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On the inertial stability of coastal flows 1978
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Title | On the inertial stability of coastal flows |
Creator |
Helbig, James Alfred |
Date Created | 2010-03-19 |
Date Issued | 2010-03-19 |
Date | 1978 |
Description | This thesis investigates two separate but related problems. In Part I a study is made of the propagation of continental shelf waves and barotropic Rossby waves in a steady, laterally sheared current of the form V + Є W, where W is a centred random function and Є << 1. If the correlation length of W is small compared with the characteristic horizontal length scale of the system; for example, the shelf width or a channel width, the waves are unstable. Their growth rate is largely determined by the magnitude of the correlation length, while the phase speed is given by the sum of weighted averages of the mean current V and the lateral gradient of potential vorticity. Application of the theory to the Brooks and Mooers (1977a) model of the Florida Straits yields wave parameters that are in accord with those measured by Duing (1975). In Part II, an attempt is made to understand the dynamics governing observed low-frequency currents in the Strait of Georgia (GS). A simple two-layer model indicates that the mean currents in GS are probably baroclinically stable. A barotropic stability model implies that a shear instability might be of some importance. However, the analysis of current meter data shows that the velocity components of the fluctuations are either nearly in phase or close to 180° out of phase; this means that the motions are not due to the type of waves considered here. Analysis of the relationship between the winds and currents in both the frequency and time domains implies that the wind may play an indirect role in forcing GS motions. It is conjectured that the wind and tide interact with the Fraser River outflow to modulate the estuarine circulation in the system and force low-frequency currents. Direct nonlinear interaction between tidal constituents produces a coherent fortnightly variation in the currents, but cannot account for the observations. |
Subject |
Waves -- Mathematical Models Ocean Currents -- Mathematical Models |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | Eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project |
Date Available | 2010-03-19 |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0053252 |
Degree |
Doctor of Philosophy - PhD |
Program |
Oceanography |
Affiliation |
Science, Faculty of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/22172 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/831/items/1.0053252/source |
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ON THE INERTIAL STABILITY OF COASTAL FLOWS by JAMES ALFRED HELBIG M. Sc., University of British Columbia, 1977 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Physics and Institute of Oceanography) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November, 1978 (c) James Alfred Helbig, 1978 In presenting t h i s thesis i n p a r t i a l f u l f i l l m e n t of the requirements f o r an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by h i s representatives. I t i s understood that copying or p u b l i c a t i o n of t h i s t hesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. I n s t i t u t e of Oceanography and Department of Physics The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, B r i t i s h Columbia V6T 1W5 Canada 27 December 1978 Abstract This thesis investigates two separate but r e l a t e d problems. In Part I a study i s made of the propagation of continental s h e l f waves and barotropic Rossby waves i n a steady, l a t e r a l l y sheared current of the form V + £W, where W i s a centred random function and £ << 1. I f the c o r r e l a t i o n length of W i s small compared with the c h a r a c t e r i s t i c h o r i z o n t a l length scale of the system; for example, the s h e l f width or a channel width, the waves are unstable. Their growth rate i s l a r g e l y determined by the magnitude of the c o r r e l a t i o n length, while the phase speed i s given by the sum of weighted averages of the mean current V and the l a t e r a l gradient of p o t e n t i a l v o r t i c i t y . A p p l i c a t i o n of the theory to the Brooks and Mooers (1977a) model of the F l o r i d a S t r a i t s y i e l d s wave parameters that are i n accord with those measured by Duing (1975). In Part I I , an attempt i s made to understand the dynamics governing observed low-frequency currents i n the S t r a i t of Georgia (GS). A simple two-layer model indicates that the mean currents i n GS are probably b a r o c l i n i c a l l y stable. A barotropic s t a b i l i t y model implies that a shear i n s t a b i l i t y might be of some importance. However, the analysis of current meter data shows that the v e l o c i t y components of the f l u c t u a t i o n s are e i t h e r nearly i n phase or close to 180° out of phase; t h i s means that the motions are not due to the type of waves considered here. Analysis of the r e l a t i o n s h i p between the winds and currents i n both the frequency and time domains implies that the wind may play an i n d i r e c t r o l e i n f o r c i n g GS motions. It is conjectured that the wind and tide interact with the Fraser River outflow to modulate the estuarine circulation in the system and force low-frequency currents. Direct nonlinear interaction between t i d a l constituents produces a coherent fortnightly variation in the currents, but cannot account for the observations. TABLE OF CONTENTS ABSTRACT i i LIST OF TABLES v i LIST OF FIGURES v i i ACKNOWLEDGEMENT x i Section 1 Introduction to Thesis . . . 1 PART I 2 Introduction to Part I 2 3 Formal Theory for Shelf Waves i n a Channel 7 4 The V o r t i c i t y and Energy Balances 17 5 The Channel Mode 25 6 The Continental Shelf Model 61 7 Rossby Waves i n a Random Zonal Flow 67 8 Summary and Concluding Remarks to Part I 71 PART II 9 Introduction to Part II 72 10 Physical Oceanography of the S t r a i t of Georgia 76 11 I n e r t i a l I n s t a b i l i t y Models 93 12 Analysis of Data 114 13 Nonlinear T i d a l Interactions 145 14 Summary of Part II 158 BIBLIOGRAPHY 160 APPENDIX A: Order of Magnitude Estimates of the Integral Terms i n (3.25) 166 i v APPENDIX :B: The First-Order Solutions 170 APPENDIX C: Evaluation of the Integral Terms for a Simple Flow Model 173 APPENDIX D: B a r o c l i n i c I n s t a b i l i t y i n a 2-Layer System 178 v LIST OF TABLES Table I Order of magnitude estimates of terms in the vorticity balance equations .19 II The characteristic growth times T = 1/0, for a 200-km wave (k/27T = .15) 59 III Calculated coherence squared and phase between velocity components for the 136-day period of analysis 133 IV Results of the harmonic analysis of tida l elevations at Point Atkinson for the 38-day period beginning 6 April 1976 146 V Relative magnitudes of the terms in (C.9) and (C.10) 176 vi LIST OF FIGURES Figure 5.1 The Brooks and Mooers model of bottom topography and mean current 26 5.2 Graphical solution of (5.14) for b = 3.0 and I = 2.5 32 5.3 Behaviour of the f i r s t mode nondimensional growth rate 0.^ as a function of O and k 37 5.4 Behaviour of the nondimensional growth rate fi^ as a function of k for the f i r s t three modes for 0=5 and e = 0.5, (A) channel model 38 (B) shelf model 39 5.5 Dispersion curves for the f i r s t three modes (A) channel model 40 (B) shelf model 41 5.6 Behaviour of the nondimensional phase speed as a function of k for the f i r s t three modes (A) channel model 42 (B) shelf model 43 5.7 The mass transport stream function for (A) channel mode 1 45 (B) channel mode 2 46 (C) channel mode 3 47 (D) shelf mode 1 48 (E) shelf mode 2 49 (F) shelf mode 3 50 5.8 Profiles of u and v for: (1) channel mode 1 taken along the line 6/2TT = 0.8 in Fig. 7A, (2) channel mode 2 taken along 6/2TT = 0.9, and (3) channel mode 3 taken along 6/2TT = 0.37 51 5.9 Plan view of the Florida Straits showing lines I and II along which the sections in Fig. 5.10 are taken 53 v i i 5.10 Sections along lines I and II of (A) a t 54 (B) alongshore velocity 55 9.1 Plan view of the west coast of British Columbia and adjoining waters 73 10.1 Plan view of the Strait of Georgia showing lines of topographic cross sections (1-10) presented in Fig. 10.2 . . 77 10.2 Topographic cross sections: (A) Upper panels: 1-9; (B) Lower panel:~10 78 10.3 Longitudinal section of a for (A) December 1968 . 79 (B) July 1969 80 10.4 Rotary spectrum of the winds at Sand Heads for the 600-day period beginning 3 January 1969 83 10.5 Cross section H showing placement of current meters 84 10.6 Mean currents along line H for the 533-day period beginning 16 April 1969 85 10.7 Current spectra for line H for the 533-day period beginning 16 Apri l 1969 87 10.8 Rotary coherence and phase between currents from (A) vertically seperated locations 88 (B) horizontally seperated locations . . . . . . 89 11.1 The baroclinic ins t a b i l i t y model 95 11.2 Mode 1 st a b i l i t y boundaries for the baroclinic model as a function of the topographic parameter T 101 11.3 Mode 1 st a b i l i t y boundaries for the baroclinic model as a function of the internal Froude number V = + F^ 102 11.4 Baroclinic model mode 1 dispersion curves for S = 0.5 and S = 1.5 104 11.5 The baroclinic model mode 1 phase speed as a function of topographic parameter for k/2iT = 0.1, 0.5, and 1.0 . . . . 105 11.6 The barotropic inst a b i l i t y model 107 11.7 The region in (k,S) space in which continental shelf waves exist I l l 11.8 Computed barotropic mean currents along line H for the 18-month period beginning April 1969 112 v i i i 11.9 Barotropic model dispersion curves for S=0.5 113 12.1 Plan view of the Strait of Georgia showing current meter locations 116 12.2 Periods of existent current meter records 117 12.3 Current spectra for the 26-day period beginning (A) 2 May 1969 119 (B) 29 August 1969 120 12.4 Mean currents and the 6-32-day band current ellipses for the 26-day period beginning (A) 2 May 1969 122 (B) 29 August 1969 123 12.5 Spectrum of the wind stress at Sand Heads for the 500-day period beginning 4 April 1969 126 12.6 Line H current spectra for the 136-day analysis period. . . . 128 12.7 Coherence and phase between the wind stress and currents at H26, 50m and H16, 50m 129 12.8 Coherence and phase between line H currents and the wind stress for (A) the 13-day band 130 (B) the 34-day band 131 12.9 Mean currents along line H for the 136-day analysis period. . 135 12.10 Line H current ellipses for: (A) the 13-day band, upper layer 136 (B) the 13-day band, lower layer 137 (C) the 34-day band, upper layer 138 (D) the 34-day band, lower layer 139 12.11 Computed barotropic and upper layer baroclinic mean currents for the 136-day analysis period 141 12.12 Low-pass filt e r e d time series of wind stress at Sand Heads and currents along line H 143 13.1 Daily barotropic residual t i d a l flow along line H 148 13.2 The residual barotropic t i d a l flow averaged over 10 days. . . 151 13.3 Time series of (A) predicted tidal height and t i d a l range at Pt. Atkinson 152 (B) calculated residual current magnitude and direction along line H 153 i x The Fraser River discharge approximately 60 miles upstream at Agassiz, British Columbia 1 5 5 Low-pass filtered time series of river speed at the Fraser River mouth x ACKNOWLEDGEMENT Many people contribute to a t h e s i s . My foremost gratitude goes to Professor L. A. Mysak f o r his steady encouragement, p e r s i s t e n t i n t e r e s t , p r o f i c i e n t advice, and, most of a l l , h i s inordinate patience and kindness. I would also l i k e to thank Professors G. L. Pickard, P. H. LeBlond, and G. S. Pond f o r t h e i r c r i t i c i s m s of an e a r l i e r version of t h i s t h e s i s . Each found a s p e c i f i c area that needed c l a r i f i c a t i o n or further development. My warmest appreciation extends to Dr. P. B. Crean f o r allowing me free access to the r e s u l t s of the numerical model of the Juan de Fuca-Strait of Georgia system; Mr. P. J . Richards was most h e l p f u l i n making these r e s u l t s accessible to me. Dr. Crean also requires recognition for encouragement of a more moral nature. I would also l i k e to thank Dr. J . A. Stronach f o r many useful conversations and for providing me with some r e s u l t s of h i s work. Mr. P. Y. K. Chang demands s p e c i a l thanks f o r allowing me use of h i s edited versions of wind and current data, and for providing me with enlarged copies of many of the figures i n h i s t h e s i s . Above a l l , I must r e g i s t e r my most h e a r t f e l t appreciation to my wife, N e l i a , f o r her s a c r i f i c e s during my long tenure as a graduate student and f o r her constant understanding and thoughtful advice. I want to thank her and my daughter E r i k a f o r the consideration extended to me during the f i n a l stages of t h i s t h e s i s . F i n a l l y , I express my gratitude to my parents f o r t h e i r constant f a i t h i n me. Much appreciated f i n a n c i a l support has been provided by the National Research Council of Canada and the Univ e r s i t y of B r i t i s h Columbia. x i 1 1. Introduction to Thesis Two separate but r e l a t e d problems i n p h y s i c a l oceanography are examined i n t h i s t h e s i s . In Part I, the i n e r t i a l i n s t a b i l i t y of currents which contain a small, randomly f l u c t u a t i n g component i s examined. While t h i s i s p r i m a r i l y a t h e o r e t i c a l i n v e s t i g a t i o n , the theory i s applied to observations made i n the F l o r i d a S t r a i t s with encouraging r e s u l t s . The l a t t e r part of t h i s thesis summarizes an attempt to understand the low-frequency currents observed i n the S t r a i t of Georgia. To t h i s end, i n e r t i a l i n s t a b i l i t y , wind forcing, r e s i d u a l t i d a l flow, and modified estuarine c i r c u l a t i o n are considered. Separate, more d e t a i l e d , introductions are provided for each part. 2 2. Introduction to Part I Under c e r t a i n conditions a planetary wave propagating through a region of mean current shear i s capable of extracting energy from the flow. This was pointed out i n the pioneering work of Kuo (1949) for Rossby waves in a zonal current. In p a r t i c u l a r , he showed that an extremum i n the po t e n t i a l v o r t i c i t y d i s t r i b u t i o n was a necessary condition for the existence of unstable waves. Since then, numerous models of unstable flows on a 3-plane have been considered, p a r t i c u l a r l y for the atmosphere (see the review by Kuo, 1973). By comparison, scant attention has been paid to the study of the modification of another class of planetary motions, namely continental shelf waves (CSWs), by sheared mean flows, although the theory of CSWs i n the absence of mean currents has been extensively investigated (see LeBlond and Mysak, 1977, for a review). Since mean coastal flows always e x i s t , t h i s represents a serious gap i n our knowledge of continental shelf dynamics. N i i l e r and Mysak (1971) considered a s t e p - l i k e continental shelf and a piecewise l i n e a r v e l o c i t y p r o f i l e and showed the existence of two classes of motions, modified CSWs and "shear" waves whose existence depends on the mean current shear. For a c e r t a i n short wavelength range, the two modes coalesce into a sing l e unstable wave t r a v e l l i n g i n the d i r e c t i o n of the current. In a more formal study Grimshaw (1976) extended many of the re s u l t s of barotropic i n s t a b i l i t y theory to include unstable CSWs f o r quite general depth and v e l o c i t y p r o f i l e s . McKee (1977) has calculated the stable response of the continental shelf to t r a v e l l i n g atmospheric disturbances and showed that i t i s comprised of a superposition of di s c r e t e 3 normal modes, a continuous set of transient solutions possessing phase speeds i n the range of the mean current, and d i r e c t l y forced motions. Brooks and Mooers (1977a, hereinafter r e f e r r e d to as BrM), i n a model of the F l o r i d a S t r a i t s , considered the e f f e c t of an intense, l a t e r a l l y sheared current on CSWs, but they evidently did not search for any possible unstable solutions. Thus they were l i m i t e d to modified CSWs with phase v e l o c i t i e s less than the minimum of the mean current v e l o c i t y . I t should be mentioned, though, that Schott and Duing (1976) found excellent agreement between BrM model predictions and observations i n the F l o r i d a S t r a i t s f or the 10-30-day wave period band. In Part I we consider the barotropic s t a b i l i t y of (1) an along- shore current and (2) a zonal flow with respect to CSW and Rossby wave perturbations, r e s p e c t i v e l y . In each case the basic current i s assumed steady and to be composed of a sheared mean component with a small, s p a t i a l l y random part superimposed upon i t . Although t h i s choice might seem to further complicate an already d i f f i c u l t mathematical problem, i t turns out that the mathematics greatly s i m p l i f i e s , and the problem may be e a s i l y solved provided that the c o r r e l a t i o n length of the f l u c t u a t i n g current i s s u f f i c i e n t l y small. This approach was adopted by Manton and Mysak (1976) f o r the case of plane Couette flow, and Part I i s an outgrowth of that work. The r a t i o n a l e for choosing a random current i s as follows. The small-scale features of the basic current are generally unknown and c e r t a i n l y vary i n both space and time. Moreover, these features make the basic current p r o f i l e so complicated as to render a d e s c r i p t i o n of the flow by a simple mathematical expression impossible. Thus i t i s reasonable to model the current by the sum of a smooth, det e r m i n i s t i c p r o f i l e and a small 4 i r r e g u l a r part. I t i s mathematically convenient to represent the i r r e g u l a r component as a random function. Thus we ignore the a c t u a l , small-scale structure of the current p r o f i l e and concentrate on i t s s t a t i s t i c a l p r operties, i n p a r t i c u l a r , i t s variance. I f t h i s decomposition i s to be p h y s i c a l l y r e a l i s t i c , i t i s necessary that the "random" features of the basic current be d i s t i n c t from the motions predicted by the ensuing theory, since these motions are due, in large part, to the basic flow. This requires that the "waves" admitted by t h i s theory be separated i n both frequency and wavenumber space from the random component of the basic current. That i s , there must be a sp e c t r a l gap or rapid change i n slope i n the v e l o c i t y spectra. Unfortunately, adequate data,is not presently a v a i l a b l e to t e s t v a l i d i t y of t h i s representation. As a f i r s t step i n a more extensive study, we concentrate s o l e l y on the l a t e r a l s p a t i a l v a r i a t i o n s and henceforth ignore temporal fl u c t u a t i o n s i n the basic flow. In the case of the F l o r i d a Current, to which the theory developed here w i l l be applied, the decomposition of the flow into d e t e r m i n i s t i c and random components i s e s p e c i a l l y appropriate, since i n the region of the F l o r i d a S t r a i t s , the current i s s t i l l adjusting to an almost 90° northward turn i n i t s passage around the southern t i p of F l o r i d a . This i s a process which should introduce a large amount of noise into the flow. The assumption that the superimposed wave f i e l d i s barotropic i n the frequency range of i n t e r e s t i s supported experimentally by observations i n both an intense western boundary current (Diiing, 1975; Mooers and Brooks, 1977) and i n a weak eastern boundary current (Huyer, et a l . , 1975; Kundu and A l l e n , 1976; Wang and Mooers, 1977). In a t h e o r e t i c a l a n a l y s i s , 5 A l l e n (1976) has shown that the continental shelf may support both barotropic and b a r o c l i n i c motions, the l a t t e r trapped within an i n t e r n a l Rossby radius of deformation of the coast. We expect t h i s assumption to be more questionable for Rossby waves i n the open ocean where b a r o c l i n i c i n s t a b i l i t y i s l i k e l y to be an important f a c t o r . Nevertheless, the present theory represents a f i r s t step i n a study of combined b a r o t r o p i c - b a r o c l i n i c i n s t a b i l i t y i n a randomly perturbed flow. Two continental shelf models are considered, one i n which the s h e l f i s bounded by a wall p a r a l l e l to the coast (channel model), and one i n which i t i s not (shelf model). In both cases the BrM model of bottom topography and mean current i s employed. Attention i s p r i m a r i l y focused on the channel model since the observations to which the theory i s to be compared were made i n the F l o r i d a S t r a i t s . This model also has the added convenience of being less complex mathematically since the coastal trapped motion i s i s o l a t e d from the ocean i n t e r i o r and thus no coupling need be considered. A channel model i s also assumed i n the Rossby wave case. Each of the two continental shelf models admits a class of unstable modified CSWs for long wavelengths provided that the h o r i z o n t a l c o r r e l a t i o n length of the f l u c t u a t i n g basic flow i s s u f f i c i e n t l y short compared with the shelf width. These waves may propagate i n e i t h e r d i r e c t i o n along the coast depending on the strength of the mean current; t h e i r phase v e l o c i t y i s given by the weighted average of the difference between the mean v e l o c i t y and the cross-stream gradient of p o t e n t i a l v o r t i c i t y . The growth rate i s proportional to the amplitude of the random component of the current and inversely proportional to i t s c o r r e l a t i o n length. The f a c t that unstable solutions e x i s t i s e s p e c i a l l y important since the BrM v e l o c i t y p r o f i l e i s almost c e r t a i n l y b a r o t r o p i c a l l y stable; that i s , the 6 c l a s s i c a l theory predicts the existence only of stable modified CSWs. Appli c a t i o n of the channel model to the F l o r i d a S t r a i t s p r e d i c t s wave parameters that are i n good agreement with observations made by Duing (1975). The present theory may thus account, i n part, f o r meanders observed i n the F l o r i d a current. The plan of Part I i s as follows. In Section 3 the formal theory for modified CSWs i n a channel i s presented, and a complicated integro- d i f f e r e n t i a l equation i s derived f o r the mass-transport stream function. Scaling arguments are employed to reduce t h i s equation to a more manageable form. A more ph y s i c a l d e r i v a t i o n of t h i s equation based on the balance of v o r t i c i t y i n the system i s given i n Section 4. An energy equation i s also derived. A perturbation s o l u t i o n i s obtained i n Section 5 for the BrM model, and the basic r e s u l t s for the growth rates and phase speeds are given. The r e s u l t s are applied to observations made i n the F l o r i d a S t r a i t s . In Section 6, we b r i e f l y consider the continental shelf model, and i n Section 7 the s t a b i l i t y of a zonal flow on a 6-plane to Rossby wave perturbations i s examined. A summary i s given i n Section 8. 