Vi(y- V ) = v<> • 2.4 Geostrophic State over an F Canyon -65-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid Equations (2.4.19) give the same results as those of (2.4.17). Substituting (2.4.17) into (2.4.10) and then using (2.4.4), (2.4.6) and (2.4.7), a so-lution of the shallow water equations (2.2.1) under the initial condition (1.3.1) is obtained, - sgn(y) + [sgn(» + (1 - <J) sgn(x)] exp[(L - |x|) / /?,], |x| > L, -sgn(v) + , x cosh(x /R,) A sinh(x I R>) cosh(/?/2) sinh(/3/2) J x < L, (2.4.20a) u(x,y,t) = 0, (2.4.20b) v(x v t) = £y±< -sgn(x)[sgnf» + (l-o-)sgn(x)]exp[(i:-|x|)/i?1], |x| > L, 1 , . sinh(x/i?~) „ \cosh(x/ic.) — sgn(y) ^ + (1 - cr) ^ 2Z r[ cosh(j312) K ' sinh(£ / 2) x < L. (2.4.20c) Equations (2.4.20) describe a state behind the front of the longest canyon wave which is independent of t and the value of y. This state is the geostrophic state far away fromthe initial surface discontinuity, i.e., the state at / » 1/1/| and max(7?i, R2) « \y \ < c0t. The far field geostrophic solutions ~ (2.4.20) as well as (2.4.16) — are plotted out-side the dashed square in Figure 2.2. Streamline details within the dashed square, where flow turns near the canyon edges, will be calculated in the next section. Although the whole picture of the flow field has not been given yet, the partial picture that emerges is most intriguing. 2.4 Geostrophic State over an F Canyon -66-- 8 - 6 - 4 - 2 0 2 4 6 8 (a> X — CR0SS CRNY0N DIRECTI0N Figure 2.2 Contours of surface elevation, 77, which are also the streamlines, for an F canyon model in the steady state. Far field solutions apply outside the dashed square. Thick lines represent the position of the canyon edges. Solid lines represent positive rj, while dotted lines represent negative r/. Arrows represent the direction of flow in the northern hemisphere. The position of the initial surface discontinuity is along the line.y = 0. The length scale is R2. y2 = 2, (a) 3= 2 and (b) 8= 30. -67-- 2 5 - 2 0 - 1 5 - 1 0 - 5 0 5 10 15 20 25 (b) X — CR0SS CRNY0N DIRECTI0N -68-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid First, a canyon acts as a complete barrier to an approaching geostrophic flow, which is completely deflected at the canyon edges. Second, as a geostrophic flow approaches a canyon in the northern hemisphere, most of it will be deflected to the right (looking downstream). Note also that the solution issymmetric about the origin. Therefore, if the sign of the initial condition (1.3.1a) is changed so that the geostrophic flow direction is reversed, the flow fields will be those of Figure 2.2 turned 180° about the z axis. Third, as shown in Figure 2.2(b), two single-steps are a limit of a canyon, i.e., if the width of a canyon is much larger than the Rossby radius over the canyon, the flow pattern at each canyon edge is identical to that at a single-step as studied by Gill et al. (1986). Flux transported in the along-canyon direction can be calculated from the far field solution but will be left for the discussion in section 2.5. 2 A. 3 Full Analytic Solution As was discussed at the beginning of section 2.4.2, it is possible to find the full geostrophic solution for the whole domain only if the far field geostrophic solution has been obtained. In this section the solution of the steady state governing equation (2.2.8) within the dashed square in Figure 2.2(a), where both x and y are not very large compared to Ri and R2, will be discussed. Equation (2.2.8) is valid only for a flat bottom, so it is appropriate to divide the 2.4 Geostrophic State over an F Canyon -69-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid domain into segments each of which has a flat bottom and solve (2.2.8) in those segments. A suitable segmentation of an F canyon domain is to divide it into two shelf segments and one canyon segment. Because the solution will be symmetric about the origin as has been shown in the far field solutions, it is only necessary to solve (2.2.8) in one shelf segment, say, that for x < -L. For the initial condition (1.3.1), the right hand side of (2.2.8) is -rjAy) according to the definition (2.2.6). Because the far field solution (2.4.16a) satisfies the steady state governing equation (2.2.8), (2.4.16a) is the boundary condition for (2.2.8) as \x | -> oo. Since the far field so-lution (2.4.20a) satisfies (2.2.8), it is the boundary condition for (2.2.8) as |jv I —> oo. By substituting (2.4.17) into (2.4.12), the values of 77 at both canyon edges in the far field are obtained as - ( 1 - 0 ) 7 7 0 at x = -L and ( 1 - 0 ) 7 7 0 at x = L. The physical meaning of (2.2.10) is that in a homogeneous, inviscid, linear fluid, no geostrophic flow can cross the edges of a canyon. Therefore, 77 will be uniform along each edge of the canyon, i.e., 77.x = - ( 1 - 0 ) 7 7 0 at x = -L and TJL = ( 1 - 0 ) 7 7 0 at x = L for any_y in the steady state. The full solution of the steady state equation (2.2.8) can be attempted now. (i) Solution on the shelf (x < -L) The solution of (2.2.8) on the shelf x < -L will be denoted 77i(x, y). Using nondi-mensional variables: x , y and 77 , : 2.4 Geostrophic State over an F Canyon -70-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid (x,y) = and Tf, = 5L (2.4.21) (2.4.16a) can be written as a function ij0(y) = - sgn ( y ) [ l - exp(-^|)], (2.4.22) while (2.2.8) can be written as *rtx<?*y) + *Tix<?.y) _ ^ ( g ft = _ ^ o ( y ) > ( 3 f < 0 _ x < $ < ( X > y (2.4.23a) The appropriate boundary conditions are ^ ( x - > 0 J ) = - ( l - o - ) , (2.4.23b) tji(x-^-co,y) = rj0(y), (2.4.23c) ^ (x , y -> oo) = - 1 + a exp(x), (2.4.23d) 77,(x,y -oo) = 1 - (2 - a)exp(x) . (2.4.23e) Let 7 1 ( x J ) = 7oU7) + ^ (^> r ) - (2.4.24) Then substituting (2.4.24) into (2.4.23) gives 2.4 Geostrophic State over an F Canyon -71-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid + = (*<<>, -«<7<ao). (2.4.25a) O , (x -> OJ) = -(1 - a) + $gn(y )[l - exp(-|?|)] =-(1" CT) + EW> (24.25b) 0>,(x ->-oo,>') = 0,.' (2.4.25c) O 1 ( x j - > ± o o ) = ( o - - l ± l ) e x p ( x ) , (2.4.25d) Using the Fourier transform and expressed in convolution form, the solution of the problem (2.4.25) is obtained as (see Appendix B) 01(x,^) = <rexp(x)-exp(x) + - 7 ^ r £ ( ^ - ^ G 1 ( x , ^ ^ , (x<0, -oo<j7<oo), (2.4.26) where £ ( > r - ^ ) = sgnrj -^ ) [ l - exp ( - l> r - ^ | ) ] , ( -<»< ?<«>), (2.4.27) G,(x,^) = ^ ^J_ a ) exp(xV(P 2 +l)-exp(-7^)^ = - y - — ^ g 2 + ^ 2 . (* <0)> (2.4.28) in which K i is the modified Bessel function defined on page 1046, Abramowitz & Stegun (1968). 2.4 Geostrophic State over an F Canyon -72-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid Now, from (2.4.24) the solution of the (nondimensional) steady state equation (2.2.8) is obtained on the shelf x < —L. The solution ffl(x,y) is composed of two parts — the first part, rj0(y), which contains no topographic information about the canyon except the local depth on the shelf (reflected in the length scale R{) and the second part, 01(x, iy), which contains topographic information about the canyon, i.e., Band / expressed by cr. (ii) Solution within the canyon (br| <L) The solution of (2.2.8) within the canyon |x| < L will be denoted Tfrix, y). Using nondimensional variables (xj) = C f . - f ) and if2 = & (2.4.29) (2.4.16a) can be written as the same form as (2.4.22) keeping in mind that the length scale is Ri instead ofRi, while (2.2.8) can be written as ^ f f i 3 0 + - K & y ) = -5 .00, ( " f < * < f , -oo<j?<oo). (2.4.30a) The appropriate boundary conditions are ij2(x->±B/2,y) = ±(l-cr), (2.4.30b) ~ . , . , coshx . sinhx . . . . . . *7a(*,y - » oo) = -1 + + (1 - cr) . , (2.4.30c) cosh(p/2) sinh(j9/2) 2.4 Geostrophic State over an F Canyon Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid ~ /~ ~ \ , coshx N sinhx , _ r t J . cosh(/? / 2) sinh(/7 / 2) Let = V0(y) + *(*,50 • (2-4.32) Then substituting (2.4.31) into (2.4.30) gives £ 2 ^ + S r ^ - ^ * ^ ) ^ (~<*<^-, -oo<j<oo),(2.4.33a) ax oy 1 1 <D2(x->±p/ 2,y) = ±(1 - a) + sgnfj)[l - exp(-|^ |)] = ±(1 - a) + £ 0 0 ,(2.4.32b) _ , , . „ N sinhx , coshx ®2(x,y -> ±00) = (1 - a) . ± , (2.4.32c) sinhfjo / 2) cosh(/? / 2) Similar to the process used to solve problem (2.4.25), using the Fourier transform, and expressed in convolution form, the solution of the problem (2.4.32) is obtained as (see Appendix B ) ®2(x,y) = (l-cr) S m h * +^{mE(y-QG2(x,QdZ, (-^<x<^, -oo<y<ao) 2 K y ) V sinh(pV2) 42K]- 2 2 2 (2.4.34) where E(y - £) is given by (2.4.27), and 2.4 Geostrophic State over an F Canyon -74-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid cosh^y V P + 1 [2~f°° cosh(xV^ 2 + l) V c o s ( ^ ) ^ , ( -y<x<^-) . (2.4.35) Now, from (2.4.31) the solution of the (nondimensional) steady state equation (2.2.8) is obtained within the canyon |x| < L. The solution ff2(x,y) is also composed of two parts — the first part, J]0(y)t which contains no topographic information about the canyon except the local depth over the canyon (reflected in the length scale R2) and the second part, Q>2(x,y), which contains the topographic information about the canyon, i.e., 0 and cr. With respect to £ (see Appendix B), the typical shapes of G2(x,%) (for examples with 0=2 and x = 0.2 or x = 0.5) and E(y - £) (for examples with y - 0.5 or y = -1.0) are shown in Figure 2.3. It can be seen that G2(x,%) decreases very quickly as |£| in-creases. Therefore, G2(x,£) can be easily evaluated numerically (it is sufficient to take |£| < 10 in the numerical evaluation). The shape of G,(x,£) vs £ i s similar to that of G2(x,%). Therefore, G\(x,£) can be easily numerically evaluated as well (it is also sufficient to take \g\ < 10 in the numerical evaluation). When analyzing the structure of the wave solutions in section 2.3.3, I speculated that the steady state solution could be considered as two parts, a part containing no canyon information and a part reflecting the effects of the canyon. This speculation is confirmed now. The steady state solution, (2.4.24) (out of the canyon) or (2.4.31) (within the canyon) 2.4 Geostrophic State over an F Canyon -75-Figure 2.3 Shapes of G 2 (x, | ) (for the examples of B = 2 and x = 0.2 or x = 0.5) and E(y - £ ) (for the examples of y = 0.5 or y = -1.0) vs £ Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid has two parts. The first part, rj0(y), is the steady state solution for the geostrophic ad-justment over an open ocean with the local depth (either hx or h2). So the first part of the solution looks like it is determined by the portion of the superinertial waves propagating in the>> direction. The second part of the solution, (i(x,y), is determined by canyon waves, because it contains some properties of canyon waves [e.g., ®(x,y) attenuates away from the closer canyon edge]. The second part of the solution can be further separated into two parts one of which is contributed from each of the canyon edges. For example, in ^(x,^), the contribution from the canyon edge at x = L is which will be zero if the canyon is infinitely wide, i.e., o —> 0, while the contribution from the canyon edge at x = -L is Obviously, for an infinitely wide canyon the solution on the shelf x < -L is consistent with the solution for a single-step [see (2.45) in Allen (1988)]. The analytic geostrophic solution over an F canyon has been obtained, and some properties of the steady state have been revealed. However, it can be seen that the solution is quite complicated even for the simplest canyon model. The process used to solve the problem shows the complexity and difficulty of extending the process to more complicated o-exp(x) 2.4 Geostrophic State over an F Canyon -77-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid and hence more realistic canyon models, say, an S canyon. It is necessary to find an easy, general method to solve the steady state governing equation (2.2.8). Numerical integration of (2.2.8) which will be introduced next is one method to achieve this goal. 2.4.4 Numerical Integration of the Steady State Equation The steady state governing equation, (2.2.8), is an elliptic equation. Consider a square domain with uniform grid space, d, in both the x and y dimensions. Multiplying through by Ogives a finite difference form of (2.2.8): where m and n are the grid indices in the x and the y dimensions, respectively. The coeffi-cients e„,„ and fm,„ are determined by d and the position of the grid point. For example, consider the case where the length scale of the domain is the Rossby radius within the can-(2.4.36) yon, i?2, the initial condition is (1.3.1) and the size of TJ is 770, then for grid points on the shelf, for grid points within the canyon. (2.4.37) 2.4 Geostrophic State over an F Canyon -78-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid d2y2, for grid points on the shelf and for y>0, d2, for grid points in the canyon and for v > 0, fmn = \ ' , (2.4.38) -dy, for grid points on the shelf and fory <0, -d2, for grid points in the canyon and for^ < 0. If the left and right domain (±x) boundaries are far enough, say, several Rossby radii, away from the closer canyon edge, the far field solution (2.4.16a) forms the appro-priate left and right boundary conditions, while for a large enough domain top to bottom, the far field solution (2.4.20a) forms the suitable top and bottom (±y) boundary conditions. As has pointed out before, (2.2.8) is valid only for a flat bottom ocean. Therefore, it is still necessary to divide the domain into segments each of which has a flat bottom. As before, an F canyon is divided into two shelf segments and one canyon segment. Equation (2.2.8) is solved separately for each segment, and solutions at the canyon edges are matched to - ( l - o ) at the left edge and (l -o) at the right edge. Using Chebyshev acceleration and Jacobian iteration, the steady state governing equation, (2.2.8), is integrated using the numerical method simultaneous over-relaxation [see Press et al. (1986), pg. 657-659] for its precision as well as its simplicity. With 401 x 401 mesh points (note that the domains are different sizes), the equation (2.2.8) is inte-grated for two different canyons with different widths and the results are plotted in Figure 2.2. Figure 2.2(a) is for a canyon with a width 2R2 in a domain of size 16R2 x 16R2 (thus, the grid space is O.O4R2); Figure 2.2(b) is for a canyon with a width 30R2 in a domain of size 50i?2 x 50i?2 (thus, the grid space is 0.\25R2). Because the flow is barotropic and 2.4 Geostrophic State over an F Canyon -79-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid geostrophic, the contours of surface elevation are also the streamlines. If the sign of the initial condition (1.3.1a) is changed so that the direction of the geostrophic flow reverses, the flow patterns will be those of Figure 2.2 turned 180° about z axis. The precision of the solution was checked by comparing the results from different spatial resolutions. In other words, (2.2.8) was also integrated with 201 x 201 mesh points (a quarter of the original number of mesh points) and 101 x 101 mesh points (one eighth of the original number of mesh points), and the results were compared. The solutions for which the number of mesh points was quartered were compared to those with the original number of mesh points. The discrepancy was of the order 10'3, i.e., the domain averaged root mean square error between the solution for the 201 x 201 mesh points and the solu-tion for the 401 x 401 mesh points was not greater than 10'2. In this section, the geostrophic state around an F canyon was calculated. This was done by first finding the far field solution for a general initial condition and then using the far field solution as the boundary conditions to determine the solution for the whole do-main with both analytic and numerical methods. Some of the nature of Rossby adjustment around a canyon has been revealed and briefly discussed. Further discussions will be presented in the next section. 2.4 Geostrophic State over an F Canyon -80-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid 2.5 Discussion Canyon waves and the geostrophic state around an F canyon were the two main subjects discussed in this chapter. The study of canyon waves laid the foundation for the analysis of the geostrophic state around an F canyon. Some properties revealed by the dis-persion relation for canyon waves were discussed in section 2.3. Discussion in this section will focus mainly on the results obtained from the geostrophic studies and applications of the results. 2.5.1 Flux in an F Canyon Sometimes the details of a flow pattern are of less interest than the volume trans-port. A calculation of the volume fluxes in an F canyon follows. If the initial condition imposed on an F canyon is (1.3.1), the volume flux approach-ing/leaving the canyon (in the +x direction) in the geostrophic state can be obtained by in-tegrating (2.4.16b) with respect toj^ e{-oo, oo}, i.e., which is independent of x and, thus, equal at the two sides of the canyon. If there was no canyon, there would be no flux in the.y direction in the steady state. However, an F canyon induces a volume flux in the along-canyon direction. Integrating (2.5.1) 2.5 Discussion -81-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid (2.4.20c) with respect to x e {-oo, 00} gives the flux in the +y direction Fy = h\\vdx + h, jydx + h, fydx = Fx(y2 - 1)(1 - a), (2.5.2) where Fx is the flux approaching and leaving the canyon expressed by (2.5.1), cr is the Can-yon Number defined by (2.4.15) and y was defined by (2.3.13). In the northern hemisphere (f> 0), Fx > 0 based on (2.5.1) and thus Fy > 0 from (2.5.2). So, a volume flux approaching a canyon in the northern hemisphere induces a net volume flux in the along-canyon direction. This flux is to the left of the approaching flow and is proportional to the flux approaching the canyon. If y is held constant, the wider the canyon (the smaller a), the larger Fy. Although cr< 1, (y2 - 1)(1 - 0 ) can be greater than unity for a very large y. Therefore, it is possible that Fy> Fx, i.e., a small shelf flux can possibly generate a large flux in the along canyon direction. Explanation of the origin of the flux in the along canyon direction will be given in chapter 4 when analyzing the distribution of mass around the canyon. 2.5.2 Extrapolation to Other Topography The most frequently applied result obtained from the F canyon is the combination of (2.4.12) and (2.4.14), i.e., the calculation of the surface elevation, 77±z,(y, 0. a t the canyon edges x = ±L, 2.5 Discussion -82-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid VL = ^ K(y+cj) + (i - j)w,(y - c0t), (2.5.3a) = V-^Wn(y + c0t) + ^Wp(y-c0t). (2.5.3b) As noted in section 2.4, once the surface elevation at the canyon edges is known, the geostrophic far field solution and hence the geostrophic solution in the whole domain can be found. Consequently, calculating the surface elevation at the canyon edges is usually the first step in analyzing circulation around a canyon. The results for Rossby adjustment in the simple F canyon model, such as the Can-yon Number and (2.5.3), can be extrapolated to many other topographies. Some examples of the application of the results will be discussed below. Single-Step Bottom Rossby adjustment over a bottom with a single-step was first studied by Gill et al. (1986). A single-step is actually the limit of a canyon whose width is much larger than the Rossby radius over the canyon. So, all the results obtained for Rossby adjustment over a single-step in Gill et al. (1986) can be derived by allowing the (nondimensional) canyon width to be much larger than unity in the results for Rossby adjustment over a canyon. One example was given in section 2.3 showing the derivation of the double Kelvin wave speed at a step from the group/phase speed of canyon waves. The circulation at each edge of a very wide (theoretically, "infinitely wide") canyon, 2.5 Discussion -83-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid say, Figure 2.2(b), is consistent with that given by Gill et al. (1986). Ridees For the homogeneous case, a ridge is actually a "negative canyon" with the under-standing that the "bottom" of such a "negative canyon" is above the adjacent shelf. Until section 2.4.3, no limitation was imposed on the ratio of the depth differential, y2, which was defined by (2.3.13). Therefore, all the results for a canyon, before section 2.4.3, are retained for a ridge except for the interpretation of the origins of Wn(y + c0t) and Wp(y-c0t) in (2.4.14). The "negative canyon" effects for a ridge are reflected not only in the reverse of the propagation direction for Wn(y + c0t) and Wp(y-cQt) and hence the reversal of the circula-tion around a canyon but also in y< 1 for a ridge in contrast to y> 1 for a canyon. For ex-ample, ifF x > 0, the right hand side of (2.5.2) will be negative for a ridge, i.e., Fy < 0. Thus in the northern hemisphere (/> 0), a flow approaching a ridge will generate a net flux along the ridge and to the right of the approaching flow. This conclusion is qualitatively consis-tent with observations around the Juan de Fuca Ridge where the flow along the ridge is found to be mainly southward as the Subarctic Current approaches the ridge from the west (Thomson, private communication). Straits Rossby adjustment in a strait was studied by Gill (1976). The results were described 2.5 Discussion -84-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid in terms of Poincare waves and Kelvin waves. If the depth on the shelf in the F canyon model diminishes, the canyon degenerates into a strait. As h\—> 0, (2.3.13) gives / —><», therefore (2.4.15) reduces to which may be named the "Strait Number", and, furthermore, the phase/group speed of the long topographic waves, (2.3.15), becomes [note that the subscript "s" in (2.5.4) and (2.5.5) is to differentiate from the corresponding parameters for a canyon]. Equation (2.5.5) is in fact the phase/group speed of Kelvin waves as well as short Poincare waves. This result is as expected because when a canyon degenerates into a strait, topography no longer exists, and topographic waves degenerate into boundary waves. As analyzed in section 2.4.3, the initial condition (1.3.1) gives a steady state in which the (nondimensional) surface elevations are +(1 - cr) at the two edges of the canyon. Similarly, the surface elevation in the steady state is ±(1 - <JS), i.e., ± tanh (y ) , at the two walls of a strait. This result is consistent with (7.3) in Gill (1976). Moreover, substituting (2.5.4) into (the part of |JC| < L in) (2.4.20a) gives a far field geostrophic solution which cosh 6-1 Vcosh/? + h = l - t a n h ( ^ ) , (2.5.4) (2.5.5) 2.5 Discussion -85-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid matches that expressed by (7.7) in Gill (1976). The flow patterns within the canyons shown Figure 2.2 are actually the same as those within the straits shown in Figure 3 in Gill (1976). Certainly, the research method used in this thesis is more straightforward and simpler than the one used by Gill (1976). The extrapolation of the results from canyons to straits is important for later studies of the S canyon model in which a strait joins a canyon. 