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Mixed baroclinic-barotropic instability with oceanic applications Wright, Daniel Gordon 1978

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MIXED BAROCLINIC.-BAROTROPIC INSTABILITY WITH OCEANIC APPLICATIONS by DANIEL GORDON WRIGHT B.Sc, Laurentian University, 1975 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Mathematics Institute of Applied Mathematics and Statistics Institute of Oceanography We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November, 1978 Daniel Gordon Wright, 1978 In presenting th i s thes is in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary shal l make it f ree ly ava i l ab le for reference and study. I further agree that permission for extensive copying of th is thes is for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t ion of this thes is for f inanc ia l gain sha l l not be allowed without my written permission. Department of Mathematics The Univers i ty of B r i t i s h Columbia 207S Wesbrook Place Vancouver, Canada V6T 1W5 D a t e November 10. 1978 i i ABSTRACT A brief introduction to the general subject of baroclinic-barotropic instability i s given in chapter I followed by a discussion of the work done in the following chapters. In chapter II a three-layer model i s derived to study the stability of large-scale oceanic zonal flows over topography to quasi-geostrophic wave perturbations. The mean density profile employed has upper and lower layers of constant densities and respectively ( p ^ <p^) and a middle layer whose density varies linearly from p^ to p ^ . The model includes vertical and horizontal shear of the zonal flow in a channel as well as the effects of 8 (the variation with latitude of twice the local vertical component of the earth's rotation) and cross-channel variations in topography. In chapter II the effects of density s t r a t i f i c a -tion, vertical curvature in the mean velocity profile, 8, constant slope topography and layer thicknesses H\ (i=l,2,3) are studied. The following general results with regard to the sta b i l i t y of the flow are found: (1) curvature in the mean velocity profile has a very strong destabil-izing influence (2) density stratification stabilizes (3) the 8-effect stabilizes (4) topography stabilizes one of two possible classes of ins t a b i l i t y (a bottom intensified instability) and (5) increasing either H, or relative to H 9 stabilizes. i i i Finally, the model is compared with two-layer models and results clearly indicate the importance of having at least three layers when curvature of the mean velocity profile i s present or when i s significant. In chapter III, mixed baroclinic-barotropic instability in a channel i s studied using two- and three-layer models. The equations appropriate to the two-layer model used have been derived previously by Pedlosky (1964a). This model consists of two homogeneous layers of fl u i d with upper and lower layers of densities and respectively ( P 2>P^) and the corresponding mean velocities are taken as = UQ. (1-cos Tr(y+1)), U2 = £U^(e=constant) . The choice of a cosine jet allows the possibility of barotropic i n s t a b i l i t y (Pedlosky, 1964b) while the possibility of baroclinic instability i s introduced by considering values of e other than 1. In the study of the three-layer model, whose governing equations were derived in chapter II,the mean velocities are chosen in the form = U Q(l-cos ir(y+l), = eU^ and U 3 = 0 and to simplify the interpretation of results, the effects of $ and topography are neglected. Again the study of mixed baroclinic-barotropic i n s t a b i l i t y i s studied by varying e. The study of pure baroclinic or pure barotropic i n s t a b i l i t y 2 2 in either model i s j u s t i f i e d for the cases ( L / r ^ « 1 or ( L / r ^ » 1, respectively (L i s the horizontal length scale of the mean currents and Tj i s a typical internal (Rossby) radius of deformation for the system). iv 2 For the case (L/r^) ^ 1 i t i s found that the properties of the most unstable waves vary with the long-channel wavenumber. For each model, i t i s found that below the short wave cut-off for pure barotropic instability there are generally two types of i n s t a b i l i t i e s : (1) a baroclinic instability which generally loses kinetic energy to the mean currents through the mechanism of barotropic instability and (2) a "barotropic i n s t a b i l i t y " which in some cases extracts the majority of i t s energy from the available potential energy of the mean state. The latter type of instability i s most apparent in the study of the three-layer model although i t i s also present in the two-layer case. It i s a very interesting case since i t s structure is largely dictated by the mechanism of barotropic instability even when i t s energy source is that of a baroclinic instability. Beyond the short wave cut-off for pure barotropic ins t a b i l i t y , only the former of these two types of i n s t a b i l i t i e s persists (i.e. the baroclinic i n s t a b i l i t y ) . Qualitative results for the three-layer model are also derived in chapter III (section 3). The energy equation is discussed, bounds on phase speeds and growth rates of unstable waves are derived and the. condition for marginally stable waves with phase speed within the range of the mean currents is presented. Chapter IV i s concerned with oceanic applications. Low frequency motions (- 0.25 cpd) have recently been observed in Juan de Fuca Strait. The three-layer model developed in chapter II is used to show that at least part of this activity may be due to an in s t a b i l i t y (baroclinic) V of the mean current to low-frequency quasi-geostrophic disturbances. Recent sat e l l i t e infrared imagery and hydrographic maps show eddies in the deep ocean just beyond the continental shelf in the north-east Pacific. The wavelength of these patterns i s about 100 km and the eddies are aligned in the north-south direction paralleling the continental slope region. A modification of the three-layer model derived in chapter II i s used to study the sta b i l i t y of the current system in this area. It i s found that for typical vertical and horizontal shears associated with this current system (which consists of a weak flow to the south at shallow depths, a stronger poleward flow at intermediate depths and a relatively quiescent region below), the most unstable waves have properties in agreement with observations. v i TABLE OF CONTENTS ABSTRACT i i LIST OF TABLES v i i i LIST OF FIGURES ix ACKNOWLEDGEMENTS xix I INTRODUCTION 1 II BAROCLINIC INSTABILITY AND CONSTANT BOTTOM SLOPE. . . . 14 1. Introduction 15 2. Formulation 17 3. Linear Perturbation analysis 30 4. Results of Independent Parameter Variations 40 5. Two-layer Models 71 6. Conclusions 81 III MIXED BAROCLINIC-BAROTROPIC INSTABILITY IN TWO- AND THREE-LAYER MODELS 86 1. Introduction 87 2. The Two-layer Model 89 3. The Three-layer Model (Qualitative Results) 112 4. Baroclinic-barotropic Instability in the Three-layer Model (A numerical study) 124 5. Conclusions 141 IV OCEANIC APPLICATIONS 150 1. Introduction 151 2. Juan de Fuca Strait 153 3. The California Undercurrent off Vancouver Island 169 4. Conclusions 193 V CONCLUDING DISCUSSION REFERENCES CITED . . . . APPENDIX (GLOSSARY OF SYMBOLS) v i i i LIST OF TABLES CHAPTER IV Table 2.1: P r o p e r t i e s of the most unstable wave found i n the t h r e e - l a y e r model of Juan de Fuca S t r a i t 161 Table 3.1: Comparison of the i n s t a b i l i t i e s found by Mysak (1977) and the corresponding i n s t a b i l i t i e s found using the analogous t h r e e - l a y e r model 181 Table 3.2: P r o p e r t i e s of the most unstable wave corresponding to the system shown i n f i g u r e 3.7 , 181 Table 3.3: P r o p e r t i e s of the most unstable wave corresponding to the system shown i n f i g u r e 3.8 188 Table 3.4: P r o p e r t i e s of the most unstable waves found using two-layer models to study the s t a b i l i t y of the upper l a y e r s of the C a l i f o r n i a current system , 192 ix LIST OF FIGURES In this thesis, figure, table and equation numbers are local to each chapter. When a figure, table or equation i s referred to which is found in a chapter other than the current one, the appropriate chapter is explicitly noted. CHAPTER I Figure 1.1: Different trajectories along which flu i d elements are displaced in the discussion of baroclinic instability 6 Figure 1.2: The displacements of flu i d elements through a velocity and vorticity distribution considered in the discussion of barotropic instability 9 CHAPTER II Figure 2.1:A cross-section of the three-layer model studied here..... 18 ys Figure 4.1: Stability boundaries (a) for = H 2 = E^, T = 0, 8 = -6 and f i r s t mode (m-1) dispersion curves corresponding to these parameter values with F 2 = 1 and S 2 =-0.75(b), 0.00(c), and 0.75(d) 45 Figure 4.2: Stability boundaries (a) for ^  = H 2 = H 3 > T = 0,6 = -3 and f i r s t mode (m=l) dispersion curves corresponding to these parameter values with F 2 = 1 and S 2 = -0.75(b), 0.00(c), and 0.75(d). Statistics for positions of maximum growth rate (marked by plus signs in the figures) are given in the table. The signs of and are given in brackets following 6^  and 6^  respectively... 46 r r Figure 4.3: As in figure 4.2 with g = 0. Also shown in part (a) i s the corresponding result of Davey (1977). (broken curve) 47 Figure 4.4:As in figure 3 with 6 =6 • ..48 Figure 4.5: As in figure 3 with 6 = 12 49 Figure 4.6:Stability boundaries (a) for = H 2 = T = " 3 0» 6 = 0 and f i r s t mode (m=l) dispersion curves corresponding to these parameter values with F 2 = 1 and S 2 = -0.75 (b), 0.00 (c), and 0.75 (d). Statistics for positions of maximum growth rate (marked by plus signs) are given in the table in order of increasing k 51 Figure 4. 