7 3. Formal Theory f o r Shelf Waves i n a Channel In t h i s section the equations governing the propagation of small-amplitude, free, modified CSWs i n a l a t e r a l l y sheared, barotropic current are derived. The basic current i s assumed to flow along a continental boundary of i n f i n i t e length and i s composed of a sheared mean component and a small s p a t i a l l y random part. Only the channel model i s treated here; consideration of the shelf model i s deferred to Section 6. The dynamics of CSWs derive from the conservation of p o t e n t i a l v o r t i c i t y , but rather than to proceed d i r e c t l y from the conservation law, i t proves convenient f o r l a t e r purposes to begin with the equations of motion. I t i s also desirable to work i n terms of nondimensional quantities and the following scale factors which are representative of continental s h e l f conditions are chosen: the shelf width L (which may be less than the channel width, see Figure 5.1) for the ho r i z o n t a l coordinates (x,y), the maximum channel depth H for z, a v e r t i c a l l y averaged v e l o c i t y U f o r the ho r i z o n t a l v e l o c i t i e s (u,v), an advective time L/U, and fLU/g f o r the sea surface elevation. Here f i s the C o r i b l i s parameter and g i s the acceleration due to g r a v i t y . A basic state which exactly s a t i s f i e s the nonlinear, f r i c t i o n l e s s equations of motion i s s p e c i f i e d by the basic current V (x) and i s r e l a t e d to the basic sea surface slope by v B(x) = n B x ( x ) . (3.1) 8 The shallow-water equations of motion l i n e a r i z e d about the basic state are: Ro(u. + V Ru ) - v = - n (3.2) x. a y x Ro(v t + V B v y + uV B x) + u = - r) y (3.3) (hu) x + h ( v ) y = 0. (3.4) Here h(x) i s the nondimensional depth and Ro = U/fL i s the Rossby number for the basic flow and i s not assumed small. To obtain (3.4) we 2 2 invoked the nondivergent approximation, which i s good to order f L /gH (10 ^ i n F l o r i d a S t r a i t s ) , and t h i s allows the d e f i n i t i o n of a mass- transport stream function given by (3.5) hu = - T y hv = ¥ x In terms of ¥ the l i n e a r i z e d p o t e n t i a l v o r t i c i t y equation i s Ro ( 9 t + V B 9 y ) [h X V 2 , F - ( h ' / h 2 ) ^ ] - 1 [Roh _ 1V" B - (1 + RoV' B)h'/h 2] = 0 (3.6) where a prime denotes d i f f e r e n t i a t i o n with respect to x. For t r a v e l l i n g wave disturbances of the form 9 f = * ( x ) e i k ( y - C t ) (3.7) with p o s i t i v e k and possibly complex phase speed c, (3.6) reduces to (V_ - c)D$ - Q $ = 0. (3.8) hi X Here V = h V / d x 2 - (h'/h 2)d/dx - h \ 2 so that P$ i s the r e l a t i v e v o r t i c i t y of the perturbation and Q = (Ro 1 + V' B)/h (3.9) i s the basic state p o t e n t i a l v o r t i c i t y scaled by Ro. The boundary conditions are obtained by requiring that there be no flow through the sidewalls; hence . . $(x) •= 0 at x = 0,1. (3.10) As the basic current i s taken to be a stationary random v a r i a b l e , i t may be separated into i t s mean and f l u c t u a t i n g parts as V B(x) = V(x) + eW(x) such that E(V B) = V and E(W) = 0 where E represents the average over an ensemble of r e a l i z a t i o n s of W. The nondimensional parameter e i s assumed to be small and i s r e l a t e d to the variances of V 0 and W by e 2 = var v B / v a r W. (3.11) 10 In the present case we choose var W = 1 although a different choice w i l l be made for the shelf model. Since the basic state is random and the disturbance interacts with i t , i t necessarily follows that the perturbation must also contain a random component; we decompose $ as $'(x) = ip(x) + £<M*) (3.13) with E 0|>) = and E (<})) =0. Although i t is not necessary to scale the random part of $ by £, | w i l l generally be large compared with £(f>. We are primarily concerned here with deriving a closed form equation for . With these definitions the vorticity equation may be cast into the form (L + e.M) 0> + e<J>) = 0, (3.14) where L and M are deterministic and random differential operators respectively defined by L = (V - c)V - Q x (3.15) M = WP - q x ; (3.16) Q and q are the respective gradients of mean and fluctuating potential vorticity (scaled by Ro), Q x = h _ 1V" - (Ro 1 + V'Vh'/h2 (3.17) 11 q = h 1W" - W'h'/h2. (3.18) x The boundary conditions become UJ = 0 f at x = 0,1. (3.19) <j> = 0 Stochastic boundary value problems of the type defined by (3.14)-(3.19) have been investigated by a number of workers, and several techniques are available to deal with them (see Mysak, 1978, for a review). I t proves useful to decompose (3.14) into i t s mean and f l u c t u a t i n g components. By ensemble averaging (3.14) and subtracting the res u l t a n t expression from i t we obtain as follows: A formal so l u t i o n of (3.20)-(3.21) was f i r s t given by T a t a r s k i i and Gertsenshtein (1963) and i s Lip + e2EM(j) = 0 (3.20) L<j> + Mip + e[M(f> - E(M<(>) ] = 0. (3.21) 00 Li|> = - eEM I [ - £ ( / - E ) L X M ] n + V (3.22) n=0 Here 7 i s the i d e n t i t y operator and L i s the operator inverse to L . 12 The sum i n (3.22) i s convergent provided that e||L "*"M|| < 1, where ||(*)|| denotes an appropriate operator norm. This c l e a r l y l i m i t s the amplitude of the f l u c t u a t i n g part of the basic flow, and i t i s henceforth assumed that £ « 1. In the present analysis we r e t a i n only the f i r s t term i n (3.22) giving This constitutes the " f i r s t - o r d e r smoothing" or " l o c a l Born" approximation and i s equivalent to ignoring the bracketed terms i n (3.21). Howe (1971) has given a c l e a r p h y s i c a l i n t e r p r e t a t i o n of t h i s approximation. E s s e n t i a l l y , the neglected terms involve the i n t e r a c t i o n of the f l u c t u a t i n g component of the basic f i e l d with the random part of the perturbation at distances exceeding t h e i r mutual c o r r e l a t i o n length, whereas the other terms i n (3.20)- (3.21) are determined by the l o c a l values of the two f i e l d s . I t i s convenient to rewrite (3.13) i n terms of the Green's function G(x,^) for (V - c) which s a t i s f i e s b\> = £ 2E[ML _ 1M]^. (3.23) PG(x,5) - (V - c) - 1Q xG(x,£) = 6(x - ?) G(0,£) = G U 7 £ ) = 0. (3.24) Substitution of (3.15) and (3.16) into (3.23) and expression of L i n terms of G(x,£) r e s u l t s i n an i n t e g r o - d i f f e r e n t i a l equation f o r , 13 [(V - c ) 2 - e2R(0)]lty - (V - c)Qxi> - e 2 h _ 1 [ (hVh)R'(0) I - R"(0)]Tp = £ 2Q X / (V - c) _ 1G(x,£) [R(x - 0 - (h'/h2)R'(x - ^)UJ - h _ 1R"(x - £ ) M d £ + £2(h'/h2)(V - c) / (V - c)_1G(x,£)[R'(x - £)tty 0 - (h'/h2)R"(x - £)IJJ - h _ 1R , M (x - 04>]dE, £ - e 2h _ 1(v - c) / (V - c) - 1G(x,£) [R"(x - OVi> 0 - (h'/h2)R"' (x - - h~1R,v (x - 5 ) M o £ . (3.25) Here the correlation function R(£) is defined as R(£) = E[W(x)W(x + O]. This equation also holds for the shelf model i f the upper limit of integration i s extended to i n f i n i t y . For the channel model we assume that W is a homogeneous random function so that R1(o) vanishes; further, we 2 choose R(o) = 1 and define O = - R"(o). Then I/O is representative of the nondimensional correlation length L c/L of the fluctuating current. 2 (In fact, R"(o) = - 2/X for a Gaussian process described by R(?) = ex P(-C 2/X 2).) Certainly, (3.25) is much too complex to be dealt with directly, and, indeed, G(x,£) w i l l generally be unknown analytically, expressing our ina b i l i t y to solve the associated deterministic problem. Hence some approximate analysis must be adopted, and i t clearly would be advantageous 14 to eliminate the i n t e g r a l terms i n (3.25). We w i l l presently demonstrate that the rhs of (3.25) i s an order of magnitude smaller than the lhs and thus, to a f i r s t approximation, may be ignored. I t i s shown i n Appendix A that a l l the i n t e g r a l s i n (3.25) are of 0(1) provided that R(x) and i t s derivatives are also 0(1). Hence, away from the singular 2 points of the equation where V = c ± £, the rhs of (3.25) i s 0(£ ) and i s thus n e g l i g i b l e compared with the lhs which contains terms of 0(1) 2 (and also terms of 0(£ ) ) . Near the singular points, however, the analysis i s considerably more complicated, but we claim that e i t h e r Q x or that both ijj and G(x,£) are 0(£) there and hence that the rhs of (3.25) may be saf e l y neglected. To see t h i s , consider the l i m i t i n g case of £ = 0 for which (3.25) reduces to (V - c)tty - Qxi> = 0. (3.26) Since (3.26) must hold pointwise i t follows that there are two p o s s i b i l i t i e s at points x c where V(x c) = c which must be considered. E i t h e r the gradient of p o t e n t i a l v o r t i c i t y Q x or the cross-stream v e l o c i t y vanishes there. We disregard the t h i r d p o s s i b i l i t y of solutions with singular d e r i v a t i v e s (that i s , those corresponding to a vortex sheet at x = xc)[ f o r two reasons. F i r s t , McKee (1977) has demonstrated f o r stable mean v e l o c i t y p r o f i l e s that, i n t h i s case, c belongs to the continuous part of the eigenvalue spectrum of L and the corresponding eigenfunction represents only a transient component of the complete s o l u t i o n . McKee's arguments are r e a d i l y extended to cases with unstable v e l o c i t y p r o f i l e s . Second, and more importantly, L i n (1961) has elegantly demonstrated that the i n c l u s i o n of a small amount of (molecular or eddy) v i s c o s i t y precludes 15 the existence of a continuous spectrum. That i s , i n the l i m i t of vanishing f r i c t i o n , the singular solutions of the i n v i s c i d theory are not obtained as l i m i t s of v i s c i d s o lutions. Now i f e i s s u f f i c i e n t l y small, i t follows that at a distance £ away from V = c, either Q x or ij; i s of 0 ( E ) . A s i m i l a r argument shows that i n the l a t t e r case G(x,£) i s also 0(£) : at x = x c and away from £ = x c , the rhs of (3.24) vanishes and hence so does G(x c,^) since Q x i s nonzero by assumption. As G(x,£) i s a continuous function of x and £, i t follows that G(x c, x c) =0. We again argue that at a distance £ from x = x c , G(x, E,) i s 0 ( E ) . (The above comments concerning the continuous spectrum of L also imply that we may exclude the p o s s i b i l i t y of G(x,£) having singular derivatives at x = x c , E, ? x c ; see McKee, 1977.) Consider the f i r s t case i n which Q x = 0(£); then the terms on 2 3 the lhs of (3.25) are a l l 0(£ ) while those on the rhs are 0(£ ). In the second case, where and G are 0(£), the f i r s t three terms on 2 3 the lhs are 0(£ ) and a l l the remaining terms i n (3.25) are 0 ( E ) . Hence i n a l l cases the lhs contains terms at le a s t 0 (£""'") greater than the rhs which may thus be ignored. An e n t i r e l y analogous argument holds i f c i s complex with a small imaginary part. We are therefore led to the consideration of the s i m p l i f i e d equation [(V - c ) 2 - £2]£ty - (V - c ) Q x ^ - £ 2 0 2 h - 1 ^ = 0, (3.27) which i s subject to the boundary conditions (3.19). A more ph y s i c a l d e r i v a t i o n of (3.27) i s given i n the next section. In the analysis presented i n the following sections, approximate 16 solutions are obtained for the case of large 0 " . Thus one of the assumptions made i n the preceding s c a l i n g argument i s v i o l a t e d , namely that R(x) and i t s f i r s t four derivatives are a l l 0(1). An evaluation of the r e l a t i v e sizes of the terms i n the analogue of (3.25) for a zonal random flow on a $-plane i s made i n Appendix C. I t turns out that the i n t e g r a l terms are generally smaller than the other terms, although i n some cases they are of s i m i l a r magnitude f o r c e r t a i n regions of the channel. As k increases and 0 decreases, the neglect of these terms i s better j u s t i f i e d . Thus O must be large enough to permit a perturbation expansion but not so large that the i n t e g r a l terms become overly s i g n i f i c a n t . In the l a t t e r case, the solutions obtained i n t h i s thesis are best regarded as approxima- tions to the f i r s t i t e r a t i v e s o l u t i o n of the f u l l i n t e g r o - d i f f e r e n t i a l equation. Of course, i f O i s not large, then the i n t e g r a l terms may be saf e l y ignored to a good approximation. Comparison of (3.27) with the corresponding equation f o r the deterministic case (3.26) shows that the randomness of the basic current manifests i t s e l f i n two ways. F i r s t , the s i n g l e c r i t i c a l point of (3.26) at V = c i s b i f u r c a t e d into the p a i r V = c ± £. In any given r e a l i z a t i o n of V , there would no doubt e x i s t numerous points at which c = V + £W (provided that c i s r e a l ) , and t h i s i s expressed, i n the mean, by the existence of two points each removed by the rms value of £W from V = c. 2 2-1 The second e f f e c t appears i n the term £ a h iL» and depends not only on the strength of the f l u c t u a t i n g current but also on i t s c o r r e l a t i o n length. The p h y s i c a l s i g n i f i c a n c e of t h i s term i s more f u l l y discussed i n the next section where the v o r t i c i t y balance i s considered. 17 4. The V o r t i c i t y and Energy Balances In t h i s section the various v o r t i c i t y and energy balances present i n the system are examined. Consider f i r s t (3.20) and (3.21) rewritten i n a more e x p l i c i t form. To an observer moving with the wave speed, the terms i n (4.1) correspond res p e c t i v e l y to: (1) alongshore . advection of mean disturbance v o r t i c i t y P i j j by the mean basic flow, (2) cross-stream advection of mean basic v o r t i c i t y by the mean disturbance, and the correlated parts of (3) advection by the f l u c t u a t i n g b a s ic flow of the random disturbance v o r t i c i t y D<j), and (4) the cross-stream advection of random basic v o r t i c i t y by the f l u c t u a t i n g disturbance. S i m i l a r l y , the terms i n (4.2) are interpreted as: (1) advection by the mean basic flow of f l u c t u a t i n g disturbance v o r t i c i t y , (2) advection of mean basic v o r t i c i t y by the random disturbance, (3) advection of mean disturbance v o r t i c i t y by the random basic flow, (4) cross-stream advection by the mean disturbance of f l u c t u a t i n g basic v o r t i c i t y , and (5) the alongshore advection of random disturbance v o r t i c i t y by the f l u c t u a t i n g basic current. The l a s t term i n (4.2) i s the only one quadratic i n the random f i e l d s and thus might be expected to be small. In f a c t , i t consists of the difference of the t o t a l advection of V$ by W and that part of Wpcf) which i s correlated. Since i t p r i m a r i l y involves (v - c)Vi> - QJ) + e2E[wP(f)] - e 2 E [ g d>] = o (4.1) (v - c)V<t> - Q (J> + wpTp - q u> = - e(wD<j) - E[wP<j)]). (4.2) 18 the i n t e r a c t i o n of the two f i e l d s at distances exceeding t h e i r mutual c o r r e l a t i o n length, i t plays an i n s i g n i f i c a n t r o l e i n the v o r t i c i t y balance expressed by (4.2) and i s henceforth neglected. We now consider the r e l a t i v e magnitudes of the various terms i n (4.1) and (4.2) and give a h e u r i s t i c d e r i v a t i o n of (3.27). Substitution of T?<J>, as determined by (4.2), into (4.1) y i e l d s the analogue of (3.25), (v - c)Vi> - Qj - e 2(v - c ) - 1 l t y - e 2a 2h - 1(v - c)~\ + e 2Q x(v - c)-1E[wcj>] - e 2h _ 1E [w"<j>] + e 2 (h'/h2)E[W'cj)] = 0. (4.3) In order to determine the magnitudes of <J> and D<j>, we use the expression cf>(x) = - / (V - c ) - 1 G ( x , 5 ) [Wtty - qxljJ]d£ (4.4) 0 which i s obtained from (3.21) i n the l o c a l Born approximation. Consider now the cases examined i n the preceding section corresponding to (V - c) = 0(1) and V - c = 0(e). (If c i s complex with small imaginary part, then the l a t t e r case i s equivalent to V - c r = 0(e), and c^ = 0(e).) The r e s u l t s are summarized i n Table I. In the f i r s t case the random f i e l d s play an i n s i g n i f i c a n t r o l e i n the balance of mean v o r t i c i t y , and Vty exceeds eVty by an order of magnitude. On the other hand, the alongshore advection of f l u c t u a t i n g disturbance v o r t i c i t y by the random basic flow (4.1, term 3) i s important i n the l a s t two cases, but the cross-stream advection of the random background v o r t i c i t y by the f l u c t u a t i n g disturbance (4.1, term 4) remains unimportant. Near the c r i t i c a l point the random component of 19 Table I. Order of magnitude estimates of terms i n the v o r t i c i t y balance equations. The rhs of (4.2) i s neglected, and the magnitudes of (j> and V§ are calculated from (4.4). A. V - c = 0(1) (4.1) 1 l e 2 2 e Vi>,ty = o(i) (4.2) 1 l 1 1 t?<|>,4> = o(i) (4.3) 1 l e 2 Jl „2 2 2 e e e e B. V - c = 0(E) Qx = 0(e) (4.1) e e e 2 e = 0(1) (4.2) l e 1 1 t t y = 0(i/e) (4.3) e e e 2 2 2 e ê e ê c. v - c = 0(e) = 0(e) (4.1) e e e 3 e Tty = 0(1) (4.2) l e 1 e P<J> = 0(l/e) (4.3) e e e 2 2 3 3 e e e e 20 disturbance v o r t i c i t y £#<}> i s the same order of magnitude as the mean component, although £<f> remains small. Thus the f l u c t u a t i n g v o r t i c i t y tends to concentrate near points where V - c i s small. In a l l cases, the l a s t three terms i n (4.3) may be disregarded, and hence (4.3) reduces to (3.27). Note that neglect of these terms i n no way requires the corr e l a t i o n s between <f> and i t s d e r i v a t i v e s to be small, as suggested by Manton and Mysak (1976). An energy equation for the mean component of the perturbed f i e l d i s now derived. I t i s easiest to proceed d i r e c t l y from the nonaveraged equations of motion (3.2) - (3.4) and then to average the r e s u l t i n g expression. In the usual fashion we take the scalar product of the momentum equations with u_ to obtain (h/2)(u 2 + v 2 ) t + (h/2)V B(u 2 + v 2 ) y = - V' Bhuv - Ro _ 1h(u(; x + vC y) - (4.5) An in t e g r a t i o n over the region R defined by the channel width and one wavelength i n the y - d i r e c t i o n with a p p l i c a t i o n of the boundary conditions and the assumed p e r i o d i c nature of the disturbance gives I 0 / 9 t ) / (h/2) (u 2 + v 2) = / V B ( x ) T 1 2 d x . (4.6) R 0 Here the Reynolds stress T.. „ i s defined by y+A = ~ ! huvdy. Y (4.7) 21 Thus i f a wave i s unstable, the Reynolds stress must extract k i n e t i c energy from the shear of the basic current. By expressing the r e a l quantities u, v and T i n (4.6) i n terms of the deterministic and random components of the stream function (3.7), and by ensemble averaging the resultant expression we f i n d , n / h - 1 ( | H 2 + k 2 M 2 ) d x 0 + e?Q J h _ 1(|cf)'| 2 + k2|4>|2)dx i 0 I I = + k J h" 1V'F(i)j R,^ I)dx + e 2k / h" V E l P t ^ ^ j J l d x 0 0 2 ^ 1 + e k E / h W'ImF(i|;*,cf))dx. (4.8) 0 Here the subscripts R and I (or r and i) r e f e r to the r e a l and imaginary parts of a quantity, Q = kc^ i s the growth rate of the disturbance, and F(f,g) = f g ' - f'g i s the Wronskian of f and g. The f u l l Reynolds stress assumes the form T 1 2 = (k/2h) [F 01^ , 1 ^ ) + F(<$>RAX) + ImF(lp*,4>) 1 • ( 4 . 