2.5.3 Dilemma of Terminology It may not be appropriate to denote a defined by (2.4.15) as a "Canyon Number", nor to coin phrases such as "canyon waves". These terms may cause confusion when ap-plied to topography other than a canyon. For example, the parameter a is as significant for a ridge or a strait as for a canyon. A general denotation, such as, "double step/wall interaction number", may be a little more suitable for the important parameter discovered in this study. Similarly, it may be better to replace the name "canyon waves" with "modified double Kelvin waves" for canyons or ridges, or, "modified Kelvin waves" for straits. However, for convenience, I will continue to use terms such as "Canyon Number", "canyon waves", etc.. 2.5 Discussion -86-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid 2.6 Summary of Rossby Adjustment over an F Canyon Waves which exist over an F canyon and the geostrophic state around an F canyon are the two subjects studied in this chapter. Analyzing the properties of the waves, espe-cially those of canyon waves, lays the foundation necessary in order to find the geostrophic solution using Rossby adjustment. Both superinertial waves (Poincare waves) and subinertial waves (canyon waves) exist around a canyon. In the process of Rossby adjustment, the superinertial waves domi-nate the short time scale processes, whereas the canyon waves control the long time scale processes. Longer canyon waves travel faster than shorter ones. The superinertial waves ~ Poincare waves ~ may propagate in any horizontal direction. The canyon waves which are concentrated at each edge of the canyon and whose amplitudes attenuate away from that edge propagate infinitely long distances but only in the along-canyon direction. In the geostrophic state, a canyon acts as a complete barrier to an approaching flow and deflects most of the flow to the right (looking downstream) in the northern hemi-sphere. A geostrophic flow in the across-canyon direction can induce a net transport along the canyon and to the left of the approaching shelf flow in the northern hemisphere. The geostrophic state for the F canyon is described by a geometric parameter ~ the Canyon Number, a e {0, 1}. The Canyon Number determines the interactive strength of one canyon edge on the circulation induced by the other edge. The wider the canyon, the 2.6 Summary of Rossby Adjustment over an F Canyon -87-Chapter 2 Geostrophic State over an F Canyon in a Homogeneous Fluid smaller a. The results obtained for a canyon can be extrapolated to a single-step (an infinitely wide canyon), a ridge (a "negative canyon"), or, a strait. The two most important discoveries in this chapter are the dispersion relation for canyon waves and the Canyon Number, a. By now the significance of studying the F canyon can be seen to some extent. A l -though an F canyon looks simple and ideal, Rossby adjustment over it reveals many power-ful results which can be applied to other topographic problems. The extension of the results to a more complicated topography ~ the S canyon ~ will be presented in the next chapter. 2.6 Summary of Rossby Adjustment over an F Canyon -88-Chapter 3 The Geostrophic State over an S Canyon in a Homogeneous Fluid 3. The Geostrophir State over a Sloping Bottom Canvnn with a Coast and a Shelf Break in a Homogeneous Fluid 3.1 Introduction In chapter 2 the geostrophic state around an F canyon was studied. The steady state was obtained without solving the transient process by using Rossby adjustment. The steady state was determined by long canyon waves transmitting information along the canyon. The results will be applied in this chapter to a more realistic geometry, an S canyon. In section 3.2, the geostrophic state over an S canyon will be studied. In particular, the surface elevation at the depth changes, as well as at the boundaries such as the strait walls and the coast, will be determined first in section 3.2.1. The full geostrophic solution will be obtained by numerical integration of the geostrophic governing equation. A discussion of the geostrophic state around an S canyon will be presented in sec-tion 3.3. A summary will be given in the last section. 3.1 Introduction -89-Chapter 3 The Geostrophic State over an S Canyon in a Homogeneous Fluid 3.2 The Geostrophic State over an S Canyon The definition of an S canyon was given in section 1.5.2 and an example of the ge-ometry can be found in Figure 1.7. The two dimensional (top) view of an S canyon is shown in Figure 3.1. Corresponding to (2.3.13), the depth ratios for the S canyon model are defined as yJ=RJ/R,=Jh~rh~, 0 = 0 , 2 , 3 ) (3.2.1) where Rj is the barotropic Rossby radius corresponding to a depth hj. Specifically, h0 is the depth of the strait; hi is the depth on the shelf; h2 is the depth of the middle canyon portion and h3 is the depth of the deep canyon portion and the deep ocean. For convenience, the initial condition in this chapter is chosen with the fluid at rest and with a surface discontinuity along a line in the across-canyon direction. The axes are chosen with the x-axis perpendicular to the canyon and along the line of the initial surface discontinuity, and with the j -axis along the central axis of the canyon. It is assumed that the coast is at y = dc, the shelf break is at y = dsB, the lower bound of the canyon head slope is at y = du while the upper bound of the slope is at y ~ d>w> a n d the lower bound of the canyon mouth slope is at y = dmi while the upper bound of the slope is at y - drm. The initial condition is expressed by (1.3.1). 3.2 The Geostrophic State over an S Canyon -90-Land y = d Land Coast, y = dSB Shelf Break X=-L X-L Shelf K Shelf Break Figure 3 . 1 Top view of an S canyon. The shaded regions represent the canyon bottom slopes. The dotted line represents the position of the initial surface discontinuity. The width of the canyon as well as the strait are 2L. The depth in the inner strait, on the shelf, over the middle canyon and over the deep canyon (as well as in the deep ocean) are ho, hi, h2 and h3, respectively. Surface elevation in the geostrophic state at all depth changes and boundaries is indicated. See text for other notations. -91-Chapter 3 The Geostrophic State over an S Canyon in a Homogeneous Fluid The governing equations are the shallow water equations, (1.4.1), for a homogene-ous fluid. As in the previous chapter, only the linear problem is considered in this chapter. Therefore, as discussed in section 2.2, the general governing equation for Rossby adjust-ment over an S canyon is (2.2.7), and the governing equation for the geostrophic state is (2.2.8) except at depth changes (such as the canyon bottom slopes, the canyon edges and the shelf break). Only the geostrophic state is considered in this chapter. 3.2.1 Analytic Solutions at Depth Changes and Boundaries In order to solve the geostrophic governing equation, (2.2.8), the surface elevation at depth changes and boundaries must be determined first. The surface elevation in the steady state at these places will be found in this section. As discussed in the previous chapter, Rossby adjustment near a boundary, say, a coast, involves Poincare waves and Kelvin waves. The geostrophic state (the long time scale process) is dominated by Kelvin waves whereas Poincare waves mainly affect the short time scale process [see Gill (1976)]. Similarly, Rossby adjustment over topography, such as a canyon, involves Poincare waves and canyon waves, but the geostrophic surface height over depth changes is determined by the canyon waves. Therefore, it is only neces-sary to study Kelvin waves, double Kelvin waves and canyon waves to determine the geostrophic state for an S canyon. 3.2 The Geostrophic State over an S Canyon -92-Chapter 3 The Geostrophic State over an S Canyon in a Homogeneous Fluid Double Kelvin waves and Kelvin waves propagate only in the direction keeping the shallow water (for double Kelvin waves) or the boundary (for Kelvin waves) to their right in the northern hemisphere, and canyon waves propagate along a particular canyon edge in the direction keeping the other edge to their left in the northern hemisphere. By using the properties of these waves and using conservation of mass, the following results can be ob-tained. (i) Qualitative Analysis From (2.2.10), surface elevation at depth changes must be uniform in the steady state (except at some singular points that will be explained below). Surface elevation at solid boundaries must also be uniform in the steady state because of the solid boundary condition. The Kelvin waves induced by the coast and the double Kelvin waves which exist at the shelf break can only propagate in the -x direction; thus the existence of the canyon and the strait cannot affect the adjustment process for x -> oo. Rossby adjustment as x co is the same as that for a single step parallel to a vertical coast (see Figure 3.2). Solution of the geostrophic governing equation, (2.2.8), for the geometry given in Figure 3.2 is presented in Appendix C. In the steady state, the surface elevation at the coast (denoted TJK) is given by (C.23), and the surface elevation at the shelf break (denoted TJLC) is given by (C.22). The surface elevation at the coast as x - » oo, rjK, is transmitted by Kelvin waves propagating in the -x direction, along the coast towards the junction point, Ps, where the 3.2 The Geostrophic State over an S Canyon -93-Figure 3.2 Side view of a coastal region where the vertical shelf break is infinitely long and parallel to the infinitely long vertical coast. The line of the initial surface elevation is aty = 0; the shelf break is at.y = dSB, the coast is at.y = dc. -94-Chapter 3 The Geostrophic State over an S Canyon in a Homogeneous Fluid coast meets the canyon edge at x = L. Similarly, because double Kelvin waves propagate only in the -x direction, the sur-face elevation at the shelf break as x -> oo, TJLC, is transmitted by these waves along the shelf break towards the mouth of the canyon. So, the uniform surface elevation in the geostrophic state at the shelf break for x > L is TJLC-Note that since the shelf break affects the Kelvin waves propagating along the coast, it may be more appropriate to use the term "modified Kelvin waves", and since the coast affects the double Kelvin waves propagating along the shelf break, it may be appro-priate to use the term "modified double Kelvin waves". However, for simplicity, we will use Kelvin waves to refer to the waves propagating along the coast and double Kelvin waves to refer to the waves propagating along the shelf break. At the canyon edge at x = L, because u = 0 in the geostrophic state, the surface ele-vation along the edge is a constant. At the part of the canyon edge x = L between the shelf break and the canyon mouth slope, the uniform surface elevation must have the same value as that at the shelf break for x > L, TJLC. The canyon mouth slope will not affect the information at the canyon edge at x = L because the topographic waves induced by the canyon mouth slope propagate only in the -x direction (keeping shallow water to their right in the northern hemisphere). The surface elevation at the canyon edge where x = L adjoining the mouth slope is TJLC- Thus elevation 3.2 The Geostrophic State over an S Canyon -95-Chapter 3 The Geostrophic State over an S Canyon in a Homogeneous Fluid is continuously transmitted by the canyon waves along the canyon edge in the +y direction and transmitted by the slope-induced topographic waves towards the canyon edge at x = -L. Therefore, no matter what shape the slope is, the surface elevation over it is a constant, r/Lc Thus, fluid over the slope in the geostrophic state is stagnant [see Allen (1988) for details]. By using the same arguments, the surface elevation in the geostrophic state at the canyon edge where x = L outside the strait is also TJLC-Beyond the canyon head slope, because u = 0 at the strait walls, the surface eleva-tion is uniform along the strait walls. Waves that propagate along the strait walls are modi-fied Kelvin waves, in the same way that canyon waves are modified double Kelvin waves. Modified Kelvin waves propagate in the same direction as canyon waves, i.e., keeping the other strait wall to their left in the northern hemisphere (or, in other words, keeping the closer wall to their right in the northern hemisphere). If the surface elevation along the strait wall at x = L in the lower strait is assumed rjis, the modified Kelvin waves will transmit this information along the strait wall at x = L in the +y direction. The canyon head slope does not interfere with the transmission of the information along the strait wall at x = L, by analysis similar to that used for the canyon mouth slope. The information J]LS will be transmitted continuously along the whole length of the strait wall at x = L and transmitted by the slope-induced topographic waves towards the strait wall at x = -L. Consequently, the surface elevation at the strait wall where x = L 3.2 The Geostrophic State over an S Canyon -96-Chapter 3 The Geostrophic State over an S Canyon in a Homogeneous Fluid is the constant TJLS. Note that TJLS is equal to neither r\iC nor TJK. So Ps is a singular point at which the value of the surface elevation has no meaning. In practice, the flow at the singular point (or the singular lines that will be introduced below) will form a thin boundary layer. At Ps, where the coast meets the canyon edge at x = L, the incoming Kelvin waves carrying the information TJK confront the incoming canyon waves carrying the information TJLC- The out-going modified Kelvin waves from Ps propagate along the strait wall at x = L and.transmit the information TJLS in the +y direction. Along the strait wall at x = -L, the modified Kelvin waves that propagate in the -y direction towards the head slope force the surface elevation to a constant denoted by TJ.LO. However, in the region where the strait wall at x = -L adjoins the canyon head slope, the incoming modified Kelvin waves carrying the information TJ.LO confront the incoming slope-induced topographic waves carrying the information TJLS. The outgoing modified Kelvin waves which propagate towards the mouth of the strait force the surface elevation along the strait wall at x = -L in the lower strait to a constant denoted by 77.1.2. Mathematically, the region where the canyon head slope meets the strait wall at x = —L is a singular line at which the value of the surface elevation has no meaning. When the modified Kelvin waves carrying the information t].a. reach the junction of the coast and the canyon edge at x = -L, the task of transmitting the information TJ.L2 is handed over to Kelvin waves propagating in the - x direction forcing the surface elevation 3.2 The Geostrophic State over an S Canyon -97-Chapter 3 The Geostrophic State over an S Canyon in a Homogeneous Fluid along the coast for x < -L to n.L2, and the canyon waves propagating towards the canyon mouth slope forcing the surface elevation along the canyon edge at x = ~L (the part be-tween the coast and the canyon mouth slope) to 77.Z.2 as well. In the region where the canyon edge at x = -L adjoins the canyon mouth slope, the incoming canyon waves carrying the information 77.^ ,2 confront the incoming slope-induced topographic waves carrying the information nLc, and the outgoing canyon waves which propagate to the deep ocean force the surface elevation along the canyon edge at x = -L in the deep canyon portion to be a constant denoted by TJ.L3. Mathematically, the region where the canyon mouth slope meets the canyon edge at x = -L is a singular line at which the value of the surface elevation has no meaning. The double Kelvin waves that propagate in the -x direction along the shelf break for x<-L force the surface elevation at the shelf break to be the constant TJ.L3. The geostrophic solution of the steady state governing equation, (2.2.8), at all depth changes and boundaries has been analyzed qualitatively and indicated in Figure 3.1 for easy reference. (ii) Quantitative Analysis The procedure to determine the surface elevation at the depth changes and the boundaries will be shown step by step. 3.2 The Geostrophic State over an S Canyon -98-Chapter 3 The Geostrophic State over an S Canyon in a Homogeneous Fluid Step 1: solving for nLC and nK For x > L, the surface elevation along the shelf break, TJLC, and the surface elevation at the coast, nK, are given by (C.22) and (C.23), respectively, in Appendix C. Step 2: solving for T]LS As given by (2.2.12), the flux across the line y = dc-e (e ->• 0, x e { -X, X} and X —> oo) shown in Figure 3.3(a) is jHv-L2-V(rX,dc -e)]+4vLC -77-J+%(*,4: - ^ ) - ^ c ] } > (3-2.2) in which \imrj(-X,dc-e) = T]_L2, (3.2.3) X->oo \\mr1<<X,dc-£) = 7jK. (3-2.4) « - > 0 jr->«> The flux across the line y = dc + e (e - » 0, x e {-L, L}) shown in Figure 3.3(a) is J ^ I L S - V - L I ) - (3-2.5) As e -> 0, the fluxes equating (3.2.2) and (3.2.5) must match because there is no source or sink in the whole domain. After substituting (3.2.3) and (3.2.4), gives 3.2 The Geostrophic State over an S Canyon -99-Figure 3.3 Close-up views of an S canyon near (a) the mouth of the strait, (b) the canyon head slope, and (c) the canyon mouth slope. Surface elevation in the geostrophic state at all depth changes is indicated. -100-Chapter 3 The Geostrophic State over an S Canyon in a Homogeneous Fluid "LS = 7r[(r22 - l)Vuc + VK] • (3.2.6) 7 2 The surface elevation along the strait wall at x = L and over the canyon head slope has been obtained. Step 3: solving for 77^ ,0 As discussed in section 2.5, the results obtained for the F canyon in particular (2.5.3), can be extrapolated to a strait. In other words, the left hand side of (2.5.3) can be replaced by the surface elevation at the two walls of a strait; the vehicle to transmit infor-mation changes from canyon waves to modified Kelvin waves; on the right hand side of (2.5.3), a can be replaced by the "Strait Number", o>, defined by (2.5.4) and c 0 can be re-placed by the long modified Kelvin wave speed in the strait, c0s, defined by (2.5.5). Therefore, the surface elevation at the two walls of the inner strait beyond the can-yon head slope is given by Vu=^-K(y + c0st) + (\-^)Wps(y-c0st), (3.2.7a) y-Lo = Q-^Wns(y + c0st) + ^-Wps(y-cQJ), (3.2.7b) where Wm(y + c0st) is the information carried by the modified Kelvin waves from the far-thest end of the strait towards the canyon head slope; Wps(y-c0st) is the information car-3.2 The Geostrophic State over an S Canyon -101-Chapter 3 The Geostrophic State over an S Canyon in a Homogeneous Fluid ried by the modified Kelvin waves towards the +y direction. Eliminating Wps(y-c0st) between (3.2.7a) and (3.2.7b) gives V-u> = Tr—[?.Vu + 2 ( l - o - , ) ^ ] (3.2.8) where as is given by (2.5.4) and is solely determined by the nondimensional width of the strait; TJLS is the constant given by (3.2.6) and Wns = -TJ0 for the chosen initial condition, (1.3.1). The surface elevation along the strait wall at x = -L in the inner portion of the strait has been obtained. Step 4: solving for 77^ ,2 From (2.2.12), the flux across the line,y = du - s (e —> 0, x e {-L, L}) shown in Figure 3.3(b) is JMVLS-V-U), (32.9) and the flux across the liney = dhu +e {e -> 0, x e {-L, L}) shown in Figure 3.3(b) is JMVIS-V-LO)- (32.10) The fluxes expressed by (3.2.9) and (3.2.10) must match because there are no 3.2 The Geostrophic State over an S Canyon -102-Chapter 3 The Geostrophic State over an S Canyon in a Homogeneous Fluid sources or sinks in the study region. Thus v-L2=-\[(r2-ro2)VLs+ro2n-Lo], (3.2.11) 12 where / is defined by (3.2.1) and the constants TJLS and.77^0 are given by (3.2.6) and (3.2.8), respectively. The surface elevation along the outer strait wall at x = —L, along the middle canyon edge at x = -L and at the coast for x < —L has been obtained. Step 5: solving for 77^ 3 Consider the rectangle shown in Figure 3.3(c) which includes the canyon mouth slope. The four vertices of the rectangle are A(-X, dmi - s), B(X, dmi - s), C(X, dm, + s) and D(-X, dmu + e) where s -> 0 and X -> 00. Using (2.2.11) and (2.2.12) gives the fluxes entering the rectangle ABCD via the line AD: jh\r]{-X,dml-e)-i1(-X,dnm+e)], (3.2.12) and via the line AB: ^Hn-u - rj(-X,dml-S)} + h \ n L C - 77_L3] +hl[rj(X,dml - * ) - / 7 J } , (3.2.13) 3.2 The Geostrophic State over an S Canyon -103-Chapter 3 The Geostrophic State over an S Canyon in a Homogeneous Fluid as well as the fluxes exiting the rectangle ABCD via the line BC: g h\r1(X,dml-e)-r1(X,dmu+e)\, (3.2.14) and via the line DC: ^Hl-L2 -Vi-X,dmu +e)] - +^V(X,dmu +e) - i /J} . (3.2.15) The flux entering the rectangle ABCD, i.e., the sum of (3.2.12) and (3.2.13), must match the flux exiting the rectangle, i.e., the sum of (3.2.14) and (3.2.15), because there are no sources or sinks in the domain. Thus where / is defined by (3.2.1); the constant nLC can be obtained from (C.22), and the con-stant 77^2 is given by (3.2.11). The surface elevation along the deep canyon edge at x = -L and at the shelf break for x<-L has been obtained. Solutions of steady state governing equation, (2.2.8), have been obtained at all the depth changes (the canyon bottom slopes, the canyon edges and the shelf break) and at all the internal boundaries (the coast and the strait walls). Equation (2.2.8) can now be solved, either analytically or numerically, for each flat bottom segment in the S canyon following (3.2.16) 3.2 The Geostrophic State over an S Canyon -104-Chapter 3 The Geostrophic State over an S Canyon in a Homogeneous Fluid the procedure introduced in section 2.4.3 and section 2.4.4. In this chapter, only the numerical integration of (2.2.8) will be discussed. 