7: As in figure 7 with T = -10 52 Figure 4.8: As in figure 7 with T =-5 53 Figure 4.9: As in figure 7 with T = 0 .54 Figure 4.10: As in figure 7 with T = 5 55 Figure 4.11: As in figure 7 with T = 10 56 Figure 4.12: As in figure 7 with T = 30 57 Figure 4.13:Stability boundaries (a) for H^ /H = 0.5, H^/^ = 1> T = 0, 8 = 0 and f i r s t mode (m=l) dispersion curves corresponding to these parameter values with =1 a n o^ ^2 = "0.75(b), 0.00(c), and 0.75(d). Statistics for positions of maximum growth rate (marked by plus signs) are given in the table 61 Figure 4.14: As in figure 14 with B^/H^ = 1, H3/H2 = 1 62 Figure 4.15: As in figure 14 = 5, H3/H2 = 1. Values are given in order of increasing k 63 Figure 4.16: As in figure 14 with Hj/H^ = 1, H3/H2 = 0.5. Values are given in order of increasing k .-64 x i Figure 4.17: As in figure 14 with I^/H^ = 1, H3/H2 = 1 65 Figure 4.18: As in figure 14 with H /H2 = 1, H3/H2 = 5 66 Figure 5.1: A cross-section of the two-layer model obtained by letting H 2 -> 0 72 Figure 5.2: Dispersion curves for the f i r s t mode (m=l) for = 1, f = IO - 4 s" 1, fS = 0, T = -10, L = 10 km, g' = 0.02 m s~ 2, H T = 180 m and ^ = H 3 = (HT~H2)/2 for (a) H 2 = 60 m, (b) H2 = 6m, and (c) H 2 = 10"5 m 73 Figure 5.3: A cross-section of the two-layer model obtained by letting •+ 0 76 Figure 5.4: Dispersion curves for the f i r s t mode (m=l) for S 2 = 1, —A — 1 9 f = 10 s , t3 = 0, T = -10, L = 10 km, g' = 0.02 m s" , 1^=180 m, and H 2 = H 3 = (HT-H1)/2 for (a) H = 60 m, (b) E± = 6 m, and (c) ^  = 10 5 m 77 Figure 5.5: A cross-section of the two-layer model obtained by letting H 3 -• 0 80 x i i Chapter III Figure 2.1: (a) Mean potential vorticity gradients in the upper layer (solid line) and the lower layer (dashed line), (b) Complex amplitude for the stream function in the upper layer (<f>2=0). (c) Transfer of Available Potential Energy (T.A.P.E.). (d) Transfer of Kinetic Energy (T.K.E.) in the upper layer. Since cf^-O, the transfer in the lower - A _2 layer is very small. fo=10 , 8=0, L=57.735 km, g'=0.66 ms , H =H2= 5 km (F^F^O.01), 1^ = l-cos1I(y+l), U2=-U 93 Figure 2.2: Stability boundary corresponding to the parameter values f o=10" 4s _ 1, 3=1.5, L=1,000 km, g'=0.66 ms"2, H =H2=5 km (F «F -3), U1=U0(l-cos1I(y+l), U2=0 96 Figure 2.3: Stability boundary corresponding to the parameter values f o=10 _ 4s - 1, 3=0.0, L=1,000 km, g'=0.66 ms"2, H^H -5 km, (F =F =3), U l = U 2 = U 0 ( 1 " C O S l I ( y + 1 ) 9 7 Figure 2.4: Approximate stability boundaries in e vs. k space for the 4^ -1 two-layer model with parameter values: fQ=10 s , 3=0.0, L=2,000 km, g'=0.66 ms"2, H^H.^5 km (F1=F2=12), U =l-cos1I(y+1) , U2=eU . Also shown 2 here is the stability boundary appropriate to (L/r/) «1 (dashed l i n e ) . Note: these boundaries are qualitative and details should not be taken seriously 99 Figure 2.5: (a) Mean potential vorticity gradients in the upper layer (solid line) and lower layer (dashed 1 ine). (b) Transfer of available potential energy, (c) Transfer of kinetic energy in the upper layer ( ^transfer in the lower layer which is indicated by a dashed line in this and future figures), (d) Compex amplitude of the stream function in the upper layer, (e) Complex amplitude of the stream function in the lower x i i i layer. Parameter values as in figure 2.4; e=-1.0, k=1.5. 101 Figure 2.6: As in figure 2.5 but e=-0.5 1 0 2 Figure 2.7: As in figure 2.5 but e=0.0 1 0 3 Figure 2.8: As in figure 2.7 - a second significant instability at the same position in parameter space 104 Figure 2.9: As in figure 2.9 but £=0.5 106 Figure 2.10: As in figure 2.9 - a second significant instability at the same position in parameter space 107 Figure 2.11: (a) q ( E q 2 y ) (b) T.K.E. (T.A.P.E.=0) (c) ^ (Hc^). Parameter values as in figure 2.4; e=1.0, k=l. 5 108 Figure 2.12: Parameter values as in figure 2.4; k=3.5, e=0.0 HO Figure 2.13: Parameter values as in figure 2.4; k=3.5, e=0.5 H I Figure 4.1: Approximate stability boundaries in e vs. k space for -4 -1 -2 the three layer model. fo=10 s , 6=0.0, g'=l. 0 ms , H^H^H^ 3333.33 m; L=57.735 km - approximately vertical line; L=1,000 km -dashed line; L=2,000 km -solid line 126 Figure 4.2: (a) The mean potential vorticity gradient in the upper layer (solid l i n e ) ; the middle layer (long dashes); the lower layer (short dashes), (b) Transfer of available potential energy due to shear between the upper layers (solid line) and that due to shear between the lower layers (dashed line), (c) The transfer of kinetic energy in the upper layer (solid line) and in the middle layer ( dashed lin e ) . The transfer in the bottom layer i s zero since U^HO. (d) The compex amplitude for the stream function in the upper layer. xiv (e) The compex amplitude for the stream function in the middle layer. (f) The complex amplitude for the stream function in the bottom layer. fo=10"V^=0.0, L-1,000 km, g'=1.00 ms"2, H^H^Hg-3333.33 m ( F 1 = F 2 = F 3 =3), U =l-cosH(y+l), U ^ e l ^ ; e=0.5 129 Figure 4.3: As in figure 4.2 but L=2,000 km 130 Figure 4.4: As in figure 4.2 - the instability at larger k 131 Figure 4.5: As in figure 4.3 - the instability at larger k 132 Figure 4.6: As in figure 4.2 but e=0.75 133 Figure 4.7: As in figure 4.2 but L=2,000 km, e=0.75 135 Figure 4.8: As in figure 4.7 136 Figure 4.9: As in figure 4.6 138 Figure 4.10: As in figure 4.10 but L=2,000 km, e=1.0 139 XV Chapter IV Figure 2.1: (top) Juan de Fuca S t r a i t and nearby geographical features, (bottom) Juan de Fuca S t r a i t with positions at which Fiss e l ' s data (•) and one section of Cannon and Laird's data (x) were collected. (from Cannon and Laird, 1978) 154 Figure 2.2: The cross-section of Juan de Fuca S t r a i t at which Fi s s e l ' s data were collected. Numbers above each station are used to indicate the station considered i n figure 2.5. Along-channel v e l o c i t i e s are given i n cm s The dashed l i n e indicates the topography used i n the model of the s t r a i t . (from F i s s e l , 1976) 155 Figure 2.3: The mean density p r o f i l e on June 14-15, 1975 across the section at which F i s s e l ' s data was collected (from F i s s e l and Huggett, 1976) 156 Figure 2.4: Along-channel total-record average currents (cm/sec.) through the cross-section marked by x's i n figure 2.1 (from Cannon and Laird, 1978) 157 Figure 2.5: The power spectral density of the residual currents for the current meter stations of F i s s e l (1976). Stations 130-136 are shown i n figure 2.2 and station 137 i s the additional station a few kilometers to the west of Sheringham Point seen i n figure 2.1 (bottom), (from F i s s e l , 1976) 159 xvi Figure 2.6: (a) Approximation to the mean currents used to model Juan de Fuca Strait; (b) the mean potential vorticity gradients corresponding to the three layers of our model: upper layer-solid line, middle layer-long dashes, lower layer-short dashes; (c) the eigenfunction corresponding to the most unstable wave in the model; (d) the transfer of available potential energy corresponding to this wave: the solid line corresponds to the transfer of energy due to the shear between the upper layers and the dashed line to that due to the shear between' the lower layers; (e) the transfer of kinetic energy for this wave: solid line corresponds to the upper layer, long dashed line to the middle layer and the short dashed line to the bottom layer 160 Figure 3.1: Enhanced infrared image of sea surface showing spatial structure of surface temperature on 10 September, 1975 off the west coast of British Columbia and Washington. The dark areas are warm water and the grey-white, cold water (after Gower and Tabata, 1976) 171 Figure 3.2: Geopotential topography (dyn m) of the 10, 150, 300 and 500 db surfaces (referred to 1000 db), 7-20 Septebmer, 1973. Open circles refer to time-series stations. The 100 and 500 fathom (1 fathom = 1.829m.) isobaths are also shown ( from Reed and Halpern, 1976) 172 Figure 3.3: The two-layer model studied by Mysak (1977) 174 Figure 3.4: Temperature, salinity and a profiles, 10 September, 1973, at 49°N, 127° 19'W (right-hand open circle in top line of time series stations shown in Fig. 3.2). (from Holbrook, 1975) H 6 x v i i Figure 3.5: Isobaths (m) off Vancouver Island and Washington. Topographic cross-sections at lines A - E are plotted i n figure 3.6. (from Mysak, 1977) 178 Figure 3.6: Topographic cross-sections at lines A - E shown i n figure 3.4. (from Mysak, 1977) 179 Figure 3.7: The three-layer, channel model analogous to the two layer model of Mysak (1977) 180 Figure 3.8: The extension of the channel model to include effects of horizontal shear and reduce the influence of the a r t i f i c i a l l y imposed western boundary 182 Fiqure 3.9: Graph of c r (non-dimensional phase speed) i n the wave-number range k = 0 - 5 (wavelength = °° - 47.1 km) for the f i r s t two cross-channel modes. The unstable regions corresponding to those studied by Mysak (1977) are labelled as such. The position at which the most unstable wave i s found i s indicated by a plus sign 183 Figure 3.10: Transfer of available potential energy (in arbitrary units) corresponding to the most unstable 1st mode i n s t a b i l i t y analogous to that considered by Mysak (1977). The position of this wave i n figure 3.9 i s marked by a c i r c l e . (The s o l i d l i n e corresponds to the transfer of energy due to the shear between the upper layers and the dashed l i n e to that due to the shear between the lower layers) 185 Figure 3.11: Transfer of available potential energy corresponding to the most unstable wave found using the three-layer model (plus sign i n Fig. 3.9) (The s o l i d l i n e corresponds to the energy transfer due to the shear between the upper layers and the dashed l i n e to that due to the shear between the lower layers.) 