9 ) The i n t e r p r e t a t i o n of (4.8) i s d i r e c t and i s not pursued here; however, we note that i t i s not energy conserving since no provision was made for the modification of the basic f i e l d which therefore acts as an i n f i n i t e r e s e r v o i r of energy for the disturbance. I t i s not c l e a r how to approximate (4.8) i n order to obtain an equation consistent with (3.27); i . e . , i t i s d i f f i c u l t to t r a n s f e r s c a l i n g 22 arguments from the v o r t i c i t y domain to the energy domain. To avoid t h i s problem we form another energy equation from (3.27) rewritten as (V - c)tty - £ 2(V - c ) " ^ - 0xu> - £2a2h_1(V - c ) " 1 ^ = 0. (4.10) M u l t i p l i c a t i o n of (4.10) by ty* and int e g r a t i o n over the channel width y i e l d s an expression whose imaginary part i s £ £ ft / h~ (|TJJ'| + k 2|^| 2)dx = k / h - 1V'F ( i ^ , ^ ) dx 2 £ [(V " o ) 2 - c. 2] + £ k / ~ ~ VF(lJJ R,^ I)dx 0 hlv - cl 2 £ (V - c r)V'|l|,'| 2 2 2 £ \xl)l2 + e Q / 7 dx + £ o Q j — 1 — 1 j dx 0 h|v - c| 0 h|v - c| 2 / | 2 + k 2|i|^ 2 ~ f. — 0 hlv - cl - £ / ; ^ dx. (4.11) A c a r e f u l comparison of (4.8) and (4.11) with p a r t i c u l a r regard to the form of rf) as given by (4.4) reveals that terms 1 and 2 i n (4.11) correspond exactly to terms 1 and 3 i n (4.8), and that terms 3 and 4 and terms 5 and 6 i n (4.11) represent terms 4 and 5 i n (4.8), r e s p e c t i v e l y . In the case of large 0", which w i l l be considered l a t e r , i t i s clear from (4.11) that the energy source for the unstable perturbation i s the shear of the random part 23 of the basic flow. One could proceed, at t h i s point, to derive the equivalents of the formal r e l a t i o n s h i p s developed by Grimshaw (1976). These concern necessary conditions f o r i n s t a b i l i t y and bounds to growth rates and phase speeds. I t i s s u f f i c i e n t here to derive only a generalized Rayleigh condition f o r i n s t a b i l i t y . Rather than use the techniques employed by Grimshaw, since they are tedious i n the present case, we proceed from the unaveraged Rayleigh condition <i J VB " C Q dx = 0 . *x (4.12) This r e l a t i o n s h i p i s the imaginary part of the expression obtained by multiplying (3.26) by ip*/(V - c) and i n t e g r a t i n g over the channel width. If a system i s unstable (c^ ^ 0 ) , the i n t e g r a l must vanish. This implies that Q x must vanish at l e a s t once i n the i n t e r v a l ( 0 , £ ) . To generalize (4.12), we express $ i n terms of if a n d $ a n c ^ expand (V B - c) 1 = (V - c + £W) ^ i n a binomial s e r i e s . Ignoring t r i p l e c o r r e l a t i o n s i n the 3 ensemble average of the resultant expression we obtain to 0 ( £ ) C i / Q> V - c dx r v - c 2Q (V - c ) T E M * I ] - -T E[WRe(#)] V - c EEW"Re(#)] 20"2 | ̂ | 2 (V - c ) + h V - c h V - c -g [ dx = 0 . (4.13) 24 2 2 In the l i m i t of large 0 ( i . e . , €0 = 0(1)), t h i s reduces to V - c 2 e 2 0 2 ( v - c ) r Q + V - c dx = 0 (4.14) Equation (4.14) shows that i n s t a b i l i t y does not demand that the mean p o t e n t i a l v o r t i c i t y d i s t r i b u t i o n vanish, since the random part of the basic flow may possess s u f f i c i e n t r e l a t i v e v o r t i c i t y to render the basic flow unstable. In t h i s sense, the i n s t a b i l i t y described more f u l l y i n the next section i s e s s e n t i a l l y a shear i n s t a b i l i t y . In more ph y s i c a l terms, the random component introduces l o c a l extrema into the p o t e n t i a l v o r t i c i t y d i s t r i b u t i o n ( i . e . , points at which Q x = 0). L i n (1945) has demonstrated that the existence of such points i s necessary for i n s t a b i l i t y ; otherwise a p a r t i c l e displaced from i t s equilibrium p o s i t i o n w i l l always be subject to a net r e s t o r i n g force. As O increase, the r e l a t i v e v o r t i c i t y associated with the random flow increases and therefore so does the l i k e l i h o o d of f i n d i n g l o c a l maxima and minima i n Q. If c^ ^ 0 and Q x > 0 everywhere, (4.14) implies that (V - c r) < 0 somewhere and hence that c r must l i e i n the range of the mean current. However, i f Q x < 0 everywhere then (V - c r) > 0 somewhere and c r i s simply bounded above by the maximum value of V; unstable waves could then, i n p r i n c i p l e , propagate against the mean flow. 25 5. The Channel Model The boundary value problem defined by (3.27) and (3.19) encompasses a l l and more of the d i f f i c u l t i e s inherent i n a d e t e r m i n i s t i c , b a r o t r o p i c a l l y unstable system, for not only does (3.27) have va r i a b l e c o e f f i c i e n t s but i t also possesses a p a i r of singular points at V(x) = c ± £. Since £ « 1, an obvious approach to (3.27) would be to attempt a perturbation s o l u t i o n i n £. However, the r e s u l t i n g equations would contain a l l the mathematical d i f f i c u l t i e s of the nonrandom problem, and the solutions, as i n t e r e s t i n g as they might be, would represent small deviations from the deterministic solutions. We are interested i n a d i f f e r e n t class of solutions to (3.27) which does not e x i s t i n the nonrandom case. I f the parameter 0 i s large ( i . e . , i f the c o r r e l a t i o n length of W i s small) then we may t r y a perturbation expansion i n the l i m i t of 0 -»- 0 0; we s h a l l show that the r e s u l t i n g solutions are unstable. Hence even though c l a s s i c barotropic i n s t a b i l i t y theory may indicate a given system to be stable, unstable waves may s t i l l e x i s t i f there i s s u f f i c i e n t "noise" i n the mean current. In order to make the following r e s u l t s more s p e c i f i c , we adopt the Brooks and Mooers (1977a) model of the F l o r i d a S t r a i t s (Figure 5.1): 1-x V = xe (5.1) 2b(x-l) e 0 5 x 5 1 h = (5.2) I1 1 5 x 5 I, 27 with the parameters b = 1.385, I = 2.5 and Ro = 0.3. This model i s chosen since i t employs n o n t r i v i a l but r e a l i s t i c v e l o c i t y and bathymetric p r o f i l e s , and since we wish to apply our r e s u l t s to the F l o r i d a S t r a i t s . Although V s a t i s f i e s a l l the necessary conditions for i n s t a b i l i t y , i t i s extremely u n l i k e l y that t h i s model i s unstable as the subsequent argument shows. The Rayleigh condition (4.12) requires that <=i / QXH'2/|V - c| 2dx = 0. (5.3) 0 This requirement i s usually stated i n the form: i f c^ ^ 0, then Q x must vanish at l e a s t once i n the i n t e r v a l (0,£). In f a c t , Q x does vanish at x = 2, but t h i s i s not s u f f i c i e n t to ensure that (5.3) i s s a t i s f i e d . A p l o t of Q x (Brooks and Mooers, 1977a, Figure 2) shows that i t i s extremely small i n the i n t e r v a l (2,& = 2.5) compared with the i n t e r v a l (0,2). This means that \i>\ must be extremely large i n the former i n t e r v a l which i s not l i k e l y as the boundary conditions require <p(J£) = 0. That i s , i f (5.3) were s a t i s f i e d with c^ ^ 0, very large alongshore currents trapped against the outer wall would be necessary. With respect to a p p l i c a t i o n of the present theory to the F l o r i d a S t r a i t s , we note, however, that a more r e a l i s t i c bathymetric p r o f i l e r e s u l t s i n a p o t e n t i a l v o r t i c i t y d i s t r i b u t i o n that i s probably unstable (Brooks and Mooers, 1977a). Unfortunately, Brooks and Mooers apparently d i d not search f o r unstable solutions. If a perturbation expansion of (3.27) i s to lead to n o n t r i v i a l 2 2 —1 r e s u l t s , the term £ O h must be balanced by another term. For f i x e d k t h i s requires that c be 0 (CT) . I t might appear that t h i s could lead 28 to a contradiction of the semicircle theorem (Grimshaw, 1976) which states that f o r each r e a l i z a t i o n of the basic flow V_ , [ c r - 1/2(V B M + V B m ) ] 2 + c ± 2 < [1/2(V B M - V B m ) - 1/2 c w ] 2 where V Bj^ and v g m are, r e s p e c t i v e l y , the algebraic maximum and minimum values of V"B , and c w i s the phase speed of the f i r s t mode CSW i n the case of V = 0. Thus i f W i s bounded, c i s bounded above independent of 0. Hence, once again the magnitude of 0 i s l i m i t e d . Nevertheless, we assume that cr i s large enough to permit i t s use as an expansion v a r i a b l e . On the other hand, the values of c^ and c r computed i n t h i s section f a l l well within the bounds of the semicircle theorem. Moreover, i t turns out that successive corrections to c r and „• • • , -2 "I c^ diminish as 0 not 0 In the following development i t i s convenient to expand both the r e a l and imaginary parts of c separately; we take m=0 m=0 -m rm c ± = 0 00 I m=0 -m 0 c. lm (5.4) The choice of eit h e r c^ = 0 ( 0 ) , c^ = 0(1) or c r ' c i = 0 ( a ) leads only to t r i v i a l s olutions. The boundary conditions (3.19) become = 0 at x = 0,£. If the system under consideration contains d i s c o n t i n u i t i e s i n the gradient 29 of p o t e n t i a l v o r t i c i t y , then the equations must be solved i n each region and matched across the point of d i s c o n t i n u i t y . In general, the matching conditions are [i>] = o (5.6) [{(V - c) - e2}ip'/h] - [(V - c)Qj] = 0 where [(*)] represents the jump i n (•). P h y s i c a l l y , these conditions ensure the continuity of mass flux and sea surface elevation across the di s c o n t i n u i t y . Their v a l i d i t y requires that V be continuous. Both h and Q are continuous i n the present case and (5.6) reduces to m = o > at x = 1. (5.7) W ] = o The three lowest order equations are VtyQ + h _ : L ( e / c i 0 ) 2 i | ) o = 0 (5.8) Vipi + h 1(e/c±0)\1 = - c . 0 " 1 { 2 [ c i l + i ( V - c x Q ) W 0 ~ iQ x^ 0} 30 + h~1(Z/c±Q)\2 = c i 0 2{ (V - c^)2 - 2c, 0c, n - C j l 2 - e 2 rO' 'i2 I O i l - 2i[(V - c r 0 ) c i ; L - c . 0 c r l ] } ^ 0 -1 2c ± 0 [ C i l + i(V - c r Q ) ] ^ 1 c i Q 2 2 x [ ( V " Cr0> - i c i l ^ 0 + i c i 0 _ 1 Q x ^ l (5.10) The zeroth order equation (5.8) defines a Sturm-Liouville problem for which , (n) an infinite number of solutions y Q exist such that the nth mode has (n) 2 exactly n zeros, and the corresponding eigenvalues ( s / c i o ' a r e ordered and tend to in f i n i t y as n -*• 0 0. The superscript (n) is henceforth dropped. Solutions to (5.8) satisfying both the boundary and jump conditions are given by: b ( x _ 1 ) • w • 1 e sm Ax/sm A 0 5 x < 1 (5.11) sin a(x - £)/sin a ( l -I) 1 5 x < I . 2 2 / 2 2 2 A = £ / c i Q - k - b (5.12) 2 2 / 2 , 2 "\ 2 ,2 a = e / c i 0 ~ = ^ (5.13) 31 provided that A and a s a t i s f y b + A/tan A = a/tan a ( l - £) . (5.14) I f A i s negative, the solutions over the shelf are hyperbolic and are obtained by replacing A with iA. For a given choice of k, b, and H, the admissible values of e/c^ are determined i m p l i c i t l y by (5.14) together with e i t h e r (5.12) or (5.13). A graphical s o l u t i o n of (5.14) i s shown i n Figure 5.2 f o r a case i n which hyperbolic solutions are found. The growth rates of the hyperbolic modes, i f they e x i s t , exceed those of the trigonometric solutions. For the values of b and £ appropriate to the F l o r i d a S t r a i t s only trigonometric modes are found. a c t u a l l y solving f o r and ^ 2 . Since these equations a l l have i d e n t i c a l homogeneous parts, the Fredholm a l t e r n a t i v e implies that must be orthogonal to the respective inhomogeneous terms. We thus obtain Further information may be extracted from (5.8)-(5.10) without c i 0 = e / ( X + k + b ) (5.15) = <V> + (c i 0 2/2£ 2)<hQ x> (5.16) c. _ , c i l r l = 0 (5.17) = - (3/2) <(V - c r Q ) 2 > - e 2 / 2 - (c i 0 2/2£ 2)<(V - c r Q)hQ x> (5.18) - i c i 0 < ( V " croHV " i(c i Q 3/2£ 2)<hQ x|V 32 Figure 5.2 Graphical s o l u t i o n of (5.14) f o r b = 3.0 and SL = 2.5. The ( l i g h t , heavy) s o l i d l i n e i s the locus of b + A/tan A for (r e a l , imaginary) A , while the dotted l i n e represents a/tan a ( l - Z). Intersections with the ( l i g h t , heavy) l i n e corre- spond to (trigonometric, hyperbolic) s o l u t i o n s . For F l o r i d a S t r a i t parameters (b = 1.385, I = 2.5) the f i r s t three solutions are a = 1.650, a„ = 2.809, and a, = 3.891. 3 3 where £ £ <f (x)| g(x)> = / h - 1 i j ; 0 ( x ) f (x)g(x)dx// h - 1 ^ 2 (x) dx (5.19) 0 0 and <f(x)> E <f(x)|^ 0(x)> (5.20) are weighted cross-channel averages. These r e s u l t s are completely general and are not l i m i t e d to the BrM model. For t h i s model, however, c r Q may be rewritten as c r 0 = <V> + <hQx>/2(X2 + k 2 + b 2) (5.21) which i s s t r i k i n g l y s i m i l a r to the expression f o r a CSW i n a constant current V over an exponential s h e l f , c = V - 2b/(A 2 + k 2 + b 2) = <V> + <hQx>/(A2 + k 2 + b 2) (5.22) where hQ x = - 2b (cf. Buchwald and Adams, 1968, f o r the case V = 0). We see that the weighted average <V> replaces V, while the p o t e n t i a l v o r t i c i t y term i n (5.21) i s diminished by 1/2. This means that the speed of the Doppler s h i f t e d wave c rQ - <V> i s reduced by the presence of random i r r e g u l a r i t i e s i n the .basic current. The explanation i s c l e a r : the disturbance must traverse a longer path length i n t r a v e l l i n g from one point to another since i t i s buffeted about and scattered by the f l u c t u a t i n g current. This phenomenon i s common to wave propagation i n random media 34 (Howe, 1971). In order to determine c^ 2 i t is necessary to f i r s t evaluate i|> ̂ . This i s a straightforward, although tedious task, and the complete results are summarized in Appendix B. The solutions in the onshore and offshore regions take the form r Ai^o + i Ao pi Vo + i A0 P2 0 5 x ^ 1 1 5 x 5 I (5.23) where and P 2 are particular solutions that satisfy the boundary conditions; the factor of i ensures that P ^ and P 2 are real. These solutions must be matched across x = 1, and the matching conditions (5.7) in matrix form are A l l " _ P 2 - P 1 M = i A Q = P (5.24) A12_ P ' 2 - P ^ where s i n sin a ( l - l) M = (5.25) b sin X + A cos a cos a( l - &) and a l l quantities are evaluated at x = 1. However, M is singular, 35 since i n matching the zeroth order solutions we required that det M = 0. If a s o l u t i o n to (5.24) i s to e x i s t , i t i s necessary that P be orthogonal to each l i n e a r l y independent s o l u t i o n of the associated homogeneous adjoint equation, M' = 0. (5.26) Since M i s of rank one, there i s only one independent s o l u t i o n of (5.26), and i t leads to the a u x i l i a r y condition P - P = a~ x tan a ( l - I) (P' - P',) 2 1 A X . (5.27) The f u l f i l l m e n t of t h i s r e s t r i c t i o n was used as a check of the numerical r e s u l t s presented i n Figures 5.3-5.7. The one independent equation represented by (5.24) serves to f i x A^2 , A12 = A l l s i n ^ / s i n ct (1 - 5,) - i A Q ( P 2 - P 1)/sin ot(l - I) , (5.28) but A-^ remains indeterminate. A c a r e f u l examination of (5.10) reveals that the part of proportional to I|JQ does not contribute to c ^ and we choose A-^ = 0. A l t e r n a t i v e l y , A ^ could be chosen so that IJJQ and ^ were orthogonal. S i m i l a r considerations apply to the higher order eigenfunctions and eigenvalues. F i n a l l y , although i t i s not immediately obvious, one finds that c r 2 n + l a n c ^ c i 2 n + l v a n i s n a n& that ^ n ^ O A N < ^ ^2n+l / / A0 a r e P u r e l v r e a l a n d imaginary q u a n t i t i e s , r e s p e c t i v e l y . This 36 2 means that successive corrections to c r and c^ diminish by 0(1/0 ). We now examine the r e s u l t s i l l u s t r a t e d i n Figures 5.3-5.7 which were computed f o r the parameters c h a r a c t e r i s t i c of the F l o r i d a S t r a i t s . The dependence of the growth rate Q-̂ = k(0c^g + 0 ^ c ^ ) on 0 for the f i r s t mode i s shown i n Figure 5.3. As 0 increases, 0,^ increases and the wave number range over which i t i s nonzero widens. There i s a threshold value of 0 above which unstable waves e x i s t . For mode 1 i t l i e s between 0 = 3 and 0 = 4 ; for a given k t h i s value decreases with increasing mode number. Figure 5.4a exhibits the dependence of on k for the f i r s t three modes. I t i s p a r t i c u l a r l y i n t e r e s t i n g that the regions of i n s t a b i l i t y overlap and that for short wavelengths the higher modes may be the more unstable. We note, however, that f o r large wavenumbers, higher order terms which are neglected here may be s i g n i f i c a n t . The frequencies and phase speeds as functions of k are p l o t t e d i n Figures 5.5a and 5.6a, re s p e c t i v e l y , for modes 1-3. The most s t r i k i n g feature i n them i s the existence of points where the curves i n t e r s e c t , which implies the possible existence of a "resonance" i n t e r a c t i o n (this p o t e n t i a l l y was also inherent i n the Couette flow model of Manton and Mysak, 1976, for which the phase speed was constant and equal for a l l modes; see Section 7). I t has been documented i n the l i t e r a t u r e (see Mysak, 1978) f o r a v a r i e t y of p h y s i c a l systems, that modes which are uncoupled i n the deterministic case may become coupled when randomness i s introduced into the problem. Here, however, we are dealing with disturbances which do not e x i s t i n the d e t e r m i n i s t i c case; moreover, i t i s c l e a r that the higher order terms i n the perturbation expansion w i l l not lead to mode coupling. Hence a more c a r e f u l analysis of (3.27) i s required to resolve t h i s point, perhaps using and ^ as a basis set for the r e a l and imaginary parts of We also note the 0.5 k / 2 7 T Figure 5.3 Behaviour of the f i r s t mode nondimensional growth rate ti^ as a function of O and k. The dimensional values given correspond to F l o r i d a S t r a i t parameters (b = 1.385, I = 2.5, Ro = 0.3) . The ( curves are labeled by the value of 0 " . WAVELENGTH (KM) k / 2 7 7 " Figure 5.4A Behaviour of the nondimensional growth rate ft^ as a function of k f o r the f i r s t three modes for 0 = 5 and £ = .5; (A) channel model (b = 1.385, I = 2.5, Ro = 0.3), (B) shelf model (b = 1.385, U = 1.0, Ro = 0.3). w oo 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 k / 2 7 T 0.8 0.9 1.0 Figure 5.4B U) 40 W A V E L E N G T H ( K M ) 3 0 0 1 5 0 1 0 0 7 5 6 0 Figure 5.5A Dispersion curves f o r the f i r s t three modes, (A) channel model, (B) s h e l f model. Beyond k/2ir = 0.6 the curves are e s s e n t i a l l y l i n e a r . 3 WAVELENGTH (KM) 600 300 150 100 75 60 50 40 30 1 » • • ' i i « • • « 1 • 1 o 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 k / 2 ? r Figure 5.6A Behaviour of the nondimensional phase speed as a function of k for the f i r s t three modes, (A) channel model, (B) shelf model. to ' I I 1 1 I 1 1 1 L - 0.0 0.1 0.2 0.3 0 . 4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 k/2 7T Figure 5.6B 44 related study by Allen (1975) of coastal trapped waves i n a s t r a t e f i e d ocean where i t was shown that CSWs may be coupled to internal Kelvin waves. Figure 5.6A indicates that the phase speed i s positive for a l l k and thus that the waves propagate northward; i . e . , i n the di r e c t i o n of the mean flow. However, i f the Rossby number were small enough, the v o r t i c i t y term i n (5.16) would predominate, and the disturbances could travel southward. (Consider, for example, the case of V = 0, hQx = - 2b.) This i s i n marked contrast to the stable CSW's admitted by the BrM model which propagate only southwards. From the slopes of the dispersion and phase speed curves, we infer that the group velocity i s positive and exceeds crQ for a l l k. F i n a l l y , we note that crQ -»• <V> as k -*• °° and thus that the waves are simply advected by the mean current i n this l i m i t . Contour plots of the mass transport stream function for the f i r s t three modes are shown i n Figure 5.7. An important feature i s the t i l t i n g of the gyre axes toward the coast since i t i s related to the sign of the mean disturbance Reynolds stress T 1 2 = (k/2h) F (ij; , 4 ^ ) . In terms of the phase 0 = k(x - c t) , the stream function i s given by 41 = tyR cos 9 - 4>j s i n 0 . The line-along which i t vanishes i s determined by tan 0 = 4 J R/^ JI / with slope d0/dx = - (cos 2 8/4^ 2 ) F ( ^ R / ^ j ) . Hence T 1 2 a - d0/dx and Figure 5.7 shows that i t i s everywhere negative. Since V* < 0 for x < 1, (4.11) reveals the rather surprising r e s u l t that over the shelf, T 1 2 acts to remove energy from the nonrandom part of the disturbance and to strengthen the mean shear. This i s i n concert with the finding of N i i l e r and Mysak (1971) that the continental shelf acts as a s t a b i l i z i n g factor. Of course, the largest source term i n (4.11) i s the one proportional to a which shows that the disturbance energy i s extracted primarily from the fluctuating part of the basic current. Plots of the mass transport stream function are of limi t e d value 0.6 0.5 0.4 9 / 2 IT 0.0 Figure 5.7A The mass transport stream function for (A) channel mode 1, (B) channel mode 2, (C) channel mode 3, (D) shelf mode 1, (E) shelf mode 2, (F) shelf mode 3. Here 9 = k(y - c r t ) . The amplitude i s a r b i t r a r y . en 0/2 TT Figure 5.7B Figure 5.7C Figure 5.7D 0 / 2 TT UI O Figure 5.7F 1.0 0.5 0.0 -0.5h -1.0 1.0 0.5 0.0 M O D E I U x io 0.0 0.5 1.0 X 1.5^^20^25 - 51 0.5 h -1.0 1.0 0.5 0.0 -0.5 -1.0 M O D E 2 x s — v. 0.0 0 . 5 K 10___^--t5 / 2.0 25 - x io \ V M O D E 3 X 0.0 V)5 Sfe 2.0N / J * 10 3 of u and v f o r : (1) channel mode l i n e 9/2TT = 0.8 i n F i g . 7a, (2) channel mode 2 taken along 6/2TT = 0.9, and (3) channel mode 3 taken along 9/27T = 0.37. In each case the values are normalized by V(x = 0). 52 i n v i s u a l i z i n g the v e l o c i t y structure over the she l f . Figure 5.8 shows selected p r o f i l e s of u and v corresponding to Figure 5.7. In a l l cases the motion i s trapped against the coast. These r e s u l t s are now compared with observations made i n the F l o r i d a S t r a i t s . The occurrence of flu c t u a t i o n s i n the F l o r i d a Current with periods ranging from a few days to several weeks i s well known, and the following review i s not intended to be exhaustive; the reader i s refe r r e d to the papers referenced here f o r a more extensive discussion. A plan view of the F l o r i d a S t r a i t s i s shown i n Figure 5.9. I t reveals the channel-like topography and i l l u s t r a t e s the sharp turn the F l o r i d a Current must make on i t s northward passage. Cross sections of O and alongshore v e l o c i t y are presented i n Figure 5.10. In addition,to showing the highly b a r o c l i n i c nature of the mean flow, i t also indicates some of the lo n g i t u d i n a l v a r i a t i o n s i n bathymetry and i n the density and v e l o c i t y f i e l d s . The s t r a t i f i c a t i o n i s compressed over the s h e l f , and a t y p i c a l value of the _2 -1 Brunt-Vaisala frequency i n the pycnocline i s 2 x 10 rad s (Mooers and Brooks, 1977). In a marked contrast to low-frequency motions i n the open ocean that are characterized by a red spectrum, there appears to be a sp e c t r a l gap between motions with periods of about 25 days and 1 year (Brooks and N i i l e r , 1977; Diiing et a l . , 1977; Wunsch and Wimbush, 1977). Duing et a l . (1975) estimate from mid-channel v e l o c i t y measurements that approximately 80% of the nontidal variance occurs at periods exceeding 8 days. In general, the low-frequency motions may be broken i n t o three time scales, 8-25 days, 4-5 days, and 2-3 days (Diiing et a l . , 1977) each of which i s treated separately here. Seasonal f l u c t u a t i o n s i n the F l o r i d a Current have also been observed ( N i i l e r and Richardson, 1973). Figure 5.9 Plan view of the F l o r i d a S t r a i t s .^showing l i n e s I and II along which the sections i n F i g . 