3.2.2 Numerical Integration of the Steady State Equation The steady state governing equation, (2.2.8), will be integrated in an x-y square domain. The left (-x) and the right (+x) boundaries of the domain are far away (several Rossby radii) from the closer canyon edge; the bottom (-y) domain boundary is several Rossby radii away from the shelf break and the top (+y) domain boundary is several Rossby radii away from the canyon head slope. The initial condition (1.3.1) will be used and the axes will be taken as the same as those in the previous section, i.e., the x-axis is along the line of the initial surface disconti-nuity and, furthermore, lies at the middle of the domain, and the j>-axis is along the central axis of the canyon and also lies at the middle of the domain. The left (-x), right (+x), bot-tom (-y) and top (+y) domain boundaries are at y = di, y = dr, y = db and y - dt, respec-tively. The domain boundary conditions can be calculated as following. (i) . The surface elevation along the boundaries on land are forced to be zero. (ii) . the solution at the right (+x) boundary, rjr(y), is set equal to the solution for Rossby adjustment over a single-step parallel to a coast for the initial condition (1.3.1). The 3.2 The Geostrophic State over an S Canyon -105-Chapter 3 The Geostrophic State over an S Canyon in a Homogeneous Fluid solution is given by (C.6) with coefficients calculated from (C.14) to (C.21) in Appendix C (iii) . The solution at the left (-x) boundary, r//(y), is set equal to the solution for Rossby adjustment over a single-step parallel to a coast for the initial condition (1.3.1) with the surface elevation at the coast and at the shelf break forced to be 7^2 and 7^3, respectively. The solution is given by (C.6) with coefficients calculated from (D.9) to (D.13) in Appen-dix D. (iv) . The solution at the bottom (-y) boundary, t]b(x), can be calculated from the known surface elevation at the left and the right bottom corners, i.e., rji(db) and t]Adb). It is as-sumed that 77fc(x) varies linearly from rji(db) to rjr(db) along the bottom boundary (the error of this assumption is negligible provided \db - dSB\ » R3). So Vb(x) = fZ^(x-dl), d,<x<dr. (3.2.17) (v). The solution at the top (+y) boundary, nfa), can be calculated from (2.4.13) (the part for within a canyon) as well as the known surface elevation at the inner strait walls, i.e., TJLS and tj-io- Therefore, provided that the initial condition is (1.3.1) and \dt -dhu\ » R0 where Ro is the Rossby radius over the inner strait and y = df,u is the upper bound of the canyon head slope, the surface elevation at the top boundary is . cosh(x/^) sinh(x/i?o) r/0, -L<x<L, (3.2.18) 3.2 The Geostrophic State over an S Canyon -106-Chapter 3 The Geostrophic State over an S Canyon in a Homogeneous Fluid Now that solutions at the domain boundaries as well as at the internal boundaries and the depth changes have been determined, the steady state governing equation, (2.2.8), can be integrated in each flat bottom segment using the simultaneous over-relaxation nu-merical method introduced in section 2.4.4. The results of the numerical integration for an S canyon are shown in Figure 3.4 using 401 x 401 grid points. The depths over the inner strait, on the shelf, over the middle canyon portion and over the deep canyon portion as well as the deep ocean are 100m, 150m, 300m and 1800m, respectively. If the Rossby radius on the shelf, R\, is the horizon-tal length unit, the domain size is 12 x 12 (thus the grid space is 0.03); the width of the canyon is 1; the coast and the shelf break are at y = 3 and y = - 1 , respectively; the two canyon bottom slopes are located at 3.75 <y < 4.5 and -0.6 <y < 0.2 and their shapes are chosen to be linear (actually, it does not matter what the shapes of the slopes are, because the fluid over a slope is always stagnant in the steady state). The distribution of surface elevation for the whole x-y domain is shown in Figure 3.4(a). Close-up views of the surface elevation and of the geostrophic velocity field in the region around the canyon are given in Figure 3.4(b) and Figure 3.4(c), respectively, be-cause the flow pattern around the canyon is of greatest interest. Comparing the geostrophic flow pattern over an S canyon (Figure 3.4) to that over an F canyon (Figure 2.2), the obvious difference is that the S canyon is not a complete bar-rier to an approaching geostrophic shelf flow. Some of the shelf flow enters the canyon via 3.2 The Geostrophic State over an S Canyon -107-- | 1 1—•—' 1 1 1—;—> 1 -> i : 1 r - 6 -4 -2 0 2 4 6 (a) X — CR0SS CRNY0N DIRECTI0N Figure 3.4 Geostrophic state around an S canyon (see text for configuration), (a) Contours of surface elevation, rj. (b) A close-up view around the canyon, (c) Veloc-ity field around the canyon. Thick lines represent the positions of the canyon edges, shelf break, coast, strait walls and boundaries of canyon bottom slopes. Solid lines represent positive 77, while dotted lines represent negative TJ. The length scale is Ri. The range of rj contoured is from -TJ0 to 770 and the contour interval is 0.14770 where 770 is half the height of the initial surface discontinuity which was taken, as 0.2m in this example. -108--109-Land t / / / ; t t / / / / r 1 / /> s / f / - r . — _ Land Head Slope t i i / i / / / Y ' r ; / t f / f t f > '/// t t t t f t ^ * N \ \ \ V s • ' v VJ \ \ \ \ >• ' f V \ \ \ >> v " '' ( \ V \ \ ^ ' \ \ \ \ ^ N l { \ \ \ v. " ' ^ { \ \ \ s ' \ \ \ \ >> s ' \ \ \ \ \ N [ \ \ \ \ N N \ \ \ \ N. ^ ^ \ \ \ N > — — \ \ s ^ — —• V \ s >. ^. -» — — Mouth Slope r (c) X ~ CR0SS CANY0N DIRECTI0N MTXTSUTTE^R -110-Chapter 3 The Geostrophic State over an S Canyon in a Homogeneous Fluid the singular line where the canyon mouth slope adjoins the left canyon edge (looking in-canyon), and some of the flow in the canyon "squeezes" out of the canyon via the singular point where the coast meets the right canyon edge (looking in-canyon). The precision of the solution was checked using the method presented in section 2.4.4, i.e., by comparing the results for different spatial resolutions. The solution found when the number of grid points was quartered was compared to that found with the origi-nal number of mesh points. The discrepancies was of order 10 - 3 . Therefore, the flow pat-tern presented in Figure 3.4 is reliable. In this section, the geostrophic state around an S canyon has been calculated. First, the analytic solutions at the depth changes and at the boundaries were found, and then the steady state governing equation, (2.2.8), was numerical integrated over each flat bottom segment. The properties of the geostrophic state over an S canyon will be discussed further in the next section. 3.2 The Geostrophic State over an S Canyon -111-Chapter 3 The Geostrophic State over an S Canyon in a Homogeneous Fluid 3.3 Discussion The main purpose of this chapter is to extend the results obtained from the primary F canyon model to a more complicated and, hence, more realistic canyon model ~ the S canyon. Rossby adjustment over an S canyon leads to some new features. The discussion following will focus on these new features. 3.3.2 Reversal of the Shelf Flow For the same configuration as the example shown in Figure 3.4 but with the sign of the initial condition (1.3.1a) reversed, the geostrophic state has the same pattern as that shown in Figure 3.4 except all flows reverse their direction. The velocity field for this situation is shown in Figure 3.5 for comparison to that in Figure 3.4(c). 3.3.2 Validity of the Solution The prominent feature of Rossby adjustment over an S canyon is the existence of singularities. The singularity is a mathematical concept. In the time-dependent numerical studies that will be shown later, it can be seen as a thinning boundary layer, or, "boundary-like" layer if the singularity is not at the coast, arising in the region of the singularity. In practice, as described by Gill et al. (1986) for a similar singularity in their studies, inertial and viscous (and finite-difference) effects set up a narrow boundary layer, or, a boundary-like layer. A detailed description of this kind of boundary layer can be found in Allen (1988). 3.3 Discussion -112-LU ca >-z rx o ca _i cr i i >-X — CR0SS CRNY0N DIRECTI0N 0.423E* MAXIMUM VECTOR fC  Figure 3.5 Velocity field around the canyon whose configuration was given in Fig-ure 3.4. Thick lines represent the positions of the canyon edges, shelf break, coast, strait walls and boundaries of canyon bottom slopes. -113-Chapter 3 The Geostrophic State over an S Canyon in a Homogeneous Fluid In the previous section, (2.2.8) was assumed to apply to all flat bottom segments in the S canyon. In practice, this equation may not apply in every segment because of the possibly substantial advection of potential vorticity across the depth discontinuity, further modified by boundary (or boundary-like) effects. The solution found by integrating (2.2.8) should be regarded as a first approximation to a complicated situation. 3.3.3 Volume Flux in an S Canyon For the geostrophic state shown in Figure 3.4, the flux approaching the canyon can be calculated by integrating (C.6) with coefficients given by (D.9) ~ (D.13) with respect to y. The flux leaving the canyon can be calculated by integrating (C.6) with coefficients given by (C.17) ~ (C.21) with respect to,y. The discrepancy between these two fluxes is the net transport induced in the inner strait. Another method to calculate the net flux in the inner strait is to integrate (3.2.18) with respect to x e {-L, L). In an F canyon, the flux in the canyon is uniform, whereas, in an S canyon, the flux in the canyon depends on position along the canyon because flow can cross the canyon edges. From the flow fields shown in Figure 3.4 and Figure 3.5, one can conclude that left-boundflow (flow with the coast on the left looking downstream) leads to an in-canyon flux in the northern hemisphere, and, on the contrary, right-boundflow (flow with the coast on the right looking downstream) leads to an out-canyon flux in the northern hemisphere. 3.3 Discussion -114-Chapter 3 The Geostrophic State over an S Canyon in a Homogeneous Fluid 3.3.4 Barotropic Circulation around Juan de Fuca Canyon If the geostrophic flow shown in either Figure 3.4 or Figure 3.5 is at mid-latitude where/ ~ 10~4, the horizontal length scale is the Rossby radius on the shelf, R\ ~ 380km; the width of the canyon is R\ ~ 380km; the width of the shelf is 4Ry ~ 1500km. These scales are convenient to demonstrate the flow pattern in the steady state but impractical to simulate the real situation. As discussed in section 1.5, the S canyon model represents a type of canyon, e.g., the Congo Canyon (see Figure 1.2) or Juan de Fuca Canyon (see Figure 1.3). In order to compare the theoretical results to observations around, say, Juan de Fuca Canyon, more realistic geometric parameters must be used. For an S canyon simulation of Juan de Fuca Canyon, suitable parameters are Coriolis parameter: /= 1.09 x lO^/s, depth over Juan de Fuca Strait: h0 = 100m, depth on the shelf: / i i = 150m, and hence i ? i « 3 5 0 k m , depth over the main canyon body: h2 = 300m, depth in the deep ocean: h3 = 1800m, width of the canyon: 0.04/?i = 14km, 3.3 Discussion -115-Chapter 3 The Geostrophic State over an S Canyon in a Homogeneous Fluid width of the shelf: 0.2Ri = 70km, domain size: 6Ri x 6Rx = 2100km x 2100km. A sketch of the domain in which the steady state equation, (2.2.8), will be integrated is shown in Figure 3.6. The initial condition is (1.3.1) where the line of the initial surface dis-continuity is along the shelf break (and thus there is a shelf break current in the steady state). A multi-grid, re-calculation technique is used to integrate the steady state equation in the region in which we are interested. If the number of grid points is 401 x 401 (and thus the grid space is 5.3km for the whole domain), there are only two grid points in the canyon and the strait. The resolution of the solution is low within the dashed square in Figure 3.6 where the solution is of greatest interest. Increasing the number of grid points can improve the precision of the solution but it is very expensive in computer time and memory. Alter-natively, the steady state equation is first integrated with 401 x 401 grid points over the whole domain, to give the solution along the boundaries of the dashed square. Then the steady state equation is integrated again with 401 x 401 grid points but only within the dashed square using the solution obtained at the boundaries of the dashed square from the first calculation as the boundary condition for the new calculation. The results of the inte-gration of the steady state equation, (2.2.8), in the dashed square is shown in Figure 3.7. In the region where Juan de Fuca Canyon is located, the actual circulation is de-termined by a competition among several different physical processes, e.g., estuary effects, 3.3 Discussion -116-_2L J L 2 -1 -Land Land 0 -- 1 - 2 -Deep Ocean - 3 "SET 0 X -3 - 2 - 1 Figure 3.6 The domain for the S canyon. Configuration is given in the text. Distance is in units of the Rossby radius on the shelf. Thick lines represent the positions of the canyon edges, shelf break, coast, strait walls and boundaries of canyon bottom slopes. Dashed square is the region of most interest. -117-Juan de Fuca Strait Deep Ocean X — CR0SS CANY0N DIRECTI0N Figure 3.7 The results of the integration of the steady state equation, (2.2.8), in the dashed square shown in Figure 3.6. Given are contours of surface elevation, 77, (also the streamlines) in the geostrophic state as the shelf break current flows over Juan de Fuca Canyon. Thick lines represent the positions of the canyon edges, shelf break, coast, strait walls and boundaries of canyon bottom slopes. Arrows represent the direction of flow in the northern hemisphere. The range of 77 contoured is from 0.5;7o to O.98770 and the contour interval is O.OI770 where 770 is half the height of the initial surface discontinuity. -118-Chapter 3 The Geostrophic State over an S Canyon in a Homogeneous Fluid topographic effects and wind effects. Also, stratification in this region is strong. The pres-ent study mainly focuses on the interaction of barotropic currents with a canyon. So, Fig-ure 3.7 only pinpoints the effects of the canyon on the shelf break current, which must be considered when comparing the results shown in Figure 3.7 to the observations. If the canyon in Figure 3.7 is assumed to be Juan de Fuca Canyon, the flow pattern given in Figure 3.7 predicts some observed effects of the canyon on the shelf break current. First, at the mouth of Juan de Fuca Strait, the flow around the corner of the Olympic Pen-insula is strong and multi-directional. Second, the left-bound shelf break current leads to an in-canyon flow. Therefore, in the summer, there should be a mainly in-canyon flow in the Juan de Fuca Canyon because the shelf flow is mainly left-bound during this season. Further discussion of the circulation in this region will be presented in chapter 6. 3.3.5 Extrapolation to Other Topography In this chapter, an example has been used to show how to apply the properties of topographic waves and the results obtained from the primary canyon model to a more complicated topography. The S canyon model represents a type of canyon observed in the ocean. This type of canyon usually has a sharp bottom rise near the canyon mouth and a relatively smooth bottom in the main canyon body. These are well represented by the "canyon mouth slope" and the "flat bottomed main canyon body" in the S canyon model. However, extrapolation of the results from an S canyon to other topographies must 3.3 Discussion -119-Chapter 3 The Geostrophic State over an S Canyon in a Homogeneous Fluid proceed carefully. For example, if a canyon has a continuous bottom slope, there will be no geostrophic flow within the canyon (2.2.10). This unlikely result is due to the assumption of a homogeneous fluid. If stratification is considered, flow will be allowed to cross depth changes and, therefore, there will be currents in a continuously sloping canyon. The effects of stratification on flow around canyons will be studied using a numerical model in chapter 5. Results from the S canyon model can be easily extrapolated to a canyon whose bottom is segmented by steps instead of slopes. A singular line in an S canyon where a can-yon bottom slope joins the left canyon edge (looking in-canyon) converts to a singular point. Other features of the flow pattern will not change. The distribution of surface eleva-tion over a stepped bottom canyon in the geostrophic state is shown in Figure 3.8, where the width of the shelf is 3Ri, the initial surface discontinuity is along the shelf break; the canyon bottom step is at.y = 0.3R\, the canyon ends in the strait at.y = 4.5i?i and the other parameters are the same as those in Figure 3.4. 3.3 Discussion -120-SI LU >-cr o cs _ i cr i i >--6 - 4 - 2 0 2 X — CR0SS CRNY0N DIRECTI0N Figure 3.8 Contours of surface elevation around a stepped bottom canyon. Configu-ration is given in the text. Thick lines represent the positions of the canyon edges, shelf break, coast, strait walls, the canyon bottom step and the canyon head. Solid lines represent positive TJ, while dotted lines represent negative rj. The length scale is -121-Chapter 3 The Geostrophic State over an S Canyon in a Homogeneous Fluid 3.4 Summary of Rossby Adjustment over an S Canyon This chapter shows by example how to use Rossby adjustment over a complicated topography. The steady state for an S canyon model was analyzed. First, the surface elevation at the boundaries and the depth changes was founded analytically using the properties of Kelvin waves, double Kelvin waves and canyon waves introduced in the previous chapter and by using conservation of mass. Then, the steady state governing equation, (2.2.8), was numerically integrated over the flat bottom segments. The configuration is fairly general. The position and the length of the canyon bot-tom slopes can be adjusted according to the circumstance. The core of the geostrophic shelf current can be positioned at any location (e.g., on the shelf, at the shelf break, or, even off the shelf) by changing the location of the line of the initial surface discontinuity. In the steady state, depending on the direction of the shelf current, flow can enter or leave the canyon via the singular regions at the canyon edge where the canyon bottom slopes join the left canyon edge (looking in-canyon) or the singular point where the coast meets the right canyon edge (looking in-canyon). In the northern hemisphere, left-bound flow (flow with the coast on the left looking downstream) leads to an in-canyon flux, while right-bound flow (flow with the coast on the right looking downstream) leads to an out-canyon flux. 3.4 Summary of Rossby Adjustment over an S Canyon -122-Chapter 3 The Geostrophic State over an S Canyon in a Homogeneous Fluid Using linear theory, the problems of Rossby adjustment over an F canyon and an S canyon have been studied analytically in a homogeneous fluid. Only the geostrophic solu-tion has been given because of the complexity of the problem. To complete the problem of Rossby adjustment, transient solutions will be determined numerically in the next two chapters. 3.4 Summary of Rossby Adjustment over an S Canyon -123-Chapter 4 Numerical Transient Solutions in a Homogenous Fluid 4. N u m e r i c a l T r a n s i e n t Solut ions in a H o m o g e n o u s F l u i d 4.1 I n t r o d u c t i o n The problem of Rossby adjustment over a canyon has been studied analytically in the previous two chapters. Only the geostrophic solution was found. The transient solution should be studied in order to complete the problem. The main purpose of this chapter is to present, numerically, the evolution of the adjustment process for an initially non-geostrophic system over a canyon. The numerical study not only provides a complete pic-ture of the process of Rossby adjustment but also allows the study of the effects of nonlin-ear flow. The numerical simulations will show that the analytic solutions are approached in finite time. The original numerical code was written by Dr. Michael Davey [Gill et al. (1986)] to produce the numerical one-layer simulations of Rossby adjustment over a single-step. A modified version of the code was kindly given to me by Dr. Susan Allen. The modified code is based on an explicit leap-frog scheme; it uses the C-grid; nonlinear effects are in-cluded, and it uses the enstrophy and energy conserving Jacobian of Arakawa & Lamb (1981). The code can be used for a multi-layer fluid. There is no explicit viscous or fac-tional damping and the forcing, as well as the boundary conditions, can be varied. A brief introduction to the code will be presented in next section. A more complete description of 4.1 Introduction -124-Chapter 4 Numerical Transient Solutions in a Homogenous Fluid an earlier version of the code can be found in Allen (1988). In section 4.2, the numerical scheme will be introduced briefly. The numerical ex-periment for the F canyon will be presented in section 4.3, and the numerical experiment for the S canyon will be given in section 4.4. The effects of a nonlinearity on the flow will be discussed in section 4.5. Results of the numerical simulation for a homogeneous fluid will be summarized in the last section. 4.1 Introduction -125-Chapter 4 Numerical Transient Solutions in a Homogenous Fluid 4.2 Description of the Numerical Code 4.2.2 The Numerical Scheme Because stratification is not considered in this chapter, the code will be discussed only for a homogeneous fluid. Equations The code uses the primitive, modified shallow water equations, (1.4.1), which in-clude a mass input/output in the equation of conservation of mass, namely, du du da • dn - = -(A + V ) - - „ _ - „ - - ( A + , ) - - v - - v - + ( ? , (4.2.1c) where q is the rate of the continuous forcing (see explanation below), and the other sym-bols have the same definitions as those in (1.4.1). The scheme has no explicit viscous or frictional damping. Continuous Forcing < As analyzed in chapter 2, Rossby adjustment includes the physics of Poincare 4.2 Description of the Numerical Code -126-Chapter 4 Numerical Transient Solutions in a Homogenous Fluid waves. Thus, releasing an initial surface height discontinuity produces sharp wave fronts and large amplitude inertial waves [see Gill (1982) or Allen (1988)], both of which make it difficult to interpret the results of the numerical simulations (Similar problems occur in laboratory experiments as well). First, because the numerical simulations must be done in finite domains, the sharp wave fronts generated by the initial surface discontinuity are reflected from the domain boundaries back towards the region of interest, and make it difficult to interpret the results of the experiments. Second, because of their small group velocities, the inertial waves re-main near the initial surface discontinuity and cause oscillations in surface height and ve-locity, which makes interpreting the steady state difficult. A method to reduce the amplitude of the inertial waves, and the wave fronts as well, is to use a different form of forcing. Consider the surface to be flat at t < 0. Then the effect of releasing an initial surface discontinuity, (1.3.1), referred to as a "barrier release forcing", is equivalent to instantaneously adding fluid to y < 0 and removing fluid from y > 0 at t = 0. Consider instead that the fluid is added gradually like rain to the region^ < 0 and is removed gradually like evaporation from the region >> > 0, and the surface is allowed to adjust through the forcing period. It was demonstrated [see Allen (1988), chapter 2],.that if the total net amount of the fluid added or removed is the same, no matter how the fluid is added or removed, the different.forms of forcing will change the transients but not the steady state in the linear case. It can be seen that continuous forcing makes the wave fronts smooth and the trailing waves, including the inertial waves, to be so small that they are not 4.2 Description of the Numerical Code -127-Chapter 4 Numerical Transient Solutions in a Homogenous Fluid seen on the scale of the asymptotic solution. In order to decrease the inertial waves further, the continuous forcing can be in-creased slowly from zero instead of being abruptly started. Two types of forcing were used for the most frequently used forcing type 1, the forcing rate, q, increases linearly over an inertial period, is held constant over three inertial periods, and then decreases linearly back to zero over an inertial period. A sketch of surface change rate vs time is given in Figure 4.1 for forcing type 1. A purpose of the numerical simulations is to compare the evolution of the numerical solution to the analytic geostrophic results obtained in the previous two chapters. Changing from a barrier release forcing to a continuous forcing will not undermine this goal, but make the experiments easier to interpret. Staggered Grids Equation (4.2.1) was integrated over a square grid whose spacing was d. If the surface height, h+rj, was calculated at a point, (x, y), the velocity component, u, was evaluated at (x+d/2, y), and the velocity component, v, was evaluated at (x, y+d/2). In other words, a staggered grid was used as the velocity and surface height depend on single derivatives of each other, and there would be two independent grids (connected only through the Coriolis term) if each variable was evaluated at every grid point. 