185 x v i i i Figure 3.12: (a) Approximation to the mean currents used to model the C a l i f o r n i a undercurrent o f f Vancouver I s land (see f igure 3 . 8 ) ; (b) The mean p o t e n t i a l v o r t i c i t y gradients corresponding to the three layers of our model: upper l a y e r - s o l i d l i n e , middle l a y e r - l o n g dashes, lower l a y e r - s h o r t dashes; (c) The Eigenfunct ions corresponding to the most unstable wave i n our model; (d) the t rans fer of a v a i l a b l e p o t e n t i a l energy corresponding to the above wave; (e) The t r a n s f e r of k i n e t i c energy i n the three l a y e r s , (see f igure 2.7 for the meaning of the d i f f e r e n t l i n e s i n (d) and (e) .) xix ACKNOWLEDGEMENTS Although discussions with many people have contributed to the work presented in this thesis, I am particularly grateful to Dr. Lawrence Mysak for his efforts as my thesis supervisor. His encourage-ment, enthusiasm, interest, and good advice through every aspect of the preparation of this thesis have been practically unlimited and are greatly appreciated. I would also like to thank Drs. Paul LeBlond, Robert Miura, and Steve Pond for their helpful criticisms of a prelim-inary draft of this thesis. The financial support of the National Research Council of Canada through a post-graduate scholarship, and that of the University of British Columbia through a teaching assistantship are also gratefully acknowledged. In particular i t i s a pleasure to express my gratitude to the Department of Mathematics for affording me the opportunity to proofread the book "Waves in the Ocean" by P.H. LeBlond and L.A. Mysak. Finally, I would like to thank my wife, Donna, my good friend, Dan F r a i l , and my parents, Fred and Connie, for their encouragement, without which this work may never have been undertaken, let alone completed. CHAPTER I INTRODUCTION 1 2 Introduction This chapter i s designed to introduce the uninitiated to the processes of baroclinic and barotropic instability, and briefly outline the contributions made in this thesis. Our intro-duction w i l l be brief and draws heavily on the references cited here. The ocean basins are f i l l e d with a slightly compressible fl u i d subjected to the influences of the earth's rotation and gravity, as well as atmospheric forces. Of the many types of waves which exist in such a system (see LeBlond and Mysak, 1978) we are concerned with the free, low-frequency, quasi-geostrophic waves which exist due to an intrinsic i n s t ability in the system. Since the aspect ratio H/L, (H and L are vertical and horizontal scales for the motion) for such waves i s generally much smaller than unity and the period i s much longer than one day, i t may be shown that vertical accelerations can be neglected relative to the acceleration due to the earth's gravity. The motions are thus in hydrostatic balance. Further since the length scale of density variations ln the ocean i s much larger than the vertical scale of motions, the continuity equation may be approximated by the incompressibility condition ^'u = 0. With exception of sound waves (which are fil t e r e d out under the assumption of incompressibility), this approximation i s valid for most oceanic motions (see P h i l l i p s , 1969). Finally since the relative density variations are very small (-10 at most), their influence through the inertia and Coriolis accelerations are also small and can be neglected in these terms. These observations allow use of the Boussinesq approximation in which the actual density i s replaced by a constant reference density exept when assoc-3 iated with the gravity where a bouyancy force is introduced due to the density difference. This approximation may be shown to be valid i f we have 2 2 N H/g « 1 (N =-(g/Po)3po/3z, where p o is the potential density in the absence of motion), (see LeBlond and Mysak,. 1978,p.15; and Pond and Pickard, 2 1978, chapter 5, for methods used to calculate N in practice). LeBlond and Mysak have also given a detailed derivation of the B-plane equations in which the natural spherical coordinate system of the earth is approx-imated by a local cartesian coordinate system. It i s readily seen from their derivation that these equations are valid for H/L << L/R << 1 (R is the radius of the earth). For the motions under consideration here, these conditions are satisfied. In particular, for Juan de Fuca Strait, Ht200m, L-20km (H/L - 1 x 10~2, L/R~3 * 10~ 3) +and for the California Undercurrent, H£3km; L-50km (H/L£5 * 10~ 4, L/R-8 * 10~ 3). Finally the influence of the horizontal component of the earth's rotation (which I shall neglect) on long period waves has been considered by Needier and LeBlond (1973). The neglect of this effect i s j u s t i f i e d for H/L << 1. The effect of viscosity on these motions w i l l be assumed negligible.. This assumption w i l l generally be true for the- i n i t i a l growth, of the perturba-2 tions i f the Ekman number (v/f^H ;v i s the kinematic [eddy] viscosity*and f Q i s twice the local v e r t i c a l component of the earth's rotation) is small in It must be noted here that the condition H/L << L/R arises as a condition under which the horizontal component of the earths rotation is negligible. wR The actual condition i s — << 1 . Scaling w with U H/L, y with L, and u with U, we get the condition H/L << L/R which i s only weakly satisfied for Juan de Fuca Strait. However, we w i l l find that for the motions considered here, w should be scaled with R Q U H/L SO that this condition may be replaced by R H/L « L/R (note that we s t i l l need H/L << 1 and L/R << 1) o which i s strongly satisfied in our applications. 4 comparison to the square of the Rossby number. However, for f i n i t e amplitude motions, this approximation w i l l not, i n general, be valid (see Pedlosky, 1970). Finally, we assume a simple equation of state of the form p =p^ (1- aT + BS) (1.1) where p^, a ,3 are constants and T and S are the temperature and salinity respectively. Restricting our attention to motions whose time scales are s u f f i c i e n t l y short that the diffusion of heat and salt may be neglected, the equations for the conservation of internal energy and salt (neglecting external sources) reduce to the statement that density i s conserved for individual f l u i d elements, i.e., &-<>• <'-2> Under the approximations made above, the governing equations reduce to equations (2.2), chapter II. These equations w i l l be taken as the starting point for our analysis. We shall now turn our attention to the question of where the growing perturbations obtain their energy from. To answer this question we shall consider the special cases of baroclinic and barotropic i n s t a b i l i t i e s seperately. Baroclinic I n s t a b i l i t y We begin with a discussion of the "wedge of i n s t a b i l i t y " . The basis of this argument was presented by Fj ^ r t o f t (1951) and has since been repeated in various forms by several authors. The discussion 5 given here i s largely a reproduction of work presented by Pedlosky (1971b)and Orlanski and Cox (1973) and is given here solely for the readers convenience. Let P 0 =P 0 ( z) and P 0 =P 0 ( z) D e t n e pressure and density in the absence of motion and let 6p and 6p be the deviations from these values due to a mean flow, U(z), in the x-direction. Then, as we shall show in chapter I I , i f the Rossby number (=U/f L) is small o relative to unity, the mean state w i l l be in hydrostatic and geostrophic balance. Thus, under the Boussinesq approximation we have: ?* f o U =' 6Py 6p z = -6p.g where subscripts y, and z indicate partial differentiation. Eliminating .the pressure we obtain: 6 p y = p* fo V8 Hence the isopycnals have a slope given by: ,3^ f y ?*y_ p * f o U z _ f U /N2 (——) — - — — = — o z 3 y P "z - 8 P o z 2 where we have used £p/p « 1 in the second equality and N = - gp /p is the square of the Brunt-Vaisala frequency. Such a t i l t of the isopycnals supplies a possible source of potential energy for the growth 6 of perturbations. To examine the possibility of a wave extracting energy from this mean state, we consider the consequences of inter-changing flu i d elements constrained to move in a plane along the various trajectories indicated in figure 1.1. A f l u i d element displaced from A to A' w i l l find i t s e l f in a region of lighter f l u i d and thus w i l l experience a downward restoring force proportional to the density difference. This fl u i d element w i l l thus return to i t s original position and no in s t a b i l i t y results. (Note that the flu i d element w i l l generally overshoot i t s original position and oscillate about this position u n t i l the motion i s damped out by viscous effects. This type of motion i s essentially an internal gravity wave). Fluid elements displaced in the plane through SB' experience no restoring force along BB since gravity acts perpendicular to this plane. No instability may result from this interchange either. Similarly a f l u i d element displaced in the plane DD feels no restoring force since a l l f l u i d elements in this plane are of equal density. Finally, consider the interchange of flu i d elements originally positioned at C and C . The particle displaced from C to C finds i t s e l f in a region of lighter f l u i d and thus the effect of gravity tends to accelerate the fl u i d element beyond C . Similarly the f l u i d element displaced from C to C experiences a bouyancy force l i f t i n g i t Figure 1.1: Different trajectories along which f l u i d elements are displaced in the discussion of baroclinic instability. 