5.10 are taken (from Mooers and Brooks, 1977). I. n. DISTANCE ( k m ) Figure 5.10A Sections along l i n e s I and II of (A) G t and (B) alongshore v e l o c i t y (from Mooers and Brooks, 1977). n. DISTANCE (kir.) 40 60 80 0 20 40 60 80 MEAN AXIAL FLOW (V) IN cm s ' 1 Figure 5.10B 56 The 8-25-Day Band From the analysis of year-long records of sea l e v e l , sea temperature, and atmospheric pressure, Brooks and Mooers (1977b) demonstrated the existence of southward t r a v e l l i n g waves with periods of 7-10 days i n winter and 12—14 days i n summer and speeds of 100 cm s ^ or greater. Strong coherence between sea l e v e l and temperature and the atmospheric variables showed that these disturbances were wind generated; a f i t of the BrM CSW model to these observations was only p a r t i a l l y successful as the model predicts wave speeds le s s than 50 cm s Schott and Duing (1976) applied a single barotropic wave model to the r e s u l t s obtained from the analysis of 65 days of current measurements taken concurrently at stations separated i n the alongshore d i r e c t i o n and found a s t a t i s t i c a l l y s i g n i f i c a n t f i t for the 10-13-day wave period band and a marginal f i t f o r the 7-10-day band. In the former case they calculated a wavelength of 270 km, a southward -1 -1 phase speed of 17 cm s , and an amplitude of 14 cm s , values which are i n e x c e l l e n t agreement with the BrM model. The wave parameters were s i m i l a r for the 7-10-day band. Duing et a l . (1977) concluded that 9-20-day o s c i l l a t i o n s possessed amplitudes ranging from 15-25 cm s-"*"; they also showed that disturbances i n the 10-14-day band occurred i n t e r m i t t e n t l y as phase-coherent wave packets c o n s i s t i n g of 4-6 cycle s . Since high coherence was observed between the v e l o c i t i e s and the atmospheric v a r i a b l e s , e s p e c i a l l y the wind stress c u r l , i t appears l i k e l y these motions are, i n i t i a l l y at l e a s t , atmospherically forced and represent stable, modified s h e l f waves. The 4-5-Day Band In the 4-5-day wave period band, Diiing (1975) described a nearly 57 barotropic wave, 160-240 km i n length, that propagates northward with a mean speed of 45 cm s and an amplitude of about 10 cm s \ An i n t e r e s t i n g manifestation of t h i s disturbance i s the r e v e r s a l of the b a r o c l i n i c mean flow at depth on the western side of the channel that accompanies i t s passage. Based on the analysis of s i x months of current, temperature, and bottom pressure measurements taken i n 1974, Wunsch and Wimbush (1977) have also described a northward t r a v e l l i n g 4-7-day wave about 60 km i n length. Diiing et a l . (1977) showed that l i k e the 10-14-day motion, a 4-5-day disturbance occurs i n t e r m i t t e n t l y as a wave packet of about 4 cycles and that i t i s s i g n i f i c a n t l y correlated with the wind stress c u r l and other atmospheric v a r i a b l e s . No i n d i c a t i o n of the d i r e c t i o n of wave propagation was given.. The 2-3-Day Band Lee (1975) and Lee and Mayer (1977) have documented the existence of wave-like meanders of the mean flow and the transient occurrence of cyclonic " s p i n - o f f " eddies i n the 2-3-day band. These eddies are trapped against the continental boundary, have a l a t e r a l length scale of about 10 km and a l o n g i t u d i n a l one 2 to 3 times greater. They occur at approximately weekly periods, propagate northward at speeds ranging between 20-40 cm s 1 and p e r s i s t for up to 3 weeks. A kinematical model of a p a i r of v o r t i c e s superimposed on the mean flow gave a good representation of the observed near-surface current. The meanders also propagate northward but at speeds between 65 and 100 cm s~^~. I t i s for the motions i n the 4-5-day band that the present theory might o f f e r a possible explanation. Indeed, barotropic i n s t a b i l i t y of the 58 mean flow has been suggested by Duing (1975) as a l i k e l y mechanism f o r these motions. We should note, however, that Brooks and N i i l e r (1977) determined that, i n the mean, the F l o r i d a Current i s i n an equilibrium state and that the net interchange of energy between the mean current and the fl u c t u a t i o n s superimposed upon i t i s extremely small. Of course, t h i s does not rule out the p o s s i b i l i t y that disturbances i n some frequency ranges may extract energy from the mean flow. Furthermore, i t has been shown that the primary source of energy f o r the motions described i n t h i s paper i s the small, sheared, f l u c t u a t i n g component of the basic current. Nevertheless, Brooks and N i i l e r ' s work indicates that along-stream v a r i a t i o n s i n the flow, as well as i t s b a r o c l i n i c nature, may be s i g n i f i c a n t . For the parameters appropriate to the F l o r i d a S t r a i t s , Figure 5.6a indicates phase speeds of about 40 cm s-"'" for modes 1 and 3, and 25 cm s~^~ f o r mode 2 f o r the wavelength range of 160-240 km. These r e s u l t s are independent of £ and 0 " . However, the corresponding growth rates are strong functions of these factors as i s i l l u s t r a t e d i n Table I I . In general, the higher modes are more unstable, and i n p a r t i c u l a r , i t i s seen that the t h i r d mode could grow s i g n i f i c a n t l y within one wave period for a wide range of values of £ and 0 ~. Diiing's (1975) p l o t s of the eastward v e l o c i t y component imply the existence of a second or t h i r d (or higher) mode; unfortunately, h i s measurements extend only over 2/3 of the channel width. Wunsch and Wimbush (1977) have calculated v e l o c i t y cross-spectra, and the phase differ e n c e at 5 days between northward v e l o c i t i e s measured at the continental boundary and the shelf break, and at the shelf break and the eastern wall i s approximately 180° i n each case. This i s consistent with a t h i r d mode unstable wave but not a second mode disturbance. On the basis of these r e s u l t s we conclude that a mode 3 f l u c t u a t i o n as described by the 59 Table I I . The c h a r a c t e r i s t i c growth times T = l/ft f o r a 200-km wave (k/27T = . 1 5 ) . The threshold values a „ are also given. Mode 1 Mode 2 Mode 3 £ . 1 . 2 5 . 5 0 . 1 . 2 5 . 5 0 . 1 . 2 5 . '50 a 1 6 . 0 6 . 6 0 3 . 5 0 1 3 . 0 5 . 5 0 3 . 3 0 5 . 0 3 . 3 0 3 . 0 0 T a T (days) 3 . 0 - - - - - - - - 4 . 4 0 4 . 0 - - 2 . 4 0 - - 2 . 6 0 - 7 . 0 0 2 . 5 0 5 . 0 - - . 8 6 • - 1 . 2 0 - 3 . 2 0 1 . 4 0 7 . 5 - - . 3 7 - 2 . 6 4 . 5 6 5 . 5 1 . 5 0 . 7 2 1 0 . 0 - - . 2 5 - . 7 8 . 3 8 3 . 1 1 . 0 0 . 5 0 60 present theory provides a possible explanation of Diiing's observations. I t i s p a r t i c u l a r l y i n t r i g u i n g that the mode 1 and mode 3 dispersion curves cross i n the range of i n t e r e s t , but i t would be improper to draw any conclusions from t h i s observation. I t i s i n t e r e s t i n g to speculate that some r e l a t i o n s h i p might e x i s t between a mode 3 wave i n the 2-3-day band and the spinoff eddies described by Lee. The propagation speeds are s i m i l a r and the "inner gyre" i l l u s t r a t e d i n Figure 5.7 i s approximately the same size as the observed eddies. (Although Figure 5.7 corresponds to k / 2 7 T = .15, i t s form changes r e l a t i v e l y l i t t l e as k i s increased.) 61 6. The Continental Shelf Model We now turn to the continental shelf model mentioned i n the introduction. In order to examine a coastal phenomenon such as the modification of CSWs by a mean current, i t i s necessary that any mechanisms capable of a l t e r i n g the p o t e n t i a l v o r t i c i t y d i s t r i b u t i o n be l o c a l i z e d with respect to the coast. In a l l studies of CSWs i n the absence of a basic 2 flow, the gradient of the background p o t e n t i a l v o r t i c i t y , h'/h , has always died out away from the coast so that t h i s requirement was automatically f u l f i l l e d . I f one includes a sheared mean current, however, -1 -1 2 the p o t e n t i a l v o r t i c i t y gradient becomes h V" - (Ro + V')h'/h , i t 2 i s no longer s u f f i c i e n t that only h'/h decay as x -*• °°. N i i l e r and Mysak (1971) avoided t h i s problem by employing a piecewise l i n e a r current that became constant at a small distance offshore. McKee (1977) and Brooks and Mooers (1977a) have avoided t h i s problem by adopting a channel model, although Brooks and Mooers b r i e f l y discussed a continental s h e l f model u t i l i z i n g an exponentially decaying current. Grimshaw (1976) also required the mean current to diminish exponentially. Our approach i s to require V"B to be bounded away from the s h e l f edge. S p e c i f i c a l l y , we choose V(x) = s(x)v(x) (6.1) W(x) = s(x)w(x) where s (x) i s a deterministic function that s a t i s f i e s s (x) ->• 0 as x -* oo, and we assume w(x) i s a homogeneous random function. In terms of 62 the c o r r e l a t i o n function r (E,) = E[w(x)w(x + E,) ] one finds that 2 R(o) = s r(o) 2 R'(o) = s r'(o) + ss'r(o) (6.2) 2 R" (o) = ss"r(o) + 2ss'r'(o) + s r"(o) 2 With the choice of r(o) = 1 and C = - r"(o) we f i n d the equation equivalent to (3.27) to be [(V - c) - £2s2]Vty - (V - c)Q x^ - £ 2h 1[s2a2 - ss" + ss'h'/h]^ = 0. (6.3) The boundary conditions are ty(x) = 0 at x = 0 (6.4) i>(x) -»- 0 as x -»• °°. The i n t e r p r e t a t i o n of (6.3) i s i d e n t i c a l to that of (3.27), but we note 2 2 that the term corresponding to e a i n (3.27) i s more complex here and involves a contribution from R'(0). The appropriate matching conditions are [{(V - c ) 2 - e 2s 2}^'/h] - [(V - c)p>] - £ 2[ss'^/h] = 0. (6.5) The general form of the BrM model i s retained and we choose 63 r l-x x e V(x) = s(x)v(x) = < 0 5 x 5 1 U(l-x) x e x > 1 (6.6) r s(x) = < y ( i - x ) 0 5 x 5 1 x > 1 (6.7) 2b(x-l) h(x) = < 0 5 x 5 1 x > 1 (6.8) where y i s p o s i t i v e . As before, we seek a perturbation s o l u t i o n of the form (5.4) The three lowest order equations are Vi>Q + h 1 ( £ s / c i 0 ) 2 ^ ( ) = 0 (6.9) -1 2 V^>1 + h (es/c i 0) i j ^ = - c ± 0 1 { 2 [ c i l + i ( V - c r 0 ) ] ^ 0 - iQ x V (6-10) 2 2 V4>2 + h (es/c .0) i f 2 = c i 0 {(V - c r 0 ) - 2 c i 2 c i 0 - c n - e s 2i[(V - c r 0 ) C i l - c i 0c r l]W 0 - 2 c i 0 " [ C i l + i ( V - c ^ ) ] ! ^ -2 2. -1 lO --x' {Q Yt(v - c r Q ) - i c . J - e h ss" + e^ss'h'/h^ifj ' i l (6.11) 64 For x < 1, (6.9)-(6.11) are i d e n t i c a l with (5.8)-(5.10). The zeroth order so l u t i o n s a t i s f y i n g both the boundary and matching conditions i s : ^0 = A0< b(x-l) . y , . y e s i n Ax/sin A J v ( p C ) / J „ ( p ) (6.12) .2 2 , 2 ,2 ,2 A = £ /C^Q - k - b (6.13) v = k/y p = £/yc iO C = e y(1-x) (6.14) provided that b + X/tan X = - k + ( e / c i 0 ) J v + 1 ( p ) / J v ( p ) (6.15) I f X i s negative, the solutions are hyperbolic over the shelf and may be obtained by replacing X with iX. The graphical s o l u t i o n of (6.15) c l o s e l y resembles that of (5.14), but note that (6.15) must be solved separately for each choice of £, y, and k . This s o l u t i o n i s not v a l i d for k = 0 since J Q cannot s a t i s f y the boundary condition at x = 0 0 ( i . e . , at 5 = 0). The f i r s t order s o l u t i o n i s of the form r An% + i A o p i 12 0 0 2 (6.16) 65. and the p a r t i c u l a r solutions are s p e c i f i e d i n Appendix B. In order f o r il>2_ t o s a t i s f y the matching conditions, the consistency condition (5.27) must again be s a t i s f i e d ; i t was used as a check on the numerical r e s u l t s derived here, and i t proved extremely s e n s i t i v e to the accuracy of the roots of (6.15). We again choose A-^ = 0 and A 1 2 i s s p e c i f i e d by (5.28) App l i c a t i o n of the Fredholm a l t e r n a t i v e to (6.9)-(6.11) implies that c r Q = <V> + (c i 0 2/2e 2)<hQ x/s 2> (6.17) c i l ' c r l = 0 (6.18) where and C i 2 C i O = ~ ( 3/2)<(V - c r Q ) 2 > - (e 2/2)<s 2> (c i 0 2/2£ 2)<(V - c r Q ) h Q x / s 2 > - (c i 0 2/2)<s"/s> + (c i 0 2/2)<h's'/hs> - i c i 0 < ( V - c r Q ) | T ) ; ^ - i ( c i 0 3 / 2 £ 2 ) < h Q x / s 2 | \l>±> (6.19) <f (x) | g(x)> = / h ^-s2^ (x ) f(x ) g(x)dx// h - 1 s V > 2 (x) dx (6.20) 0 0 <f(x)> = <f(x) | ̂ Q(x)> (6.21) 66 by comparison with (5.17)- (5.19) , we see that the terms inv o l v i n g the p o t e n t i a l v o r t i c i t y gradient are emphasized since s < 1 o f f the s h e l f . Otherwise the i n t e r p r e t a t i o n of (6.17)- (6.19) remains unchanged. A comprehensive study of the dependence of the various r e s u l t s on y was not c a r r i e d out, and the choice ]i = 1 was made to f a c i l i t a t e comparison with the channel mode. The dependence of 0,^ on o~ i s s i m i l a r to that shown i n Figure 5.3, and i s not shown here. The threshold values of 0" are smaller for modes 1 and 3 and larger for mode 2 than they are for the corresponding channel modes. P l o t s of fi^ and c rQ as functions of k are i l l u s t r a t e d i n Figures 5.4b and 5.6b and a d e t a i l e d comparison with t h e i r channel counterparts reveals nearly as many differences as s i m i l a r i t i e s . Two general conclusions may be drawn, however. As e i t h e r k or the mode number increases, the d i s p a r i t i e s between the two models increase for and decrease f o r c r Q . This i s p h y s i c a l l y reasonable since as k or the mode number increases, the e f f e c t i v e wavelength decreases; thus, the wave should become less s e n s i t i v e to the outer boundary and more s e n s i t i v e to the basic current p r o f i l e . Now fi^ depends intimately on t h i s p r o f i l e and so the two models should be i n c r e a s i n g l y disparate at small wavelengths. S i m i l a r l y , the wave frequency does not depend on the d e t a i l s of the basic current but i t c e r t a i n l y depends strongly on the p o s i t i o n of the channel w a l l . Hence the phase speeds predicted by the two models should d i f f e r f o r long wavelengths. We also note that the growth rates for the d i f f e r e n t models no longer overlap. Figure 5.5b shows that the group v e l o c i t y i s always p o s i t i v e , and from the slope of the curves i n Figure 5.6b we see that, with the exception of the f i r s t mode at small wavenumbers, i t exceeds the phase v e l o c i t y . F i n a l l y , the mass transport stream function i s shown i n Figure 5.7; i t generally resembles i t s channel counterpart although the axis t i l t i s increased. 67 7. Rossby Waves i n a Random Zonal Flow In t h i s section we examine the i n t e r a c t i o n of small-amplitude, nondivergent, free barotropic Rossby waves with a sto c h a s t i c , sheared zonal current. For convenience we assume that the flow i s confined to an i n f i n i t e l y long channel with side walls at y = 0,L. The theory i s also applicable to topographic Rossby waves provided that the bottom slope a i s small enough so that aL/H i s of the order of the Rossby number. The following analysis i s e n t i r e l y analogous to that of the preceding sections, and i t i s therefore presented i n as succinct a fashion as po s s i b l e . We note the rel a t e d study of K e l l e r and Veronis (1969) who examined the propagation of Rossby waves i n a weak random current on an i n f i n i t e $-plane. For a zonal flow of zero mean they found that the waves were damped and the wave speed reduced. In terms of the v e l o c i t y stream function defined by u (7.1) v x the nondimensional, l i n e a r i z e d v o r t i c i t y equation i s 0 T + u B 3 x)? 2T + V B T U ' V = 0 (7.2) where the zonal current Ug(y) defines the basic state. As scale factors we choose an average current U f o r (u,v), the channel width L f o r 68. 2 (x,y), and L/U f o r the time; 3 i s nondimensionalized by L /U so that the dimensionless C o r i o l i s parameter i s f = 1 + Ro3y. For a t r a v e l l i n g wave s o l u t i o n of the form ik(x-ct) V = <Hy)e (7.3) (7.1) reduces to (U B - c)($" - k 2$) + (3 - U" B)$ = 0 (7.4) which i s p r e c i s e l y the equation f i r s t considered by Kuo (1949) . The boundary conditions require no flow normal to the channel w a l l , hence <3?(y) = 0 at y = 0,1. (7.5) We decompose U B into i t s mean and f l u c t u a t i n g components as U B = U(y) + £W(y) (7.6) where E(U_,) = U and E(W) = 0, and we choose E(W ) = 1. Then a p p l i c a t i o n of the theory developed i n Section 3 leads to an equation f o r the mean part of $, [(u - c ) 2 - e2]RJj + (u - c)Qy4> - e2o24> = o. (7.7) Here F = d 2/dy 2 - k 2 , cr2 = - R" (0) , and Q y = (3 - U") . A perturbation expansion of the form (4.4) leads to the following 69 r e s u l t s , Here c i 0 = e 2/K 2 (7-8) c r Q = <U> + <Qy>/2K2 (7.9) c , ,c , = 0 (7.10) i l r l c i 2 c i 0 = - (3/2XCU - c r Q ) 2 > - £ 2/2 + (1/2K 2)<(U - c r 0)Qy> - i c i 0 < ( U - c r Q ) | tyf + i ( c i 0 / 2 K 2 ) < Q y | tyf . (7.1D 1 1 2 <f(y) | g(y)> = / (y) f (y) g (y) dy/J iJJ0(y)dy (7.12) 0 0 and <f (y)> = <f (y) | 4>Q(y)> (7.13) 2 2 2 2 where tyQ = A Q sin(nTry) and K = k + n TT . Again one finds that c 2 n + i vanishes and that ^ n ^ O a n < ^ ^2n+1^0 a r e rea-'- a n c ^ imaginary q u a n t i t i e s respectively< These expressions are generalizations of those found by Manton and Mysak (1976) for plane Couette flow; t h e i r r e s u l t s may be recovered by putting U = y and Q v = 0. In p a r t i c u l a r , they showed that a l l modes t r a v e l l e d with the same constant phase speed; the i n c l u s i o n of a nonzero 70 v o r t i c i t y gradient i n the present model serves to separate the dispersion curves as i s revealed by (7.9) . Somewhat more s p e c i f i c r e s u l t s are given i n Appendix C f o r a parabolic flow model, U = 3y (y - D/2. F i n a l l y , we note that the present theory i s not i n c o n f l i c t with the study of K e l l e r and Veronis which predicts wave damping. Their r e s u l t s require two-dimensional,' t r a n s l a t i o n a l invariance of the basic state (Keller, 1967), a condition which cannot be s a t i s f i e d by a system with a sheared mean current or by one confined to a channel. Hence t h e i r r e s u l t s do not apply to the present system, even i n the l i m i t i n g case of U = 0. 71 8. Summary and Concluding Remarks to Part I I t has been demonstrated that shelf and Rossby waves propagating through a region of basic sheared current of the form Vg = V + £W where W i s a centred random function may be unstable i f the l a t e r a l c o r r e l a t i o n length of W i s small compared to the c h a r a c t e r i s t i c length scale of the problem. This i s true whether or not V s a t i s f i e s the well-known necessary conditions for barotropic i n s t a b i l i t y . The growth rate of these disturbances i s p r i n c i p a l l y determined by the inverse of the c o r r e l a t i o n length. The phase speed i s the sum of weighted cross-stream averages of the mean current and the mean gradient of p o t e n t i a l v o r t i c i t y . Depending on the Rossby number of the system, the waves may t r a v e l with or against the mean flow. When t h i s theory i s applied to a model of the F l o r i d a S t r a i t s , unstable CSWs are found with properties that are i n good agreement with observations made by Duing (1975). I t may, therefore, o f f e r an explanation for some of the observed meanders of the F l o r i d a Current. The present theory could obviously be extended i n many ways. A de t a i l e d comparison i s needed with an i n t r i n s i c a l l y unstable system i n order to compare growth rates. One could also introduce a small, random cross- stream v e l o c i t y i n t o the basic flow. The problem of mode-coupling has yet to be resolved as does the e f f e c t of temporal or along-shore v a r i a t i o n s i n the basic current. Of course, the present theory represents only a f i r s t step i n a more comprehensive examination of the e f f e c t s of random p o t e n t i a l v o r t i c i t y d i s t r i b u t i o n s on b a r o t r o p i c - b a r o c l i n i c i n s t a b i l i t i e s . 72 9. Introduction to Part II The waters l y i n g between Vancouver Island, the mainland coast of B r i t i s h Columbia, and the State of Washington (see Figure 9.1) are important from economic, environmental, navigational, and r e c r e a t i o n a l points of view. Oceanographically i t i s a complex estuarine system. In addition to the major influences of t i d e s , fresh water inflow, topography, C o r i o l i s force, winds and other atmospheric v a r i a b l e s , one must consider the intense mixing that occurs i n the channels that separate the S t r a i t of Georgia (GS) from Juan de Fuca S t r a i t i n the south, and Queen Charlotte Sound i n the north. Part II of t h i s thesis represents an attempt to understand some of the r e s u l t s presented by Chang et a l . (1976; see also Chang, 1976). From the analysis of 18 months of current records c o l l e c t e d along l i n e H i n GS (see Figure 10.1), Chang showed that nearly one-half of the k i n e t i c energy associated with h o r i z o n t a l motions i s contained i n broad-banded, low-frequency current f l u c t u a t i o n s characterized by periods ranging from 4 to over 200 days. No f o r c i n g mechanisms were evident from Chang's analysis as the coherences between the currents and the wind, atmospheric pressure, sea l e v e l , and water temperature were a l l calculated to be very small. In an e a r l i e r attempt to understand the low-frequency dynamics of GS, Helbig and Mysak (1976) constructed an a n a l y t i c model that included both bottom topography and density s t r a t i f i c a t i o n . This model admits northward-travelling topographic planetary waves with periods that l i e i n the observed range, but i t i n c o r r e c t l y predicts the v e r t i c a l d i s t r i b u t i o n of h o r i z o n t a l energy. Helbig and Mysak (1976) suggested that b a r o c l i n i c i n s t a b i l i t y of the mean flow was a l i k e l y mechanism to account f o r observed f l u c t u a t i o n s , and i t i s from t h i s premise that the present study commenced. Figure 9.1 Plan view of the west coast of B r i t i s h Columbia and adjoining waters. 