4.2 Description of the Numerical Code -128--129-Chapter 4 Numerical Transient Solutions in a Homogenous Fluid Explicit Scheme Equations (4.2.1) are suited to an explicit scheme, i.e., the solution at each new time is calculated only from solution at previous times. The time derivatives in (4.2.1) were expressed as central differences and the leap frog time stepping scheme was used. The calculation was started using the rest state as the solution at time step 0. The solution at time step 1 was predicted using the solution at time step 0. The solution at time step 1/2 was found by averaging this prediction with the solution at time step 0. Then the solution at time step 1 was corrected by evaluating the right-hand side of (4.2.1) at time step 1/2 and then adding it to the solution at time step 0. As two separate solutions can de-velop (at the even time steps and at the odd time steps), the two solutions were averaged every 101 time steps and the calculation restarted. When the calculation is restarted, the average solution is used. Boundary Conditions Boundary conditions were used to calculate the solution at the domain edge points after each time step. The code allows the following boundary conditions: free-slip, no-slip, periodic, radiation, set values, sponge region attached, as well as combinations of these. The most frequently used boundary conditions were the free-slip (wall) boundaries in the along-canyon direction and periodic boundaries in the across-canyon direction. 4.2 Description of the Numerical Code -130-Chapter 4 Numerical Transient Solutions in a Homogenous Fluid Outputs The code produces various outputs such as plots showing the state of the flow field (the surface/interface height, the potential vorticity distribution and tracer positions), time series data as well as a data file containing the main variables to be used for post-run analysis or as a basis for a continuing run. Qualitative analysis used these plots. A precise analysis was done by comparing cross-sections of the (analytic) geostrophic surface height to the surface height at different times during the numerical simulation. 4.2.2 The Parameters and the Domain Used For both the F and the S canyons, the domain is a 40a x 40a square where a is the barotropic Rossby radius over the deepest water. The number of grid points is 161 x 161. Thus the grid size is 0.25a. The canyon is centered in the middle of the domain and orien-tated in the >> direction. For the F canyon, the numerical experiment was forced over a period of time by adding fluid continuously, like rain, in the region y < 0, and removing fluid continuously, like evaporation, in the region y > 0. The rate of rainfall equaled the rate of evaporation. Therefore, the forcing produced a jet whose core was at.y = 0 in the regions beyond those affected by the canyon. The typical forcing was forcing type 1. The effects of changing the forcing type will be discussed in section 4.3.2. For the S canyon, the forcing was similar to that for the F canyon except that the division line between rain and evaporation was not always aXy = 0. 4.2 Description of the Numerical Code -131-Chapter 4 Numerical Transient Solutions in a Homogenous Fluid The boundary conditions were periodic along the left and right domain boundaries, and free-slip along the top and bottom domain boundaries. The experiments were conducted for a polar region where the Coriolis frequency /= 1.453703 x lO^/s (4.2.2) so that the inertial period was 0.5 of a pendulum day. There are two reasons to choose a large Coriolis frequency. First, the barotropic Rossby radius needs to be reduced to run the code for a stratified fluid (see explanation in section 5.2.2) and increasing/is a method to do so. Second, the code is usually forced over certain number of inertial periods, e.g., 5 inertial periods for forcing type 1 and a large/corresponds to a short inertial period. Be-cause what happens after the forcing stops is of greatest interest, a shorter inertial period saves computer time. The numerical simulation was run for a long time (e.g., 4 or 5 days) in order to ob-serve the solution far behind the wave fronts and to analyze the evolution of the solution. The experiment was shut down as soon as there was a possibility that the canyon waves reflected from the domain boundaries could influence the flow within the areas of interest, or, for the S canyon that the waves, which exited a periodic boundary and therefore en-tered from the other periodic boundary, could influence the area of interest. 4.2.5 Stability and Accuracy of the Code The advantage of an explicit leap-frog method is its accuracy and the fast execution 4.2 Description of the Numerical Code -132-Chapter 4 Numerical Transient Solutions in a Homogenous Fluid of each time step. The main disadvantage is the short time step required for stability of the code. For stability, the time step, At, and the grid size, Ax, must satisfy the two dimensional C F L (Courant-Friedrichs-Lewy) condition [see Press et al. (1986), pg. 627], i.e., Ax At <-;==. (4.2.3) Because the code is most unstable where the depth is deepest, i.e., the wave speed is high-est, h in (4.2.3) was chosen to be the depth over the canyon for the experiments running over the F canyon and the depth in the deep ocean for the experiments running over the S canyon. For the domain given above, Ax = 0.25a, therefore, At < / 4V2 . As the Corio-lis frequency was given by (4.2.2), the time step limit was 1216s, but the actual limit was much shorter due partially to the fact that the most unstable waves (those seen when the code goes unstable) are 45° to the axes. The time step used in the experiments for the barotropic case was 540s. The precision of the numerical simulation was checked by halving the time step or the grid size or both and comparing the new solution with the original. When just the time step was halved, or just the grid size was halved, or both were halved simultaneously, the differences between the new solutions and the originals were extremely small for either lin-ear or nonlinear cases. 4.2 Description of the Numerical Code -133-Chapter 4 Numerical Transient Solutions in a Homogenous Fluid 4.3 Transient Flow Patterns Observed over an F Canyon This section describes the flow field produced by the numerical simulation of linear Rossby adjustment over an F canyon. First, a qualitative description of the flow will be given. Second, effects of changing the forcing type will be presented in section 4.3.2. F i -nally, a direct comparison between the numerical solution and the analytic one, as well as the transport around the canyon will be discussed in section 4.3.3. The depths over the canyon and on the shelf are 400m and 100m, respectively, and therefore y-2. The width of the canyon is la where a is the Rossby radius over the can-yon. The configuration of the canyon makes it compatible, in geometry, to the F canyon in Figure 2.2(a), and, thus, it is appropriate to compare the trend of the transient solution to the analytical steady state solution. As indicated in chapter 2, a very wide F canyon is ac-tually a limit of two single-steps. The numerical simulation for this situation was presented by Gill et al. (1986), and will not be repeated here. 4.3.1 Description of the Flow In Figure 4.2, the surface elevation contours are plotted at four different times dur-ing the adjustment: after forcing 1 day, 2, 3 and 4 days; a velocity stick diagram at day 4 and a three dimensional view of the surface elevation at day 3 are also plotted. By day 2, as shown in Figure 4.2(b), the flow on the shelf far away from the canyon has reached the long time asymptotic solution, (2.4.16), and does not change further. 4.3 Transient Flow Patterns Observed over an F Canyon -134-S u r f a c e E l e v a t i o n 20 1 5 -1 0 -5 -Y 0 - 5 -- 1 0 -15 - 2 0 T= 1 .00 L y r = 1 CI i p = 1 V l V I • I - 2 0 - 1 5 - 1 0 - 5 (a) A l A 0 X T 5 10 15 20 Figure 4.2 The results of the simulation for an F canyon for forcing type 1. The configuration of the canyon is given in the text. Distance is in units of the Rossby radius over the canyon. The contours of surface elevation, 77, are shown after (a) 1 day, (b) 2 days, (c) 3 days and (d) 4 days. Thick lines: positions of the canyon edges; solid lines: positive 77; dotted lines: negative 77; range of 77: -0 .4m ~ 0.4m; contour interval: 0.06m. The velocity field at day 4 in the central region [dashed square in (d)] is shown in (e). The three dimensional view of 77 is shown in (f) after 3 days. -135-2 . 0 0 L y r = 1 CI i p = 1 Y 0 - 1 0 H - 1 5 -20 -136--137--138-V e I o c i t y F i e l d 7 T= 4 .00 7 Lyr=1 CI ip=2 s > \ / ; / / \ ' V \ ' V \ ' V V u V \> \' \t \' y< v " v v \' \i \> \t U if < f " •t 'I (e) X 8.i43E»aa MAXIMUM VECTOR -139--140-Chapter 4 Numerical Transient Solutions in a Homogenous Fluid Rossby adjustment produces tongues along both canyon edges and the tongues propagate in opposite directions. Ahead of the two wave fronts, the fluid remains undis-turbed; behind the wave fronts, the flow around the canyon gradually adjusts towards the analytic long time asymptotic solution shown in Figure 2.2(a). By day 4, as shown in the dashed square in Figure 4.2(d) or Figure 4.2(e), the flow far behind the wave fronts is in-distinguishable from the analytic solution. It can be seen that when a shelf flow approaches a canyon most of the flow will turn to the right in the northern hemisphere. Most of the flow crosses the canyon edges af the wave fronts as they move away from the line, y = 0, which separates the rain and evapora-tion regions. Considering the tongues along the canyon edges, it can be seen that the length scale over which the surface height changes is smaller on the shelf than over the canyon. From Figure 4.2(f) ~ the three dimensional view of the surface elevation at day 3, it can be seen that the very gentle, continuous forcing (forcing type 1) has made the wave fronts smooth and the amplitudes of the trailing waves very small so that the numerical so-lution has been little affected by the trailing waves, including the inertial waves. 4.3.2 Effects of Changing the Forcing Type It has been discussed in section 4,2.1 that gentle forcing not only smoothes the wave fronts but also makes the amplitudes of trailing waves, including inertial waves, so small that the interpretation of the numerical solution will not be contaminated. Gentle forcing also makes the interpretation of the numerical solution easy and straightforward 4.3 Transient Flow Patterns Observed over an F Canyon -141-Chapter 4 Numerical Transient Solutions in a Homogenous Fluid using the surface elevation. To illustrate this point a relatively abrupt forcing will be chosen and the effects will be compared with those generated by the smooth forcing (type 1). Instead of using forcing type 1, now the forcing ramps up (increases linearly) over the first half inertial period and then ramps down (decreases linearly) back to zero over the second half inertial period. This type of forcing is called forcing type 2. After 3 days, the surface elevation is shown in Figure 4.3. Comparing Figure 4.3(a) to Figure 4.2(c), the striking difference between the two results is that the forcing type 2 produces intermittent cells over the canyon edges. The cells are cyclonic along the upstream canyon edge in the lower half domain and anticyclo-nic along the downstream edge in the upper half domain. The cyclonic circulation along the upstream canyon edge could be due to stretching of a water column as it moves down the edge, and the anticyclonic circulation along the downstream canyon edge could be due to compressing of a water column as it moves up the edge. Thus, there is flow entering or leaving the canyon via many spots (the centers of these cells) at each canyon edge instead of only via the wave fronts. When comparing Figure 4.3(c) to Figure 4.2(f) it is obvious that forcing type 2 re-sults in amplitude waves trailing the fronts and the dispersion of these waves causes the observed intermittent cells. In summary, if the numerical simulation is forced by a relatively abrupt forcing, such as forcing type 2, interpretation of the results is not as straightforward as the numeri-4.3 Transient Flow Patterns Observed over an F Canyon -142-20 S u r f a c e E l e v a t i o n . T= 3.00 i • • i I . 7 1 Y • 1 15-1 0 -Y 0 - 5 --10 15--20 L y r = 1 C l i p = 1 . I . % — u 1 1 1 1 A I. A f -20 -15 -10 -5 0 5 X 10 15 20 (a) Figure 4.3 The results of the simulation after 3 days of forcing (type 2) for an F canyon. The configuration is same as that in Figure 4.1. The contours of surface ele-vation, rj, are shown in (a) the whole domain and (b) the central region [dashed square in (a)]. Range of 77: -0.03m ~ 0.03m; contour interval: 0.006m. The three dimensional view of the surface elevation is shown in (c) after 3 days. In (a) and (b), thick lines: positions of the canyon edges; solid lines: positive 77; dotted lines: nega-tive 77. -143-X ( b ) -144--145-Chapter 4 Numerical Transient Solutions in a Homogenous Fluid cal simulation forced by forcing type 1. Now that the advantage of using forcing type 1 has been demonstrated, all following numerical experiments will be forced by forcing type 1 except as indicated. 4.3.3 Discussion Direct Comparison between the Numerical and Analytic Solutions The numerical simulation of Rossby adjustment over an F canyon has been analyzed qualitatively. By viewing the plots of the solution, it can be seen that the evolution of the numerical solution is consistent with the analytic long-time asymptotic solution presented in chapter 2. In order to make the analysis more precise, the numerical solution in an across-canyon section will be plotted at different times to compare with the analytic long-time asymptotic solution. The configuration of the F canyon is the same as that in Figure 4.2 except that the width of the canyon is l a instead of 2a. All other parameters, e.g., the grid size, the time step, the forcing type, /, etc., are same. A cross-section through the canyon showing the surface elevation and the v-component of the velocity at,y = 5a after forcing 1 day, 2, 3, 4 and 5 days are plotted in Figure 4.4. The analytic, far field, long time asymptotic solutions given by (2.4.20a) and (2.4.20c), are plotted in Figure 4.4(a) and Figure 4.4(b), respec-tively. Obviously, the trend of the numerical solution is to approach the analytic long time 4.3 Transient Flow Patterns Observed over an F Canyon -146-East-West Sec t ion of Eta at Layer=1 Clip=1 Figure 4.4 A cross-section across the canyon of the numerical solution for the F canyon whose configuration is the same as that in Figure 4.1 except that the width of the canyon is l a . The horizontal axis represents the distance in the across-canyon direction with its median marking the central axis of the canyon. The vertical axis represents the relative value at y = 5a of (a) the surface elevation and (b) the v-component of velocity. Thick line: the analytic long time asymptotic solution; thin lines: the numerical solutions after 1 day, 2, 3, 4 and 5 days. -147-Chapter 4 Numerical Transient Solutions in a Homogenous Fluid asymptotic solution studied in chapter 2. Interpretation of the Mass Transport in the Canyon As indicated in chapter 2, when a shelf flow approaches an F canyon, it induces a net flux in the along-canyon direction. Now the physics of this phenomena can be easily interpreted with the numerical simulation, e.g., the results plotted in Figure 4.2. In order to accommodate the discontinuity in flux as well as to conserve potential vorticity at a single-step bottom, there must be a depletion of fluid when a jet flows down a step, and, on the contrary, there must be an accumulation of fluid when a jet flows up a step [see Gill et al. (1986) and Allen (1988) for details]. The depletion or accumulation of fluid will occur at the place where the flow crosses the step, i.e., at the wave front. The above properties pertain to the two edges of an F canyon. The flux approaching the canyon equals that leaving the canyon, i.e., the shelf jet does not leave or take any net amount of mass from the canyon because the depths on the shelf at the two sides of the canyon are equal,. However, there is a depletion of fluid in the vicinity of the upstream canyon edge at the wave front where the jet flows down this edge. Exactly the same amount of fluid must be accumulated in the vicinity of the downstream canyon edge at the wave front where the jet flows up this edge. The depletion occurs at the wave front which moves progressively in the -y direction, while the accumulation occurs at the wave front which moves progressively in the +y direction. As the wave fronts move further and further apart, the net effect of the mass movement along the canyon is equivalent to 4.3 Transient Flow Patterns Observed over an F Canyon -149-Chapter 4 Numerical Transient Solutions in a Homogenous Fluid mass being depleted from the vicinity of the upstream canyon edge at large -y, traveling all the way along the canyon to large +y to be deposited in the vicinity of the downstream canyon edge. In other words, a shelf jet induces a net transport along the canyon. As a re-sult of the mass redistribution during Rossby adjustment, after a long time, the surface height at the downstream canyon edge tends to the theoretical constant, ( 1 - 0 ) 7 7 0 , and the surface height at the upstream canyon edge tends to the theoretical constant, - ( 1 - 0 ) 7 7 0 , in which a e {0, 1} is the Canyon Number defined by (2.4.15) and 770 is half of the maximum surface height change. Note that, strictly speaking, the depletion and accumulation of fluid happen simul-taneously at the two canyon edges. For instance, by viewing (say) Figure 4.2(d), it can be noted that depletion of fluid occurs at both canyon edges in the lower half (-y) domain but at different rates, while accumulation of fluid occurs at both canyon edges in the upper half (+y) domain but at different rates. Depletion of fluid occurs only in the lower half (-v) do-main and is stronger at the upstream canyon edge, where the surface elevation needs to be lowered from 770 to - ( 1 - 0 ) 7 7 0 than at the downstream edge, where the surface elevation needs to be lowered only from 770 to ( 1 -0 ) 7 7 0 . On the contrary, accumulation of fluid oc-curs only in the upper half (+y) domain and is stronger at the downstream canyon edge, where the surface elevation needs to be elevated from -770 to ( 1 - 0 ) 7 7 0 than at the upstream edge, where the surface elevation needs to be elevated only from -770 to - ( 1 - 0 ) 7 7 0 . In summary, when a jet approaches an F canyon in the northern hemisphere (looking downstream), the net effect of the mass redistribution looks like an amount of 4.3 Transient Flow Patterns Observed over an F Canyon -150-Chapter 4 Numerical Transient Solutions in a Homogenous Fluid mass is being scraped from the right hand side of the jet in the vicinity of the upstream canyon edge and deposited on the left hand side of the jet in the vicinity of the down-stream canyon edge. 4.3 Transient Flow Patterns Observed over an F Canyon -151-Chapter 4 Numerical Transient Solutions in a Homogenous Fluid 4.4 Transient Flow Patterns Observed over an S Canyon This section describes the flow field produced by the numerical simulation of linear Rossby adjustment over an S canyon. First, a qualitative description of the flow will be given. Then, a discussion of transport around the S canyon will be presented. The parameters for the numerical simulation of the S canyon are same as those used for the F canyon, i.e., domain size: 40a x 40a where a is the largest Rossby radius in the domain; number of grid points: 161 x 161; boundary conditions: periodic in x and free-slip in y; Coriolis frequency: 1.453703 x 10"4/s and forcing type: 1. The depths in the strait, on the shelf, over the main canyon portion and over the deep canyon portion as well as in the deep ocean are 100m, 150m, 300m and 400m, re-spectively. The width of the canyon mouth slope and the head slope are 1.4a and 2a, re-spectively, where a is the largest Rossby radius in the domain, i.e., that over the deep canyon portion as well as in the deep ocean. The width of the shelf is 8a. The width of the canyon as well as the strait is la . Note that the S canyon has a width of one Rossby radius over the deep ocean (400m deep) for the numerical simulation in contrast to a width of one Rossby radius on the shelf (150m deep) for the analytic calculation given in chapter 3. It is expected that this slight change of the canyon width will not change the general flow pattern significantly. Therefore, qualitative comparison between the numerical results for the geometric configu-4.4 Transient Flow Patterns Observed over an S Canyon -152-Chapter 4 Numerical Transient Solutions in a Homogenous Fluid ration indicated above and the analytic results presented in chapter 3 is not inappropriate. A line, y = df is set on the shelf, 0.85a away from the shelf break, and the simula-tion is forced by rain in the region y < df and evaporation in the region y> df. The rate of rainfall equals the rate of evaporation. 4.4.1 Description of the Flow In Figure 4.5, the surface elevation contours are plotted at four different times dur-ing the adjustment (after forcing 2, 3, 4 and 5 days) and a velocity stick diagram is plotted at day 5. By day 2, as shown in Figure 4.5(a), the flow far enough away from the canyon has reached the long time asymptotic solution (C.6) with coefficients given by (C.17) ~ (C.21). On the downstream side far enough away from the canyon, this state will change. How-ever, the propagating Kelvin waves (near the coast) and the double Kelvin waves (over the shelf break), each of which originates at the canyon, will ultimately readjust the state eve-rywhere upstream of the canyon. One of the analytic assumptions has been verified: effects of the S canyon are radi-ated only to the left side of the canyon (looking in-canyon) in the northern hemisphere. The Rossby adjustment produces a tongue centered at the right edge of the canyon (looking in-canyon), and the tongue heads towards the coast. The tongue keeps growing until it hits the coast. 4.4 Transient Flow Patterns Observed over an S Canyon -153-Deep Ocean - 2 0 - 1 5 - 1 0 (a) "zrnr 0 x 10 15 20 Figure 4.5 The results of the simulation for an S canyon. The configuration of the canyon is given in the text. Distance is in units of the Rossby radius in the deep ocean. The contours of surface elevation, 77, are shown after (a) 2 days, (b) 3 days, (c) 4 days and (d) 5 days. Thick lines: the positions of the canyon edges, boundaries of canyon bottom slopes, shelf break, coast and strait walls; solid lines: positive 77; dotted lines: negative 77; range of 77: -0 .4m ~ 0.4m; contour interval: 0.06m. The close up view of 77 and the velocity field in the central region [dashed square in (d)] are shown in (e) and (f), respectively. -154-S u r f a c e E l e v a t i o n . T= 3.00 Lyr=1 C l i p = 1 2 0 -I ' 1 ' 1 1 1 L . , I i I , L 1 5 H 10H 5H 0 g o - 5 -- 1 0 -15H - 2 0 H • 1 1 1 1 1 1 JJS " 1 1 1 1 1 1 4" - 2 0 - 1 5 - 1 0 - 5 0 5 10 15 20 X ( b ) Y - 2 0 - 1 5 -156-S u r f a c e E l e v a t i o n . T= 5.00 L y r = 1 CI i p = 1 Y X -157--158-V e l o c i t y F i e l d . T= 5 . 0 0 L y r = 1 •CI ip = 1 0.244 MAXIMUM VECTOR -159-Chapter 4 Numerical Transient Solutions in a Homogenous Fluid It can be seen, say, from Figure 4.5(e) and (f), that there are three special places. The first one is the junction point of the coast and the right edge of the canyon (looking in-canyon). The second one is the line where the canyon mouth slope meets the left edge of the canyon. Another is the line where the canyon head slope meets the left wall of the can-yon (looking in-strait). Flow at these places is accelerated faster and faster and squeezed into a thinner and thinner layer. The evolution of the numerical solution is consistent with the analytic long time as-ymptotic solution shown by Figure 3.4. 