8 beyond C. Hence motions of this type (exchanging f l u i d between the wedges BOD and BOD' ) w i l l be amplified. Note, however, that such an interchange of f l u i d elements results in a decrease in the potential energy of the system. This i s the basic mechanism of baroclinic in s t a b i l i t y by which potential energy i s released from the mean state to the perturbations. Further details of the nature of this type of instability may be found in Bretherton (1966b), Pedlosky (1971b), and ©rlansky and Cox (1973). Barotropic Instability The basic mechanism of barotropic instability was described in 1945 by Lin. The presentation given here i s simply a reproduction of part of Lin's paper and as in the previous section i s given solely for the readers convenience. The argument i s kept as simple as possible by considering wave perturbations to a horizontally sheared mean current in a homogeneous, inviscid f l u i d . Since the effect of the earth's rotation i s not essential to the in s t a b i l i t y process,(This i s not to say that the effects of rotation are negligible. Rather i t i s f e l t that the inclusion of rotation i s not necessary to understand the nature of the instability.) we consider a two-dimensional parallel flow on a non-rotating plane. Consider the interchange of f l u i d elements between lines Lj 9 and , and lines and in figure 1.2. If one envisions the flu i d as being f u l l of vortex filaments, then when flu i d elements are interchanged in an inviscid f l u i d , the f l u i d maintains i t s v o r t i c i t y and the interchange of fl u i d elements implies the interchange of vortex filaments. Lin has shown that an element of fl u i d displaced in the y-direction (perpendicular to the direction of the mean flow) by the component v' of the perturbation velocity experiences an acceleration in the positive y-direction given by T Ml { v'(x y) r 2 \" (y)dxdy, f l u i d L 3 ° where § q (y) I s the gradient of vorticity of the mean flow and V = Jj J ( f»7 ) df^T i s t n e strength of the superposed perturbation f l u i d vortex (this restoring force i s due to the distortion of the mean vortic-i t y f i e l d ) . Now, consider the consequences of this acceleration on the fl u i d elements labelled a, b, c and d. The element a carries an excess of vorticity (relative to the surrounding f l u i d in i t s new position on L ^ with i t so that Y > 0. Since in this region $ q < 0, the f l u i d element experiences an acceleration in the negative y-direction thus restoring i t to i t s original position. Similarly the fl u i d element b has T < 0 and J Q < 0 so this element experiences a positive acceleration and i s also restored to i t s original position. Thus for the case in which f l u i d elements are not displaced across an extremum in the mean vorticity,the motion must be stable. Now consider the forces acting on c after being displaced. V E L O f T T Y V O R T T C T T Y o Figure 1.2: The displacements of fluid elements through a velocity and vorticity distribution considered in the discussion of barotropic instability. 11 Since J O changes sign between and the restoring force i s clearly impaired by the presence of an inflection point in the mean velocity profile and i f the integrated value of j*^ when weighted with v' i s of the opposite sign to P , the motion may actually be amplified. Thus, due to the conservation of vorticity following a f l u i d element a perturbation may under certain circumstances extract energy from the kinetic energy of the mean flow. This exchange of energy i s , of course, only possible in the presence of a non-zero mean vorticity and, in particular, requires a local extremum in this quantity. The process by which this occurs i s essentially the mechanism of barotropic i n s t a b i l i t y . Further insight into this mechanism may be found in Lin (1945, Part II), Rossby (1949) and Brown (1972). The physical arguments discussed here (for both baroclinic and barotropic in s t a b i l i t i e s ) are unquestionably inadequate for a detailed investigation of the sta b i l i t y of a given system. A detailed study of the i n i t i a l growth' of wavelike disturbances to a given mean flow generally requires the solution of a singular non-separable partial differential equation (see Pedlosky 1964a, and chapter II). In the literature several methods have evolved to study these equations. The earliest studies were made under the assumption that the mean state was either vert i c a l l y or horizontally uniform. Under the assumption of vertical homogeneity the mechanism of baroclinic i n s t a b i l i t y 12 i s rendered inactive (see for example Kuo, 1949), and when the mean currents are assumed horizontally uniform, the mechanism of barotropic in s t a b i l i t y i s rendered inactive (Charney, 1947; Eady, 1949; Fj^ r t o f t , 1951). Probably the most natural extension of the study of pure baroclinic instability i s the extension to include weak horizontal shears in which case the problem may be attacked using the WKB technique (Stone, 1969; Gent, 1974, 1975). Other important contributions have been made through the numerical integration of the governing equations (see for example Brown, 1969, G i l l , et a l , 1974), integral methods involving the use of generalized Green's functions (Mclntyre, 1970), integral equations (Miles 1964a,b), and asymptotic methods (Miles, 1964 a,b,c,; Killworth, 1978). However, the most widely used method to study mixed baroclinic-barotropic in s t a b i l i t y i s probably via the introduction of layered models in which the non-separable partial differential equation mentioned above i s replaced by a system of n (where n i s the number of layers) coupled, singular, ordinary differential equations. The study of layered models was initiated by Phil l i p s (1951) who considered a two-layer model. This model has since been extensively studied, most notably by Pedlosky (1963, 1964 a.b.c; 1970, 1971a,b; 1972, 1974, 1975, 1976). This thesis i s mainly concerned with a model in which the density stratification i s approximated by three layers. Davey (1977) 13 i has studied pure baroclinic instability in a three-layer model similar to the two-layer model of Pedlosky. Here, on the other hand, we derive a three-layer model from the equations for a specialized continuously stratified f l u i d (see figure 1, chapter II). In deriving our equations from those appropriate to a continuously strat i f i e d f l u i d i t i s possible to circumvent one of the major d i f f i c u l t i e s in using layered models. That i s , i t is no longer necessary to "guess" (though an educated guess may be quite good) what the appropriate density differences between the layers should be. This choice i s automatically built into the model when the actual density stratification is approximated by a simpler continuously stratified f l u i d . The new three-layer model i s then used to study the effects of density stratification, vertical curvature in the mean velocity profile, variation of the Coriolis parameter with latitude, bottom slopes and relative layer thicknesses, a l l in the absence of horizontal shear. A brief discussion of the limiting cases of two-layer models i s then given. In chapter III a detailed study of mixed baroclinic-barotropic in s t a b i l i t y in two- and three-layer models i s made, and in chapter IV, two case studies are considered. Further introductory comments on each of these studies may be found in an introductory section at the beginning of each chapter. Finally, some general concluding remarks are made in chapter V. CHAPTER II BAROCLINIC INSTABILITY AND CONSTANT BOTTOM SLOPE 14 15 1. Introduction As discussed in the previous chapter, baroclinic in s t a b i l i t y is the process by which the kinetic energy of a quasi-geostrophic disturbance may increase with time by extracting energy from the available potential energy of the mean state, and barotropic instability is the process by which perturbations may extract energy from the kinetic energy made available by horizontal shear in the mean currents. One method of simplifying the study of baroclinic (or mixed baroclinic-barotropic) instability i s to consider layered models. That i s , one considers two or more layers of f l u i d , each of uniform but different densities, lying one above the other. In this way the essential dynamics of the baroclinic problem are retained while the possible modes of prop-agation are reduced to a small number. Most of the earlier work has been with two-layer models (see for example Pedlosky, 1963,1964a,b,c) but some work has recently been done on three-layer models. Holmboe (1967) has generalized Eady's baroclinic model of the atmosphere with constant entropy gradient to a vertically symmetric three-layer model with constant entropy gradient in each layer and Davey (1977) has generalized the two-layer model of Ph i l l i p s (1951) to a three-layer model with each layer of uniform but different density. 