74 As i t turns out, t h i s conjecture i s probably i n c o r r e c t as the a n l y s i s presented i n Sections 11 and 12 shows. Two simple s t a b i l i t y models were constructed of a purely b a r o c l i n i c and barotropic system, r e s p e c t i v e l y . For the b a r o c l i n i c system the r e s u l t s indicate that the mean flow i s unstable for only a narrow band of wave numbers. An unstable shear wave exis t s at a l l wavelengths i n the barotropic system and for a 15-day period, has an e-folding time of about 8 days. However, as i s shown i n Section 12, the observed Cartesian v e l o c i t y components are generally i n phase i n d i c a t i n g that the motions are not composed of the types of waves studied here. These r e s u l t s imply that i n e r t i a l i n s t a b i l i t y plays only a minor r o l e , at most, in the dynamics of GS. A d d i t i o n a l current data c o l l e c t e d by the Canadian Hydrographic Service at points not along l i n e H (see Figure 10.1) were also examined. Since these records were of l i m i t e d length (about 30 days) t h e i r analysis i s subject to severe s t a t i s t i c a l l i m i t a t i o n s . I t i n d i c a t e s , however, that during the observation period an a n t i c y c l o n i c gyre existed i n the southern h a l f of GS. I n t e r e s t i n g l y , t h i s c i r c u l a t i o n was of the opposite sense to that postulated by Waldichuck (1957). The longer-term records investigated by Chang were also reexamined with the object of gaining fresh i n s i g h t s . In p a r t i c u l a r , Chang calculated rotary spectra which y i e l d no d i r e c t information about the i n d i v i d u a l Cartesian v e l o c i t y components. As mentioned, the present r e s u l t s i n d i c a t e the motion i s not comprised of simple waves. Cross-spectra between the currents and wind stress were computed. While generally low coherence was found, a consistent phase pattern seemed to emerge. In the frequency range of i n t e r e s t , the along-channel currents are 180° out of pria.se with, trie wind.. The conjecture i s made that the f o r c i n g of the low-frequency motions i s not 75 d i r e c t but rather that the winds i n t e r a c t nonlinearly with the t i d e s and Fraser River outflow to modulate the estuarine c i r c u l a t i o n of the system. An examination i n the time domain of winds and currents suggests that the water column responds most d i r e c t l y to the wind along the eastern side of GS with a lag of about f i v e days. The response elsewhere i s not c l e a r . Residual t i d a l currents were calculated from the time ser i e s of barotropic t i d a l streams generated from the numerical model of the Juan de Fuca-Strait of Georgia system developed by Crean (1976, 1978). A coherent pattern of residuals that varied with the f o r t n i g h t l y t i d a l cycle was found. These currents were i n s u f f i c i e n t l y large and of the wrong d i r e c t i o n to explain the observations, however. The o u t l i n e of Part II i s as follows. A b r i e f d e s c r i p t i o n of the p h y s i c a l oceanography of GS i s given i n Section 10 and includes a discussion of the possible character of the observed low-frequency currents and an enumeration of various f o r c i n g mechanisms that might be important. Two simple i n e r t i a l i n s t a b i l i t y models are considered i n Section 11, while the data analysis i s presented i n Section 12. In Section 13, t i d a l r e siduals are calculated, and a b r i e f development of the concept of modulated estuarine flow i s given. In Section 14, the key points of Part II are summarized. 76 10. Physical Oceanography of the S t r a i t of Georgia Although the p h y s i c a l oceanography of GS has received comprehensive treatment elsewhere (cf. Waldichuck, 1957), i t i s important to summarize here some of i t s p r i n c i p a l features to provide a motivation for the following study. Some parts of t h i s d e s c r i p t i o n are abstracted from Helbig (1977) . A plan view of GS i s shown i n Figure 10.1. I t reveals that the average width of GS i s about 30 km while i t s length i s s l i g h t l y l e s s than 250 km. Thus, the aspect r a t i o of channel length to width i s approximately 8:1. Bathymetric sections along the l i n e s 1-10 are presented i n Figure 10.2, and were extracted from a topographic map of GS compiled by Dr. P. B. Crean (personal communication) giving average depths over 2-km squares throughout the S t r a i t . Even though small-scale features are i m p l i c i t l y smoothed, the bathymetry exhibits great i r r e g u l a r i t y , p a r t i c u l a r l y i n the northern sector. In general, extremely steep slopes characterize GS -2 along i t s western boundary, while slopes nearly as steep (exceeding 10 ) are common along the east. North of l i n e 4, two channels e x i s t : a narrow one to the east of Texada Island and a much wider one on the western sid e . South of l i n e 4 the topography becomes progressively smoother; l i n e s 7 and 8 i l l u s t r a t e the marked e f f e c t of Fraser River sedimentation as extensive banks on the east. The l o n g i t u d i n a l section 10 reveals that although the a x i a l bathymetry i s somewhat smoother than the transverse bathymetry, i t s t i l l possesses a high degree of i r r e g u l a r i t y and exhibits slopes that often -2 exceed 10 Figure 10.3 shows l o n g i t u d i n a l sections of density f o r winter and summer. In the upper 50 metres near l i n e 7 (see Figure 10.1) there e x i s t s a strong seasonal v a r i a t i o n which i s associated with the outflow of fresh Figure 10.1 Plan view of the S t r a i t of Georgia showing l i n e s of topographic cross sections (1-10) presented i n F i g . 10.2. 78 • — 3 0 0 4 0 0 DISTANCE FROM NORTHERN BOUNDARY (Km) 400 i Figure 10.2 Topographic cross sections: (A) Upper panels: 1-9; (B) lower panel: 10. The v e r t i c a l exaggeration i s 30:1 i n (A) and 150:1 i n (B). The insets i n d i c a t e slopes of 10~ 2. C. FLATTERY BOUNDARY C.MUDGE Figure 10.3A Longitudinal section of Ot for (A) December 1968; (B) July 1969 (from Crean and Ages, 1971). Figure 10.3B 03 o 81 1/2 water from the Fraser River. The Brunt-Vaisala frequency N = [-gpz/p^] generally l i e s i n the range of 3 x 10~ 3 to 3 x i o ~ 2 rad S - 1 throughout the water column which i s thus well s t r a t i f i e d . Here P z i s the v e r t i c a l density gradient, p^ i s a representative value of the density, and g i s the acceleration due to g r a v i t y . The winds i n GS are strongly affected by the surrounding mountainous t e r r a i n , and they are predominantly up- or down-strait; that i s , to the northwest or southeast, r e s p e c t i v e l y (Kendrew and Kerr, 1955). During the winter months of November through March the p r e v a i l i n g wind i s u p - s t r a i t while i n the summer months of June to September i t i s down-strait. In a l l seasons the strongest winds are from a southerly d i r e c t i o n . There i s of course a great deal of v a r i a t i o n about t h i s average pattern. Although Waldichuck (1957) indicates that a cyclonic gyre e x i s t s over the southern s t r a i t during the winter, the evidence f o r t h i s seems weak. We do note, however, that the p r e v a i l i n g wind at Vancouver i s usually to the west. The rotary spectrum of the winds from Sand Heads computed by Chang (1976) i s shown i n Figure 10.4. A rotary spectrum of a vector process, say u, i s obtained by r e s o l v i n g each frequency component of the d i s c r e t e Fourier transformed vector u into two other components, one of which rotates with a p o s i t i v e frequency (anti-clockwise) and the other with a negative frequency (clockwise). This gives a p a i r of spectra representing the respective tendency of the process to move i n an anti-clockwise or clockwise sense. (The reader i s r e f e r r e d to Chang (1976) or Mooers (1973) for a comprehensive discussion of rotary current spectra.) Notice that a p l o t of the spectrum m u l t i p l i e d by the frequency (f o S) against the logarithm of the frequency i s variance preserving; that i s , the area under the curve i s d i r e c t l y proportional to the variance. 82 In the present case, Figure 10.4 indicates the cyc l o n i c tendency of the wind. The spectra are broadly peaked about 3-5 days but the wind has s i g n i f i c a n t energy to periods as large as 25 days. Approximately 10 percent of the variance i s contained i n the 10-20-day band and one-third of the variance i s i n periods exceeding 7 days. Some r e s u l t s of Chang's analysis of GS currents are presented i n Figures 10.5-10.8. The current records examined were c o l l e c t e d a t st a t i o n s H06, H16 and H26 as shown i n Figures 10.1 and 10.5. Meters were positioned at 3, 50 and 200m at the western (H06) and ce n t r a l (H16) locations and at 3, 50 and 140 m i n the east (H26). Chang did not analyze records from the near surface instruments. Most of the current records were obtained with Aanderaa Model 4 current meters, but several Geodyne Model 850 meters were employed. The currents were sampled ei t h e r every 10 (Aanderaa) or 15 (Geodyne) minutes. A subsurface buoy mooring was used f o r the i n i t i a l year of the experiment, but was replaced thereafter by a surface buoy, taut-rope mooring. Although the threshold l e v e l of these meters i s 1.5 cm s \ t h i s presents minimal d i f f i c u l t i e s i n the detection of small, low-frequency currents since stronger t i d a l currents were superposed on these f l u c t u a t i o n s . The mean currents computed over the 18-month period are shown i n Figure 10.6. There are two s i g n i f i c a n t features. The f i r s t i s the strong, cross-channel flow at the 50-m cen t r a l l o c a t i o n , and the second i s the very strong current found at 140 m i n the east. The mean speed there i s f i v e times greater than that found at the other deep l o c a t i o n s , while the root mean square v e l o c i t y i s twice as large. In the east and the west, both shallow and deep currents are c l o s e l y aligned with the l o c a l topography. Figure 10.7 shows the current spectra obtained by summing the respective p o s i t i v e and negative parts of the rotary spectra computed by PERIOD (DAYS) 510 2550100200500 —i r-1 \ \ \ \* 1' H i l l F—•—i 1 i f J j • • CM 0) CM £ 4.0- i i i 1.0 0.0 -1.0 "2.0 -3.0 -4.0 -3.0 -2.0 -1.0 0.0 ~ f log (f req /1 cpd) + f 1.0 Figure 10.4 Rotary spectrum of the winds at Sand Heads for the 600-day period beginning 3 January 1969 (from Chang, 1976). oo I Figure 10.5 Cross section H showing placement of current meters. ^ The moorings are spaced 10 km apart. The deep meters ' are (from west to east) 50, 80, and 25 m from the bottom (from Tabata et a l . , 1971). Figure 10.6 Mean currents along l i n e H for the 533-day period beginning 16 A p r i l 1969. 86 Chang. The area under the curve i s thus proportional to the t o t a l variance of the s i g n a l . Examination of t h i s f i g u r e reveals the complex nature of the low-frequency currents i n GS, but i t must be emphasized that most of the f i n e structure i s not s t a t i s t i c a l l y s i g n i f i c a n t to 95%. The s i g n i f i c a n t features i n Figure 10.7 are: (1) the spectra are broadbanded and appear to peak about 15-25 days, and (2) i n contrast to the 140-m s i g n a l from H26, the 200-m records contain l i t t l e energy i n comparison with the 50-m records. Chang (1976) found that coherences between currents at p o s i t i o n s separated both h o r i z o n t a l l y and v e r t i c a l l y were generally small at low frequencies as i s shown i n Figure 10.8. The highest value of the squared coherence between v e r t i c a l l y separated currents was observed i n the east and was only about 0.3. There the upper- and lower-layer rotary v e l o c i t i e s were nearly i n phase which may be i n d i c a t i v e of a barotropic motion. At the other locations the v e r t i c a l coherence was very small and the phases were scattered; t h i s r e s u l t suggests l i t t l e or no coupling between the upper and lower layers and hence implies mainly b a r o c l i n i c motions there. In a l l cases the h o r i z o n t a l coherences were below the 95% noise l e v e l . Chang also analyzed sea l e v e l , atmospheric pressure, wind, and water temperature records for the 18-month period. The temperatures were co l l e c t e d by the Aanderaa meters which were equipped to sample currents and temperatures concurrently. In a l l cases these quantities were e s s e n t i a l l y uncorrelated with the currents. The highest value of the squared coherence was found between the currents and the wind at the eastern l o c a t i o n (about .3 for both 50 and 140 m), which suggests that the surface wind stress may be a possible f o r c i n g mechanism. I t i s not clear from Chang's analysis what other forcing mechanisms are important. I t i s apparent, however, that the low-frequency currents must 87 -. 2001-0050251510 5 H 06,50 m PERIOD (DAYS) 50.0 200IQQ 50 251510 5 11* r 500 20010050251510 5 I - i 1 — i i i t i H 26,50 m 0.0 -3.0 -2.0 -1.0 0.0 log(f/lcpd) •3.0 -2.0 -1.0 0.0 Figure 10.7 Current spectra for l i n e H for the 533-day period beginning 16 A p r i l 1969 (adapted from Chang, 1976) . 88 ( A ) E A S T E R N -2 .5 -1.5 -1 FREQUENCY (CPD) -0.5 0 . 0 7 -1 0 . 9 5 H 0 . 9 0 . 8 0 . 7 O . S A i . 0 . 3 • o.i -i 1 o J 1 8 0 9 0 O ' - 9 0 - 1 8 0 4> 95V. 0.5 1 1.5 FREQUENCY (CPD) T 5 ( B ) CENTRAL - 2 . 5 •1.5 -1 — 1 — •0.5 0 . 9 7 . 0 . 9 5 - 0 . 9 • 0 . 8 • 0 . 7 - 0 . 5 : 0 . 3 : 0 . 1 -o - — i — 0.5 1.5 2.5 o - 9 0 (C ) WESTERN -2 .5 •1.5 -1 •0.5 0 . 9 7 -1 0 . 9 5 • 0 . 9 H 0 . 8 0 . 7 0 . 5 0 . 3 3 - 0 . 1 s o 1 8 0 - 9 0 • O • • - 9 0 • - 1 8 0 ; 0.5 1.5 T 5 Figure 10.8A Rotary coherence and phase spectra between currents, from (A) v e r t i c a l l y separated locations and (B) h o r i z o n t a l l y separated loca t i o n s . Notice that the frequency here extends to much higher values than are d i s - cussed i n the text. The s o l i d l i n e indicates a 95% noise l e v e l (from Chang, 1976). 89 (A) CENTRAL- 50m EASTERN •2.5 -1.5 -1 F R E Q U E N C Y ( C P D ) -0.5 0.97 0.95 0.9 0.8 0.7 0.5 0.3 -. 0.1 T 0 . 90 O - 9 0 - 180 -95V. 0.5 1 1.5 F R E Q U E N C Y ( C P D ) 2.5 (B) CENTRAL-WESTERN 50 m . 2.5 •1.5 -1 -0.5 0.97 • 0.95 ' 0 . 9 • 0.8 • 0 .7 • 0 . 5 : 0 . 3 • 0.1 - o- T 180 • 90 • O - 9 0 —r-~ 0.5 — i — 1.5 2.5 (C)CENTRAL 200m WESTERN 2.5 "2 •1.5 1 -0.5 0.97 • 0.95 - 0.9 - 0.8 - 0.7 - 0.5 - . 0 .3 - 0.1 -i o n u 180 90 • O • - 9 0 - 1 8 0 — i — 0.5 1.5 2.5 Figure 10.8B 90 r e s u l t from more or less continuous f o r c i n g of some kind; otherwise, f r i c t i o n would quickly damp out the motions. P r i o r to the enumeration of various possible f o r c i n g mechanisms, i t i s useful to consider what character the low-frequency motions might possess. F i r s t they could be wavelike. This c l a s s i f i c a t i o n includes both a superposition of plane waves (as i n Part I) i n which the dependence on the h o r i z o n t a l coordinates i s separable and more complex wavetypes (eddies) i n which i t i s inseparable. Waves could be d i r e c t l y forced, for example, by the wind, and move with the phase speed of the atmospheric disturbance, or they could be free and have a c h a r a c t e r i s t i c frequency. Moreover, waves could occur i n t e r m i t t e n t l y i n wave packets or e x i s t almost continuously. Possible s u b i n e r t i a l waves include i n t e r n a l K e l v i n and topographic planetary waves. Second, the low-frequency currents might be manifestations of transients that could be i n i t i a t e d by a v a r i e t y of d r i v i n g mechanisms. Third, they could consist of a superposition of any of these types. F i n a l l y , the motions might be describable only i n s t a t i s t i c a l terms. Any mechanism capable of a l t e r i n g the d i s t r i b u t i o n of momentum, v o r t i c i t y , or mass i n the system might force the low-frequency currents. Such mechanisms include the wind stress and wind stress c u r l which impart momentum and v o r t i c i t y , r e s p e c t i v e l y , to the system through the sea surface. In addition, the wind stress may introduce anomalies into the v o r t i c i t y d i s t r i b u t i o n by f o r c i n g water columns across bathymetric contours, thus stretching or compressing vortex l i n e s . Atmospheric pressure differences act i n a s i m i l a r manner at the sea surface. The momentum, v o r t i c i t y , and mass d i s t r i b u t i o n s may be altered i n t e r n a l l y i f the mean flow of the system i s i n e r t i a l l y unstable. In addition, nonlinear i n t e r a c t i o n s between t i d a l constituents may r e s u l t i n r e s i d u a l flows and produce t i d a l stresses 91 analogous to the usual Reynolds stresses (Heaps, 1978). S i m i l a r l y , the tides may i n t e r a c t with the topography to generate i n t e r n a l motions. F i n a l l y , freshwater influxes or intrusions of s a l i n e oceanic water r e s u l t i n density differences which i n turn drive currents. I f the magnitude of any of these mechanisms varies i n time, the resultant motions should vary i n a s i m i l a r fashion. Thus one might expect the spectrum of currents driven d i r e c t l y by the winds to be peaked about 3 to 5 days as i s the wind spectrum. This notion ignores the possible importance of the s p a t i a l c h a r a c t e r i s t i c s of the wind f i e l d , however, and i t i s conceivable that these motions might peak at some other frequency for which the length scales of the winds and currents were comparable. I t seems u n l i k e l y , however, that the s p a t i a l scale of the wind decreases with decreasing frequency. On the other hand, a r e l a t i v e l y modest s p e c t r a l component of the wind might be capable of e x c i t i n g a free wave at i t s c h a r a c t e r i s t i c frequency. In the case of t i d a l f o r c i n g , any process dependent upon the strength of the t i d a l streams should vary with a f o r t n i g h t l y period. This includes the turbulent mixing that occurs i n the constricted channels separating Juan de Fuca S t r a i t from GS (Figure 10.1). Thus int r u s i o n s of intermediate density water into GS r e s u l t i n g from the mixing of more dense, r e l a t i v e l y deep Juan de Fuca water with outflowing, comparatively l i g h t GS water could generate currents of f o r t n i g h t l y period. In f a c t , Herlinveaux (1957, 1969) has noted that semimonthly v a r i a t i o n s occur i n the surface s a l i n i t y and temperature at various locations i n the Juan de Fuca-Strait of Georgia system, and that these vari a t i o n s are most evident near the connecting passages. Webster and Farmer (1976) have substantiated t h i s observation from the analysis of a long timeseries of lighthouse s t a t i o n 92 data. These findings suggest that the degree of mixing depends on the t i d a l range and hence varies with a f o r t n i g h t l y period. I t i s also possible that Juan de Fuca S t r a i t and GS are dynamically coupled and that influences i n one may d i r e c t l y or i n d i r e c t l y force motions i n the other. I n t e r e s t i n g l y , F i s s e l and Huggett (1976) have shown that low-frequency current f l u c t u a t i o n s of about a 15-day period also occur i n Juan de Fuca S t r a i t . Motions could also be driven by one or more of the above-mentioned mechanisms. For example, the Fraser River outflow (Figure 9.1) might i n t e r a c t nonlinearly with the t i d a l currents r e s u l t i n g i n a f o r t n i g h t l y modulation of the basic estuarine flow. F i n a l l y , i t i s possible that a s i g n i f i c a n t f r a c t i o n of the observed low-frequency currents i n GS can only be c l a s s i f i e d as geostrophic turbulence. That i s , nonlinear interactions between both large- and small-scale motions, i r r e s p e c t i v e of t h e i r source, may be a predominate influence. Rhines (1975) has demonstrated that in a geostrophically turbulent system, small-scale f l u c t u a t i o n s tend to evolve into l a r g e r - s c a l e , more well-defined, planetary wavelike motions. C l e a r l y , t h i s discussion of the character of the observed low-frequency currents and possible f o r c i n g mechanisms i s not exhaustive. Perhaps many or a l l of the mentioned mechanisms play a s i g n i f i c a n t r o l e i n GS dynamics. In Part II of t h i s t h e s i s , the i n e r t i a l s t a b i l i t y of the mean flow and the r e s i d u a l t i d a l c i r c u l a t i o n are examined i n greater d e t a i l . A few comments are also made concerning the possible i n t e r a c t i o n of the t i d e s , Fraser River outflow, and the winds. 93 11. I n e r t i a l I n s t a b i l i t y Models As mentioned i n Section 9, at the conclusion of a previous study (Helbig and Mysak, 1976), i t was strongly suspected that i n e r t i a l i n s t a b i l i t y of the mean flow within GS was an agent responsible f o r a major proportion of the observed low-frequency energy. This b e l i e f was based on two f a c t s . F i r s t , the phase speed of a low-frequency wave of moderate wavelength would be comparable to mean current speeds as a purely kinematic argument shows. Consider a 14-day wave of length X. I f A i s expressed i n kilometres, the phase speed i n cm s ^ i s given by .08A; t h i s gives, for example, a value of 8 cm s **" f o r a 100-km wave, a speed within the range of the currents. Second, based on the findings of Chang (1976), there was no apparent f o r c i n g mechanism for the f l u c t u a t i o n s . In p a r t i c u l a r , i t was f e l t that the i n s t a b i l i t y would be p r i m a r i l y b a r o c l i n i c , the l a t e r a l shear of the currents playing a r e l a t i v e l y minor r o l e . This hypothesis was based on two premises. F i r s t , v e r t i c a l shears were generally observed to be larger than ho r i z o n t a l shears (with the exception of the deep eastern s t a t i o n ) . Second, Helbig and Mysak (1976) showed that for an i d e a l i z e d model of GS, topographic planetary waves e x i s t with frequencies that l i e i n the observed range for reasonable choices of the wave length. The v e r t i c a l d i s t r i b u t i o n of h o r i z o n t a l k i n e t i c energy ( i . e . , that associated with the h o r i z o n t a l motion) for these waves was opposite to that observed. That i s , the waves were bottom trapped. I t was f e l t that perturbations of t h i s form, perhaps i n i t i a t e d by the winds, might grow i n time by extracting p o t e n t i a l energy from the mean flow with a resultant enhancement of upper-layer k i n e t i c energy. 