4.4.2 Discussion Interpretation of the Mass Transport around the Canyon Similar to the case for the F canyon, it is of greatest interest to analyze the net ef-fect of mass redistribution during the Rossby adjustment process. As discussed in the previous section, the state far enough away from the canyon downstream remains unchanged at all times after, say, 2 days. Therefore, the flux far enough away from the canyon downstream is constant. In other words, the flux leaking from the canyon is constant. As explained in section 4.3.3, there must be a depletion of fluid when a jet flows down a canyon edge (or, a shelf break), and, on the contrary, there must be an accumula-tion of fluid when a jet flows up a canyon edge; depletion or accumulation of fluid occurs 4.4 Transient Flow Patterns Observed over an S Canyon -160-Chapter 4 Numerical Transient Solutions in a Homogenous Fluid at the wave fronts. During the initial phase of Rossby adjustment, an out-canyon tongue is formed at the upstream canyon edge if the shelf flow is not exactly along the shelf break, and fluid is depleted at the head of the tongue. As the tongue reaches the mouth of the canyon, it can go nowhere but continue to extend upstream along the shelf break. Thus depletion of fluid continues along the upstream shelf break as the surface elevation along the shelf break is lowered to the theoretical constant given by (3.2.16). Similarly, during the initial phase of Rossby adjustment, an in-canyon tongue is formed at the downstream canyon edge; fluid accumulates along the edge at the head of the tongue, and an in-canyon flow is induced within the canyon. As the flow moves up the canyon mouth slope, fluid accumulates over the slope, and, as shown by Figure 4.5(e), an anticyclone is formed and it pushes the flow crossing the slope into a thinner and thinner layer against the upstream canyon edge. The surface elevation along the downstream canyon edge and the mouth slope is being raised to the theoretical constant given by (C-22). Since there are no sources or sinks in the numerical simulation after the forcing stops, the fluid depleted from the upstream shelf break must be deposited somewhere, and similarly the fluid accumulated along the downstream canyon edge and the mouth slope must be from somewhere in the domain. Logically, from the point of view of the net effect, the mass deposited along the downstream canyon edge and the canyon mouth slope are depleted from the upstream canyon edge and the shelf break upstream. 4.4 Transient Flow Patterns Observed over an S Canyon -161-Chapter 4 Numerical Transient Solutions in a Homogenous Fluid As the double Kelvin waves propagate along the upstream shelf break, more and more fluid is depleted. Also, because there is a jet continuously flowing down the upstream canyon edge in the region where canyon mouth slope adjoins the canyon edge, fluid is con-tinuously injected into the canyon in this region. However, the accumulation of fluid at the downstream canyon edge and at the canyon mouth slope cannot continue forever. As soon as the surface height at the downstream canyon edge and the mouth slope reach the same level as the downstream shelf break, fluid is no longer deposited in these places. Even though a substantial amount of fluid leaks out of the canyon, mostly via the junction point of the downstream canyon edge and the coast and some fluid bypasses the canyon mouth in the deep ocean, the flux exiting the canyon becomes constant after adjustment after a cer-tain time (about 2 days). The rest of the fluid continuously depleted from the upstream shelf break and injected into the canyon where the canyon mouth slope adjoins the up-stream canyon edge is deposited along the strait and at the canyon head slope. The net effect of the mass movement is equivalent to the mass in the vicinity of the upstream shelf break being taken away and deposited along the canyon and the strait. Reversal of the Shelf Flow For the same configuration as shown in Figure 4.5 but for reversed forcing, i.e., the rain region is changed to be an evaporation region, and vice versa, the surface elevation contours are plotted in Figure 4.6 at four different times during the adjustment: after forc-ing 2, 3, 4 and 5 days, a velocity stick diagram is plotted at day 5. 4.4 Transient Flow Patterns Observed over an S Canyon -162-Chapter 4 Numerical Transient Solutions in a Homogenous Fluid The flow field given in Figure 4.6 is almost the same as that given in Figure 4.5 ex-cept that all the flow reverses its direction. Consequently, a right-bound flow (flow with the coast on the right looking downstream) leads to an out-canyon flow in the northern hemisphere. In comparing the transient numerical solution, Figure 4.6(f), to the analytic long time asymptotic solution, Figure 3.5, it can be seen that the trend of the transient solution is consistent with the analytic asymptotic solution. The transport for reverse flow is also reversed. Mass is depleted from the canyon and the strait and deposited in the vicinity of the downstream shelf break, i.e., at the right of canyon mouth (looking out-canyon) in the northern hemisphere. 4.4 Transient Flow Patterns Observed over an S Canyon -163-0 S u r f a c e E l e v a t i o n . T= 2.00 L y r = 1 C l i p = 1 7i7 • i l 0 - Land Land 5 -Figure 4.6 The results of the simulation for an S canyon. Al l specifications are the same as those in Figure 4.4 except that the forcing is reversed and thus the flow is in the opposite direction. The contours of the surface elevation, 77, are shown after (a) 2 days, (b) 3 days, (c) 4 days and (d) 5 days. The close up view of 77 and the velocity field in the central region [dashed square in (d)] are shown in (e) and (f), respec-tively. Range of 77: -0 .4m - 0.4m; contour interval: 0.06m. -164--165--166-S u r f a c e E l e v a t i o n . T= 5 . 0 0 Lyr=1 C l i p = 2 7 7 Y -168-V e l o c i t y F i e l d . T = . 5 . 0 0 Lyr=1 C l i p = 1 7 V • • i — — — , • i t • i I • i i • i i < i l ' i i ' ' i 7S 1 (f) X •JJiL* 8.241E»Bg HAXInun VECTOR -169-Chapter 4 Numerical Transient Solutions in a Homogenous Fluid 4 .5 Description of Nonlinear Flow As discussed by Allen (1988), the strength of the nonlinearity can be measured by a parameter, 8 = n0/ Ah, where 770 is the range of possible fluid column height changes and Ah is the range of possible depth changes. The parameter, 8, gives the ratio between the possible fluid column height variations produced by the forcing to those produced by the topography. In linear flow, 770 is infinitesimal, while in strongly nonlinear flow, 770 is equal to, or, even greater than the depth change, Ah. This section describes the numerical simulations of a strongly nonlinear flow over an F canyon with the parameter 8=2. Because of the difficulty of analytic study, the non-linear effects on the numerical solution will be discussed qualitatively by directly comparing to the linear results. The numerical simulation is run in a square domain: 40a x 40a where a is the Rossby radius over the canyon; the number of grid points is 161 x 161; the boundary condition are periodic in x and free-slip in y; the Coriolis frequency is 1.453703 x lO^/s and the forcing is of type 1. The geometry is chosen so that the depths over the canyon and the shelf are 80m and 76m, respectively, and the maximum surface elevation difference is 8m. Therefore, 8 = 2. In Figure 4.7 the velocity field for strongly nonlinear flow over the F canyon is plotted at four different times during the adjustment: 7, 11, 15 and 19 days, and the surface 4.5 Description of Nonlinear Flow -170-20 1 5 -1 0 -5 -Y 0 > » » 3 5 -- 1 0 --15-- 2 0 V e l o c i t y F i e l d . T= 7 . 0 0 L y r = 1 C l i p = 1 i . i • 7 i 7 • i , i > > > » > » » » » » >»>>>>>>>>>>>>>>>>>>>>>>>»>>>> • * » » > » » > » » > » » > > » » » > > > » » » • » » » > » » ] - 2 0 - 1 5 - 1 0 (a) - 5 0 X 10 15 20 MAXIMUM VECTOR Figure 4.7 The results of Rossby adjustment over an F canyon for 5=2. Distance is in units of the Rossby radius over the canyon. The velocity fields are shown after (a) 7 days for the whole domain, and the close up view of the velocity fields in the cen-tral region [dashed square in (a)] are shown after (b) 7 days, (c) 11 days, (d) 15 days and (e) 19 days. The contours of the surface elevation, TJ, are plotted after 19 days in (f) where the range of n: -4.0m ~ 4.0m; contour interval: 0.4m. Thick lines: the positions of the canyon edges; solid lines: positive TJ; dotted lines: negative TJ. -171-00 Lyr=1 CI ip = 2 ( b ) B.154E+B1 MAXIMUM VECTOR -172-V e l o c i t y F i e l d . T= 1 1 . 0 0 L y r = 1 C I i p = 1 V (C) X v ' MAXIMUM 01 VECTOR V e l o c i t y F i e l d . T= V 1 5 . 0 0 Lyr=1 CI ip=1 V - » V N I N >. N i S N N. » k \ \ N . \ v -\ \ N N N , v A N N ^ " -ill \ * ' * \ \ \ \ ^ " ™ ^ - - / 1 \ \ \ ^ - . - » v « - • / I \ \ g,177Ej>ai MAXIMUM VECTOR -174-/ V e I o c 1 9 . 0 0 Lyr=1 C I i p = 2 7 (e) B.181E»B1 MAXIMUM VECTOR -175--176-Chapter 4 Numerical Transient Solutions in a Homogenous Fluid elevation contours at day 19 are also plotted. In Figure 4.8 the vorticity and the potential vorticity are plotted at day 7 and 19. Note that because the fluid is shallow (80m maxi-mum), the canyon wave speed is quite slow so that it takes a long time for the flow to ad-just over the canyon. The importance of nonlinearity relative to geostrophy is indicated by the Rossby number which is defined as < 4 5 » -where U is the typical horizontal velocity which is usually chosen to be the maximum ve-locity in the domain studied,/is the Coriolis frequency and L is a typical horizontal length scale. In the numerical problems studied in this thesis, there are several horizontal lengths, e.g., the widths of the domain, the canyon and the shelf (for an S canyon). What should be chosen as the typical length scale depends on the flow. As will be demonstrated below, the nonlinear effects are prominent at depth changes, i.e., at the canyon edges in our problem. Consequently, the width of a canyon edge, which is one grid space in the numerical con-figurations, will be chosen as the typical horizontal length scale to calculate the Rossby number. For the current numerical simulation, the maximum velocity is about 1.8m/s and the typical length scale (one grid space) is 0.25a giving a Rossby number of about 0.26. This value is not negligible, so a flow with 5=2 represents a nonlinear situation. 4.5 Description of Nonlinear Flow -177-V o r t i c i t y . T= 7 . 0 0 Lyr=1 C l i p = 2 V V (a) X Figure 4.8 The results of Rossby adjustment over an F canyon for S= 2. Near the canyon, the contours of vorticity are shown after (a) 7 days and (b) 19 days, and the contours of potential vorticity are shown after (c) 7 days and (d) 19 days. Thick lines: positions of the canyon edges; solid lines: positive values; dotted lines: nega-tive values. -178-V o r t i c i t y . T= 19 .00 Lyr=1 C l i p = 2 IV V ( b ) -179-P o t e n t i a l V o r t i c i t y . T= 7 . 0 0 Lyr=1 C l i p = 2 A A (C) X P o t e n t i a l V o r t i c i t y . T= 1 9 . 0 0 Lyr=1 C l i p = 2 -181-Chapter 4 Numerical Transient Solutions in a Homogenous Fluid The dynamics of the nonlinear flow pattern can be illustrated by analyzing, in se-quence, the distribution of the potential vorticity, vorticity and velocity at a certain time. Figure 4.8(c) in conjunction with the velocity field shows that advection of potential vor-ticity occurs at both canyon edges in the vicinity of the jet. Because of the advection of potential vorticity, a cyclone forms in the canyon and an anticyclone forms on the down-stream shelf as shown in Figure 4.8(a). Therefore, as given by Figure 4.7(b), as the flow is cyclonic in the canyon, it turns to the left a little bit, and as the flow is anticyclonic on the downstream shelf, it turns to the right. In order to make a comparison between the nonlinear and linear cases, an experi-ment was run for the same geometry and the same parameters including the boundary conditions, the Coriolis frequency, etc., but the magnitude of forcing was reduced so that 8 = 0.02. In Figure 4.9 the velocity field for the case 8= 0.02 is plotted at two different times during the adjustment, 7 and 19 days, and the surface elevation contours at day 19 are also plotted. In Figure 4.20 the vorticity is plotted at day 7 and 19. The potential vorticity field is unchanged for this case and therefore is not presented in the figure. Note that because the maximum velocity is about 0.02m/s, the Rossby number is about 0.0029 which is very small. Consequently, a flow with 5= 0.02 can be treated as linear. Comparing the linear flow pattern with the nonlinear one, some striking differences can be noted. The linear flow crosses the upstream canyon edge mainly at the right hand side of the shelf flow (looking downstream), where the vorticity is positive because of stretching 4.5 Description of Nonlinear Flow -182-20 V e l o c i t y F i e l d . T= 7 . 0 0 Lyr=1 C l i p = 1 l , I , I i \7 I 7 • i . I 1 5 -1 0 -5 -Y 0 - 5 - 1 0 -- 1 5 - 2 0 - 2 0 - 1 5 - 1 0 (a) - 5 A l A 0 5 10 15 20 g.156E-B1 MAXIMUM VECTOR Figure 4.9 The results of Rossby adjustment over an F canyon for S = 0.02. Dis-tance is in units of the Rossby radius over the canyon. The velocity field is shown after (a) 7 days, and the close up view of the velocity field in the central region [dashed square in (a)] are shown after (b) 7 days and (c) 19 days. The contours of the surface elevation, rj, are plotted after (d) 19 days where range of r\: -4 .0m ~ 4.0m; contour interval: 0.4m. Thick lines: positions of the canyon edges; solid lines: positive TJ; dotted lines: negative TJ. -183-V e l o c i t y F i e l d . T= 7 . 0 0 L y r = 1 C I i p = 2 Y (\)\ X r i M * « V"/ MAXIMUM VI VECTOR -184-L y r = 1 (C) X B.238E-81 MAXIMUM VECTOR -185--186-V o r t i c i t y . T= 7 . 0 0 Lyr=1 C l i p = 2 7 V / J 1 1 I -V, 11 • 1 4 i \ » 1 l» 1 _ ii _ v jF^ <•-- -i / • / / / / * * X \ * A A (a) X Figure 4.10 The results of Rossby adjustment over an F canyon for 8 = 0.02. Near the canyon, the contours of vorticity are shown after (a) 7 days and (b) 19 days. Thick lines: positions of the canyon edges; solid lines: positive value; dotted lines: negative value. -187--188-Chapter 4 Numerical Transient Solutions in a Homogenous Fluid of the water column, and crosses the downstream canyon edge mainly at the left hand side of the shelf flow, where the vorticity is negative because of compression of the water col-umn (see Figure 4.20). The regions where the linear flow crosses the canyon edges extend along both edges but in opposite directions. Also the cyclone and anticyclone confine themselves to the canyon edges. Comparing Figure 4.8(a)/(b) with Figure 4.20(a)/(b), it can be argued that the non-linear flow adjusts over the canyon in a pattern which looks like a linear pattern modified by advection. Similar to the linear flow, the nonlinear flow is cyclonic over the upstream canyon edge at the right hand side of the shelf flow and anticyclonic over the downstream canyon edge at the left hand side of the "shelf flow [see Figure 4.8 (a) and (b)]. However, there is a new feature that does not show up in the linear flow pattern: a cyclonic circula-tion is formed over the canyon at the left hand side of the shelf flow, and an anticyclonic circulation is formed on the shelf downstream at the fight hand side of the shelf flow. These two gyres are generated by nonlinear effects as explained previously. Because of the advection of potential vorticity, the two gyres are advected by the flow and two additional tongues are formed [see Figure 4.8 (b) and (d)]. A cyclonic tongue extends towards the upper right and an anticyclonic tongue extends towards the lower right. The cyclonic tongue will eventually move onto the shelf downstream [see the tip of the upper right tongue in Figure 4.8 (b) and (d)]. Since the cyclonic tongue originates from the shelf upstream, water parcels cross the canyon from the shelf upstream to the shelf downstream. On the contrary, because there is no advection of the potential vorticity for 4.5 Description of Nonlinear Flow -189-Chapter 4 Numerical Transient Solutions in a Homogenous Fluid the adjustment of the linear flow, there is no water parcel transport in the domain. The net transport in the along canyon direction, Fy, is proportional to a constant ra-tio times the flux approaching/leaving the canyon, Fx, and, therefore, for a certain canyon-shelf system FylFx is a constant regardless of the level of nonlinearity. The formula for the calculation the flux in the steady state has been given in (2.5.2). To substantiate the argu-ment for the transient situation, the flux in the +y direction was calculated for both the nonlinear case, Figure 4.7 (e), and the linear case, Figure 4.9(c). It was found that the ratio FylFx was the same for both cases, which is as expected because the geometries are identi-cal. Because the greatest nonlinearity occurs at the edges of the canyon, Fx, the volume flux approaching the canyon, can be still calculated with (2.5.1) for the nonlinear cases. In other words, calculation of Fx and, hence, Fy do not depend on the level of nonlinearity for these cases. We have briefly described Rossby adjustment of a nonlinear flow over an F canyon. The numerical nonlinear solution retains the properties of that for a linear solution but is modified by advection. In order to simplify interpretation of the numerical results, further discussions will mainly concern linear flows. 4.5 Description of Nonlinear Flow -190-Chapter 4 Numerical Transient Solutions in a Homogenous Fluid 4.6 Summary of the Numerical Simulations for a Homogeneous Fluid In this chapter, a numerical scheme for the simulation of Rossby adjustment was introduced, and the numerical transient solutions for both the F and the S canyon models were presented. Also nonlinear effects were briefly discussed. The numerical code is nonlinear without explicit viscous or frictional damping. The time derivatives are calculated by using the leap frog scheme, and the spatial derivatives are calculated on an Arakawa "C-grid". The domain boundary conditions and the forcing of flow can be varied according to the situation. The code is stable as long as the CFL condi-tion is satisfied. The evolution of the numerical transient solution for either an F or an S canyon is consistent with the analytic long time asymptotic solution given in chapter 2 or 3. The numerical simulation makes it easy to analyze the net effect of mass movement during the process of Rossby adjustment. In the northern hemisphere: (1) . for an F canyon, the net effect of the mass transport is equivalent to depleting mass from the right hand side of the shelf flow (looking downstream) in the vicinity of the up-stream canyon edge, and depositing it on the left hand side of the shelf flow in the vicinity of the downstream canyon edge; (2) . for an S canyon and a left-bound shelf flow (flow with the coast on the left looking downstream), the net effect of the mass transport is equivalent to depleting mass in the vi-4.6 Summary of the Numerical Simulations for a Homogeneous Fluid -191-Chapter 4 Numerical Transient Solutions in a Homogenous Fluid cinity of the upstream shelf break, and depositing some of it along the canyon and the strait; (3). for an S canyon and a right-bound shelf flow (flow with the coast on the right looking downstream), the net effect of the mass transport is equivalent to depleting mass from the canyon and the strait, and depositing it in the vicinity of the downstream shelf break. " Rossby adjustment of a nonlinear flow over an F canyon gives a flow pattern that is similar to that for a linear flow but the pattern is advected downstream. Advection adds a new features: a cyclonic circulation is formed over the canyon and an anticyclonic circula-tion is formed on the shelf downstream. The ratio between the net flux along the canyon and the flux approaching/leaving the canyon is determined only by the geometric parame-ters and is independent of the strength of the flow. Up to now all the discussion has been for a homogenous fluid. In order to make the research more realistic, stratification should be considered. In next chapter, numerical simulation in a stratified (two-layer) fluid will be introduced. 4.6 Summary of the Numerical Simulations for a Homogeneous Fluid -192-Chapter 5 Numerical Transient Solutions in a Stratified Fluid 5. Numerical Transient Solutions in a Stratified Fluid 5.1 Introduction 5.1.1 Outline The preceding chapters discussed Rossby adjustment of a homogeneous fluid over a canyon. The neglect of stratification, while revealing many fundamental aspects of the ad-justment process, excludes the possibility of more complex baroclinic responses which are likely to be relevant to geophysical flows. As a first step towards understanding the ad-justment in a stratified fluid, the adjustment of a two-layer fluid over a canyon is considered in this chapter. This form of stratification is the simplest possible that allows the fluid to have both a baroclinic and a barotropic response which are, of course, inseparable over to-pography. In this chapter the results of the two-layer numerical simulation of Rossby adjust-ment over a canyon are presented. The theoretical results obtained by previous researchers will be summarized in the sub-section 5.1.2 to allow comparison with the numerical results. The numerical scheme is an extension of the one-layer scheme introduced in chapter 4. A brief introduction to the code, its stability and details of the domain for the simulations given in this chapter will be presented in section 5.2. 5.1 Introduction -193-Chapter 5 Numerical Transient Solutions in a Stratified Fluid One experiment, in which the interface is above the level of the shelf and the flows in the two layers on the shelf are in opposite directions, will be done for an F canyon. De-scription of the flow pattern and discussion will be given in section 5.3. Four experiments, in each of which the interface is below the level of the shelf, will be done for the S canyon. Flows in the two layers beyond the canyon are in the opposite directions for two experiments and in the same direction for the other two. Description of the flow patterns and a discussion will be given in section 5.4. In section 5.5 the transient solution obtained for the S canyon will be compared, qualitatively, with the observed circulation pattern around Juan de Fuca Canyon. Results of the numerical simulations in a stratified fluid will be summarized in the last section. 