16 By introducing density discontinuities, the usual layered models overemphasize the stabilizing Influence of density s t r a t i f i c a t i o n which must be compensated for by decreasing the density difference between successive layers in the model. By approximating a continuously st r a t i f i e d model with specialized density p r o f i l e , this thesis attempts to make use of the simplifications of the layered approach without intro-ducing discontinuities i n density and velocity profiles^ The motivation for considering such a density profile i s the desire to model situations in which the transition of density from i t s surface value to i t s value at depth occurs gradually. Such a situation occurs in Juan de Fuca Strait which i s the subject of Chapter IV, Section 2 of this thesis. The baroclinic problem for flows over constant slope topography i s considered in this chapter. Later chapters are concerned with the combined case of mixed baroclinic-barotropic i n s t a b i l i t y in the presence of more r e a l i s t i c topography. Following the completion of this thesis (October 30, 1978) i t was brought to my attention that the choice of density s t r a t i f i c a t i o n employed here had been considered earlier by Savithri Narayanan (1973). Her approach to the problem was, however, very different than that taken here. Emphasis is placed on the study of free waves and the study of pure baroclinic instab-i l i t y in the absence of vertical curvature in the middle layer (a case which we shall find to be rather more stable than the more general case including vertical curvature in this layer). No side walls are included so that the model is directly applicable to studies of the open ocean. For the case of no horizontal shear in the mean currents i t is found that by transforming to density coordinates the continuously s t r a t i f i e d problem is made analytically tractable and i t is the resulting analytical solutions which are studied. 17 In section 2 of this chapter, the stability problem i s formu-lated for a channel flow with mean velocities containing both horizontal and vertical shear under the assumption that the Rossby number is small. Section 3 i s concerned with the study of small perturbations propagating on a steady mean flow. Necessary conditions for i n s t a b i l i t y are discussed and an analytic solution i s found for the s t r i c t l y baroclinic problem in the presence of small, constant bottom slope. In section 4 some general results concerning parameter variations are discussed and in section 5 two-layer models are considered as limiting cases. Finally a brief summary of the results of this chapter i s given in section 6. 2 . Formulation We w i l l formulate the st a b i l i t y problem for quasi-geostrophic disturbances of oceanic channel flows containing both ve r t i c a l and latera l shear. The beta effect i s included but the f l u i d i s assumed inviscid and non-diffusive. The basic state density s t r a t i f i c a t i o n i s modelled * by three layers with upper and lower layers of constant densities and Pg respectively (p^<p^) and a middle layer whose density varies linearly it if from p^ to p^ (Figure2.1). This choice of density s t r a t i f i c a t i o n i s convenient as well as being a reasonable approximation to many real situations. The basic state density st r a t i f i c a t i o n i s thus given by: Figure 2.1: A c r o s s - s e c t i o n of the three-layer model studied here; 19 '4 * * p = 1 p2 * J . P-. + z* + H, ( P 1 - P 3 ) < z* < 0 - B ^ - H 2 < z* < -H-L -IL, + bh < z* < -H, - H„ (2.1) H^ > H 2 and are the thicknesses of the upper, middle and lower layers In a state of no motion, H^ , = + H 2 + , and bh i s the height of the bottom above z* = -H^ , (b i s a typical amplitude of the topographic variations). *'s are used to indica t e dimensional v a r i a b l e s . We choose our coordinate system such that y* = 0 i s at the centre of the channel, with y* increasing to the north, x* points along the channel to the east and z* is vertical with z* = 0 as the position of the free surface in the absence of motion. If the effect of B i s negligible the orientation of the x*, y* axis i n the horizontal plane may, of course, be varied. Under the hydrostatic approximation, the equations of motion for an inviscid,incompressible,nonTdiffusive f l u i d on the ^ -plane are: (a) u* , + u*u* * + v*u* . + w*u* . v ' t* x* y* z* -p* x #/p* + fv* (b) v* t* + u*v*xit + v*v* y A + w*v*z* - -P * v * / P * - fu* (c) P*z* (d) u * . + v * , + w * . x* y* z* (e) p * t * + u * p * ^ + v *p * v J k + w*p* z* _ p * g 0 0 (2.2) Where (u*, v*. w*) i s the velocity of the f l u i d , p* i s the total pressure and p* the density. 20 We now introduce the following nondimensional (unstarred) variables (a) (x*,y*) (b) z* (c) (u*,v*) (d) w* (e) t* (f) f (g) p* (h) p* ( 2 . 3 ) o B* = B(U/L2) p Q(z) + p 3Uf oLp p*(z) + p*R o 2 2 f L o I gH J * * where p (z) and p (z) are the pressure and density in the state of o o no motion, U and L are typical horizontal velocity and length scales, H i s a vertical scale (= BL^  say), and R Q ( = u / f D L ) i s t n e Rossby number which we shall assume to be small. Making these substitutions into (2.2), and invoking the Boussinesq approximation, we have: (a) Ro(ut+uux+vuy+wuz) - -p^ + vU+R^y) (b) R (v +uv +w +wv ) «= -p - u(l+R By) v / o t x y z r y o (c) P*z - P * 8 H » P z * "P (d) (e) u + v + w =0 x y z oz _w - - p £ o 2 2 f * V o I gH J"-f ""x-^y [p -hip +vp J ( 2 . A ) 21 Now expressing each of the dependent variables as a power series expansion in R , e . g . p - n £ g R 0 P » v e f i n d t 0 l o w e s t order in R o (a) V p ( 0 ) X (b) (0) _ u • - p < 0 ) y (c) * Poz = - p > (d) w<°> = z 0 (0) _ (0) (2-5> p z = -p (Note that p^^ i s a stream function for the zeroth order problem.) * * Now, since fluid elements on the surface (at z = n^; see figure 2.1) remain there, we have: T) * * DT* ( Z-V = 0 • at the surface. We now scale by: 2 2 * f 1 n, = H R — - T) . 1 1 o gH^ 1 This choice of scaling is consistent with (2.3g) and the hydrostatic approximation. Thus in terms of the non-dimensional variables we have: -2.2 • R -1 surface „ | . R f ! f c 2 f ( » + u » ) | _ + T ( 0 ) l ^ (0)" 1 c o gH I 3t 3x 3y 1 surface 2 2 f L Expanding w in a power series in R q and using — — << 1 (this condition states that the external Rossby radius of deformation is much greater than the channel width and is valid in many oceanic applications), we thus have: w ( 0 ) - w^> - 0 22 at the surface. Combined with (2.5d), w ^ = 0 at the surface gives = 0, while w ^ = 0 at the surface is the basis of the rig i d l i d approximation. To the next order in R we have: o (.) u ( 0 ) • u<°>„<°> + v ( 0>u<°> t x y (b) v ( 0> + u<°>v(°> + v« V 0 ) (c) (d) -pW + v ( 1 ) + v ( 0 ) p y X .(1) u ( 1 ) + v ( 1 ) + w x y z y (1) P - - -u ( 1> - u ( 0 ) 6 y y p* w(1> oz -"5 2 2 f Z l T o I gH J (2.6) [,<»*.»>P»>^ (»P< 0 )I Cross-differentiating (2.6 a,b) and using (2.6c) gives: 1_ ^(0)<L ^ 0 ) 1 3t 3x 3yJ v( 0 ) - u ( 0 ) + e y _x y II w (1) (2.7) (2.1), (2.5), (2.6d) and (2.7) are the basic equation for the following analysis. These equations shall now be used to derive a consistent approximation to the continuously s t r a t i f i e d flow which has three degrees 23 of freedom in the v e r t i c a l . In the following work a subscript w i l l be used to denote which of the three "layers of flu i d defined by (2.1) i s being considered. Since p* is constant in the upper and lower layers, p^^ = p^^ = 0 and hence from (2.5) u ^ and v ^ are depth inde-pendent in these layers. Hence we may immediately integrate (2.7) over these layers. Using w ^ - 0 at the surface and requiring the ve r t i c a l velocity to be continuous across the interface between the layers, we have: 1 H 9t (0)3_^ 1 9x (0)9 1 3yJ H_ (1) H W2 z*=-H +n* (2.8) where n* is as shown in figure 2.1. (1) Using (2.6d) and (2.1) may be expressed as .(1) where F 0 = f 2L 2/g'H 0 , i. o / H„ H (0)9 ,,,(0)9 |.(0) 9t Z 9x 2 9yT2 z t * *\ p 3 " p l ( 2 . 9 ) Using this expression in (2.8) and using (2.5a,b) we have: H 9 n ( 0 ) L u j 0 ) i J r v 2 (0). n _ 9T _ p l yFx +Pl x9yj [ V l + B y ^ H, ] = 0 (2.10) z*=-H1+n* From (2.7) and (2.9) the equation for the second layer i s simply: 24 3 _ o(0) 1 . +_(0) 9_ 3t p2 y 3x ^ 2 x 3y vHP2+By+ H 2 r 2 z z = 0 (2.11) Finally integrating over the depth of the lower layer, using the fact that the vertical velocity i s continuous across the upper sur-face of the layer and that particles on the bottom remain on the bottom , (1) f3 . (0)3 . (0)8 } bh (this gives w3 = +u3 — +v3 ^ j — ) we have: H3-bh| f l _ n(0) iL+^CO) l l r o Z (0) , [ 3 t T - p 3 y 1x^3 x 3yJ [ VH p3 + B y ] L_ JO) S_ (0) i_ [3t " p3 y 9x "^3 x 3yJ b , . % , (0) R H„ n lH3 P 2 z o 3 z*=-H1-H2+n* = 0 (2.12) Appropriate boundary conditions are that each of the stream functions p^^(i=l,2,3) remain constant on the boundaries of the chan-nel. Taking L to be the half channel width,the boundary conditions are: 3p(0> -*=— =0 on y = ±1 (1=1,2,3) »x We now expand P^^ * n a P o w e r series about the middle of the second layer (2.13) .