94 B a r o c l i n i c I n s t a b i l i t y Therefore, the f i r s t step i n the analysis of the i n e r t i a l s t a b i l i t y of GS was to extend the model of Helbig and Mysak (1976). A two-layer system confined to a channel with a sloping bottom and with a constant mean v e l o c i t y i n each layer was adopted (Figure 11.1). For parameters c h a r a c t e r i s t i c of GS, the r e s u l t s below indicate that i n s t a b i l i t y can occur only f o r a narrow band of wavelengths. The primary reason f o r t h i s i s the strong s t a b i l i z i n g e f f e c t that the narrow channel has on the system, as i t l i m i t s the e f f e c t i v e wavelength of any perturbations. I t i s i n t e r e s t i n g that t h i s e f f e c t was also l a r g e l y responsible f o r the high degree of bottom trapping found by Helbig and Mysak. This model has been applied by Mysak and Schott (1977) and Mysak (1977) to the Norwegian current and the C a l i f o r n i a undercurrent, r e s p e c t i v e l y , with considerable success. Although the present development of t h i s model was c a r r i e d out independently, i t s d e t a i l s are r e s t r i c t e d to Appendix D since the model has appeared i n the l i t e r a t u r e . The dynamics of b a r o c l i n i c i n s t a b i l i t y derive from the conserva- t i o n of p o t e n t i a l v o r t i c i t y . The governing equations expressing t h i s were f i r s t derived by Pedlosky (1964), and the de r i v a t i o n presented i n Appendix D i s s i m i l a r although i t d i f f e r s i n some respects. The b a s i c state i s s p e c i f i e d by the constant currents V and (see Figure 11.1) which are i n geostrophic balance with the mean surface and i n t e r f a c i a l d i s p l a c e - ments. A perturbation with i n i t i a l v e l o c i t i e s small compared with mean currents i s applied to the system. I f i t grows i n time the system i s said to be unstable. The nondimensional equations governing the perturbed state are 95 Z / / 2 0 / / / / / / / * A upper layer p lower layer pz h 2 (x) / / / / / / / / / / / / / / Figure 11.1 The b a r o c l i n i c i n s t a b i l i t y model. The sloping surface elevation and i n t e r f a c i a l displacement are i n geostropic balance with the mean currents. 96 9t + V 1 3 y ] [ V 2 $ 1 + F 1 ( $ 2 - - $ly[Vn1 - F 1 ( V 1 - V 2 ) ] = 0 (11.1) [dt + V 2 3 y ] [ V ^ 2 " F 2 ( $ 2 " " $ 2 y [ V " 2 + F 2 ( V 1 _ V 2 } + T] = 0. (11.2) Here and $ 2 a r e stream functions f o r the perturbation v e l o c i t i e s , U i = " $ i y (11-3) (11.3) v i = $ i x ( l l - 4 ) where i = 1,2 s p e c i f i e s the layer, and $ and ^ are defined i n terms of the perturbation surface and i n t e r f a c i a l displacements and ? 2 , re s p e c t i v e l y , }2 = h + h (11.4) The following scale factors were used i n the nondimensionalization: L fo r the h o r i z o n t a l coordinates (x,y), a t y p i c a l current U for the v e l o c i t i e s (u,v), an advective time L/U, and fUL/g and fUL/g' for £^ and £ 2 , r e s p e c t i v e l y . Also appearing i n (11.1) and (11.2) are the layer i n t e r n a l Froude numbers F^ = f 2L 2/g'h-^ and F 2 = f 2L 2/g'h 2Q where 9"' = 9"(P2 ~ Pl^/P2 :"'s t' i e r e (3uced ac c e l e r a t i o n due to gr a v i t y ; T i s a 97 topographic parameter defined by T = - ( L / R Q I ^ Q ) ( d h 2 / d x d ^ m ) , and Ro = U/fL i s the Rossby number for the flow. The negative sign ensures that T i s of the same sign as the bottom slope, and the subscript "dim" denotes a dimensional v a r i a b l e . ik(y-ct) For wavelike perturbations of the form <J)̂ e , (11.1) - (11.2) reduce to ( V X - c) [<(>»! - k 2 ^ ] - (^[V'-L - F 1 ( V 1 - V 2 ) ] + - c) (o>, - <j>n) = 0 2 (11.5) (V 2 - c) [ c f ) " 2 - k^ ( f ) 2 ] - c j) 2[V" 2 + F 2 ( V 1 - V 2) + T] - F 2 ( V 2 - c) (<J>2 - f^) = 0. 2 (11.6) For constant and V 2 , the solu t i o n i s (j)-̂ = A n s i n nTTx (J>2 ~ B n s:"-n n 7 r x n = 1,2, . . . > (11.7) where K 2 + F x V x c - V, A„ (11.8) and K = k + n TT i s the " t o t a l " wave number. If the solutions are unstable, A n and B n w i l l be complex and the v e l o c i t i e s i n the upper and 98 lower layers w i l l be out of phase. The phase speed i s given by c = V, + „- 2 T { (K 2 + F T ) (T - SK 2) - F , SK 2 1 2K Z(K^ + F± + F 2) 1 1 ± ([(K Z + F x ) ( T SK2) + Fi SK ] 2 n2 4F 1F 2sr ![T - S K 2 ] ) 1 / 2 } (11.9) where S = V-̂ - V 2 i s the v e r t i c a l "shear." With no loss of generality, T i s r e s t r i c t e d to be p o s i t i v e while S may have either sign. The r a t i o of the h o r i z o n t a l k i n e t i c energy (HKE) per u n i t depth i n the upper layer to that i n the lower layer averaged over the area defined by the channel width and one wavelength i n the y - d i r e c t i o n i s R = A (V-, - c j ^ + cV (K 2 + F±)2 [(V± - c r) - F XS/(K 2 + F x ) ] 2 + c ± 2 . (11.10) By HKE we mean the k i n e t i c energy- associated with the h o r i z o n t a l components of motion. An int e g r a t i o n of (11.10) over the layer depths gives R = ( h 1 / h 2 0 ) R (11.11) so that R represents the r a t i o of the t o t a l HKE i n the upper layer to that i n the lower layer. Two l i m i t i n g cases are of i n t e r e s t . In the f i r s t we set 99 V x = V 2 = 0 to obtain T(K 2 + F ) K 2 ( K 2 + F1 + F 2) 2 F l (11.12) R = 7~2 72 < !• (11.13) (IT + These r e s u l t s are e s s e n t i a l l y those obtained by Helbig and Mysak (1976) f o r bottom i n t e n s i f i e d , topographic planetary waves i n a 2-layer channel. In the second case, we put T = 0 to obtain the 2-layer analogue of the c l a s s i c a l Eady (1949) so l u t i o n , c = V n - = {(K 2 + 2F-,) ± (K 4 - 4 F , F 9 ) 1 / 2 } . (11.14) X 2(K Z + F± + F 2) X R = h 2 0 / h 1 . (11.15) From (11.11) we see that R = 1 and the two layers contain equal amounts of HKE i r r e s p e c t i v e of t h e i r thicknesses. In the general case, the stable solutions l i e between these two extremes. For s u f f i c i e n t l y large T, one root of (11.8) corresponds to a shear modified topographic wave while the other represents a topographically modified shear wave. Whether or not these waves are more intense i n the upper or lower layer depends upon the choice of parameters. In the present case at wavelengths f o r which the system i s stable, one wave i s concentrated i n each layer. Unstable waves are found to be more energetic i n the upper 100 layer. On the other hand, f o r Norwegian current parameters, Mysak and Schott (1977) found unstable waves to be bottom i n t e n s i f i e d . In a recent study, Wright (1978) has treated t h i s question i n much greater d e t a i l . From (11.9) we see that i f S i s p o s i t i v e a necessary but not s u f f i c i e n t condition for i n s t a b i l i t y i s S > T/K 2. (11.16) The bathymetry thus acts to s t a b i l i z e the flow i f the bottom slopes upward to the east and i s a d e s t a b i l i z i n g influence i n the opposite case. This agrees with the findings of Blumsack and Gierasch (1972) f o r a continuously s t r a t i f i e d system. However, t h i s notion must be q u a l i f i e d , f o r the presence of weak topography may render a flow unstable. To see t h i s consider a system with F± = F 2 and K 4 = 4 F 1 F 2 corresponding to a neutral Eady wave (see 11.14). Then (11.9) becomes 1/2 V2 3T - 8F 1S T X / Z [ 9 T - lSF^S] C = V l + - l 6 F i ; ± 1 ^ ' ( 1 1 - 1 7 ) and i f 16F 1S > 9T ( i . e . , S > 9T/8K ), the system i s unstable with a 1/2 growth rate proportional to T . DeSzoeke (1975) has studied the i n t e r - action of Eady waves with topography. Numerical r e s u l t s corresponding to the general r e l a t i o n (11.9) are now presented. Figures (11.2) and (11.3) i l l u s t r a t e t y p i c a l s t a b i l i t y boundaries f o r the system. The former shows the s t a b i l i z i n g e f f e c t of a p o s i t i v e bottom slope. As T increases, the region of i n s t a b i l i t y shrinks i n size and s h i f t s to smaller wavelengths. There are no unstable waves for Figure 11.2 Mode 1 s t a b i l i t y boundaries f o r the b a r o c l i n i c model as a function of the topographic parameter T. 103 negative S (cf. 11.9). Figure 11.3 i l l u s t r a t e s the dependence on the parameter V = F^ + F 2 which may be rewritten as the square of the r a t i o of the channel width L to the i n t e r n a l Rossby radius of deformation r ^ , f 2 L 2 ( h + h ) 2 V = F, + F 2 =•• _ = (J±) . (11.18) * ' h l h 2 0 For small V, i . e . , for narrow channels or strong s t r a t i f i c a t i o n , the region of i n s t a b i l i t y i s r e l a t i v e l y narrow. As V increases, corresponding to an increase i n the channel width, a decrease i n the s t r a t i f i c a t i o n , or a thinning upper layer, t h i s region spreads out and s h i f t s to higher wave- numbers . Parameters c h a r a c t e r i s t i c of GS are V = 7.5, A = h-^/(h-^ + h2Q> = .14, and T = 7.4 (Helbig and Mysak, 1976). (Note that F = Av and F 2 = (1 - A)V.) For these parameters, the first-mode dispersion curves are shown i n Figure 11.4 for the cases S = 0.5 (V"2 = O.SV^) and S = 1.5 (V 2 = - 0.5V-^). In the f i r s t case, which i s generally representative of GS, the system i s unstable only i n the wavelength band of 40-46 km. The most unstable wave has an e-folding time of 78 days, a period of 11 days, and i s i n t e n s i f i e d i n the upper layer (R = 1.3). In the second case, i n which the currents are d i r e c t e d i n opposite d i r e c t i o n s , the f i r s t mode i s unstable f o r a l l wavelengths exceeding 93 km. The most unstable wave has a period of 70 days and an e-folding time of 39 days, and i s strongest i n the upper layer (R = 2.8). The higher modes are stable i n each case. Figure 11.5 dramatically i l l u s t r a t e s the disparate nature of the two roots of (11.9) for GS parameters. While the phase speed of the shear wave i s almost independent of T, that of the topographic wave varies 250 100 50 WAVELENGTH (km) 25 Figure 11.4 B a r o c l i n i c model, mode 1, dispersion curves fo r S = 0.5 and S = 1.5. The s o l i d c i r c l e s i n d i c a t e the most unstable waves. The wave- length and period are calculated using the scale factors of U = .5 cm s - x and L = 25 km. 105 Figure 11.5 The b a r o c l i n i c model, mode 1, phase speed as a function of topographic parameter f o r k/2TT = 0.1, 0.5, and 1.0. 106 l i n e a r l y with T. Barotropic I n s t a b i l i t y A simple barotropic model i s now considered i n order to gain some idea of the importance of h o r i z o n t a l shear. The model i s i l l u s t r a t e d i n Figure 11.6. Although an exponential bottom p r o f i l e i s chosen for s i m p l i c i t y , t h i s choice does not severely l i m i t the conclusions drawn here. Indeed, the sloping topography has l i t t l e e f f e c t on the unstable waves. The governing equations are abstracted d i r e c t l y from Part I; with £ = 0, (3.8) reduces to (V - c) [ ( ^ 1 ) • V 1 + V ' (j) = 0. (11.19) We specify fv. V(x) =< 0 < x < d d < x 5 1 (11.20) and h(x) = e 2b(x-l) (11.21) and thus obtain a constant c o e f f i c i e n t d i f f e r e n t i a l equation i n each region. As before, the boundary conditions are (cf. 3.19) cf> = 0 at x = 0,1. (11.22) 107 z 1 > y x h v, [x) x ^ ^ ^ x = 1 = d x = 0 Figure 11.6 The barotropic i n s t a b i l i t y model. 108 The s o l u t i o n i s given by r <f> (x) b(x-lW A^ s i n XjK 0 < x < d A 2 s i n A 2 ( x - 1 ) d < x 5 1 (11.23) where 2b i R 0 ( v i " <=) (k 2 + b 2) i = 1,2. (11.24) In order that the normal fluxes of mass and momentum be continuous at the material i n t e r f a c e centred at x = d, <f> (x) must s a t i s f y V - c = 0 [ (V - c)<j>' - V<1>] = 0 at x = d. (11.25a) (11.25b) (These r e l a t i o n s h i p s are derived i n LeBlond and Mysak, 1978; p. 429. However, (11.25b) d i f f e r s from t h e i r (45.9) since they e f f e c t i v e l y assumed that $ i s continuous. In the present case i t i s not, and one must proceed from the i n t e g r a l r e l a t i o n s h i p preceding t h e i r (45.9).) A p p l i c a t i o n of these conditions leads to the i m p l i c i t d ispersion r e l a t i o n 2-, 2 (V, - c) A, (v- - c) A, j ~ ± k 1 2 _ b [ ( v _ c)Z _ ( v c)Z] = Q_ (H.26) tan A-̂ d tan A2 (d - 1) ^ 109 Several l i m i t i n g cases are of i n t e r e s t . In the f i r s t we put = V 2 to obtain tan A 2 d = tan A 2 ( d - 1) which has the s o l u t i o n A 2 = nir independent of d. This gives 2b c = V„ - p p (11.27) 1 R Q (k + bz + n TT ) which i s the dispersion r e l a t i o n f o r a free continental s h e l f wave i n a mean current V 2 (cf. 5.21) . In the second case we put b = 0 to obtain V l + Q V 2 i 0 1 / 2 i I (1 + Q) 1 + Q 1 1 21 (11.28) where Q = - tanh kd/tanh k(d - 1) and i s p o s i t i v e . These waves t r a v e l at a speed given by a weighted average of the mean currents and grow i n time at a rate proportional to the shear. F i n a l l y , i n the short wave l i m i t of k -> «>, (11.26) reduces to V, + V- (V, - V 9) c = - ^ - r — £ ' ± i n . (11.2-2 2 Equations (11.28) and (11.29) represent a p a i r of shear waves, one of which i s unstable and another which decays i n time. As these s p e c i a l cases suggest, there are at most three solutions to (11.26). For nonzero b and S, where S = V 2 - , there e x i s t a p a i r of complex roots f o r a l l k, corresponding to amplified and damped modified shear waves. Provided that both A ^ and A 2 are r e a l , a t h i r d , r e a l root e x i s t s and represents a shear modified CSW. This r e s t r i c t i o n may be expressed as 110 2b c > V 1 / 2 R Q ( k 2 + b 2) . ( 1 1 . 3 0 ) The region i n k, S space i n which CSWs e x i s t i s shown i n Figure 11.7. Along the l i n e = V,, only a CSW i s found. To determine a value of S appropriate to GS, the mean currents along l i n e H were depth averaged as BT f h l H l + h 2 ( x ) u 2 ^ [h± + h 2(x)] (11.31) h-̂ was chosen as 50 m and h 2 was determined f o r each mooring. The re s u l t s are shown i n Figure 11.8. A reasonable choice i s V = 1.0, V 1 = 0.5 (with a scale v e l o c i t y of 5 cm s - 1 ) , d = 0.66, and b = - 0.3. The dispersion r e l a t i o n f o r these values i s shown i n Figure 11.9. I t i s seen that a CSW e x i s t s f o r wavelengths greater than 55 km. An unstable shear wave of 15-day period has an e-folding time of about 8 days, a phase speed of about 4 cm s-"*" and a wavelength of 44 km. I t i s possible therefore that a shear i n s t a b i l i t y of t h i s type might play some ro l e i n GS dynamics. However, i n the next section i t i s shown that the motions i n the 15-day band are predominantly nonwavelike i n the sense of the waves studied here. This implies shear i n s t a b i l i t y i s of l i m i t e d importance i n GS. Figure 11.7 The region i n (k,S) space i n which continental shelf waves e x i s t . 112 Figure 11.8 Computed barotropic mean currents along l i n e H for the.18-month period beginning A p r i l 1969. WAVELENGTH (km) 1 1 3 250 100 50 25 15 I U i — i 1 1 1 — — 1 Figure 11.9 Barotropic model dispersion curves f or S = 0.5. The growth time i s defined as the e-folding time, u> i s the frequency, and 9. i s defined as k • Im c. The wavelength and period are computed using the scale factors of u = 5 cm s _ x and L = 25 km. 116 Figure 12.1 Plan view of the S t r a i t of Georgia showing current meter locations. These l i n e s should not be confused with those of F i g . 10.1. Winds were recorded at Sand Heads. 1969 1970 .A. M . J . J . A . S . 0 . N . D J F . M A M . J . J . A . S STATION DEPTH l_J I I I I I I I I I I I I I I I I (m) H 2 6 5 0 140 3 |H16 5 0 200 3 |H06 5 0 200 9 9 9 9 J] 35 £ 1 A 7 -] 35 [ 10A ] 29 [ 108 ] E — 1 0 0 — 3 -] 28 £ 158 ] 5 2 5 62 t H5 0 " £ Q 7 9 £ •258 382 [-36-311 [- 3 9 - J f - 2 9 ] 196 3 [-42 - 3 49 E - 5 9 - 3 — 3 33 E 1 5 0 3 3 Figure 12.2A Periods of existent current meter records. STATION DEPTH(m)| 1969 APR | MAY | JUN | JUL | AUG | SEP I OCT H 06 H 16 H 26 31 33 36 42 43 44 45 46 47 51 52 53 54 55 56 61 62 63 64 65 50 200 50 200 50 140 100 100 100 50 84 100 300 100 300 100 200 100 200 100 50 100 50 130 100 200 250 100 200 100 180 50 100 50 100 50 100 50 100 100 I 5 2 5 - 2 5 8 3 8 2 I 9 9 9 9 - 3 7 - - 3 7 - E—29 E-I5--3 """"1-15-3 • 3 6 ... -41 E - 2 7 — 3 F 3 6 l = 8 = l _ E-I7H 2 9 — - j E - i e - 3 1 4 5 - 1 4 7 - 1 0 4 b 3 5 J — 3 5 3 U H r , t - 1 5 - d , E-15-3 - 3 8 - 3 - 3 0 - 2 0 - 3 Figure 12.2B 10^ 10 10°lr 16 854 2 — i — r — n — | — 31 100 m C P ^Q*2I i i i m i l l i I I i n n 102| 1 — r - n — r ~ 101 CvJ CvJ o 10% 10"1 10 ,-2 41 84 m o o o PERIOD (DAYS) 16 854 2 A - northward O - eastward 119 10"2 10"1 10° 10"2 10"1 10° FREQ (cpd) Figure 12.3A Current spectra for the 26-day period beginning (A) 2 May 1969 and (B) 29 August 1969. 10' 16 8 54 2 — i i—n—|— PERIOD (DAYS) 16 854 2 16 854 2 H06 50 m °- 4 1 0 2 1 — i i i mill • 1 0 2 120 FREQ (cpd) Figure 12.3B 121 course, none of the peaks are s t a t i s t i c a l l y s i g n i f i c a n t to 95 percent but these spectra do indicate the existence of low-frequency energy at locations north of l i n e H. The comparatively quiet spectrum from Station 41 may be due to the f a c t that the meter l i e s i n the "shadow" of a topographic high j u s t to i t s south. Spectra computed from l i n e s 5, 6 and H records f o r the 26-day period beginning 29 August 1969 are shown i n Figure 12.3b. These time series were treated as above, and the spectra a l l indicate low-frequency energy. The record from Station 64, 50 m i s of dubious q u a l i t y and thus i s of l i m i t e d value f o r comparison with H26, 50 m. Mean currents were calculated d i r e c t l y from the time s e r i e s , and current e l l i p s e s (see, f o r example, Stone, 1963) were constructed from the average of the lowest two s p e c t r a l bands. The e l l i p s e s are i l l u s t r a t e d i n Figure 12.4 by t h e i r major and minor axes, although i n several cases, the l a t t e r i s too short to be v i s i b l e . Due to the s t a t i s t i c a l l i m i t a t i o n s of the data, no i n d i c a t i o n i s given of eit h e r the d i r e c t i o n of r o t a t i o n of the o s c i l l a t i n g current vector around the e l l i p s e or of the r e l a t i v e phases between e l l i p s e s . Of course, the e l l i p s e parameters are subject to contamination by trends during the period of analysis since these a f f e c t the lowest-frequency s p e c t r a l estimates, and thus the e l l i p s e s c a l c u lated here must be viewed with caution. Of p a r t i c u l a r i n t e r e s t i n Figure 12.4A i s the cross-channel o s c i l l a t i n g flow suggested at H16, 50 m and Station 43, 100 m. The pattern of mean currents i l l u s t r a t e d i n Figure 12.4B i s extremely i n t e r e s t i n g since i t indicates a closed, clockwise, mean c i r c u l a t i o n i n the lower s t r a i t . While t h i s may not be true f o r longer periods, i t lends support to Waldichuck's (1957) conjecture that a gyre e x i s t s . I t i s , however, of the opposite sense to that indicated by Waldichuck. During t h i s 122 Figure 12.4A Mean currents and the 6-32-day band current e l l i p s e s for 26-day period beginning (A) 2 May 1969 and (B) 29 August The e l l i p s e s are indicated by t h e i r major and minor axes. the 1969. Figure 12.4B 124 period, the flow at H26, 50 m i s southward whereas the 18-month mean flow i s northward (Figure 10.6). This implies that the gyre may not be a permanent feature. A closed c i r c u l a t i o n i s not indicated f o r the c e n t r a l s t r a i t during May 1969 (Figure 12.4A). The strong a x i a l current present at H26, 50 m i s not observed at Station 47, 100 m. Wind-Driven Motions The dynamics of low-frequency, large-scale motions are due i n large part to the conservation of p o t e n t i a l v o r t i c i t y , and i t i s the wind stress c u r l that enters the v o r t i c i t y equation as a f o r c i n g function. In addition, i f the system under consideration possesses s i g n i f i c a n t bottom topography, the wind stress i t s e l f may induce v o r t i c i t y by fo r c i n g water columns across bathymetric contours thus squeezing or stretching vortex l i n e s . Indeed, i n a barotropic system the v o r t i c i t y input by t h i s mechanism may far exceed that due to the wind stress c u r l ( G i l l and Schumann, 1974). Chang (1976) calculated cross-spectra between the winds and currents along l i n e H and found the coherence to be generally small at low frequencies. However, the use of rotary spectra does not reveal r e l a t i o n - ships between the various rectangular components of the currents and the winds. Moreover, the f a c t that the current spectra are peaked at about 14 days (at l e a s t f o r the eastern and western s t a t i o n s ) , that the motion may be barotropic at H26, and that the period of a free CSW for GS parameters i s about 14 days f o r a v a r i e t y of bottom p r o f i l e s (see, e.g., LeBlond and Mysak, 1977; or Csanady, 1976) suggests very strongly that the wind stress might force modified CSW's. I t i s therefore sensible to examine the re l a t i o n s h i p between the wind stress and the currents by computing components spectra. S u f f i c i e n t data do not e x i s t to adequately determine the wind 125 stress c u r l , which at any rate i s a d i f f i c u l t task due to the e f f e c t the complex orography of the B r i t i s h Columbia coast has upon the winds. That i s , measurements taken at land-based stations are not n e c e s s a r i l y representative of conditions at sea. One could attempt to evaluate the wind stress c u r l from atmospheric surface-pressure maps, but i t i s a tedious exercise and i s not pursued here. We note as before, however, that the ensuing analysis implies that the low-frequency currents are not simple wavelike motions of the type studied i n t h i s t h e s i s . In t h i s respect, therefore, i t i s u n l i k e l y the wind stress c u r l plays a s i g n i f i c a n t r o l e . Figure 12.5 shows the spectrum of the wind stress at Sand Heads for the 500-day period beginning 4 A p r i l 1969. Sand Heads i s located i n shallow water at the mouth of the Fraser River (see Figure 12.1) adjacent to a r e l a t i v e l y f l a t region. Thus winds measured there should be compara- t i v e l y free of l o c a l topographic influences. The wind stresses were computed using a quadratic law with a value of 1.5 * 10 for the drag c o e f f i c i e n t . Its exact value i s unimportant i n t h i s discussion since i t enters only as a scale f a c t o r . Figure 12.5A shows the true northward and eastward components of the wind s t r e s s . Both are peaked at about 3 days, but s i g n i f i c a n t variance i s present to periods up to at l e a s t 20 days. The mean axis of GS i s dir e c t e d approximately 50° west of north, and the spectra rotated by 50° are shown i n Figure 12.5B. Since s i g n i f i c a n t l y higher coherences were found i n t e s t runs using the rotated wind stress time s e r i e s , they were employed i n the following analysis. On the other hand, i n the v i c i n i t y of l i n e H, the topography runs nearly north-south. Therefore the currents were not rotated. A l l subsequent figures r e f e r to the rotated wind stress and nonrotated currents. 