5.1.2 Theoretical Results of Rossby Adjustment in a Stratified Fluid Rossby adjustment of a stratified fluid over topography has been studied analytically by a number of researchers. For example, Allen (1988) studied the Rossby adjustment of a two-layer fluid over a flat bottom basin, a single-step and a slope; Klinck (1989) analyzed the geostrophic adjustment of a stratified coastal current over an F canyon with a "two-level" as well as a "three-level" system. Traditionally, the dynamic analysis is posed in terms of a multilayer fluid, such as that used by Allen (1988). However, the layer technique is hard to handle in some circum-5.1 Introduction -194-Chapter 5 Numerical Transient Solutions in a Stratified Fluid stances around topography. For instance, as pointed out by Klinck (1989), for the problem of Rossby adjustment over a canyon, if the dynamics in the canyon is allowed to be decou-pled from that on the shelf and the canyon is taken to be a layer by itself, when water spills out of the canyon, the location of the edge of the "canyon" water must be calculated ~ a difficult problem. Therefore, in the layer model used in this chapter, the dynamics in the canyon are always coupled with that on the shelf, which requires that the interface in the two-layer fluid is either within or above the canyon but never exactly at the same level as the shelf. The two-layer stratification is the simplest form which allows the fluid to have both a baroclinic and a barotropic response which, however, are not separable over topog-raphy [see Allen (1988), chapter 4]. In contrast, the level model used by Klinck (1989) allows separate dynamics for a canyon by taking the canyon to be a level itself. The advantage of using a level model is that the flow can be separated into barotropic and baroclinic modes. Also the so called "canyon up welling/do wnwelling" (water spills out of or sinks into the canyon) can be easily demonstrated by calculating the density fluctuations at the top of the canyon. However, since the numerical code used in this thesis was based on the layer model, only the theoretical results derived from layer models will be introduced below. The linear analytic results of Rossby adjustment of a two-layer fluid over a single-step, which is the limit of an infinitely wide canyon, will be listed without derivation. Detailed discussion of these results can be found in Allen (1988). Note that the interface in Allen (1988) is always above the step, and, consequently, some of the following theoretical results cannot be used 5.1 Introduction -195-Chapter 5 Numerical Transient Solutions in a Stratified Fluid to compare to the numerical solution for the S canyon where the interface is below the canyon edges. The analytic solution for a two-layer model in which the interface is below the canyon edges was presented by Allen (1995). (1) In the absence of topography, a two-layer flow can be separated into baroclinic and barotropic modes each of which behaves in the same manner as the one-layer flow but with different length scales. However, the presence of topography couples the modes and, thus, the stratification introduces new and more complex responses to the adjustment over to-pography. (2) There is an accumulation of lower layer fluid in the step region if the flow in the lower layer is up the step, and there is a depletion if the flow is down the step. The upper layer flow pattern depends on whether the upper layer flow is in the same or the opposite direction as the lower layer flow. If the two layers move in the same direction (strong barotropic and weak baroclinic situation, which will be referred to as weak baroclinic later), the flows in both layers bend together at the step. If, however, the layers move in opposite directions (weak barotropic and strong baroclinic situation, which will be referred to as strong baroclinic later), the flows bend in opposite directions over the step. (3) The surface elevation variation is, principally, over the barotropic Rossby radius, a. However, the interface elevation variation is, principally, over a region given by the baro-clinic Rossby radius, R, which is defined as 5.1 Introduction -196-Chapter 5 Numerical Transient Solutions in a Stratified Fluid R 1 g'hH fl(H + h). 1/2 (5.1.1) where R is the baroclinic Rossby radius, g1 is the reduced gravity, / is the Coriolis fre-quency, h is the thickness of the lower layer (measured from the ocean bottom), and H is the thickness of the upper layer (measured from the interface). (4) The information carried by the two-layer double Kelvin waves travels along the step in only one direction ~ keeping the shallow water on the right in the northern hemisphere. Since a canyon is actually two coupled steps, the analytic results listed above for a single-step can be used as guidelines for analyzing the numerical solutions over a canyon. 5.1 Introduction -197-Chapter 5 Numerical Transient Solutions in a Stratified Fluid 5.2 The Numerical Code, Parameters and Domain Used 5.2.1 The Numerical Code The two-layer numerical program is an extension of the one-layer program dis-cussed in section 4.2. The two-layer code uses the same explicit leap-frog scheme and the same grid as the one-layer code, with the upper layer variables defined at the same horizon-tal positions as the lower layer variables. The number of variables is doubled compared to the one-layer program since the upper and lower depths and the horizontal components of velocity in the two layers are coupled because the surface height variation affects both the upper and lower layers. In this chapter only the general properties of the stratified flow over a canyon are studied.'Nonlinear effects on the flow may appear in some of the numerical simulations but they will not be discussed in this chapter. In the one-layer case presented in section 4.5, it was demonstrated that, for the nonlinear case, water parcels were advected downstream across the canyon but the gross properties of the flow pattern derived from the linear me-m-ory were retained, e.g., the transport along the canyon was towards the left hand side of the shelf flow in the northern hemisphere. The code was forced by continuous forcing (type 1) in order that the numerical re-sults can be interpreted easily (see explanation given in section 4.2). In the experiment for an F canyon, in the region y < 0 mass was removed from the upper layer and mass was 5.2 The Numerical Code, Parameters and Domain Used -198-Chapter 5 Numerical Transient Solutions in a Stratified Fluid added to the lower layer and vice versa in the region y > 0. The forcing was symmetric about the line y = 0 which is the core of the shelf jet. A strong baroclinic case was simu-lated because only the interface was forced producing contrary flows in the two layers,. In the experiments for an S canyon, the center of the forcing for the two layers was separated, i.e., the division line between the region of mass addition and the region of mass removal were not aligned vertically as they were for the F canyon. Four experiments with different flow regimes were conducted to simulate both strong and weak baroclinic situations. The stability of the code is controlled by the time step. It is the stability of the sur-face gravity waves that limits the size of the time step because surface gravity waves have a much faster velocity than internal gravity waves. As the grid size used in the two-layer ex-periments is based on the baroclinic Rossby radius, R, rather than the barotropic one, a, the time step must be much smaller than the one used in the one-layer experiments in order to satisfy the CFL condition. For the experiments presented in this chapter, a = 5R and hence the typical time step At = 90s for the F canyon and At = 180s for the S canyon. The small time step means that the code runs for a long time. Changing to an implicit scheme could have reduced the running times. 5.2.2 The Parameters and Domain Used The code was run in a square domain 40R x 40R for the F canyon and SOR x SOR for the S canyon, where R is the baroclinic Rossby radius defined by (5.1.1), The number of the grid points was 161 x 161. Thus the grid size was 0.25i? for the 5.2 The Numerical Code, Parameters and Domain Used -199-Chapter 5 Numerical Transient Solutions in a Stratified Fluid F canyon and 0.52? for the S canyon. One difficulty in running the two-layer model is the choice of the domain size. As has been pointed out in subsection 5.1.2, the variation of the elevation of the surface is, principally, over the barotropic Rossby radius, a, while the variation of the elevation of the interface is over a region given by the baroclinic Rossby radius, R. In a coastal region of the real ocean (e.g., the La Perouse Bank region shown in Figure 1.3), the typical baro-clinic Rossby radius is R = 20km, however, if the total depth is chosen as H+h = 400m and the Coriolis frequency is chosen as/= 1.453703 x 10_4/s (so the inertial period is 0.5 day), the barotropic Rossby radius is a « 430km, which is 21.5 times larger than the baroclinic Rossby radius, R (note: in the La Perouse Bank region where / = 1.088346 x 10~4/s, the barotropic Rossby radius is larger: a « 575km, which is 28.5 times larger than the baro-clinic Rossby radius, 2?; if the real total depth, 1800m is used, a is much larger: 1220km, which is 612?). In order to show the surface elevation variations, the domain size must be several a wide and long. However, because the interface only varies over a small region given by R, in order to resolve the interface elevation, the number of the grid points must be large. Consequently, the running time for the code may become too long to be tolerated for a large domain with a huge number of grid points. As long as the baroclinic solution is the principal concern (as it is in this chapter), the barotropic Rossby radius can be reduced artificially while retaining an acceptable pre-cision for the baroclinic solution and simultaneously reducing the running time of the code. As given by the analytic results of Rossby adjustment over a step (item 3 in section 5.1.2), 5.2 The Numerical Code, Parameters and Domain Used -200-Chapter 5 Numerical Transient Solutions in a Stratified Fluid the barotropic Rossby radius controls the barotropic solution, which is, principally, repre-sented by the surface elevation variation. Therefore, the reduction in the size of the barotropic Rossby radius has little effect on the baroclinic solution, which is, principally, represented by the interface elevation variation. There are three ways to reduce the barotropic Rossby radius: first, increase the Coriolis frequency, second, reduce the total depth of the water column and, third, artifi-cially reduce the acceleration due to gravity. The maximum value of/has been chosen, i.e., the research location has been moved to a polar region; the optimum value of the water depth, 400m, has been assumed; only the last method, i.e., to reduce the acceleration due to gravity, is left for us to reduce the barotropic Rossby radius further to make it easy to run the code. Note that the second and the third methods are equivalent in reducing the barotropic Rossby radius. For instance, if we want to conduct a numerical simulation in which the value o f/ is that given in (5.1.1), a = 100km, R = 20km, and the grid space is 0.5/?, in order that there are 11 grid points every a, the depth of the fluid is required to be 21.6m for g = 9.8m/s2 and 400m for g = 0.528313 lm/s2. The last method is adopted in the numerical simulations in this chapter, i.e., for both the F and the S canyon the acceleration due to gravity is chosen as g = 0.528313 lm/s2. If the reduced gravity is g1 = 8.80521 x 10~2m/s2, the total depth for either the F or S canyon is H+h = 400m and the baroclinic Rossby radius is R = 20km, the depth of the upper layer, H, (or for the second layer, h,) must be either 160m br 240m. The depth over the shelf for either the F or the S canyon is 200m. Therefore, if the 5.2 The Numerical Code, Parameters and Domain Used -201-Chapter 5 Numerical Transient Solutions in a Stratified Fluid thickness of the upper layer is 160m, which is the case in the experiment for the F canyon, the interface is above the level of the shelf; if the thickness of the upper layer is 240m, which is the case in the experiments for the S canyon, the interface is below the level of the shelf. 5.2 The Numerical Code, Parameters and Domain Used -202-Chapter 5 Numerical Transient Solutions in a Stratified Fluid 5.3 Transient Flow Pattern Observed over an F Canyon Only one experiment was conducted for Rossby adjustment over an F canyon. 5.3.2 Description of the Flow The width of the F canyon in the experiment is 12? and the other parameters are as given in section 5.2.2, i.e., the depth over the canyon and the shelf are 400m and 200m, respectively; the thickness of the upper layer is 160m; the .Coriolis frequency is 1.453703xl0~4/s (and thus the inertial period is 0.5 day); the gravity and the reduced grav-ity are 0.528313 lm/s2 and 8.80521 x 10"2m/s2, respectively (and thus the barotropic and baroclinic Rossby radius are a = 100km and 2? = 20km, respectively); the size of the do-main is 402? x 402?; the number of grid points is 161 x 161 (and thus the grid space is 0.252?); the time step is 90s; only the interface is forced with "forcing type 1" (see defini-tion in section 4.2.1) along the line,y = 0. In Figure 5.1, the surface and the interface elevation are plotted at three different times during the adjustment, 0.5 day, 1 day and 1.5 day; stick diagrams of the velocities in both layers and a three dimensional view of the surface and interface are also plotted at day 1.5. The range of contours shown and the contour interval are automatically chosen by the plotting program to best fit the output. One common feature of the flow pattern in the two layers is the symmetry about the origin (0, 0). The other common feature is that the time scale for the two-layer flow is, like 5.3 Transient Flow Pattern Observed over an F Canyon -203-20 S u r f a c e E l e v a t i o n . T=" l • I • I . 7 i 7 15-1 0 Y 0 0.50 L y r = 1 C I i p = 1 I i _JL_ . L -10--15-•20 Range: -0.012m ~ 0.012m Interval: 0.001m 5 -20 15 -10 -5 (a) ATS 0 X 10 15 20 Figure 5.1 The results of Rossby adjustment for a two-layer fluid over an F canyon. The configuration of the canyon is given in text. Distance is in units of the baroclinic Rossby radius over the canyon. The contours of the surface elevation and the inter-face elevation are given after (a) and (b) 0.5 day, (c) and (d) 1 day, (e) and (f) 1.5 day. Thick lines: positions of the canyon edges; solid lines: positive values; dotted lines: negative values. The velocity fields at day 1.5 in the central region [dashed square in (a) — (f)] are shown in (g) for the upper layer and (h) for the lower layer. The three dimensional views of (i) the surface and (j) the interface at day 1.5 are also given. " -204--205--206--207-Sur face . 1 5 H Y 0 - 1 5 - 2 0 -20S- , Interface Elevation j _ -| 50 \_yr=2 C l i p = 1 2 0 - i ' — . — i 1 i — — L , i -J yjT , _ i , j , -209-V e I o c i ty F i e l d T = 1 .50 Lyr=1 CI ip=2 (a) X "JUL-VfeJ MAXIMUM B.214E-ai VECTOR -210-V e l o c i t y F Id T = V 1 . 5 0 L y r = 2 C I i p = 2 * i t < < < < < • < < < < < < < < < •C-J^'v. *» N ^ \ \ ]' j 1.1, 1—r-\ \ \ t \ \ \ \ \ \ \ \ \ \ \ \ t t V (h) X 446E-B1 MAXIHUM VECTOR -211-1.5 days -212-1.5 days -213-Chapter 5 Numerical Transient Solutions in a Stratified Fluid the one-layer case, l/f. Therefore, within a few inertial periods after the forcing stops, the flow a few Rossby radii away from the canyon has almost reached the steady state. However, the difference between the flow patterns in the two layers is striking. First, the surface and the interface adjust at different length scales. Similar to the listed analytic result for Rossby adjustment over a step (item 3 in section 5.1.2), the surface elevation variation is over a length scale given by the barotropic Rossby radius, a, whereas the interface elevation is over a length scale given by the baroclinic Rossby radius, R. Therefore, even though the barotropic and baroclinic modes cannot be separated over the canyon, the barotropic component of the solution is mainly represented by the surface ele-vation, while the baroclinic component of the solution is mainly represented by the inter-face elevation. Second, the flow in the lower layer adjusts over the canyon in the same way as in an one-layer fluid, while the flow in the upper layer adjusts in a way to accommodate the changes at the interface. Similar to the one-layer case, when the shelf current in the lower layer approaches a canyon in the northern hemisphere, most of the flow turns to the right, crosses the upstream canyon edge, travels along the canyon to the left of the shelf current and then crosses the downstream canyon edge to return to the original .y-position of the shelf current. The net flux in the along canyon direction is towards the left of the shelf cur-rent. However, when the shelf flow in the upper layer approaches the canyon in the north-ern hemisphere, most of the flow crosses the canyon directly with only a minor shift in the 5.3 Transient Flow Pattern Observed over an F Canyon -214-Chapter 5 Numerical Transient Solutions in a Stratified Fluid vicinity of the canyon; an anticyclone, centered at the upstream canyon edge at x = -0.5R, is formed to the right of the shelf flow and a cyclone, centered at the downstream canyon edge at x = 0.5R, is formed to the left of the shelf flow. There is no net flux in the along canyon direction in the upper layer. 5.3.2 Discussion The dynamics of Rossby adjustment for a two-layer fluid need to be explained. Since the forcing of the experiment is only at the interface, the flows in the two layers are in opposite directions, and, hence, a strong baroclinic situation is simulated. The behavior of the lower layer flow shown in Figure 5.1 can be easily explained. Consider an extreme case where the upper layer becomes infinitely deep, jj + n -> 0 In this case the surface changes should have no effect on the lower layer flow. For this upside down one and a half layer model, the one-layer flow pattern described in chapter 3 is re-Jgh h covered with a Rossby radius given by R = —,— (Gill et al., 1986). As increases, / ii +n the effects of the surface changes on the lower layer flow increase, but the topographic ef-fects still dominate the flow pattern in the lower layer. Since the lower flow is in the - x di-rection as shown in Figure 5.1(h), the lower layer will be stretched at the canyon edge where x = 0.5R in the upper half (+y) domain as the flow enters the canyon, but will be compressed at the canyon edge where x = -0.5R in the lower half (->>) domain as the flow exits the canyon. Therefore, as shown in Figure 5.10, the interface has a trough along the 5.3 Transient Flow Pattern Observed over an F Canyon -215-Chapter 5 Numerical Transient Solutions in a Stratified Fluid canyon edge at x = 0.5R in the upper half (+y) domain and a bump along the edge at x = -0.5R in the lower half (-y) domain. The behavior of the upper layer flow, shown in Figure 5.1, can be explained by considering conservation of potential vorticity. Since the upper layer flow is in the +x di-rection as shown in Figure 5.1(g), the upper layer will be compressed at the canyon edge where x = -0.5R in the lower half (-y) domain as the flow encounters the interface bump, but be stretched at the canyon edge where x = 0.5R in the upper half (+y) domain as the flow drops into the interface trough. Consequently, as shown in Figure 5.1(i), the surface has a pit at the canyon edge where x = 0.5R in the upper half (+y) domain and a dome at the canyon edge where x = -0.5R in the lower half (-_y) domain. From the point of view of the net effect, the upper layer flow looks like it has adjusted over an interface "topography", and, consequently, changing the lower layer flow direction will change the behavior of the upper layer flow because of the changing interface shape. Rossby adjustment of two-layer fluid over an F canyon has provided a basic picture of the circulation around an idealized canyon. To get closer to the real situation, Rossby adjustment of two-layer fluid over an S canyon will be described in the next section. 5.3 Transient Flow Pattern Observed over an F Canyon -216-Chapter 5 Numerical Transient Solutions in a Stratified Fluid 5.4 The Transient Flow Pattern Observed over an S Canyon In the simulation of Rossby adjustment over an S canyon, the width of the canyon as well as the strait is \R; the width of the shelf is 3R; the depth on the shelf, in the strait, over the middle canyon part and over the deep canyon part as well as in the deep ocean are 200m, 270m, 300m and 400m, respectively; the widths of both the mouth slope and the head slope are 0.51?; the lower bound of the mouth slope is 0.5R away from the shelf break, while the lower bound of the head slope is 17? away from the coast; the thickness of the upper layer is 240m, and, hence, the interface is 40m below the shelf but 30m above the bottom of the strait; the Coriolis frequency is 1.453703 x 10_4/s (and thus the inertial pe-riod is 0.5 day); the gravity and the reduced gravity are 0.5283131m/s2 and 8.80521 x 10" 2m/s2, respectively (and, thus, the barotropic and baroclinic Rossby radii are a = 100km and R = 20km, respectively); the size of the domain is 807? x 801?; the number of the grid points is 161 x 161 (and, thus, the grid space is 0.51? = 10km); the time step is 180s; the surface, or interface, or both are forced with "forcing type 3" in which forcing ramps up over one inertial period, is held constant over the second inertial period, and then ramps back down to zero over the third inertial period. The x-y domain for the simulation is shown in Figure 5.2. What is of greatest inter-est is the circulation in the small central region indicated by the dashed square in the figure. Therefore, only the numerical results within the small square region will be presented be-low. Note that the simulation will be terminated when the flow reflected from the top 5.4 The Transient Flow Pattern Observed over an S Canyon -217-40 J I L 3 0 -2 0 -1 0 - Land Land 0 -1 0 -t> -_ 2 0 i -33 - 3 0 - Deep Ocean •40 ~S5~ 0 X - 4 0 . - 3 0 •20 - 1 0 10 20 30 4 0 Figure 5.2 The domain for the simulations of Rossby adjustment over an S canyon. Thick lines represent the positions of the canyon edges, shelf break, coast, strait walls and boundaries of canyon bottom slopes. The length scale is the baroclinic Rossby radius over the canyon. The results of the experiments will be plotted within the region of the dashed square. -218-Chapter 5 Numerical Transient Solutions in a Stratified Fluid (wall) boundary in the strait enters the small square, or, when the flow affected by the can-yon wraps due to the side periodic boundary conditions and intrudes into the small square. In practice, the simulation was terminated after 2 days to prevent contamination of the so-lution in the region of interest by the flow reflected from the end of the strait. The main purpose of the simulation over an S canyon is to examine the flow direc-tion within the canyon under different flow conditions (i.e., different combinations of the direction of the shelf flow and the off-shelf undercurrent). There are four flow regimes nec-essary to simulate all the combinations. 5.4.1 Description of the Flow Experiment One: left-bound upper layer flow and right-bound lower layer flow The layers in this experiment are forced in such a way that the flow is left-bound (flow with the coast on the left looking downstream) in the upper layer and right-bound (flow with the coast on the right looking downstream) in the lower layer. The line that separates the regions of mass addition and removal in the upper layer is on the shelf and 11? away from the shelf break. The line that separates the regions of mass addition and removal in the lower layer is in the deep ocean and 0.51? away from the shelf break. In short, a left-bound surface shelf current and a right-bound off shelf-break undercurrent are produced. A strong baroclinic situation is simulated. In Figure 5.3, the velocity fields for this experiment are plotted at two different 5.4 The Transient Flow Pattern Observed over an S Canyon -219-Chapter 5 Numerical Transient Solutions in a Stratified Fluid times during the adjustment: after 1 day and after 2 days. In both layers, in-canyon flows are generated. Experiment Two: left-bound upper layer flow and lower layer flow The positions of the lines separating the regions of mass addition and removal are the same as those for experiment one. However, the current experiment is forced in such way that both the upper and lower flows are left-bound. A weak baroclinic situation is simulated. In Figure 5.4, the velocity fields for this experiment are plotted at two different times during the adjustment: after 1 day and after 2 days. In both layers, in-canyon flows are generated. Experiment Three: right-bound upper layer flow and lower layer flow The positions of the lines separating the regions of mass addition and removal are the same as those for experiment one. However, the current experiment is forced in such way that both the upper and lower flows are right-bound. A weak baroclinic situation is simulated. In Figure 5.5, the velocity fields for this experiment are plotted at two different times during the adjustment: after 1 day and after 2 days. In both layers, out-canyon flows are generated. 