(0) - n=0 Y n (z-z_) with z_ = (-H1- y-)/H m m (2.14) The procedure is now to assume that the f i r s t few terms in this series give a reasonable approximation to P^^ • From t f t e work done on 25 continuous models i t appears that the most unstable waves have a rela-tively simple vertical structure (e.g. see G i l l , et a l ; 1973, §6) and hence this approximation may be reasonable (provided, of course, that the mean currents also have a simple vertical structure). A second problem in using this approach is the neglect of possible c r i t i c a l layers. However as pointed out by Bretherton (1966a),it appears from the work of Green (1960) that the growth rates of unstable waves associated with c r i t i c a l layers (which are not found using the crude layered theory) are markedly smaller than those which are found using the layered model. Hence, provided the c r i t i c a l layers do not play a crucial role in the dynamics of the flow, the f i l t e r i n g out of c r i t i c a l layers should not cause significant errors. [H2]3 Now, neglecting terms of order ^ i * 1 (2.10) and (2.12) * 4 and satisfying (2*11) exactly at the middle of the layer the following equations are derived. .IF - * i y h I7) t vH*i +By- ri< 3*i - * * 2 4 * 3 ) 1 = ° (ft" -<"2y fe ^ I x fe) [*H*2+«**F2<*l"2 W ] = ° H 3 " b h ' ( f r ^ 3 y f c ^ 3 x f 7 ) [ V H ^ ] (2. H, 3t ~w3y 3x "^3x 3y — h - F 3 ( * r 4 V 3 * 3 ) o 3 = 0 26 where * - p ( 0 ) - p* 0 ) = Yo + Y l 1 1 2 z=-H1/H ° 1 »2 2H *2 = P2 (0) z=z m n(0) . n (0 ) . - v - v P T p ? Y o Y l 3 2 z=-(H1+H2)7H ° l 2H and F * f 2L 2/g'H i (i-1,2,3) 2H. + 0 2H + Y- 2H + 0 Appropriate boundary conditions are: j± - 0 on y - ±1 (i-1,2,3) (Note that n e g l e c t i n g bh i n (H^-bh)/H i s valid i f bh H since this causes only a negligible perturbation to the equations. However, i f bh~H then provided u^'V h « R is satisfied (so that (2.2) is valid), 3 3 H ° i t is more accurate to retain this term. Since we w i l l consider only variations in h across the channel, the condition u^^'^h^CR^ for h ~ H^ is satisfied i f v 3/U«R 0.) The corresponding equations for a fluid with three layers of uniform densities p*^ , p*2 and p*3 (p^ = (p^+p^)/2) have been given by Davey (1977). [When comparing this paper with Davey's i t is important to note that where we have 6(= (p^-p^l/p* ) he has used <5/2 (= (p^-P^/p) and whereas we have chosen L to be the half-channel width, he chose the f u l l width]. When ( i j ^ - ^ ) = ( i ^ - ^ ) (corresponding to a linear velocity variation in the middle layer in our model and interpreted as a linear velocity variation through the total depth in Davey's model) 27 the two systems of equations are identical. When this equality i s not satisfied the two systems may be quite different. In figure 4.3, we shall see that for the same mean velocities in each layer the model derived here gives an unstable range (in the total wavenumber squared) roughly twice as wide as that found by Davey using the model consisting of three layers of uniform density. This i s not to be interpreted as an error in either model but rather as a reminder that when using a layered model care must be taken in f i t t i n g the model to the real continuous profile (see for example P h i l l i p s , 1951). The model derived here avoids this " f i t t i n g problem" for the density and velocity profiles considered here by consistently deriving the layered model from the equations for a continuously strat i f i e d f l u i d . On the other hand the usual layered models must be fitt e d by appropriate reduction of the density difference between the layers. Since both models are in agreement for the case (V-j.-1^) = ^3^' s o m e explanation is needed for the latter statement. From the form of the governing equations for the layered model of Davey (these equations are of the same general form as (2.11)) i t is clear that decreasing the density difference between the layers has basically the same effect as increasing the vertical curvature of the. Hence the close rela-tion between vertical curvature and density stratification i s expected. Now, consider the approximations to the vertical curvature in the two models. In our model, the vertical curvature in the middle layer is approximated by: ^ Z Z " \TV2HT - ay2HT}1 ( H2 / H ) (2.12) =2(H/H 2) 2(U l o-2U 2 o +U 3 o) whereas the corresponding approximation for the usual lnyered model is (for Hl = H 2 = H 3): (2.13) - ( H / H2 ) 2 ( Ulo- 2 U2o +V-Although we have written the approximation to in terms of the zz values of the velocities at the middle of each layer, the layered models have the velocities in each layer indepentent of z and hence in these For an approximately linear velocity variation in each layer the difference between (2.13) and (2.14) is small. However, in the presence of large vertical curvature, the difference is significant. Even i f we use (2.13) there is s t i l l a factor of two difference between i t and (2.12). This difference arises due to the fact that in the usual layered model the velocity difference between the layers is essentially assumed to occur over a separation of (H^+E^)/2 (the distance from the middle of the upper layer to the middle of the second) while in our model, since the density i s uniform in the upper and lower layers, the shear » between the upper layer and the middle layer occurs over a distance of H^ /2 (the distance from the bottom of the upper layer to the middle of the second layer). Clearly the approximations of the vertical curvature in the two models are quite different and i f both models are to be applied to the same situation, some f i t t i n g procedure is needed. The advantage of the model developed here is that the density stratification is chosen to closely approximate the actual density stratification whereas this is not the case for the usual layered model. If we want to use these models, 11 2zz is generally approximated by: ^2zz " ( H / H 2 ) 2 ( V 2 V U 3 ) (2.14) 29 layered models i t is clear that the density difference between successive layers must be reduced (The mean currents should not be tampered with as they also appear in the advective terms and thus changing them w i l l change more than just the approximation to the vertical curvature of the mean currents. Note, however, that i f one considers the case of large vertical curvature, U^, l ^ , and should be replaced by U„ , and U_ in the layered models.). The f i t t i n g of the model by zo jo an appropriate choice of density structure is not a l l that surprising when one realizes that the differences in the approximations to the vertical curvature of the mean currents in the two models are originally due to the different choices of density stratification in the two models. Finally we note that i f the actual density stratification is not well approximated by the model developed here, one might prefer to use the usual layered model (for which intuition through^analogy with f i n i t e difference approximations is probably better) or derive a yet another layered model (either by the method described in this section or by some other method). The latter choice is certainly preferable but generally not as convenient. 30 3. Linear Perturbation Analysis We now wish to consider the st a b i l i t y with respect to quasi-geostrophic disturbances of a steady mean flow which i s uniform along the channel. Hence we take ^ = ^ + C i (1=1,2,3) where ty^ = ^ ( y ) is a zonally uniform time independent solution of (2.15) and £ i s a perturbation stream function which we take to be of the form appropriate to waves propagating along the channel (i.e. £ = Re{<J>i(y) exptik(x-ct) ]}) The linearized equations to be satisfied by (f>^ , ^  and ^ are: (U1-c)[*lyy-k2*1-F1(3*1-A<t»2+4»3)] + • 1[e-U l y y+F 1(3U 1-4U 2 o-HI 3)] = 0 ( U2o- c ) [*2yy- k 2*2 + 4 F2 ( * r 2*2 +*3 ) ] + ¥ e ' % / 4 V V 2 U2o + U3> ] " ° (U3-c) H3-bh (• 3 y y-k 2* 3) - F 3(* 1-4* 2 +3* 3) H3-bh <8-U3yy) + F 3 ( D i - 4 D 2 o + 3 u 3 ) + TT; S where ( = - ^ i y ) » ^ 3^ = _%y^ a r e t* i e t :" n e independent velocities in the upper and lower layers and (""^y^ i s t n e corresponding velocity at the middle of the second layer.It i s convenient at this point to express U 2 o in terms of the vertically averaged velocities in the three layers. In the preceeding analysis we have essentially approximated the velocity; in the middle layer by: 31 2 n U 0(z) = Z U. (z-z ) where z = -2 n=0 2n m m 2H1+H2 2H Requiring that (^ (z^B^/H) = V± , U2(z—(B^+Hj)/!!) = U 3 and that -Hi H j^— I " (J.(z)dz = U 0 (the vertically averaged mean velocity in the *2 'HJ-WJJ H middle layer) it is easily shown that U 2 = (6U2-U1-U3)/4 (and U 2 i= ( H / H 2 ) - ( U ^ ) , U 2 2= 3(H/H2)2.(U1-2U2+U3): note that the local curvature in the middle layer is ^ 2 ^ ^ a n <^ n e n c e that the equations for c^ 2, and »^3 may be written as: (U r c)[ct x -k 2* 1-F 1(3* 1-4* 2+* 3)] + *! fe1 = 0 (3. (U3-c) H3-bh H, ^3yy-k *3> " V*r4*2+3*3> + * 3 | f 3 = 0 32 where ^ 1 = p - U, + 2F, (2U -3U 0+U„) 3y l y y 1 1 2 3' fe2 = e . - U 2 o y y - 6VV 2W. 133 = ?y H 3-bh H 3 (B-U 3 y y) + 2F 3(U 1-3U 2 +2U 3) + ^ g - h y o 3 J u 2 b = <6VVn3)/4 w i t h boundary c o n d i t i o n s cf>^  = cp^  = ^3 = 0 on y = ±1. Before d i s c u s s i n g the s o l u t i o n s of (3.1) i n p a r t i c u l a r cases, i t i s of i n t e r e s t to co n s i d e r the p o s s i b l e mechanisms of energy t r a n s f e r f o r t h i s model. To d e r i v e an energy equation we begin w i t h the r e a l form of (3.1) f o r bh << H 3 ( i . e . f o r topographic v a r i a t i o n s s m a l l compared to the depth of the lower l a y e r ) . ( f t " + U l b l ^ l y y - ' ^ V W ] + ^ ^ " 0 <!F + U 2 o b E 5 2 x X + 5 2 y y + 4 F 2 « r 2 5 2 + 5 3 ) ] + ^ ^ = 0 (3.2) (!ir + U3 fx^3xx +S3yy-VVS + 3V 3 + ^  ^ = ° * * 1 9x 3y K2 9 q 2 3x 3y »h 3q 3 3x. 3y on y M u l t i p l y i n g the 1 ^ equation by a n d i n t e g r a t i n g from y = -1 to y = +1 and over one wavelength i n the x d i r e c t i o n (a reg i o n we s h a l l r e f e r to as R) and adding,the f o l l o w i n g energy equation i s r e a d i l y d e r i v e d . 33 2 +f(V 2 52 + e3 ) 2 dxdy[ (3.3) / F 1 + U2oy*2x*2y / F2 + U 3 y « 3 * 5 3 y / F 3 l d x d * R • R • •+ ^ L X > ( ^ - U ^ ) ' +; ( 4 5 2 5 3 x + 5 3 5 l x > ( U 2 O - U 3 ) ]dxdy The l e f t hand side represents the rate of change of k i n e t i c plus p o t e n t i a l energy of the depth-averaged perturbations. The f i r s t energy transformation i n t e g r a l on the ri g h t hand side i s an expression for the hori z o n t a l Reynolds stress conversion of k i n e t i c energy i n the three layers while the second i n t e g r a l expresses the conversion of the available p o t e n t i a l energy of the mean flow. The i t h term of the f i r s t i n t e g r a l on the right hand side of (3.3) w i l l thus he referred to as the trans-th fer of k i n e t i c energy (T.K.E.) i n the i layer, and the two terms i n the second i n t e g r a l w i l l be referred to as the transfer of available p o t e n t i a l energy (T.A.P.E.) due to the shear between the upper layers and lower layers respectively. We note here that the terms involving the correla t i o n between the upper and lower layers i n the T.A.P.E, terms of (3,3) are absent i n the usual layered models, These terms ar i s e here as a direct consequence of the approx-imations to'C. and £_ at the upper and lower interfaces (see chapter V, P . 