126 ( A ) ( B ) PERIOD (days) 50 25 10 5 2 1 10 I E - . — i - r 10V 10 10 »-5 if O - eastward A - northward i i i mill i i i mill 3 . 1 0 2 p — i 1 1 — i r ia ! : A 10"'r , : Y O-cross-strait A-along-strait • i « " " i l i 11 nml 10"2 10"' 10' FREQ (cpd) Figure 12.5 Spectrum of the wind stress at Sand Heads for the 500-day period beginning 4 A p r i l 1969: (A) the north and east wind stress components; (B) the wind stress components rotated anti-clockwise by 50°. 127 The 209-day period beginning 29 August 1969 was selected f o r an a l y s i s , as records existed at a l l stations along l i n e H (Figure 12.2). Large gaps i n the time series were deleted from good as well as bad records and the time series were l i n e a r l y interpolated across short gaps of the order of a day or two. The r e s u l t i n g record was 136 days i n length. Current spectra computed from these time ser i e s are shown i n Figure 12.6; the spectrum of the wind stress i s e s s e n t i a l l y that shown i n Figure 12.5. Two exemplary plo t s of coherence and phase between the currents and the a l o n g - s t r a i t wind stress are shown i n Figure 12.7. They i l l u s t r a t e the generally low coherence observed which decreases with increasing frequency and the tendency for the currents to be e i t h e r i n phase or 180° out of phase with the a l o n g - s t r a i t component of the wind s t r e s s . A more d e t a i l e d presentation of the coherence and phase r e l a t i o n - ships i s shown i n Figure 12.8 for the 34- and 13-day bands. S t a t i s t i c a l l y s i g n i f i c a n t coherences are found i n many cases, although i n some, due respect must be paid to the amount of energy i n the given s i g n a l . Thus, 2 f o r example, the meaning of high value of y = .6 i n the 34-day band calculated f or H06, 200 m, i s unclear due to the corresponding low value of the spectrum there (Figure 12.6). The most s t r i k i n g feature, however, i s the consistency with which the phase estimates c l u s t e r about e i t h e r 0° or 180°. This tendency together with the f a c t that phase determination may be good even though the corresponding coherences are i n s i g n i f i c a n t (Schott and Diiing, 1976) i n s t i l l s some degree of confidence i n the calculated phases. More s p e c i f i c a l l y , phases between currents and the a l o n g - s t r a i t wind stress tend to be close to 180°, while those between the currents and the cross- channel wind stress l i e near 0°. The a l o n g - s t r a i t wind stress i s , of course, considerably more energetic than the cross-channel component. 128 34 13 7 3 1 PERIOD ( D A Y S ) 34 13 7 3 1 w 10 E— r 10l 10" 10 H06 200 m 4 h 4> 10 -2 • • • " " I » i • u m 34 13 7 3 1 H16 50 m A-northward O - eastward I I I I itlll I I I I M i l 10 1-1 10° 1 0"2 10T1 1 0 ° 1 F R E Q (cpd) Figure 12.6 Line H current spectra f o r the 136-day analysis period. The v e r t i c a l bars i n d i c a t e 95% confidence l i m i t s . PERIOD (DAYS) 13. 6.5 2.8 1.4 PERIOD (DAYS) -1.0 log (freq / I cpd) -2.0 270 180 0.6 34. 13. 6.5 2.8 1.4 i i i i 0.5 HI6 50m • - u 0.4 • - V 0.3 _ • • 0.2 0.1 0 A i t -* -1.0 log ( f req / I cpd) T o.c Figure 12.7 Coherence and phase between the wind stress and currents at H26, 50 m and H16, 50 m. A p o s i t i v e phase indicates the current leads the wind. Here u and v r e f e r to the eastward and northward v e l o c i t y components. The s o l i d l i n e indicates the 95% noise l e v e l . 130 O UJ cr < z> o U) UJ o 0.6 0.5 0.4 0.3 O U - T X • u-ry , A v-rx LA v-ry LxJ o ° 0.1 r 0.0 270 H 0 6 UJ Q UJ to < I 0_ 180 90 0 •90 UPPER LAYER H 16 O A 13 DAYS H26 0.6 0.5 0.3 0.2 0.1 0.0 270 180 90 h •90 - LOWER 1 LAYER ' 13 DAYS A - 1 O A - A n 1 A • 1 • • 1 o A H06 H16 H26 I A • A I # A a IO . O A Figure 12.8A Coherence and phase between l i n e H currents and the wind stress f o r : (A) the 13-day band, and (B) the 34-day band. The s o l i d l i n e s i n the coherence p l o t s i n d i c a t e the 95% noise l e v e l s . A p o s i t i v e phase indicates the current leads the wind. 131 Q L U cr < ZD a in L U o 0.6 0.5 0.4 0.3 I- L U cr 0.2 L U X o ° 0.1 h 0.0 270 O U - T X • U - T V A V - T X A v-ry o L U O L U < X 0_ 180 90 H06 •90 UPPER LAYER H16 34 DAYS A_ H26 -©• A. 0.6 0.5 0.2 0.1 0.0 270 180 LOWER LAYER 34 DAYS • - - A . , A • A I O • O A H06 - e — A - 90 h •90 H16 H26 Figure 12.8B 132 These phase r e s u l t s should not be interpreted to mean, for example, that the currents flow down-channel when the winds blow up-channel. They imply simply that the currents are i n opposition to the given s p e c t r a l component of the a l o n g - s t r a i t wind. The r e l a t i o n s h i p i n the time domain between the currents and winds w i l l be examined shortly. Table III l i s t s the coherences and phases calculated between v e l o c i t y components f o r the 13- and 34-day bands. If a motion i s composed of the types of waves studied here, then the phase difference between u and v should be somewhere near 90°. Deviations from t h i s value may be caused by f r i c t i o n (see, e.g., Csanady, 1978) or by i n s t a b i l i t y (see Section 4). With the exception of the value of <J> = 117°. f o r H26, 140 m at 34 days, examination of Table III shows that i f a s i g n i f i c a n t coherence i s found, the corresponding phase i s e i t h e r near 0° or 180°, that i s , the motion i s l i n e a r l y p o l a r i z e d . This implies, with the noted exception, that the motion i s not wavelike. This i s true, i n p a r t i c u l a r , f o r the observed 13-day current o s c i l l a t i o n s at H26 for which the previous evidence strongly suggested the contrary. On the other hand, the r e s u l t s f or 34 days for H26, 140 m, do suggest a wavelike character and i t i s e n t i c i n g to speculate that t h i s might be a bottom-enhanced wave of the type described by Helbig and Mysak (1976). There i s no d i r e c t evidence, of course, to support t h i s speculation, and the motion i s just as l i k e l y to be barotropic. We note, however, that CSWs of 34-day period should have wavelengths i n excess of the length of GS (see, e.g., Figure 11.5). The mean currents and current e l l i p s e s for the 13- and 34-day bands are shown i n Figures 12.9 and 12.10. While the mean flow i s s i m i l a r to that calculated f o r the f u l l 18 months (Figure 10.6), the deep currents i n both the east and west are considerably stronger. Indeed, at H06 the 133 Table I I I . Calculated coherence squared and phase between v e l o c i t y components f o r the 136-day period of a n a l y s i s . A p o s i t i v e phase indicates that v leads u. 13 days 34 days Station Depth Y 2 <j) (deg) Y 2 <f> (deg) H06 50 .36 - 4 .08 52 200 .24 - 11 .10 - 178 H16 50 .01 3 .05 39 200 .29 179 .69 179 H26 50 .37 5 .18 177 140 .51 2 .48 117 1 3 4 upper- and lower-layer mean flows are nearly the same. The current e l l i p s e s bear a s t r i k i n g resemblance to the respective mean v e l o c i t i e s . Although trends during the period of analysis may contaminate the 34-day band (which i s averaged over the second to s i x t h frequencies), they should exert minor influence on the 13-day band (which i s averaged over frequencies 7 - 1 4 ) . These r e s u l t s thus may imply a dynamical r e l a t i o n s h i p between the mean and f l u c t u a t i n g flows, a point which i s discussed further i n the next section. It i s evident from the figures that the channel boundaries exert a strong topographic influence on the near-shore currents. Approximate barotropic and b a r o c l i n i c time seri e s were formed by depth averaging the 136-day records. Indeed, the period of analysis and the treatment of the data records were selected f o r t h i s purpose. The v e l o c i t y time series were combined as HfiT = ( h l ^ l + h 2 ^ 2 ) / ( h l + V and HB C = h 2 ( ^ l " U 2 ) / ( h l + V to give barotropic (HBT^ a n <3 upper-layer b a r o c l i n i c Ĥgc) records, re s p e c t i v e l y . The upper-layer depth h-̂ was chosen as 50 m since t h i s corresponds to a reasonable f i t of a two-layer model to the observed density d i s t r i b u t i o n (Helbig and Mysak, 1 9 7 6 ) . The lower-layer depth h 2 was then simply obtained from the t o t a l depth at each mooring. The v e r t i c a l v e l o c i t y p r o f i l e i s , i n r e a l i t y , much more complex. I f a greater number of meters had been used at each s t a t i o n , the method of empirical orthogonal functions could have been employed to resolve the v e r t i c a l structure (see, e.g., Mooers and Brooks, 1 9 7 8 ) . No other choices of h^ Figure 12.9 Mean currents along l i n e H f o r the 136-day analysis period. Figure 12.10A Line H current e l l i p s e s f o r the 13-day band: (A) upper layer, (B) lower layer; and the 34-day band: (C) upper layer, (D) lower layer. Figure 12.10B 138 Figure 12.IOC Figure 12.10D 140 were t r i e d . Each time series was s p e c t r a l l y analysed, but the barotropic and b a r o c l i n i c spectra were almost i n d i s t i n g u i s h a b l e from the lower- and upper- layer spectra, r e s p e c t i v e l y . The r e s u l t s were inconclusive with regard to c o r r e l a t i o n between the wind and currents. In some cases higher values of 2 Y were found but i n others the c o r r e l a t i o n was diminished. I t i s d i f f i c u l t to determine, therefore, i f the separation was successful. The barotropic and b a r o c l i n i c mean currents f o r the 136-day period are shown i n Figure 12.11. While the barotropic means are s i m i l a r to the lower-layer means of Figure 12.9, the b a r o c l i n i c means are less suggestive of a gyrelike c i r c u l a t i o n than are the upper-layer means. One current meter, that at H16, 50 m, operated almost continuously over the 18-month period. Consequently, the 500-day time ser i e s of currents and winds beginning 17 A p r i l 1969 was analysed for comparison with spectra computed from shorter record lengths. The current spectrum i s very s i m i l a r to that shown i n Figure 12.6. In general, the calculated coherences between the currents and winds are decreased from those shown i n Figure 12.8 for both the 34-~ and 13-day bands. I n t e r e s t i n g l y , i n the former band, the coherence between the v e l o c i t y components increased from 0.05 to 0.23. The computed phase differences were s i m i l a r , and i n p a r t i c u l a r both the u and v components were nearly 180° out of phase with the wind s t r e s s . In order to obtain an appreciation i n the time domain of how the water column responds to the wind, the two-month period of March-April 1970 was selected for more intensive study. This period was chosen for two reasons. F i r s t , records were a v a i l a b l e for a l l meters for most of the period. Second, during t h i s time several s i g n i f i c a n t storms occurred, some with northwest and some with southeast winds. The time ser i e s of winds and 141 Figure 12.11 Computed barotropic and upper mean currents f o r the 136-day layer b a r o c l i n i c analysis period. 142 2 2 currents were f i l t e r e d with an A 2^ A2,j/(.24 • 25) f i l t e r (Godin, 1972) i n order to remove di u r n a l and semidiurnal o s c i l l a t i o n s . This i s a low-pass f i l t e r and produces a record with a twelve-hour time step. The r e s u l t s are shown i n Figure 12.12. Notice that the wind has been advanced f i v e days with respect to the currents. As before, the wind stress was rotated to bring i t into alignment with channel geometry. Seven wind events are i d e n t i f i e d i n Figure 12.12; peak values of the wind stress occur f o r (down-strait) and E^ ( u p - s t r a i t ) . For the 50-metre record at H26, the signature of the wind on the currents i s c l e a r , and the currents lag the wind by about f i v e days. S i m i l a r l y , the response at the 140-m s t a t i o n i s apparent f o r the f i r s t month. For the second month, however, some ambiguity e x i s t s i n the assignment of E^ - E^ . I f the choice indicated i s correct, then a down-strait wind does not necessa r i l y produce a down-channel current (compare events E^ , E^ , and E ^ ) . This i s opposite to the response observed at 50 m. I t i s not possible to d e f i n i t i v e l y c o r r e l a t e currents and winds at the c e n t r a l 50-m s t a t i o n , but the c o r r e l a t i o n at 200 m i s cl e a r - c u t , again with a 5-day lag. I t i s also not possible to make the assignment at H06, 50 m for the one-month record that e x i s t s . As was the case f o r the deep eastern meter, c o r r e l a t i o n of winds and currents at H06, 200 m, i s ambiguous i n the second month. Unlike the eastern s t a t i o n , however, the chosen assignment indicates d i r e c t response to the wind with a 9-day l a g . In the f i r s t month the lag i s about 7 days. In an attempt to determine i f the response to the winds observed at 50 m i s representative of the ent i r e upper water column, progressive vector diagrams of the currents at 3 m and 50 m [Tabata and Stickland, 1972a; 1972b; 1972c; Tabata et a l . , 1971] were compared with one another and the wind f o r the period beginning A p r i l 1970 (Figure 12.2). 143 Figure 12.12 Low-pass fi l t e r e d time series of wind stress at Sand Heads and currents along line H. A solid line represents either the along-strait (northwestward) component of wind stress or the northward component of current. A dashed line represents either the cross-channel wind stress (northwestward) or eastward current component. The wind stress time series is advanced by 5 days. 144 At a l l stations there were times during which the currents at both l e v e l s were obviously correlated and i n phase with each other and the wind. But there were also times during which the currents were out of phase with each other or the wind. S i m i l a r l y , the ho r i z o n t a l r e l a t i o n s h i p s between the 3-m currents were unclear. However, the currents at H06 and H16 were, at times, highly correlated and i n phase. Therefore, by comparison with the 3-m currents, i t i s d i f f i c u l t to state i f the measurements at 50 m are representative of the en t i r e upper layer. In summary, examination of the data has indicated several i n t e r e s t - ing features. F i r s t , the low-frequency fluctuations are not i s o l a t e d to the v i c i n i t y of l i n e H or the southern s t r a i t . Second, as suggested by the spectra and response to the wind, Station H26 may l i e i n an oceanographic domain d i s t i n c t from the other two stations. Third, the o s c i l l a t i n g currents bear a resemblance to the mean flow which may indic a t e that the two are dynamically lin k e d . I f t h i s supposition i s v a l i d , then three a l t e r n a t i v e s are p o s s i b l e : (1) the flu c t u a t i o n s are due to the mean currents ( i n e r t i a l i n s t a b i l i t y ) , (2) the mean flow i s a byproduct of the o s c i l l a t i o n s (transients, arrested waves; see, e.g., Csanady, 1978), or (3) they are both caused by some other agency of the type outlined i n Section 10. The f i r s t p o s s i b i l i t y may be ruled out on the basis of r e s u l t s of Section 11 and the fact that the components of the observed o s c i l l a t i n g currents tend to be i n phase. The l a s t a l t e r n a t i v e i s explored b r i e f l y i n the next section. F i n a l l y , the wind obviously plays some ro l e i n GS dynamics but i t s r o l e i s not c l e a r . 145 13. Nonlinear T i d a l Interactions In an e a r l i e r study, Helbig and Mysak (1976) discounted the p o s s i b i l i t y that the tides were responsible for the low-frequency motions i n GS. However, they were r e f e r r i n g to the f o r t n i g h t l y M^ t i d e and did not consider the p o s s i b i l i t y of nonlinear i n t e r a c t i o n s between t i d a l constituents. In a system l i k e GS with large v a r i a t i o n s i n bottom topography and channel geometry, i t i s l i k e l y that such i n t e r a c t i o n s produce s i g n i f i c a n t f o r t n i g h t l y v a r i a t i o n s i n the t i d e . These int e r a c t i o n s occur between t i d a l constituents through f r i c t i o n a l or advective terms, and the resultant o s c i l l a t i o n s are known as shallow-water constituents. The harmonic constants for d i u r n a l , semidiurnal, and higher-frequency t i d a l constituents are l i s t e d i n Table IV and were obtained from the analysis of a 38-day record of t i d a l heights observed at Point Atkinson (see, e.g. , Figure 12.1). From the frequencies given i n Table IV, one may show that interactions between the M2 and K 2 , M 2 and S 2 , 0-̂ and K-̂ , and 0-̂ and P-̂ constituents a l l produce shallow-water constituents of f o r t n i g h t l y period. For example, the M 2 - S 2 i n t e r a c t i o n gives the MS^ constituent with a 14.76-day period, while the O-̂ - K-̂ i n t e r a c t i o n r e s u l t s i n a 13.66-day o s c i l l a t i o n . In t h i s section two types of t i d a l i n t e r a c t i o n s are considered. The f i r s t I term d i r e c t , nonlinear i n t e r a c t i o n and i s that j u s t o u t l i n e d . The second or i n d i r e c t , nonlinear i n t e r a c t i o n consists of the i n t e r a c t i o n of the t i d e with another agency. In p a r t i c u l a r , I speculate upon the i n t e r - a ction of the t i d e with the Fraser River outflow. To determine the s i g n i f i c a n c e of the f i r s t mechanism, r e s u l t s generated from the Department of the Environment numerical t i d a l model of 146 Table IV. Results of the harmonic analysis of t i d a l elevations at Point Atkinson f o r the 38-day period beginning 6 A p r i l 1976. The P^ and S - L constituents are i n f e r r e d from , Nû , i s in f e r r e d from N 2 , T 2 and K 2 are i n f e r r e d from S"2 . (Dr. J . A. Stronach, pr i v a t e communication.) Constituent No. Name Frequency (cpd) Amplitude (cm) Greenwich phase (deg) 1 ZO 0.0 30.2087 0.0 2 2Q1 0.85695237 0.1155 96.73 3 Ql 0.89324397 0.6998 326.92 4 01 0.92953563 3.9706 215.33 5 N01 0.96644622 0.4666 276.69 6 PI 0.99726212 2.8014 347.70 7 SI 1.00000000 0.3242 154.08 8 Kl 1.00273705 7.9255 95.47 9 J l 1.03902912 0.4867 359.38 10 001 1.07594013 0.1736 221.21 11 MNS2 1.82825470 0.1048 322.62 12 MU2 1.86454678 0.3987 317.05 13 N2 1.89598083 2.0722 249.94 14 NU2 1.90083885 0.3795 228.27 15 M2 1.93227291 9.3629 148.55 16 L2 1.96856499 0.2994 262.76 17 T2 1.99726295 0.1426 24.32 18 S2 1.99999905 2.2687 298.19 19 K2 2.00547504 0.4967 220.37 20 2SM2 2.06772518 0.0464 1.47 21 M0 3 2.86180973 0.0132 137.41 22 M3 2.89841080 0.0187 247.17 23 MK3 2.93500996 0.0152 87.29 24 SK3 3.00273800 0.0095 203.48 25 MN4 3.82825470 0.0106 276.20 26 M4 3.86454678 0.0353 139.86 27 SN4 3.89598179 0.0083 281.85 28 MS 4 3.93227291 0.0283 341.27 29 S4 4.00000000 0.0093 291.57 30 2MN6 5.76052761 0.0569 146.10 31 M6 5.79681969 0.0709 33.69 32 MSN6 5.82825565 0.0220 266.26 33 2MS6 5.86454582 0.0826 198.62 34 2SM6 5.93227386 0.0209 334.93 35 3MN8 7.69280148 0.0086 51.41 36 M8 7.72909451 0.0177 97.38 37 3MS8 7.79681969 0.0077 327.90 38 M12 11.59364128 0.0087 99.16 147 the Juan de Fuca-Strait of Georgia system (Crean, 1976; 1978) were examined. This i s a two-dimensional, v e r t i c a l l y integrated model u t i l i z i n g an e x p l i c i t , forward-stepping, f i n i t e d ifference scheme; adjoining i n l e t s and northern passages to the open ocean are simulated as one-dimensional channels. In the l a t e s t version, a 2-km mesh siz e i s employed. The system i s driven by s p e c i f y i n g t i d a l elevations along the open boundaries; these elevations are obtained from a 61 harmonic constituent t i d e . A 13-day time series of v e l o c i t i e s and elevations with a time step of 15 minutes has been generated from t h i s model. To determine the r e s i d u a l currents, the v e l o c i t y time series was f i r s t smoothed to one hour 2 2 with an A 4 A 5/(4 • 5) f i l t e r (Godin, 1972) and then low-passed f i l t e r e d 2 2 to 12 hours with the A 2 4 ^25^ * f i l t e r previously described which e f f e c t i v e l y eliminates o s c i l l a t i o n s with frequencies exceeding 0.8 cpd. By r e s i d u a l , we r e f e r to the remaining low-frequency components. Three days of data were l o s t i n the a p p l i c a t i o n of the two f i l t e r s leaving a 10-day time s e r i e s . The r e s i d u a l flow along l i n e H i s indicated i n Figure 13.1 for 3 separate days; the average flow over the 10-day period i s shown i n Figure 13.2. The currents calculated f o r the grids encompassing stations H26, H16 and H06 are i l l u s t r a t e d i n Figure 13.3. The t i d a l elevation and range based on the predicted tides f o r Point Atkinson are also shown i n Figure 13.3 for the period of the a n a l y s i s . I t i s evident that a coherent pattern of r e s i d u a l c i r c u l a t i o n e x i s t s , and that i t i s dependent upon the t i d a l range. I t i s also c l e a r that the strongest r e s i d u a l flows occur near the eastern boundary. Unfortunately, the time series i s i n s u f f i c i e n t l y long to f u l l y resolve a f o r t n i g h t l y v a r i a t i o n , and i t i s possible that the very strong flows occur- r i n g at the beginning of the analysis period may be transients associated J i l i i • — _ i — i — 0 2 cms" H06 MARCH 13 d Figure 13.IA Daily, barotropic, r e s i d u a l t i d a l flow along l i n e H. co ? in G A| L I A N 0 I S L A| N D i i i i i 0 2 cms" . 