5.4 The Transient Flow Pattern Observed over an S Canyon -220-V e l o c i t y F i e l d . T= 1.00 Lyr=1 Clip=1 Ti 2" 8.495E*88 MAXIMUM VECTOR Figure 5.3 Velocity fields for Rossby adjustment over an S canyon in experiment one. (a) and (b) are for the upper and lower flows, respectively, after 1 day; (c) and (d) are for the upper and lower flows, respectively, after 2 days. -221-V e l o c i t y F i e l d . T= • 1.00 Lyr=2 Cl ip=1 ( b ) X B.646E*Bg MAXIMUM VECTOR -222-V e l o c i t y F i e l d . T : \7 2 . 0 0 L y r = 1 C I i p = 1 7 / S / A (c) X B.656E*B0 -I MUM MAX M VECTOR -223-V e l o c i t y F i e l d . T= 2 . 0 0 L y r = 2 CI i p = 1 ( d ) X MUM 8.187E+81 MAXI VECTOR -224-V e l o c i t y F i e l d . T= 7 1 .00 L y r = 1 CI i p = 1 7 Land / / / Land -=2_ . . j j ; ( a ) B.641E-B1 MAXIMUM VECTOR Figure 5.4 Velocity fields for Rossby adjustment over an S canyon in experiment two. (a) and (b) are for the upper and lower flows, respectively, after 1 day; (c) and (d) are for the upper and lower flows, respectively, after 2 days. -225-V e l o c i t y F i e l d . T= . 1.00 Lyr=2 C l i p = 1 Y •f-( b ) X 9E*i HUM B . 1 1 g MAXIMU  VECTOR -226-V e l o c i t y ' F i e l d ' . T = , 2 . 00 L y r = 1 C l i p = 1 \7 1 y • Y / / 1 7 T S s - - — ( c ) MAXIMUM VECTOR -227-V e l o c i t y F i e l d . T : 7 Y / t / / / 2.00 Lyr=2 CIfp=1 7 ( d ) X Su^ I 8.267E»BB MAXIMUM VECTOR -228-V e l o c i t y F i e l d . T= c v o 1 .00 L y r = 1 CI ip = 1 Land r / / Land 00 X 6E-B1 SDM VI B.646E. MAXIMU  VECTOR Figure 5.5 Velocity fields for Rossby adjustment over an S canyon in experiment three, (a) and (b) are for the upper and lower flows, respectively, after 1 day; (c) and (d) are for the upper and lower flows, respectively, after 2 days. -229-V e l o c i t y F i e l d . T= , 1 .00 L y r = 2 C l i p = 1 V V I i I -4-- 4 — I • I — i i v / -4-( b ) B.123E«B8 MAXIMUM VECTOR -230-V e l o c i t y F i e l d . T= , 2.00 Lyr=1 Cl ip=1 7 V / -/ / (C ) X BE. BUM MAXIM VECTOR -231-V e l o c i t y F i e l d . T= , 2 . 0 0 L y r = 2 C l i p = 1 v * v I I ' / / < t i \ \ 1 / / * * \ • \ is 1 ( d ) X B.324E+ag MAXIMUM VECTOR -232-V e l o c i t y F i e l d . T= . 1.00 Lyr=1 C l i p = 1 v v 7 Land / / \ Land (a) B.5g3E-g1 MAXIMUM VECTOR Figure 5.6 Velocity fields for Rossby adjustment over an S canyon in experiment four, (a) and (b) are for the upper and lower flows, respectively, after 1 day; (c) and (d) are for the upper and lower flows, respectively, after 2 days. -233-' e l o c i t y F i e l d . T= , 1 . 0 0 L y r = 2 C l i p = 1 V * V 1 I 0>) B.667E-B1 HAXIHUn VECTOR -234-V e l o c i t y F i e l d . T= , 2 . 0 0 L y r = 1 C l i p = 1 it if J / / V / ( c ) 0.8B0E-01 UH MAXIM M VECTOR -235-V e l o c i t y F i e l d . T= , 2 . 0 0 L'yr = 2 CI I p = 1 y * 7 " I : I J (<«) X HUH 8.139E*BB MAXI VECTOR -236-Chapter 5 Numerical Transient Solutions in a Stratified Fluid Experiment Four: right-bound upper layer flow and left-bound lower layer flow The positions of the lines separation the regions of mass accumulation and removal are the same as those for experiment one. However, the current experiment is forced in such a way that the upper layer flow is right-bound and the lower layer flow is left-bound. A strong baroclinic situation is simulated. In Figure 5.6, the velocity fields for this experiment are plotted at two different times during the adjustment: after 1 day and after 2 days. In both layers, out-canyon flows are generated. The numerical solutions shown in Figure 5.3 — 5.6 are transient. Will the flow pat-terns change dramatically (e.g., the flow direction in the canyon reverse) at subsequent times? I postulate here that such subsequent, dramatic changes will not happen for the given forcing pattern if the configuration of the model remains the same (e.g., the lower layer flow in the canyon will not reverse its direction if the interface between the two layers is above the bottom of the strait as it was in the four experiments). If things were to change, the adjustment would involve changed Kelvin waves, double Kelvin waves or can-yon waves. However, these waves can not suddenly form and propagate (Thomson, private communication). Comparison of the results among the four experiments and discussion of their common features will be presented in next section. 5.4 The Transient Flow Pattern Observed over an S Canyon -237-Chapter 5 Numerical Transient Solutions in a Stratified Fluid 5.4.2 Discussion Nonlinearitv The effects of nonlinearity should be considered when interpreting the results of the four experiments (for instance, the Rossby number for experiment one is about 0.33 which indicates that the flow in the experiment is nonlinear). However, for a homogeneous flow over an S canyon, the numerical simulations as well as the analytic calculation were only presented for linear situations. In order to compare with the results for a homogeneous flow, an experiment in which flow is linear was run for the two-layer fluid. Figure 5.7 gives the results of an experiment in which the configuration is exactly same as that for the experiment one but the forcing magnitude is only 10~2 of that for ex-periment one so that the magnitude of velocity is approximately 10~2 of that for experiment one (and hence, the Rossby number is about 0.0033 which indicates that the nonlinear ad-vection is not important). In both the upper and lower layers, the visual differences are not striking between Figure 5.3 and Figure 5.7. Flow in the upper layer Comparing the upper layer flow in the two-layer fluid in Figure 5.7(a) to the homo-geneous flow in Figure 4.5(f), similarity can be found. Because the thickness of the upper layer in the experiment for the two-layer fluid is chosen to be greater than the depth on the shelf, the upper flow on the shelf adjusts exactly like a homogeneous flow; while the upper 5.4 The Transient Flow Pattern Observed over an S Canyon -238-I ' o c i t y F i e l d . T= , 2 . 0 0 L y r = 1 C l i p = 1 1.58BE+BB Land B.BBBJ / / T / < Land IBE+BB 1.70BE-BB (a) 8.857E-B2 IMUM MAXIM VECTOR Figure 5.7 Velocity fields for Rossby adjustment over an S canyon which has the same configuration as the experiment one but the forcing magnitude is only 10'2 of that for the experiment one. (a) is for the upper flow and (b) is for the lower flow after 2 days. -239-1 .5B8E*BB 0.0BBE+B V e l o c i t y F i e l d . T= , 2 . 0 0 L y r = 2 CI•p=1 7 ' V f t / t / t I \ \ f / / * f V • ( b ) 0.151E-01 MAXIMUM VECTOR -240-Chapter 5 Numerical Transient Solutions in a Stratified Fluid flow over the canyon, in the deep ocean or in the strait, looks like a homogeneous flow adjusting over an interface "topography". In other words, the upper flow in a two-layer model is analogous to a homogeneous flow which adjusts over a canyon whose bottom is given by the shape of the interface of the two-layer model. Consequently, the upper flow retains the properties of a homogeneous flow over a canyon, e.g., a left-bound shelf flow leads to an in-canyon transport, while a right-bound shelf flow leads to an out-canyon transport (in the northern hemisphere). Note that the upper layer does not feel the bottom slopes and that it will take time to look like Figure 4.5(f). Flow in the lower layer In the lower layer, no matter what the direction of the off shelf-break undercurrent, the direction of the flow within the canyon is determined solely by the direction of the up-per layer flow for the given forcing pattern. A left-bound upper layer flow leads to an in-canyon flow in the lower layer, and a right-bound upper layer flow leads to an out-canyon flow in the lower layer. Consequently, no matter how strong the vertical shear, a left-bound shelf flow in-duces an in-canyon flow within the canyon throughout the water column, and a right-bound shelfflow induces an out-canyon flow throughout the water column. A qualitative comparison of the numerical solution with observations in a coastal region will be presented below. 5.4 The Transient Flow Pattern Observed over an S Canyon -241-Chapter 5 Numerical Transient Solutions in a Stratified Fluid 5.5 Circulation over Juan de Fuca Canyon The numerical simulation in this chapter is analogous to a laboratory experiment, in which the conditions are under the user's control, e.g., the parameters can be chosen arti-ficially within some limitations. The main purpose of an experiment is to reveal the basic properties of a natural phenomenon in an ideal environment, which must be taken into ac-count when extending the experimental results to a natural phenomenon. Because of the limitation of the numerical simulation, i.e., results with an acceptable resolution must be obtained in a finite time using the available computer resources, artificial parameters (e.g., the acceleration due to gravity, Coriolis frequency, etc.) are chosen. The numerical simula-tions reveal the basic dynamics of certain natural phenomena but may not be suitable to compare quantitatively to real observations. The parameters used in the numerical simulations were chosen (see section 5.2.2 for detailed explanation of the reasons) not to simulate the real circulation around a real canyon but to provide a general model. For example, in the region where Juan de Fuca Canyon is located, the depth of the deep ocean and the deep canyon part is about 1800m instead of the 400m used in the simulations; the Coriolis frequency is 1.08835 x lO^/s in-stead of 1.45370 x lO^/s (which indicates that the inertial period is 0.5 day, i.e., in polar regions); the acceleration due to gravity is 9.8m/s2 instead of 0.528313m/s2. Even if real data (say, of Juan de Fuca Canyon) are used, the numerical results are still only suitable to compare with observations qualitatively because of the neglect of other physical processes 5.5 Circulation over Juan de Fuca Canyon -242-Chapter 5 Numerical Transient Solutions in a Stratified Fluid in the experiments, such as, the neglect of the fresh water outflow from Juan de Fuca Strait. However, the results of some of the simulations provide a good general insight into the circulation around Juan de Fuca Canyon because of the similarity of the flow conditions (i.e., the positions and directions of the upper and lower flows) and the similarity of the geometric configuration (e.g., the width of the canyon is 20km and the width of the shelf is 60km). In the summer over Juan de Fuca Canyon, there is a southeastward (left-bound) surface shelf current with maximum speeds of 50cm/s [Thomson (1984), Thomson et al., (1989)] and a northwestward (right-bound) undercurrent off the shelf break with speeds of 5-15cm/s [Freeland et al., (1984), Simard & Mackas (1989)]. In experiment one the maxi-mum surface shelf flow is about 50cm/s and the off shelf-break undercurrent is in the range of 5-10cm/s [note that the lower layer flow off shelf-break is only about 1/10 of the maxi-mum speed which is observed in the strait in Figure 5.3 (b) or (d)], both of which are com-parable to the observed magnitudes in the vicinity of Juan de Fuca Canyon. According to the numerical results shown in Figure 5.3, there should be in-canyon flow throughout the water column, which is consistent with the general circulation pattern observed around Juan de Fuca Canyon in the summer (see Figure 5.8). Note that the numerical simulation gives very strong flow in ihe canyon and the strait. However, if other physical processes are taken into account, this flow should be 5.5 Circulation over Juan de Fuca Canyon -243-Figure 5.8 General summer circulation pattern in La Perouse Bank region. Open ar-rows show upper layer currents, and solid arrows indicate the sub-surface undercur-rent. [After Fig. 1, Mackas & Sefton (1982)] -244-Chapter 5 Numerical Transient Solutions in a Stratified Fluid weaker. For instance, if estuarine effects are considered, regardless of the existence of the shelf break current, there is flow out of the strait at the top and back in at depth. The modeled upper (in-canyon) flow may be greatly slowed, or, even reversed, whereas the modeled lower (in-canyon) flow may not be strengthened if bottom friction is also consid-ered. In fact, friction is expected to greatly reduce the lower-layer flow speed. In the winter, both the shelf current and the off shelf-break undercurrent are north-westward (right-bound) in the region where Juan de Fuca Canyon is located [Cannon (1972)]. The observed large out-canyon flows with speeds sufficient to transport sediment suggests a mechanism and a route for the transportation of the sediment found in the Nit i -nat fan which is located slightly north of the mouth of Juan de Fuca Canyon [Cannon (1972)], i.e., the source of the sediment could be inside Juan de Fuca Strait [Carson & McManus (1969)]. According to the numerical results of experiment three shown in Figure 5.5, there should be out-canyon flow throughout the water column if both the upper and the lower layer flow along the shelf break is right-bound. The numerical result is consistent with the observations reported by Cannon (1972). 5.5 Circulation over Juan de Fuca Canyon -245-Chapter 5 Numerical Transient Solutions in a Stratified Fluid 5.6 Summary of the Numerical Simulations for a Stratified Fluid In this chapter, Rossby adjustment of a two-layer fluid over an F and an S canyon was studied numerically. The numerical code is an extension of the one used for Rossby adjustment in a ho-mogeneous fluid. The time step and the domain size are all much smaller than those for the simulation in a homogeneous fluid. The two-layer flow adjusts over the same time scale, 1/f as the one-layer flow does. However, the surface and the interface adjust at different length scales: the surface elevation adjusts mainly over the barotropic Rossby radius and the interface elevation ad-justs over the baroclinic Rossby radius. One experiment, in which the interface was above the level of the shelf, was done for an F canyon. Because the flows in the two layers on the shelf are in opposite directions, a strong baroclinic solution is presented. The flow in the lower layer adjusts over the can-yon in the same way as in a one-layer fluid, whereas the flow in the upper layer adjusts in a way to accommodate the changes of the interface. Four experiments, in each of which the interface was below the level of the shelf, were done for an S canyon. Two experiments were for the strong baroclinic situation where the flows in the two layers beyond the canyon are in opposite directions; the other two are for the weak baroclinic situation where the flows in the two layers, beyond the 5.6 Summary of the Numerical Simulations for a Stratified Fluid -246-Chapter 5 Numerical Transient Solutions in a Stratified Fluid canyon, are in the same direction. In the northern hemisphere, for either the strong or the weak baroclinic case, a left-bound shelf current (flow with the coast on the left looking downstream) leads to an in-canyon flow throughout the water column, and a right-bound shelf current (flow with the coast on the right looking downstream) leads to an out-canyon flow throughout the water column. The numerical results for Rossby adjustment over the S canyon in the two-layer fluid are qualitatively consistent with the observed circulation patterns around Juan de Fuca Canyon. 5.6 Summary of the Numerical Simulations for a Stratified Fluid -247-Chapter 6 Conclusions, Discussion and Suggestions 6. Conclusions. Discussion and Suggestions 6.1 Conclusions Rossby adjustment over a canyon was studied using two geometric models. The basic geometric model ~ the F canyon model ~ includes an infinitely long, flat bottom can-yon. A more realistic model — the S canyon model ~ includes a variable depth canyon in-truding into a semi-infinitely long, flat-bottom strait. The waves which exist over an F canyon consist of superinertial waves and subin-ertial waves. The superinertial waves, Poincare waves, cannot be trapped over the canyon. The subinertial waves, canyon waves, were studied. The dispersion relation of the canyon waves was derived and revealed some important properties. First, the upper frequency limit r 2 - l for canyon waves is ~^T~^ f • Second, canyon waves are dispersive, but at the long wave limit they are approximately nondispersive. Third, canyon waves propagate along a particu-lar canyon edge in the direction keeping the other canyon edge to their left in the northern hemisphere and, thus, the canyon waves propagate in both directions along the canyon. The derivation of the dispersion relation for canyon waves is one of the important contri-butions of this thesis. The long time asymptotic solution over an F canyon in a homogeneous fluid was 6.1 Conclusions -248-Chapter 6 Conclusions, Discussion and Suggestions derived analytically. The canyon acts as a complete barrier to an approaching geostrophic flow. In the northern hemisphere, as a geostrophic flow approaches an F canyon, most of the flow is deflected to the right (looking downstream) and a net flux is generated in the along-canyon direction to the left of the shelf flow. The transient solutions over an F canyon in both a homogeneous and a stratified fluid were studied numerically. In the northern hemisphere and in a homogeneous fluid, a shelf flow adjusts around the canyon by depleting fluid from the vicinity of the upstream canyon edge to the right hand side of the shelf flow (looking downstream) and then de-positing fluid in the vicinity of the downstream canyon edge to the left hand side of the shelf flow. In a two-layer fluid, in which the interface is above the shelf, the lower flow adjusts over a canyon in the same way as in a homogeneous fluid but over a length scale given by the baroclinic Rossby radius, while the upper flow adjusts in a way to accommo-date the changes of the interface and over a length scale given, principally, by the barotropic Rossby radius. The long time asymptotic solution over an S canyon in a homogeneous fluid was found by a mixed method using both analytic and numerical analysis. In the northern hemi-sphere, depending on the direction of the shelf flow, the flow can enter or exit the canyon via the singular region at the canyon edge where the canyon bottom slope meets the left canyon edge (looking in-canyon) or the singular point where the coast joins the right can-yon edge (looking in-canyon). A left-bound shelf flow (flow with the coast on the left looking downstream) leads to an in-canyon flux, while a right-bound shelf flow (flow with 6.1 Conclusions -249-Chapter 6 Conclusions, Discussion and Suggestions the coast on the right looking downstream) leads to an out-canyon flux. The transient solutions over an S canyon in both a homogeneous and a stratified fluid were studied numerically. In the northern hemisphere and for a homogeneous fluid, a left-bound shelf flow approaches the final geostrophic state by depleting fluid in the vicinity of the upstream shelf break and then depositing some of the fluid along the canyon and the strait, while a right-bound shelf flow approaches the final geostrophic state by depleting fluid from the canyon and the strait and then depositing the fluid in the vicinity of the downstream shelf break. For the northern hemisphere, in a two-layer fluid in which the in-terface is below the shelf, a left-bound shelf current leads to an in-canyon flow throughout the water column, while a right-bound shelf current leads to an out-canyon flow through-out the water column. When a nonlinear flow adjusts over a canyon, the general features of the flow pat-tern are similar to that of the corresponding linear flow. However, because of the substan-tial advection of potential vorticity, a cyclonic circulation is formed over the canyon and an anticyclonic circulation is formed on the shelf downstream. The Canyon Number, a e (0, 1}, was found to determine the geostrophic state over an F canyon, cr, which is calculated from the geometric parameters of the shelf-canyon system, determines the interactive strength of one canyon edge on the circulation induced by the other edge. The narrower the canyon, the larger cr. The value ( l - o ) is the absolute value of the nondimensional surface elevation at the canyon edges. The parameter crcan be used to unify theories for canyons, steps, ridges and straits. Defining the Canyon Number is 6.1 Conclusions -250-Chapter 6 Conclusions, Discussion and Suggestions a significant contribution. 6.1 Conclusions -251-Chapter 6 Conclusions, Discussion and Suggestions 6.2 Discussion and Suggestions 6.2.2 Validity of the Results All studies before chapter 5 where for a homogeneous fluid, and thus the results obtained are only the barotropic solution. Although the oceans are never homogeneous, and, therefore, the flow patterns presented will be modified by the presence of stratifica-tion, the study of the barotropic case provides us with insight into real, complicated flow patterns. The barotropic case revealed the basic dynamics of the interaction between cur-rents and canyons. For example, the Canyon Number, which describes the strength of topographic effects, was derived; the properties of canyon waves were revealed. Efforts to generalize the barotropic solutions, or, to try to compare the theoretical results to observations must be done carefully, because of the simplification of the theoreti-cal models. However, qualitatively comparing the theoretical results with experiments and observations is reasonable. 6.12 Friction Bottom friction is excluded from this study, which is a reasonable approximation. First, the shallowest depth in this study is the order 100 m, whereas, the boundary layer is typically in the range of 2-10 m. Second, bottom friction is significantly smaller than the Coriolis force. Let us calculate the Ekman number and the spin-down time for a typical case from this study. By choosing a Coriolis parameter of/= 1.45 x 10"4 s'1, a depth of h ~ 6.2 Discussion and Suggestions -252-Chapter 6 Conclusions, Discussion and Suggestions 102 m, and a large vertical kinematic eddy viscosity ofAz ~ 10"1 mV, the maximum likely Ekman number is [see Pedlosky (1979), pg. 180] E = l\* 1.38x10-', (6.2.1) and the minimum likely spin-down time is [see Pedlosky (1979), pg. 180] T= • H «5.16 hours. (6.2.2) The maximum Ekman number for the study is well below the limit for flows strongly affected by bottom friction [see Pond & Pickard (1983), pg. 58]. The spin-down time also indicates that the bottom friction is not significant. Therefore, ignoring the effects of bottom friction is a reasonable approximation for this study. 6.2.3 Further Theoretical Analysis Although the theoretical study has provided significant insights into the interaction between currents and canyons, the models used are primitive. More advanced models need to be developed. For example, a three-layer model would improve the baroclinic studies. There could be two layers on the shelf, which is suitable to simulate a baroclinic shelf flow, and the third layer confined within the canyon. Of course, an advanced multi-layer fluid model (the num-ber of the layers greater than three) would provide a even more realistic solution. 6.2 Discussion and Suggestions -253-Chapter 6 Conclusions, Discussion and Suggestions The main concern here was the horizontal movement around canyons. Conse-quently, a multi-layer model is suitable for this purpose. If the upwelling and downwelling are of more interest, a multi-/eve/ model may be more appropriate. A series of laboratory experiments which demonstrate Rossby adjustment over can-yons is recommended for future research in order to further demonstrate the validity of the analytic and numerical results as they are, necessarily, based on a number of assumptions. 