199-200). 34 Necessary Conditions for Instability If the f i r s t and third equations of (3.1) are multiplied by ^/[(V±-c)¥±] (i=l,3) and the second equation by <!>*/[(U2Q-C)F2] (this i s j u s t i f i e d i f lm(c)^0; a .* 'indicates complex conjugation here), integrated from y=-l'to^y=+l, and the.three resulting equations added, the following equation is derived: r l 3 U . | 2 +k 2 U, | 2 9 9 1 2,1 j I 1 Y F 1 +2[U 2-» 1| 2 + |*3-*2|2 +|U1-2^3| 2][ dy -1 1 - 1 1 rl 3 I ^ J 2 a q , = J x l±ml \zc *y F i 7 (Note that since the upper and lower layers are ve r t i c a l l y uniform, the ve r t i c a l mean value of the velocities ( U ^ and U^) and the value at the middle of these layers ( U ^ Q and U ^ ) are equivalent.) Taking the imaginary part of this equation gives: (3.4) = •0 ( 3 - 5 ) We see that the expected necessary condition for i n s t a b i l i t y i s found, i.e. the potential v o r t i c i t y gradient must change sign either within a given layer or in going from one layer to another. For c^O, the real part of- (3.4), after using'(3.5) yields: f l ( 3 U . | 2 + k 2 U . | 2 9 9 , o • I I Y F 2tU2-*il + lv*2l +il*r2*2+*3l 1 i l 1 dy (3.6) I > i=l U -c ^ u i o 3 y F. y -1 - io i 35 Since the l e f t hand side of (3.6) i s positive, i t i s clear that the product of the mean velocity and potential v o r t i c i t y gradient must be positive, at least somewhere in one of the layers. Thus a sufficient condition for s t a b i l i t y i n this case i s that u l o g ^ — < °i C 1 = 1>2>3>. Analytic Solutions For the remainder of this chapter we shall be concerned with the case in which the following conditions hold! = constant ( i = 1,2,3) bh « H 3 h = constant y (3.7) Under these conditions, solutions of (3.1) exist in the form <j>^  = sin[ijp-(y+l) ] , ( i = 1,2,3) where the V^'s are constants to be determined from the following eigenvalue problem (with the Doppler shifted phase speed, C = c - U„ as the eigenvalue). z o < slo- c ) [- Km vl " V3y r4Vu 3) ] + *i "ST = ° 2 8 q2 -C[-K mu 2+4F 2( y i-2y 2+y 3)] + ^ — - 0 (3.8) (-S 2 o-C)I-K my 3-F 3(y 1-4p 2 +3y 3)] + y 3 = 0 where 2 ,2 , , IMK2 K = k + (—) m 2 36 3q 1 3y = 3 + 2F 1(2S 1-S 2) 3q 2 ay = 3 - 6F 2(S 1-S 2) a q 3 3y = 3 + 2F 3(S 1-2S 2) + (b/R oH 3)hy s i = U l " U2 -'. S2 - u 2 - u 3 8 l . = D x - U 2 o (5S i : S 2)/4 S26 - U2o " U 3 - ( 5 S 2 " S^/4 (Note that although I have expressed the equations in terras of C - U 2 Q > in the dispersion curves plotted in the following sections we have used U„ =U„-(S -S.)/4 to plot the results in terms of (c-U )«k.) zo L 1 I *-The condition for a nontrivial solution "(the vanishing of the coefficient matrix of u^, y 2 > u 3 ) gives the following dispersion relation: " ( 3 F l + K m M S l o - C ) + W 4 F l ( S l o " C ) 3q, 4F £C F 3 ( S 2 o + C ) (8F 2 +K m)C + F l ( S l o - C ) - 4F 2C 4F 3(S 2 o4€) ( 3 F 3 + K m ) ( S 2 o ^ ) ^ = 0 (3.9) For our calculations we w i l l want to consider many different 2 values of for each set of parameters so that i t i s convenient to rearrange the dispersion relation into the form given in Appendix a, at the end of this chapter. 37 If we normalize with respect to the middle layer by setting 1 , then y^ and y 3 are given by: -(3F 1 +K m)(S l o-C) + ^  - V S l o - 0 F 3(S 2 o +C) (3F3-HCm)(S2o-rC) + - 4 F l ( S l o - C ) 4F 3(S 2 o +C) (3.10) The perturbation velocity components are now readily computed by using the relations u = -% and v = E ; they take the forms 3 e n ny n *nx 3 to^t u = -sgn(u )|y | ir- e cos(kx-u> t+6 ) cos[-r— (y+1) ] n n n t r n / r to t v = -sgn(u )|y | k e 1 sin(kx-u) t+6 ) s i n t f 1 (y+1)] (3.11) n n n r n £• (n = 1,2,3) where to = ck , to = Re(to), to. = Im(to) , y = Re(y ) > and r i n n r tan 6 = y /y , -TT/2 < 6 < TT/2. sgn i s the signum function which n n. n " I r gives the sign of the argument. Following Davey (1977) we now reduce the range of parameters b to be studied. Defining T = (^rr-) h i t : i s dear from (3. 8) that i f R0 H3 y the system ( S ^ S 2, 6, T, K , H 2 > H3) has solution (C, y ^ y 2 > y 3) , then the system (aS^ aS 2, c.6, aT, H^, H 2 > H3) has solution (aC, y 2 > y 3) . Further, i f the system ( S ^ S 2 > 8, 0, B^, H2, H3) has solution (C, y^, y 2 > y 3) , then (-S2, -S^, B, 0, H 3, H2, H^) has 38 solution (C, u^, u 2» u^) . The f i r s t of these relations corresponds to a simple rescaling while the second follows by virtue of the r i g i d l i d approximation and corresponds to interchanging the top and bottom layers. From the f i r s t relation i t i s clear that we need only consider the case • 1 . This corresponds to taking the horizontal velocity scale, U , equal to the shear between the upper layers. (Note that although U i s taken to be a typical horizontal velocity for the pre-ceding analysis, once the equations have been derived i t factors out of each equation and hence we may choose i t to be S* = U* - for convenience.) If T = 0 , then using the above relations, one may show that the systems (1, S 2, g, 0, B^, H 2 > H3) and (1, 1/S2, -e/S2, 0, H3, H 2 > H1) have the same st a b i l i t y with respect 2 to Kffi . Hence in this case, the behaviour of a l l systems with T=0 can deduced from the subset ( S ^ l , - f cS^l.B, 0, H 2 > H3) . In fact, in every case considered we w i l l take = 1 and consider -2 < S 2 ^ 2 . In this manner, most cases of interest are covered simply by choosing We may also reduce the range of 3 to be considered. From (3.5) we know that for i n s t a b i l i t i e s to occur the following relation must be satisfied. Hence, i f a l l the basic state potential v o r t i c i t y gradients are of one sign no i n s t a b i l i t i e s w i l l occur. This condition w i l l thus give us U = S* . f dy = o 39 bounds on B for Instabilities to occur. For S. =1 -1-& S 4=1 a n d T=0 a l l the are positive for B = B/F2 > max(12, 2 H2/H3) and a l l are negative for B < - S m a x O^/H^, H2/H3) . It is interesting to note further that the corresponding range of B found by Davey using a model with three layers of constant density and equal thicknesses i s -2 £ & < U . The range of B for which i n s t a b i l i t i e s occurs i s reduced by a factor of three due to the over estimation of the stabilizing effect of density statification (see the last paragraph In section 2). It i s important to note here that although i t appears that waves with B < 0 (corresponding to a relative westward flow i n the surface layer) are more stable than those with B > 0 (eastward flow in the surface layer), our results do not contradict the qualitative statement made by G i l l et a l (1973) that, "because of the effect of B on s t a b i l i t y , the most favourable conditions for i n s t a b i l i t y are found where the isopycnal slope upwards towards the equator On the contrary, i f we consider • S 2 (which i s approximately satisfied in the studies mentioned above) we find that i n s t a b i l i t i e s can only occur for -2H2/H1 < B < 2H2/H3 Since we generally have « H 3 we recover the result that for the open ocean (or anywhere that 8 2 ^ 8 ^ , and « E^) i t appears that flows with a relative westward flow i n the surface layer are more favourable for Instabilities than those with a relative east-ward flow i n the surface layer. It i s , however, Important to note that 40 this conclusion relies strongly on the condition << 4. Results of Independent Parameter Variations For the purpose of considering the effect df parameter variations, i t i s useful to f i r s t divide each of the equations in (3.8) by F 2 . The resulting set of equations i s : 2 9 Q 1 ( S l o - C ) [ - ( K m / F 2 ) 4 , l " (H2/Hx) ( 3 ^ - 4 ^ 3 ) ] + ^ j± = 0 - C[-(K*/F 2H 2 + 4(cj,1-2<J,2+<(>3)] + <(,2 j-± = 0 2 9 Q 3 (-S 2 o-C)[-(K m/F 2)^ - (H2/H3) (• 1-4* 2+3# 3) ] + ^ jf = 0 9Q, . where -^=- = 8 + 2(H2/H^) (2S^-S2) 9Q2 . 9 T " e " ^ V V 3 Q 3 -^ = B + 2(H 2/H 3)(S 1 -2S 2 ) + T B = B/F2 = g*L2/F2U T = T/F 2 = (g'/fU)(H 2/H 3)bly 2 Note that i f plotting i s done against ^ m / F 2 and i f the velocity scale, U , is chosen to be equal to S* = U* - U* (so that = 1), then the parameters to be considered are S 2, H^ /H^ , ILj/H^, § and T . From the form of the equations above we may also make some general conclusions about the effect of density s t r a t i f i c a t i o n . The effect of varying each of these parameters independently shall be 41 discussed in this section. Before beginning a study of the effects of parameter variations, i t is useful to give a qualitative discussion of the roots of (3.9). There are, of course, three roots for a given wavenumber, k. In a l l of the dispersion curves presented in this chapter, I have plotted w^-I^k ^(c-lL^k) vs. k for the f i r s t cross-channel mode (m=l). The non-dimensional phase speed is thus obtained by dividing the ordinate by the absissa in the graphs and then adding the appropriate value of U^, and the non-dimensional group velocity is obtained by taking the slope of the dispersion curve and adding . When two distinct real dispersion curves cross each other so that the phase speeds and wavelengths are identical,an interaction between the two waves (and the mean state) is possible and an instability may occur. Since we have neglected horizontal shear and considered only constant bottom slopes, any such instab-i l i t y must extract i t s energy from the potential energy of the mean state (i.e. we