4H 06 MARCH 17 H 16 _TL VANCOUVER FRASER RIVER Figure 13.IB G A L I A N 0 I S L A N D i i i i i 0 2 cms H 06 MARCH 21 ' / HI6 H 26 VANCOUVER F R A S E R RIVER Figure 13.1C 7 ? _ i — i — i 2 c m s H06 AVERAGE FLOW HI6 d VANCOUVER FRASER RIVER Figure 13.2 The r e s i d u a l barotropic, t i d a l flow averaged over 10 days. 152 5.0 M A R C H ( 1 9 7 3 ) Figure 13.3A Time seri e s of predicted (A) t i d a l height and 'tidal range at Pt. Atkinson arid calculated (B) r e s i d u a l current magnitude and d i r e c t i o n along l i n e H. 153 Figure 13.3B 154 with s t a r t i n g the model from an i n i t i a l state of r e s t . However, the model was run f o r two (tidal) days p r i o r to the 13-day period i n order to avoid t h i s problem. At any rate, i t i s clear that the r e s i d u a l motions are of i n s u f f i c i e n t strength and improper d i r e c t i o n to serve as an explanation of the observed low-frequency motion along l i n e H. On the other hand, nonlinear t i d a l i n teractions may be important i n other ways, for example, i n the generation of i n t e r n a l tides or i n i n t e r a c t i o n with the Fraser River. While the r e s i d u a l flow does not resolve the present problem, i t c l e a r l y merits further i n v e s t i g a t i o n . In p a r t i c u l a r , i t i s l i k e l y to be s i g n i f i c a n t i n the southern t i d a l passes (Crean, 1978; Figures 12 and 13). F i n a l l y , I speculate on the p o s s i b i l i t y that the t i d e i n t e r a c t s nonlinearly with the Fraser River outflow to produce, i n part, the observed low-frequency currents. An examination of the Fraser River discharge at some distance upstream from the mouth indicates no consistent f o r t n i g h t l y or monthly v a r i a t i o n s (Figure 13.4); the discharge i s dominated by the large annual peak that occurs i n lat e spring and i s due to the melting of the snowpack. However, near the r i v e r mouth the ti d e modulates the r i v e r flow, indeed the region comprises a salt-wedge-type estuary. I f the magnitude of t h i s i n t e r a c t i o n varies with t i d a l range, then i t i s possible that motions that are driven by the pressure gradient due to Fraser River water l y i n g above GS water may vary with a f o r t n i g h t l y period. The hypothesis as expressed i s obviously crude and ignores e f f e c t s due to density d i f f e r e n c e s , for example, but i t i s offe r e d as a speculative p o s s i b i l i t y that could be examined i n the future. However, there i s some evidence f o r i t . Figure 13.5 shows the low-frequency r i v e r speed at the mouth obtained by low-pass f i l t e r i n g current records. Unfortunately, i t i s superimposed on an increasing FRASER RIVER DISCHARGE AT AGASSIZ Figure 13.4 The Fraser River discharge approximately 60 miles upstream at Agassiz, B r i t i s h Columbia (from Chang, 1976). Figure 13.5 Low-pass f i l t e r e d time series of r i v e r speed at the Fraser River mouth ( Stronach, 1977) t— 1 157 discharge due to the onset of freshet. Nevertheless, a s i g n i f i c a n t v a r i a t i o n i n the speed i s evident i n the f i r s t 12 days of the record. In addition, the analysis of Chang (1976; Figures 40 and 42) suggests that the mean current d i r e c t i o n at H26 turns to the south i n periods of h i g h - r i v e r runoff. This speculation has the advantage of p r e d i c t i n g that the most s i g n i f i c a n t flow should occur along the eastern boundary since the pressure head due to the r i v e r should be l o c a l i z e d there. In a period of high runoff t h i s mechanism might be o f f s e t . Moreover, the theory allows f o r i n t e r a c t i o n with the winds which would serve to modify the outflow. I t i s c l e a r , however, that t h i s hypothesis must be part of a more encompassing theory of the modulation of the estuarine flow i n the Juan de Fuca-Strait of Georgia system. Other e f f e c t s such as the influence of the strength of t i d a l mixing i n the southern t i d a l passes must be examined. Moreover, f r i c t i o n has yet to be considered. I t i s hoped, however, that the present work w i l l stimulate numerical modelers to work on t h i s system and to examine not only short-period e f f e c t s but also longer period ones. 158 14. Summary of Part II From the analysis of current and wind data taken i n the S t r a i t of Georgia and from the consideration of simple i n e r t i a l i n s t a b i l i t y models, the following conclusions may be drawn from Part I I . 1. The observed fl u c t u a t i o n s are not due to simple wavelike motions. That i s , they are not composed of fr e e , forced, or unstable plane waves of the type considered i n t h i s t h e s i s . This conclusion i s based on the f i n d i n g that the components of the f l u c t u a t i n g currents tend to be i n phase. 2. The o s c i l l a t i o n s may bear some dynamical r e l a t i o n s h i p to the mean currents. This notion i s based on the general resemblance of the mean and f l u c t u a t i n g currents. 3. As evidenced from the spectra and the response to the wind, the eastern s t a t i o n may respond to f o r c i n g d i f f e r e n t l y than the cen t r a l and western s t a t i o n s . 4. The wind plays some r o l e i n determining the low-frequency currents. This i s suggested by the fact s that (a) s t a t i s t i c a l l y s i g n i f - icant although small coherences are calculated between the currents and winds, (b) that the corresponding phases consistently l i e near 0° or 180°, and (c) i n the time domain, the response of the water column to a wind event i s often evident. 5. B a r o c l i n i c i n s t a b i l i t y of the mean flow i s an u n l i k e l y mechanism i n GS due to the narrow region of i n s t a b i l i t y i n parameter space. 6. A barotropic i n s t a b i l i t y model indicates that shear i n s t a b i l i t y might be of some s i g n i f i c a n c e but (1) m i l i t a t e s against t h i s 159 p o s s i b i l i t y . 7. 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I t i s convenient to rewrite (3.25) as fty = £ 2 { Q X I 1 + (h'/h 2)(V - c ) I 2 - h _ 1 ( V - c ) I 3 } (A.l) where H = [ ( v - c ) 2 - e2]V - ( v - c)Q x - e 2 a 2 h _ 1 (A.2) and I j , I 2 r and I^ are i n t e g r a l s defined, for example, by I I x = / (V - c) 1G(x,^)[R(x - bJVty 0 (A.3) - (h'/h 2)R'(x - 0$ - h _ 1R"(x - £)Md£. I t s u f f i c e s to consider only 1^ since the other i n t e g r a l s are s i m i l a r i n form. There are two types of points i n the range of in t e g r a t i o n that must be dealt with, namely points x c at which c = V(x c) and the integrand i s singular and points x+ where c = V(x+) ± £. At a l l other points the integrands are continuous and are assumed to be of un i t order. To i s o l a t e the s i n g u l a r i t i e s we p a r t i t i o n the i n t e g r a l as x_-6 x_+6 x c - 6 xc+<5 x + - 6 x + + 6 £ J l ~ / + / + / + / + / + / , + / , 0 X -0 X +0 X -0 x^+o x,-0 x.+o — — C C T -r (A.4) 1 2 3 4 5 6 7 167 where 6 i s some small p o s i t i v e number which we w i l l l a t e r l e t tend to zero. We need determine only J 2 , , and Jg . Consider J 2 and f i r s t ; since they are of the same form we examine only J 2 • Since i s proportional to the cross-stream v e l o c i t y i t must be continuous, and, therefore, ip" can have, at worst, a 6-function- l i k e s i n g u l a r i t y . We require a somewhat sharper r e s u l t , however. O r d i n a r i l y , one could f i n d a Probenius-type s o l u t i o n to (A.l) i n order to determine the behaviour of at x_ , but the presence of the i n t e g r a l terms prevents t h i s . We therefore look f o r an i t e r a t i v e s o l u t i o n to (A.l) of the form Hxb0 = 0 (A.5) Hty± = £ 2 { Q X I 1 ( ^ 0 ) + (h'/h 2)(V - c ) I 2 ( ^ 0 ) (A.6) - h _ 1 ( V - c ) I 3 ( ^ 0 ) } , etc. A series expansion of (A.5) about x = x_ indicates that the two l i n e a r l y independent solutions are of the form 00 ^ 0 1 = (x - x j [1 + I a n ( x - x_) n] (A.7) n=l C O ^ 0 ( 2 ) = I b n ( x - x _ ) n + al(; 0 ( 1 ) i l n | x - x_| . (A.8) n=0 That i s , behaves no worse than (x - x_)£n|x - x_| ; since ip-̂ i s determined by the integrated value Î Q > i t can be no more singular than ^ n . The same argument holds for higher 4*n , and we conclude that ijj 168 shares t h i s q u a l i t y . We are now ready to estimate J„ , J - < (l/e)max[.G(x,p ] x_+6 R(e) / tty x_ - 6 ( A . 9 ) x +6 - h 1[R' (e)h'/h + R" (e) 1 / 4> Since ijj i s continuous, the l a s t term vanishes i n the l i m i t 6 -»• 0 , and the f i r s t term gives the jump i n ^'/h and therefore also vanishes since ' i s of the form In] x - x_| near x = x_ . We conclude that , and hence Jg , both vanish as 6 -*• 0 . Evaluation of i s more straightforward. We have J 4 < max[G(x,£) ] [R(<S)tty - (h'/h 2)R'( 6 )^ ( A . 1 0 ) -1 x c + 6 . 1 - h R" (6)^] / (V - c) xd£. x c - 6 Consider only the i n t e g r a l and put £ = V(^) to obtain x c + 6 - i c + 6 ' -1 1 = / (V - c) d£ = / [ F ' ( C ) ( ? - c ) ] d? x c ~ 6 c-<51 where F(£) i s the function inverse to V(£), and 6 ' i s an appropriately defined constant which tends to 0 with 6 . Now ( A . 7 ) i s a Cauchy i n t e g r a l and i s r e a d i l y evaluated by allowing c to have a small imaginary part ^ . In the l i m i t of c^ -> 0, we f i n d 169 c+6' I = sgn(c i)/[2F'(c) ] + PV / [F' (£) (? - c) ] (A.11) c-S' Here PV denotes the Cauchy p r i n c i p l e value, and therefore i n the l i m i t of 6 ->- 0, I reduces to ± (l/2F'(c)) which i s an 0(1) quantity. I t follows then that i s 0(1) and hence that the i n t e g r a l terms i n (A.l) are also 0(1). 170 Appendix B: The First-Order Solutions In t h i s section we specify the f i r s t - o r d e r solutions f o r the channel and shelf models. For the channel case we f i n d : r A i ^ o + i A o p i k A12^0 + i A0 P2 0 5 x < 1 1 5 x 5 I (B.l) where c i 0 s i n XP± = e b ( x { (G^x - eD^cos Ax + e 1 X [ ( A x + B-^sin Ax + (CjX + D±) cos Ax]}, (B.2) c i Q s i n a ( l - £)P 2 = [G 2(x - £) - e 1 £ D 2 ] c o s a(x - I) 1-x + e {[A 2(x - I) + B 2 ] s i n a(x - I) + [c 2(x - £) + D 2]cos a.(x - I) (B.3) A l = a l ^ l B 1 = (p x + 2a 1/Y 1)/Y 1 = 2Aa 1/Y 1 D = 2A[ P ; L + a ^ l + 2 A 1 ) ] / Y 1 G l = " q l / 2 X a x = 2 £ 2 / c i 0 2 + 1 + 2b 171 p1 = - 2(1 + b) % = ~ 2 ( e 2 c r ( / c i 0 2 + b ) y = l + 4A' A 2 = a 2/Y 2 32 = - 2(1 - a 2 / y 2 ) / Y 2 C 9 = 2aa 9/Y 2 " 2 D 2 = - 2a[2 - a 2 ( l + 2/Y 2)]/Y 2 G 2 = - q 2/2a a2 = 2 e 2 / c i 0 2 + 1 2 e c r ( / c i O Y 0 = 1 + 4a For the sh e l f case we f i n d A11^0 + i A 0 P l ° " x " 1 A12^0 + 1 A 0 P 2 x > 1 (B.4) where i s given as above and P 2 = u r ( x ) J v ( p ? ) + u 2(x)Y y(p?) (B.5) 172 (1-x) u (x) = - ^[2y 2 c . J,.(p) ] 1 j J (p5)Y v(p£)F(?)d£ (B.6) 1-x e u 2(x) = T r [ 2 y 2 c i 0 J v ( p ) ] / J v 2(p?)F(Od? (B.7) F(C) = (2p 2? 2 + 1) (1 - y hn£) - 2 p 2 c r 0 ^ - 2. (B.8) 173 Appendix C: Evaluation of the Integral Terms for a Simple Flow Model The purpose of t h i s appendix i s to estimate the s i z e of the i n t e g r a l terms i n the mean v o r t i c i t y equation for the case of large O. To do so we choose the simplest possible model, a parabolic zonal flow on the 3-plane. The equation equivalent to (3.25) i s (u - c ) 2 - e 2]RJj + (u - c)gyi(> - e 2 a 2 ^ 1 - e 2Q / (u - c) _ 1G(y,?) [R(y - 5)F̂ - R"(y - ZWiaZ (c.D Y 0 1 - (U - c) / (U - c)~1G(y,5) [R"(y - K)H - R 1 V ( y - O ^ d ? , 0 2 2 2 where F = d /dx - k . Although (C.l) i s somewhat less complicated than (3.25) , the i n t e g r a l s are of the same form. In order to evaluate these terms two obstacles must be surmounted. F i r s t G(y,£) must be determined, and second, a form for R(y - K) must be s p e c i f i e d . To obtain an a n a l y t i c a l s o l u t i o n for G(y,£) we choose U such that 0 = 3 - U" vanishes, namely U(y) = 3 y(y - l ) / 2 ; (C.2) then G(y,£) i s given by fsinh ky sinh k ( l - £)/k sinh k 0 < y < § S 1 G(y,?> = i (C.3) sinh k^ sinh k ( l - y)/k sinh k 0 5 £ 5 y < 1. 174 We s e l e c t R(y) to be Gaussian, R(y) = exp(- y 2 a 2 / 2 ) . (C.4) We now take the solution IJJ = + a ^ty as determined from the perturbation expansion outlined i n Section 7, substitute i t into ( C . l ) , and determine the r e l a t i v e values of the various terms i n the r e a l and imaginary parts of the r e s u l t i n g expression. S p e c i f i c a l l y , we f i n d I J J 0 = A Q s i n nlTy (C.5) ijj = iA Q(gK /2nTTci0) { (y 2 - y) s i n nTTy/mr - [y 3/3 - y 2/2 - (2c r Q/3 + l/2n 2TT 2)y]cos nlTy] } = - (U - ^ ( ^ - I 4) - c . ( I 2 + I 3) (C.6) c i 0 = £/K (C.7) c r Q = - (3/4)(1/nV + 1/3), (C.8) where K 2 = k 2 + n 2iT 2. The r e a l and imaginary parts of (C.l) are [(u - c r ) 2 - c±2 - e 2]Bf 0 + 2(u - cr)c±o 1 F ^ 1 - e 2a 2ip 0 (C9) a~ 1[(u - c r ) 2 -. C i 2 - e 2 ] F ^ - 2(u c r ) C i F ^ 0 - e 2a^ 1 = - (u - c r) {i2 + i 3 ) + c±(i1 - i 4 ) (CIO) 175 where 1 , i 1 = / |u - c| Z(U - c r)G(y rC) [R"(y - 0 R l v ( y - O^AdK (c.ii) 1 9 i 2 = i / a / |u - c|"^(u - c r)G-(y ,5) [R"(y - o - R i v ( y - O ^ l d ? (C12) 1 , i 3 = c ± / |u - c|- zG(y,S) tR"(y - ? ) F ^ 0 0 R l v ( y - C)^ Q]d5 <c-13> i _ 9 i 4 = c ± / a / |u - c| G(y,C) [R"(y - 0H± 0 - R i v ( y - ?)^]dC- (C14) The r e l a t i v e magnitudes of the terms i n (C.9) and (C.10) are presented i n Table V for selected values of £ , 0 , and k. The r a t i o T4/T3 determines whether or not neglect of the i n t e g r a l terms i s j u s t i f i e d . One sees that the v a l i d i t y of t h i s approximation improves as k increases and 0 decreases. This e f f e c t i v e l y puts an upper bound on the choice of a. There i s r e l a t i v e l y l i t t l e dependence on £. Only i n the case of £ = .5, a = 10, and k = ir/5 i s the rhs of - ( C l ) of greater magnitude than the lhs and then, only by 16 percent. The approximation i s very good for the two cases of £ = .5, O = 10, k = 2TT and £ = .5, a = 5 V k = TT. 176 Table V. Relative magnitudes of the terms i n (C.9) and (C.10). Here T12, T3, and T4 r e f e r , r e s p e c t i v e l y , to the absolute value of the sum of terms 1 and 2, and the absolute value of terms 3 and 4 i n (C.9) and (C.10). The values given here are symmetrical about y = 0.5. £ = .5, a = 10, k = TT/5 E = .5, a = = 10, k = TT (C .9) (c: 10) (c. .9) (C. 10) y T4/T3 T12/T3 T4/T3 T12/T3 T4/T3 T12/T3 T4/T3 T12/T3 0.1 0.26 0.91 1.52* 0.95 0.21 0.85 0.21 0.92 0.2 0.89 0.91 1.22Z 0.92 0.85 0.84 0.61 0.86 0.3 1.13 0.91 0.70 0.91 0.94 0.84 0.64 0.84 0.4 1.16 0.91 0.53 0.90 0.93 0.84 0.67 0.83 0.5 1.16 0.91 0.63 0.90 0.92 0.83 0.72 0.83 £ = .5, a = 10, k = 27T £ = .1, a = 10, k = TT (C .9) (C. 10) (C .9) (C. 10) y T4/T3 T12/T3 T4/T3 T12/T3 T4/T3 T12/T3 T4/T3 T12/T3 0.1 0.30 0.76 0.02 0.97 0.08 1.00 0.17 0.48 0.2 0.60 0.75 0.41 0.80 0.74 0.86 0.66 0.73 0.3 0.61 0.74 0.50 0.74 0.89 0.76 0.69 0.75 0.4 0.61 0.73 0.53 0.72 0.92 0.69 0.66 0.75 0.5 0.62 0.72 0.55 0.72 1.00 0.64 0.69 0.74 £ = 5, a = 5, k = TT (C.9) (C.10) y T4/T3 T12/T3 T4/T3 T12/T3 0.1 0.11 0 .83 2. 3 17 2. A 44 0.2 0.27 0 .83 0. 70 4 1. 23 0.3 0.51 0 .83 0. 28 0. 87 0.4 0.67 0 .83 0. 13 0. 75 0.5 0.75 0 .82 0. 27 0. 75 ,2,3,4 I f t h e r a t j _ 0 T4/T1, where TI r e f e r s to the f i r s t term i n e i t h e r (C.9) or (C.10), i s formed the r e s u l t i n g values are 0.28, 0.60, 0.73, and 0.51, r e s p e c t i v e l y . 177 We conclude, then, that the neglect of the i n t e g r a l s i s a good approximation, but that i n some cases, namely very large O or small k, the perturbation solutions are best regarded as representing a f i r s t - o r d e r i t e r a t i v e s o l u t i o n to the complete i n t e g r o - d i f f e r e n t i a l equation. The value of T12/T3 indicates how well the two-term perturbation s o l u t i o n s a t i s f i e s the s i m p l i f i e d v o r t i c i t y equation (7.7). Since the terms represented by T12 and T3 are generally of opposite sign, t h i s s o l u t i o n represents a good approximation to the s o l u t i o n of (7.7) i n most cases. 178 Appendix D: B a r o c l i n i c I n s t a b i l i t y i n a 2-Layer System In t h i s appendix, the equations governing the model described i n Section 11 are derived. Proceeding from the f u l l nonlinear equations of motion (see, e.g., Veronis and Stpmmel, 1964, or Helbig, 1977) , we e s s e n t i a l l y follow the procedure developed by Pedlosky (1964) although the two approaches d i f f e r i n some respects. The p r i n c i p a l assumption made i n the de r i v a t i o n of the 2-layer equations i s that the ho r i z o n t a l components of v e l o c i t y are z-independent within each layer. Consider then t h i s set of equations: upper layer »lt + % ' - fv± = - g n l x (D.l) v + u • V v, + fu, = - g r u (D.2) I t -1 H I 1 l y ( \ - V t + Hi • v \ - V + ( h i + n i - V V H * \ = 0 ( D - 3 ) lower layer u 2 t + H 2 * V H u 2 - f v 2 = - g n l x - g6 0 i 2 - n x ) x (D . 4 ) V 2 t + ^2 • V 2 + f U 2 = " g n i y " g 6 ( T 1 2 " n i > y ( D ' 5 ) n 2 t + u 2 • v H n 2 + (h 2 + n 2)V H • u 2 + u^ • Vh 2 = o. (D.6) Here the subscript i = 1,2 re f e r s to the upper or lower layer, r e s p e c t i v e l y ; r)^ and n 2 are the sea surface and i n t e r f a c i a l displacements 179 (see Figure 11.1); and h 2 are the mean layer thicknesses; 6 = (p 2 - p^)/p 2 expresses the density difference between the two layer s , and V H r e f e r s to the two-dimensional Laplacian operator. As before, i t i s convenient to non-dimensionalize these equations. The following scale factors are chosen: the shelf width L f o r the ho r i z o n t a l coordinates (x,y), a t y p i c a l speed U for the v e l o c i t i e s , and an advective time L/U for t. In addition, we write the lower-layer depth as h 2 = h 2Qb(x,y) where h 2Q i s the maximum depth of the lower layer, and b(x,y) i s an 0(1) quantity. The elevations are scaled geostrophically by (fUL/g) and (fUL/g6), r e s p e c t i v e l y . In non-dimensional form the equations of motion are: upper layer R ° ( u i t + -1 ' V l » - ? v l = - ^ l x ( D - 7 ) R o ( v l t + u± • VgV^ + f u x = - n l y (D.8) R o [ ( n 2 - 6n 1) t + u± • v H ( n 2 - S n 1 ) ] (D.9) = [B1 - Ro ( n 2 - &r\±) ] V • u± lower layer R o ( u 2 t + u 2 • v Hu 2) - f v 2 = - n 2 x - (i - 6)n l x (D.IO) R o ( v 2 t + • v H v 2 ) + f u 2 = - n 2 y - ( i - 6 ) n 2 y (D-ii) R o ( n 2 t + u 2 • V n 2 ) = - [B 2b + Ron 2lV • u 2 - • Vb. (D.12) 180 /N Here Ro = U/fL i s the Rossby number, f = f / f = 1 i s retained temporarily to a id i n the i d e n t i f i c a t i o n of the C o r i o l i s term, and B-̂ and B 2 are 2 2 2 2 Burger numbers defined by B^ = g'h-^/f L and B., = g'h2/f L where g' = go i s the reduced acceleration due to gravity. We w i l l henceforth ignore the term of 0 ( 6 ) on the rhs of (D .10) and (D.ll) since 6 « 1 . I t w i l l also be assumed that b i s a function of x only. The presence of the Rossby number i n these equations, which for GS i s approximately 4 x 10 , suggests a perturbation expansion of the form (u.,^) = I R o ^ ^ ' V ^ n=0 The basic state must s a t i s f y the zeroth-order equations, (D.13) f v x ( 0 ) = n l x ( 0 ) (D.14) f U l ( 0 ) = - n l y ( 0 ) (D-15) V H • u x ( 0 ) = 0 (D.16) f v o ( 0 ) = ( n , ( 0 ) + T, 1 ( 0 )>„ (D.17) f u 2 ( 0 ) = (T,2 ( 0 )>n 1 ( 0 ,) y (D.18) n 0 (0), -1 db n , n 1 0 ) V „ • U 0 - U 9 b — - = 0. (D.19) H — I £ 3 X Equations (D.14), (D.15), (D.17) and (D.18) define stream functions f o r each layer. We note that while (D.16) i s i d e n t i c a l l y s a t i s f i e d , (D.19) 181 requires that e i t h e r u 2 ^ vanish or that db/dx be 0(Ro). Although (0) we w i l l , i n f a c t , choose u., =0, t h i s problem again arises at the next l e v e l , and so we put db/dx = - RoT(x). A minus sign i s chosen so that the sign of T coincides with that of the bottom slope; i . e . , i f the bottom slopes upward to the east, T > 0. This choice also f a c i l i t a t e s comparison with Mysak and Schott (1977). As a basic state we choose u± i0) = (0,V.(x)) n.(°> = H.(x) (D.20) where V± and H i are r e l a t e d by (D.14)-(D.19). I t i s easy to show that t h i s state a c t u a l l y s a t i s f i e s the complete nonlinear set of equations (D.7)-(D.12). The perturbation state i s governed by the f i r s t - o r d e r equations - (1) (1) f v x = n x y (D.21) (1) (1) f U l = - n x (D.22) (1) (1) . _ (1). f v 0 = - (ni + n 9 ) x (D.23) - (1) , (1) (D x f u 2 = - (n-L + n 2 ) y (D.24) which again define stream functions f o r each layer. As such (D.21)-(D.24) are indeterminate, and i t i s necessary to go to second order to obtain an equation for H]/"^ a n d ^ ^ ' T ° s e c o n d o r <3er, one f i n d s : 182 upper layer O t + V ^ y J u ^ 1 5 - f v x ( 2 ) = - T 1 l x ( 2 ) (D-25) O t + V X 3 y ) v 1 ( 1 ) + u 1. ( 1 )V' 1 + f U l ( 2 ) = - n l y ( 2 ) (D.26) O t + V ^ y ) ( T l 2 ( 1 ) - <5lli ( 1 )) + u x ( H 2 - 6 H 1 ) X = B^V • u 1 ( 2 ) (D.27) lower layer ( 3 t + V 2 8 y ) u 2 ( 1 ) - f v 2 ( 2 ) = - ( H 2 ( 2 ) + V 2 ) ) x (D.28) O t + V2Zy)v2™ - u 2 ^ > V 2 + f u 2 ( 2 ) = - ( n 2 ( 2 ) + r)l (2))Y (D.29) ( 3 t + V 2 9 y ) n 2 ( 1 ) - U 2 ( 1 ) H ' 2 = - B 2 b V • u 2 ( 2 ) + B 2 U 2 ( 1 ] T . (D.30) Here a prime denotes d i f f e r e n t i a t i o n with respect to x. By cross- (2) d i f f e r e n t i a t i n g the momentum equations and s u b s t i t u t i n g f o r V • u^ from the continuity equation we obtain the v o r t i c i t y equations f o r the perturbation state, [ 3 t + V ^ y ] [ v l x - u l y + B j ( n 2 - 6 m ) ] + (D.31) + B ^ U 1 [ H 2 " 6 H l ] x = 0 &t + V 2 3 y ] t v2x " U 2 Y ~ B ^ ' 1 1 2 1 + U2 V"2 ( ° - 3 2 ) f [u,H', + B 9u 0T] == 0. B 2 b L U 2 n 2 °2 U2 183 Here we have dropped the superscript (1). A Taylor series expansion of b(x) about the point x Q where b(x Q) = 1 gives b(x) = 1 + RoT(x 0) (X - x n) + 0(R n ) 0 (D.33) It follows then that to the present order of a n a l y s i s , 1/b may be replaced by 1. In terms of the stream functions, ? = n + n 2 2 1 (D.34) and the basic state v e l o c i t i e s , (D.31)-(D.32) may be rewritten [ 9 t + V y ] [ V 2 $ 1 + F l ( $ 2 " $ 1 ) ] ~ $ l y [ V " l " F 1 ( V 1 " V 2 ) ] ( D - 3 5 ) = 0 [ 8 t + V 2 9 y ] [ y 2 $ 2 - F 2 ( $ 2 - $ 1 ) ] - $ 2 y [ V " 2 + F 2 ( V 1 " V 2 } + T ] (D.36) = 0 , where and F 2 are i n t e r n a l Froude numbers given simply by the r e c i p r o c a l s of and B 2 , r e s p e c t i v e l y . To obtain these, a term of 0 ( 6 ) was dropped. This i s the desired set of equations governing the perturbation state. Since they express the conservation of p o t e n t i a l v o r t i c i t y i n the system, they could also have been developed d i r e c t l y from the f u l l , nonlinear, v o r t i c i t y equations.
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