6.2.4 The Sources of Nutrient Rich Water in the La Perouse Bank Region La Perouse Bank region, which is one of the most productive commercial fishing fields in the northern hemisphere (see Figure 1.4), encloses Juan de Fuca Canyon (see Fig-ure 1.3). Examination of fishery haul locations and hydroacoustic hake surface densities indicates a contiguous distribution with hake concentrated along the shelf break and within the basins of the La Perouse Bank region (see Figure 6.1). The distribution of hake matches very closely with the distribution of euphausiid biomass (see Figure 6.2); Based on scaling arguments and field data, Simard & Mackas (1989) suggest that the mesoscale ag-gregations are more likely to result from the coupling of biological and physical transport processes rather than from purely behavioral or population growth mechanisms. The physical oceanographic processes in the La Perouse Bank region are compli-cated by the unique topography. The general summer circulation pattern (upper layer) in La Perouse Bank region is summarized in Figure 6.3. Sortie physical oceanographic proc-esses have been to be identified as the major reasons for the high secondary and tertiary 6.2 Discussion and Suggestions -254-Figure 6.1 Distribution of trawl hauls during the Pacific hake fishery, 1987-1989 in La Perouse Bank region, [after Fig. 15, Saunders & McFarlane (1990)] -255-g dry wt.nr* >0.25 >0.5 >1.0 >2 .0 >4.0 m m \ Figure 6.2 Contours of the acoustical estimates of euphausiid biomass during a June cruise in L a Perouse Bank region, [after Fig. 1, Simard & Mackas (1989)] -256-Chapter 6 Conclusions, Discussion and Suggestions productivity over the commercial fishing banks. As summarized in the La Perouse Project Sixth Annual Progress Report (1990), one key feature appears to be the existence of three distinct sources of nutrient supply for primary productivity in the region: (1) direct wind-induced up welling onto the outer shelf from depths of 300m to 400m with subsequent mixing into the near surface inshore waters; (2) persistent transport of nutrient-rich water seaward from Juan de Fuca Strait; (3) intermittent topographically enhanced, wind-induced upwelling off the Scott Islands and Brooks Peninsula, with subsequent along-shore trans-port within the equatorward flowing shelf break current toward the La Perouse Bank re-gion. In addition, the canyon-enhanced upwelling, which is dynamically different from the direct wind-induced upwelling, was proposed to be a contributor to the nutrient rich water in the region [Freeland & Denman (1982)]. Canyon-enhanced upwelling/downwelling has drawn a great deal of attention in recent years [Hickey et al., (1986), Klinck, (1988, 1989 & 1995), Allen (1995)]. Among the physical oceanographic processes, the first one (direct wind-induced upwelling) and the second one (outflow from Juan de Fuca Strait) are obvious and have been well studied by many researchers, whereas the third one (across-shelf transport of the water of the shelf break current) is obscure and has not been explained satisfactorily. The biggest limitation of the third process is an explanation of the on-shore diversion of the shelf break current when it confronts Juan de Fuca Canyon. This study of Rossby adjust-ment over an S canyon provides theoretical support for the third physical process for transporting nutrient rich water to the La Perouse Bank region. Furthermore, results of this 6.2 Discussion and Suggestions -257-Chapter 6 Conclusions, Discussion and Suggestions research (e.g., flow field given in Figure 5.4) support the hypothesis that a canyon can en-hance upwelling. 6.2 Discussion and Suggestions -258-125*30' 126*00' «25*30' 125*00' 126*30' 126'OC 125*30' 125-00' Figure 6.3 General summer circulation pattern in La Perouse Bank region. Unfilled arrows denote an area of confused flow, [after Fig. la, Ware & Thomson (1991)] -259-Bibliography Abramowitz, M. , and Stegun, I. A. (1968). Handbook of Mathematical Functions. Dover, New York. Allen, S. E. (1988). Rossby adjustment over a slope. Ph. D. Thesis, Univ. of Cambridge. Allen, S. E. (1995). Topographically generated, subinertial flows within a finite length can-yon. Submitted to J. Phys. Oceanogr.. Arakawa, A., and Lamb, V . 'R. (1981). A Potential Enstrophy and Energy Conserving Scheme for the Shallow Water Equations. Monthly Weather Rev., 109, 18-36. Cannon, G. A. (1972). Wind effects on currents observed in Juan de Fuca submarine can-yon. J. Phys. Oceanogr., 2, 281-285. Carson, B., and McManus, D. A. (1969). Seismic reflection profiles across Juan de Fuca canyon. J. Geophys. Res., 74, 1052-1060. Erdelyi, A., Magnus, W., Oberhettinger, R., and Tricomi, F. G. (1954). Tables of Integral Transforms. Vol. I. McGraw-Hill, New York. Freeland, H. J., and Denman, K. L. (1982). A topographically controlled upwelling center off southern Vancouver Island. J. Mar. Res., 40, 1069-1093. Freeland, H. J., Crawford, W. R., and Thomson, R. E. (1984). Currents along the Pacific coast of Canada. Atmos. Oceans, 22, 151-172. Gill, A. E. (1976). Adjustment under gravity in a rotating channel. J. FluidMech. 77, 603-621. Gill, A. E. (1982). Atmosphere-Ocean Dynamics. Academic Press, Orlando, Fla... Gill, A. E., Davey, M : K., Johnson, E. R , and Linden, P. F. (1986). Rossby adjustment over a step. J. Mar. Res., 44, 713-738. Han, G. C , Hansen, D. V., and Gait, J. A. (1980). Steady-state diagnostic model of the New York Bight. J. Phys. Oceanogr., 10, 1998-2020. Hickey, B,, Baker, E., and Kachel, N. (1986). Suspended particle movement in and around Quinault Submarine Canyon. Mar. Geol., 71, 35-83. Hsieh, W. W., and Gill, A. E. (1984). The Rossby adjustment problem in a rotating, strati-fied channel, with and without topography. J. Phys. Oceanogr., 14, 424-437. -260-Hunkins, K. (1988). Mean and tidal currents in Baltimore Canyon. J. Geophys. Res., 93, 6917-6929. Killworth, P. D. (1989). How much of a baroclinic coastal Kelvin wave gets over a ridge? J. Phys. Oceanogr., 19, 321-341. Klinck, J. M . (1988). The influence of a narrow transverse canyon on initially geostrophic flow. /. Geophys. Res., 93, 509-515. Klinck, J. M. (1989). Geostrophic adjustment over submarine canyons. J. Geophys. Res., 94, 6133-6144. Klinck, J. M . (1995). Circulation near submarine canyons: a modeling study. Submitted to J. Geophys. Res.. LeBlond, P. H , and Mysak, L. A. (1977). Trapped coastal waves and their role in shelf dynamics. In: The sea. V o l . 6: Marine modeling. Ed. Goldberg, E. D., McCave, I. N , O'Brien, J. J., and Steele, J. M. John Willey & Sons. LeBlond, P. H , and Mysak, L. A. (1978). Waves in the ocean. Elsevier. . Longuet-Higgins, M . S. (1968a). On the trapping of waves along a discontinuity of depth in a rotating ocean. J. FluidMech., 31, 417-434. Longuet-Higgins, M . S. (1968b). Double Kelvin waves with continuous depth profiles. J. FluidMech., 34, 49-80. Mackas, D. L., and Sefton, H. A. (1982). Plankton species assemblages of southern Van-couver Island: Geographic pattern and temporal variability. J. Mar. Res., 40, 1172-1120. Mayer, D. A., Han, G. C , and Hanson, D. V. (1982). Circulation in the Hudson Shelf Valley: M E S A physical oceanographic studies in New York Bight. J. Geophys. Res., 87,95630-95678. Mysak, L. A. (1969). On the generation of double Kelvin waves. J. Fluid Mech., 37, 417-434. Pedlosky, J. (1979). Geophysical Fluid Dynamics. Springer-Verlag. Philips, N. A. (1963). Geostrophic motion. Rev. Geophys. 1, 123-126. Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterbing, W. T. (1986) Numerical Recipes: The Art of Scientific Computing. Cambridge Uni. Press. Rhines, P. B. (1967). The influence of bottom topography on long-period waves in the ocean. Ph.D. thesis, Univ. of Cambridge. -261-Rhines, P. B. (1969a). Slow oscillations in an ocean of varying depth. Part I. Abrupt to-pography. J. FluidMech., 37, 161-189. Rhines, P. B. (1969b). Slow oscillations in an ocean of varying depth. Part II. Islands and seamounts. J. FluidMech., 37, 191 -205. Rossby, C. G. (1938). On the mutual adjustment of pressure and velocity distributions in certain simple current systems. II. J. Mar. Res. 2, 239-263. Saunders, M . W., and McFarlane, G. A. (1991). Pacific hake Distribution and Recruitment. La Perouse Project Sixth Annual Progress Report 1990. 42-44. Simard, Y. , and Mackas, D. L. (1989). Mesoscale Aggregations of Euphausiid Sound Scattering Layers on the continental Shelf of Vancouver Island. Can. J. Fish. Aquat. Sci., 46, 1238-1249. Shepard, F. P. (1931). Glacial troughs of the continental shelves. J. Geol., 39, 345-360. Shepard, F. P., Marshall, N . F., Mcloughlin, P. A., and Sullivin, G. G. (1979). Currents in submarine canyons and valleys. Tulsa, Okla.. Thomson, R. E. (1984). A cyclonic eddy over the continental margin of Vancouver Island: evidence for baroclinic instability. J. Phy. Oceanogr., 14, 1326-1348. Thomson, R. E., Hickey, B. M. , and LeBlond, P. H. (1989). The Vancouver Island Coastal Current: fisheries barrier and conduit. Can. Spec. Publ. Fish. Aquat. Sci., 108, 265-296. Ware, D. M. , and Thomson, R. E. (1991). La Perouse Project Sixth Annual Progress Re-port 1990. Weaver, A. J., and Hsieh, W. W. (1987). The influence of buoyancy flux from estuaries on continental shelf circulation, J. Phy. Oceanogr., 17, 2127-2140. Willmott, A. J. (1984). Forced double Kelvin waves in a stratified ocean. J. Marine Res., 42,319-358. Willmott, A. J., and Grimshaw, R. H. J. (1991). The Evolution of Coastal Currents over a Wedge-Shaped Escarpment. Geophys. Astrophys. Fluid Dynamics, 57, 19-48. -262-Appendix A Appendix A; Solving the System of Linear Partial Differential Equations For convenience, rewrite the system of linear partial differential equations, (2.4.11), as anAiy,t) + ^ 2Dt(y,t) + bnAyiy,t) = ei, (A.la) ^ ^ ( ^ O + ^ A C v . O + ^ ^vCv.O = *2> ( A 1 b) where a„ =a 2 2 = J%cth/5 + J ^ , (A.2) g sinh/? ' a\2 = a2\ = -J— o » ( A 3 ) * i i = - ^ =:-(*.-A). (A-4) ^^-e2=-(j\-K)liy (A-5) In the system of linear partial differential equations, (A.1), y and f are independent variables except at the wave front where y and t are correlated. If cp is assumed a reference variable, we have t = 3(cp) and y = y{cp) at the wave front, and hence in (A. 1) A(y, t) = A(cp) and D(y, t) = £(<p), where J, y, A and J are functions of one variable. Therefore, at the wave front, the system of equations, (A.l), can be written as Solving the System of Linear Partial Differential Equations -263-Appendix A (Px >>'" «i , / H (y, t) - aX2y'Dy(y, t) = e,t'- auA'(y, t) - al2D'(y, t), (A.6a) W'Ay{y,t) + (but' + auy')Dy(y,t) = ext' + al2A'(y,t) + auD'(y,t), (A.6b) where a quantity with a superscript""' over head denotes the first derivative of the quantity with respect to (p. The characteristic directions of (A. 6) are given by V - t f n / -any' a 1 2 / but' + auy' = 0 (A.7) which can be written in the form y' = ±c0t' (A.8) where c n = -Vilv-Al Vau2- ai22 (h2+hl+2^h2'cthB) 1/2 (A.9) Substituting (2.3.13) into (A.9) gives the phase/group speed of the long canyon wave ex-pressed by (2.3.15). Equation (A.8) indicates that the system (A.6) is of hyperbolic type, and the two sets of characteristics are dy = -c0dt, i.e., y + c0t = Cl, (A. 10a) Solving the System of Linear Partial Differential Equations -264-Appendix A and dy = c0dt, i.e., y-c0t = C2, (A. 10b) where C1 and C2 are constants. Equation (A. 7) implies that the ratio of the coefficients of (A. 6) are constant if a solution of (A. 6) is to exist. If this constant is defined as %, then V - « u / = -<*ny' =e1t'-auA'(y,t)-ai2D'(y,t) ( A U ) aX2y' but'+auy' e,t'+anA'(y,t) + auD'(y,t) Substituting (A. 5) into the third expression in (A. 11) and rearranging gives (%an+au)dA(y, t) + (%au + an)dD{y, t) + (1 - X - h,) • ^  • dVl (y) = 0. (A. 12) There are two situations, (i) On the characteristic (A. 10a), namely, y + c0t = C l (i.e., c/y = -c0dt): From (A. 11) the constant ratio is , , = - ^ . (A. , 3 , "12C0 Thus (A. 12) becomes ( * 1 a 1 2 + a „ ) « H ( y , 0 + ( ^ (A. 14) c0 Solving the System of Linear Partial Differential Equations -265-Appendix A Integrating (A. 14) gives (*A2+auMO>0 + ( ^ (A.15) co which is valid on the characteristic y + c0t = CI . C3 is a constant in (A. 15). (ii) On the characteristic (A. 10b), namely, y - c0t = C2 (i.e., dy - c0dt): From (A. 11) the constant ratio is • * = * f f * . (A.16) MI2*'0 The equation that corresponds to (A. 15) is a-2aI2+anMtoO + ^  ( A 1 7 ) co which is valid on the characteristic y - c0t = C2. C4 is a constant in (A. 17). In summary, on each characteristic y + c0t = C1, the right hand side of (A.15) is a constant C3; on each characteristic y - c0t = C2, the right hand side of (A. 17) is a constant C4. These two constants are assumed to be C3 = - 4-^- (l-ZiW.(y + cj)- (A. 18) Co and Solving the System of Linear Partial Differential Equations -266-Appendix A C 4 = >L_A (1 _ %lWp(y- c0t), (A. 19) where W„ and Wp can be any single variable functions and are constants on the characteris-tics y + c0t = C\ and y-c0t = C2, respectively. W„ represents the information carried by the long canyon wave front y + c0t = C l propagating towards the -y direction, and Wp rep-resents the information carried by the long canyon wave front y-c0t = C2 propagating towards the +y direction. Substituting (A. 18) and (A. 19) into (A.15) and (A. 17), respectively, and rearrang-ing gives U^2+au)A(y,t) + (Z^n+cm)D(y,t) = - ( f h ^ %x)*[Wn(y + c0t)-Vl(y) ,(A.20a) U 2 a , 2 +au)A(y,t) + (Z2au + an)D(y,t) = (^ \ ) 0 Z})' [Wp(y - c0t)-7ft ly)]. (A. 20b) Now, it is straightforward to solve the system of linear equations. In the process of solving (A. 20), a parameter appears a = l _ f u l f i l « ! . \an-an -|l/2 y (cosh/? - 1 ) + sinh/5 X(cosh/3 + l) + sinh/5 The solution of (A.20) is given by (2.4.14). (A.21) Solving the System of Linear Partial Differential Equations -267-Appendix B Appendix B: The Full Geostrophic Solution over an F Canyon Part 1 Solution of (2.4.25) The Fourier transform of ^ (x,y) with respect to y is ^(x,p) = ?[<S>,(x,5>)] = ^ \^(x,y)exip(-jpy)ay (B.la) where p is real and j = V-T. The while its reverse transform is 1 f«> — *, (*. y) = -L ^ (x,p) Qxp(Jpy)dp. ( B . lb) Taking the Fourier transform of (2.4.25a), (2.4.25b) and (2.4.25c) with respect to y gives ^P)-(p2+l)Ol(x,p) = 0, (x<0), (B.2a) 'Ol(x^0,p) = (T-l + E(p), (B.2b) q"(x ->-oo,/?) = 0, (B.2c) where E(p) is the Fourier transform of E(y), i.e., £(/?) = 3[E(y)]. The solution of (B.2) is O, (x ,/>) = [o- - 1 + £ (p)] exp(xVp2+l). ( B . 3 ) F M / / Geostrophic Solution over an F Canyon -268-Appendix B An important property of the Fourier transform is that if F(p) is the Fourier transform of F(y), and G(x,p) is the Fourier transform of G(x,y), F(p) • G(x,p) will be the convolution F(y)*G(x,y), namely, the Fourier transform of 1 rfF(y-4)G(x,Z)dt. fin-' Therefore, the (reverse) Fourier transform of (B.3) is . c r - 1 r* „ J e 1 r (x,y) = ^ j=i\ G,(x,Z)d£+-7L=j E(y-Z)Gx(x,Z)dt, (x<0, -oo<J;<oo) (B.4) where E(y - Bj) and G , (x ,£) are given by (2.4.27) and (2.4.28), respectively. One must demonstrate that (B.4) satisfies the boundary condition (2.4.25d) before claiming that it is the solution of (2.4.25). Since exp(x-y/p2 + l j is the Fourier transform of Gx(x,^) from the definition (2.4.28), i.e., exp(xV7^) = - 7 = t G 1 ( x , ^ ) e x p ( / ^ ) ^ (B.5) wherep e {-oo, co} can be regarded as a referential variable. Whenp = 0, this implies zrtx) = -jyiy](x,t)d{. (B.6) Full Geostrophic Solution over an F Canyon -269-Appendix B and from the definition (2.4.27) lim£Cv-£) = ±l. (B.7) By using (B.6) and (B.7), (B.4) gives (2.4.25d) as y - > ± 0 0 . Therefore, (B.4) is the solution of (2.4.25) which can be written in the form (2.4.26) by using (B.6). Part 2 Solution of (2.4.32) Similar to part 1, if Q>2(x,p) where p is real is the Fourier transform of Q>2{x,y) with respect to y, taking the Fourier transform of (2.4.32a) and (2.4.32b) with respect to y gives ^ | ^ - ^ + l ) O 2 ^ ) = 0, ( - f <*~<f), (B.8a) <X>2(x -> ±/7/ 2,p) = .±(1 - o-) + £(/>), (B.8b) where E(p) is the Fourier transform of E(y) as in part 1. The solution of the problem (B. 8) is _ sinh(x -yj p2 +1 _ coshfx-y/p2 +1 02(x5Jp) = (1 - cr) ; ( + ,V ( • (B.9) s inh^VP 2 + lj cosh^V^ 2 + 1 Because of the convolution property introduced in part 1, the (reverse) Fourier transform of (B.9) gives Full Geostrophic Solution over an F Canyon -270-Appendix B (D2(?j1 = ^ f G 3 ( x ~ ^ + 4=r^-^(?>^ (-v<*<T' - ° ° < ^ < ° o ) (B.10) where E{y-%) and G 2 ( x ,£ ) are given by (2.4.27) and (2.4.34), respectively; 1 f°° sinhfxvV + l) /Yfo, sinhlxyjp2 +l) sinh y^ V P +1J s i n l \ Y V P + 1 J ( B . l l ) One must demonstrate that (B.10) satisfies the boundary conditions (2.4.32c) be-fore claiming that it is the solution of (2.4.32). From the definition G 2 (x ,£) and G3(x,<ff), i-e., (2.4.34) and (B . l l ) , the following Fourier transforms can be derived s i n h x J » 2 + l i f o o sinh(^yVp2 + lJ coshl xJp2 + 1) 1 ^ (0 , -\ = J^LG^ <*, 4?) « P 0 S W , (B. 13) C O S l | y V p 2 + l J wherep e {-oo, oo} can be regarded as a reference variable. Whenp = Q, these become Full Geostrophic Solution over an F Canyon -271-Appendix B sinhx s i n , ii.fi coshx = -4 -J" .G a (x ,^. (B.15) cosh(/5/2) V 2 ^ By using (B.14), (B.15) and (B.7), (B.10) gives (2.4.32c) as y - > ± 0 0 . Therefore, (BIO) is the solution of (2,4.32) which can be written in the form (2.4.33) by using (B.14). Full Geostrophic Solution over an F Canyon -272-Appendix C Appendix C: Rossby Adjustment over a Single Step Parallel to a Coast For the geometry of a single-step parallel to a vertical coast (see Figure 3.2), define the following nondimensional parameters: DSB:=^ ( C D A:=f% " (C3) where dSB and dc are the y values at the shelf break and at the coast, respectively; R\ and R3 are the Rossby radii over the shelf and over the deep ocean, respectively. Now the steady state governing equation, (2.2.8), is to be solved for the initial condition, (1.3.1), in the deep ocean and on the shelf separately. Because the initial condi-tion and the topography are all independent of x, the solution at any time should also be independent of x. Form of the Solution (i). In the Deep Ocean ( - 0 0 <y< afcs): The steady state equation, (2.2.8), and the boundary condition are Rossby Adjustment over a Single-Step Parallel to a Coast -273-Appendix C R* d2n(y) dy2 (C.4a) 77(-oo) = 7 o , (C.4b) where 770 is half the quantity of the initial surface discontinuity. Equation (C.4) is easy to solve. (ii). On the Shelf (dSB<y<dc): The steady state equation, (2.2.8), is dy2 v(y) = \ [ 770, 0<y<dc. (C.5) Equation (C.5) is also easy to solve. Consolidating the solutions of (C.4) and (C.5) gives the steady state solution V(y) = Vo [AQxp(y/R2) + \], [B exp(y IRX) + Cexp( -v I Rx) +1], [E exp(y IR,) + F exp(-j> //?,)-1], -*><y<dSB, dSB<y<o, 0<y<dc, (C.6) where coefficients A, B, C, E and F are constants to be determined. The corresponding geostrophic velocity is Rossby Adjustment over a Single-Step Parallel to a Coast -274-Appendix C — expCy//^), -co<y<dSB, u(y) = ^\-l-[Bcxp(y/R])-Cexp(-y/Rl)}, dSB <y<0, (C.7) ME expty 7Rt)-F exp(-y /*,)], 0<y<dc Determination of Coefficients By using (C.6), (C.7), the boundary conditions and conservation of mass, the coef-ficients of (C.6) can be determined. (i). At>- = ^sB (shelf break): Only 7j is continuous (u can be discontinuous because the depth is discontinuous on both sides of the line>- = dsB). Therefore, (C.6) gives where DSBI and DSB3 were defined by (C. 1) and (C.2), respectively. (ii). At y = 0 (position of initial surface discontinuity): Because the line y = 0 is over a flat bottom, either in the deep ocean or over the shelf if dSB * 0, there is no reason for either n or u to be discontinuous. Therefore, the con-tinuity requirement for n and u at y = 0 gives ^exp(D 5 B 3 ) = Bexp(Dsm) + Cexp(-Dsin) (C.8) B + C + \ = E + F-\, (C.9) Rossby Adjustment over a Single-Step Parallel to a Coast -275-Appendix C B-C = E-F. (CIO) (iii) Aty = dc (coast): du Because 77 is always independent of x, and v is always zero at the coast, ~r at the ot coast should be always zero from the momentum equation (2.2.1a), i.e., u at the coast should always keep its initial value — zero [for the initial condition (1.3.1)]. Thus, J Eexp(D c . ) - J Fexp(-Z) c ) = 0 (C . l l ) where Dc was defined by (C.3). (iv) For the whole domain: Conservation of mass and incompressibility of the fluid gives tniyt^dy=fr/OOL,*. (c-12) Substituting (C.6), (CIO) and the initial condition (1.3.1a) into (C.12) gives Ay, exp(D O T 3) -Bexp(D sm) + Cexp(-Z) 5 f l l) + £exp(Z) c ) - F e x p ( - Z ) c ) = 0 (C. 13) where y-i was defined in (3.2.1). Solving the system of linear equations (C.8), (C.9), (CIO), (C . l l ) and (C.13) with the following definitions, Rossby Adjustment over a Single-Step Parallel to a Coast -276-Appendix C A = expCD^ - Dmi(y3 - l)exp(2DJSBI) +1> 3 + l)exp(2Dc)], (C. 14) A ,=2 [ l - exp (2D c ) ] , (C.15) AE = 2 exp(Z)SB3 )[y 3 cosh(Z)SB1) - sinh(£>5 B 1)], (C. 16) allows the coefficients A, E, B, F and C to be obtained: A = ^ f, (C.17) £ = (C18) B = E-\, (C.19) F = Eexp(2Dc), (C.20) C = £ e x p ( 2 Z ) c ) - l . (C21) Now the surface elevation at the shelf break and the coast can be obtained from (C.6), namely, nLC = n0[Aexv(DSBi) + l], (C.22) • 7 j c =J7 0 [2£exp(Z ) c ) - l ] , (C23) where A and E were given by (C. 17) and (C. 18), respectively. Rossby Adjustment over a Single-Step Parallel to a Coast -277-Appendix D Appendix D: Rossby Adjustment over a Single-Step Parallel to a Coast with Two Constraints For Rossby adjustment over the topography shown in Figure 3.2, the problem is almost identical to that studied in Appendix C. However, there are two limitations imposed here, i.e., the surface elevation at the coast and the at shelf break are equal to 77^ ,2 and 77.43, respectively. Similar to Appendix C, the geostrophic solution can still be expressed by (C.6). The continuity requirements (i) and (ii) in Appendix C are effective here, i.e., 77 is continuous at dn the shelf break, and 77 as well as — are continuous at y = 0, however, (iii) and (iv) are dy substituted by n(dc) = n_L2 and n(dsm) = n_L3 here. Therefore, the system of equations that will be used to determine the coefficients of the solution is A exp(D,B 3) = Bexp(Dsm) + Cexp( -D S B 1 ) , (D. 1) B + C+\ = E + F-\, (D.2) B-C = E-F, (D.3) Aexp(DSB,) + l = Tj_L3/rjo, (D.4) £ e x p ( D c ) + F e x p ( - J D c ) - l = r7_L2 / 770, (D.5) where the parameters DSBI, D S B 3 and D C were defined by (C.l), (C.2) and (C.3), respec-Rossby Adjustment over a Single-Step Parallel to a Coast with Two Limitations -278-Appendix D tively; r/o is half the value of the initial surface discontinuity. Solving the system of linear equations (D. 1) ~ (D.5) with the following definitions: T = -2smh(Dc-DSB]), (D.6) T £ = [7-L3 / Vo ~ 1 + 2cosh(Z)SB1)] exp( -D c ) - (r/_t2 / n0 + l )exp( -D 5 S 1 ) , (D.7) r F = - [ 7 7 - L 3 / ; 7 o - l + 2cosh( JD f f i l)]exp(Z)c) + (r/_L2 / 770 + l)exp(Z)^1), (D.8) allows the coefficients ^ , E, F, B and C to be obtained: >t = ( ^ L 3 ^ o - l ) e x p ( - A « 3 ) , ( ° - 9 ) E = ^jL, (D.10) *" = ^ v (D.l l) 5 = £ - l , (D.12) C = F - 1 . (D.13) Now the geostrophic solution has been found for the topography shown in Figure 3.2 and for specified values of the surface elevation at the coast and at the shelf break. Rossby Adjustment over a Single-Step Parallel to a Coast with Two Limitations -279-

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