are considering only pure baroclinic instability here). Since vertical shear clearly plays a fundamental role in the mechanism of baroclinic instab-i l i t y (through i t s role in t i l t i n g the isopycnals), I have have chosen to classify the different waves in terms of the vertical shear of the mean current. To make this classification, we consider the case 3=T=0, to eliminate a l l but the effects of the shears. In this case we note that i f S ^ 1 ^ (no curvature in the middle layer-see the paragraph preceeding (3.1)) then the constant term in the dispersion relation (6 in appendix a) vanishes identically since q„ =0 for this case. Thus for zero curvature, one of the roots of (3.9) is C = 0 (i.e. c = U_ ). (This i s due to the fact that for 3 = 0 and S. = S 0, zo ± . z q 2y = 0. Analogous results also hold for q^ = 0, and q^y = 0, i.e. q ^ = 0 => c = U. i s one root of the dispersion relation, i=l,2,3). Since in the io absence of 3, the wave corresponding to this root depends on the curvature 42 of the mean currents to give i t a non-zero phase speed relative to the mean flow in the middle layer, I shall refer to this wave as the curvature wave. In the dispersion curves presented in figures 4.1-4.18, this root generally l i e s near u> -l^k = 0 at large k (note that at very large k, this root reduces to c = t^o' i-e. co -I^k = -k(S^-S2)/4 so i f we went to large enough k this root would be better identified by i t s slope). For ^ this root plays a fundamental role in destabilizing the flow. A careful examination of figures 4.1 to 4.18 reveals that when this curve meets either of the others, the flow is generally destabilized. The other two waves do not generally interact in this way unless is relatively small. These other waves may be classified as an upper shear wave (S^-wave) and a lower shear wave C ^ -wave). [Alternatively each of the roots may be classified by i t s dependence on the mean potential vorticity gradient - the S^-wave depends c r i t i c a l l y on d^l, the curvature wave on ^ q2, and the S2~wave on ^3.] The lower shear 3y 3y 3y wave is indicated by a dashed curve in the figures and is clearly very sen-sitive to variations in as well as variations in topography and 8. The upper shear wave i s rather insensitive to variations in and rises up to the l e f t showing relatively l i t t l e variation throughout the figures in this section (figures 4.1-4.18). It is however strongly affected by variations in S.^  holding S 2 fixed. Finally we note that although these three types of waves are quite distinct at short wavelengths (where the layers tend to de-couple) , the distinction at long wavelengths (where the layers are strongly coupled) is not nearly as clear. Also shown in figures 4.1 to 4.18 are stability boundaries (part(a) in each figure). The stability boundary for each set of parameters is a plot 2 of S„ vs. K /F. and shows the curve in parameter space on which the discrimin-2 m 2 ant of the cubic given by (3.9)vanishes. This curve separates the region in 43 which waves grow with time (Im(c) ^  0) from the region of stable waves (Im(c) = 0) and hence i s very useful in studying the sta b i l i t y of a given flow. The position of this curve was determined by numerical evaluation using the form of (3.9) given in Appendix a. Let us now consider the effect of independently varying the parameters in the model. Variation of the Coriolis Parameter with Latitude The variation of the local normal component of the earth's rotation enters (4.1) through the term: * H * 3 = (g'/f S ) 21 3 f o A negative value of 6 corresponds to < 0 , and we see that at a given latitude the effective 3 is increased by either an increase in density stratification or by a decrease in shear. (Note that i f = = 0, the only non-zero terms in q (i=l,2,3) are due to 3 and topography. In this case the only non-trivial roots of the dispersion relation correspond to Rossby waves, i.e. one of the planetary Rossby wave ( 3 ^ 0 ,T = 0 ) , the topographic Rossby wave (3=0,T^0),or the topographically modified planetary Rossby wave ( 3 ^ 0 , T / 0 ) . The case S 1 = S 2 = 0 , 3=0, T r 0 is briefly considered in appendix a. From figures 4.1 to 4.3 we see that decreasing 3 from zero stabilizes the flow and from figures 4.3 to 4.5 we see that increasing 8 from zero also stabilizes the flow. In each case the effect i s very similar with the very long waves being quickly stabilized as expected. Further, we note that the ins t a b i l i t i e s near = = 1 are quickly stabilized by 3 (either positive or negative). This result i s not very surprising as the unstable waves in 44 this region are long and strongly affected by 6. Perhaps more surprising i s the stabilizing effect of 6 on the short waves corresponding to the branch extending to the right in figure 4.3. However, the growth rates corresponding to these waves are very small even for 8 = 0. (a) 2.0 <M 00 CO -2.0 STABLE 0.0 8.0 16.0 24.0 2 (c) 5.0 rs 2= o.oo 0.0 I 3 -5 .0 -wave Curvature wave S2-wave 0.0 20 4.0 6.0 (b) I D I 3 (d) 3 50r s 2 = o.75 0.0 -5.0 -wave Curvature wave ,^5^-wave 0.0 2.0 4.0 6.0 Figure t.V- Stability boundaries (a) for H « Hj » Hj , T = 0 , B • -6 and first mode (m=l) dispersion curves corresponding to these parameter values U l with F 2 = 1 and S2 = -0.75(b), 0.00(c), and 0.75(d) (a) 2.0 -2.0 (b) UNSTABLE STABLE I i-3 5.0 r s 2 = - 0 . 7 5 OO 0.0 8.0 16.0 K m / F 2 24.0 -5.0 0.0 2.0 4.0 k 6.0 (c) 5.0 r £ OO i L. 3 -5.0 S 2 = 0 0 0 (d) ( M ZD i_ 3 5.0 r 0.0 0.0 2.0 4.0 6.0 k -5.0 S 2 = 0 . 7 5 OO 2.0 4.0 k ao Figure /f. Z: S 2 Maximum Growth Rate |M3/y2l «1 «3 -0.75 0.48 0.61 0.33 46 32(+) 35.25(+) 0.00 0.23 0.65 0.56 36 14(+) - 8.37(+) Stability boundaries (a) for H l " H 2 - H 3 , f - 0 . B - -3 and f i r s t mode (m»l) dispersion curves corresponding to these parameter values with F 2 - 1 and S 2 - -0.75(b), 0.00(c), and 0.75(d) . Statistics for positions of maximum growth rate (marked by plus signs in the figures) are given in the table. The signs of and are given in r r brackets following 6 and 5-j respectively. 0 .0 8.0 16.0 K m A z 5.0 r 0 0 5.0 S 2 = 0.00 0 . 0 2.0 4 . 0 2 4 . 0 6.0 (d) ZD I L. 3 0.0 2.0 5.0 0.0 - 5 . 0 S2 = 0.75 0 .0 2.0 Figure .^3 S2 Maximum Growth Rate |M3/P2I «1 43 -0.75 0.64 0.88 0.51 54.870+) 64.20(+) 0.00 0.38 1.19 0.30 48.700) - 2.42(+) 0.00 0.07 0.16 0.66 - 9.28C+) 87.72(+) 0.75 0.09 1.38 0.56 19.79(+) -11.40(+) As ln figure*2with 6 - 0 . Also included i n part (a) I s the corresponding r e s u l t of Davey (1977)^ (broken curve) Figure *h4: S2 Maximum Growth Rate -0.75 0.53 As in f i g u r e y t w i t h (3-6. \vx/v2\ |w 3 /u 2 | 6j_ 1.81 1.05 45.36(+) 6 3 77.20(+) F i g u r e 4-.5~: As i n f igureV:2 w i t h 6 = 12. 50 Finally, we note here that whereas we have choseh to classify our waves in terms of shears, when $ is significant a classification in terms of potential vorticity gradients, as mentioned earlier^is more appropriate. These comments also apply to the following section on topography, however we shall continue to classify our waves in terms of shears, due to the role of vertical shear in supplying an energy source for baroclinic in s t a b i l i t y . Topography The effect of topography in (4.1) is given by: b T = (R H F } h v R0 H3 F2 y g' = (j-gy) (H /H ) bh r0 1 J y as in the case of g we see that an increase in stratification or a decrease in shear increases the effect of bottom topography. Bottom topography i s , of course, also f e l t more strongly i f the depth of the lower layer is decreased. The qualitative difference in st a b i l i t y between the cases T < 0 and T > 0 is understood when one realizes that of the three roots of (3.9), only the S^wave i s significantly affected by topography. (This i s , of course, consistent with a classification in terms of potential vorticity gradients.) From figures %(, to 4-.12 we see that as T increases (b) STABLE 2L i 3 50 T /s2=-0.75 0.0 (c) (NJ I 0.0 8.0 16.0 24.0 _ K r n / F 2 50r / / s 2 =ooo 0.0 -5.0 0.0 2.0 4.0 6.0 k -5.0 (d) (NJ = > I 3 0.0 2.0 4.0 6.0 k Figure *+.b! S 0.0 2.0 4.0 k 60 Maximum Growth Rate 1-rSl 61 S3 -0.75 0.50 1.14 0.06 66.47(+) -47.18(+) 0.00 0.35 1.38 0.05 43.37(+) 23.44(-) 0.00 0.03 2.90 7.66 -71.94(+) 21.16(+)^ 0.75 0.04 1.73 0.09 ll.38(+) - 1.17(-) 0.75 0.03 0.90 6.51 77.16C+) 16.54(+) S t a b i l i t y boundaries (a) 'for H l * H2 - Hj , f - -30 , B = 0 and f i r s t Ln I—1 mode (ra-1) dispersion curves corresponding to these parameter values with F 2 - 1 and S 2 = -0.75 (b), 0.00 (c), and 0.75 (d). S t a t i s t i c s for positions of maximum growth rate (marked by plus signs) are given in the table in order of increasing k . 2 0 f .VJNSTXBLE STABLE UNSTABLE 0 . 0 8 .0 16.0 2 4 . 0 :2 5.0 0 . 0 - 5 . 0 s2=o.oo 0 . 0 2 . 0 4 . 0 6 . 0 Figure *t\7: 32 -0.75 0.00 0.00 0.75 Maximum Growth Rate 0.48 0.33 0.04 0.02 As In figure V.fcwith T - -10 k 1^1 |u 3/u 2l 61 63 1.10 0.18 61.6K+) -33.85(+) 1.48 0.22 43.47(4;) 33.8K+) 1.84 2.54 59.23(+) -37.82(+) 1.91 0.04 5.8K+) 0.78(-) I I I I . 0.0 2.0 4.0 6.0 k F i g u r e ^f.J; S 2 Maximum Growth Rate -0.75 0.49 0.00 0.26 0.00 0.23 0.75 0.12 As i n f Igure'V.fcwith f = -5. 3 _ Rf) I i - — i * 0.0 2.0 4.0 6.0 k U l / u 2 l Iw3/P2l 61 53 0.99 0.39 67.38C+) -14.24(+) 1.88 0.80 58.54(+) 69.16(-) 1.45 1.11 60.06(+) -53.07(+) 1.32 1.94 62.36C+) -32.63(+) 2.0 0.0 h 2.0 STABLE UNSTABLE 0.0 8.0 16.0 K m / F 2 5.0 r s2= o.oo 0.0 5.0 0.0 2.0 4.0 24.0 6.0 F i g u r e 4.1'• •"2 -0.75 0.00 0.00 0.75 Maximum Growth Rate 0.64 0.38 0.07 0.09 As i n f i g u r e V.fcwith T -5.01 1 1 ' 0.0 2.0 4.0 6.0 k |U3/P2I 5 1 • 4 3 0.88 0.51 54.87(+) 64.20(+) 1.19 0.30 48.70(+) - 2.42(+) 0.16 0.66 - 9.28(+) 87.72C+) 1.38 0.56 19.79C+) -11.40(+) 2.0 iv/ 0.01 -2.0 STABLE UNSTABLE 0.0 8.0 16.0 KmAz 5.0 r s 2 = o.oo 0.0 -5.0 0.0 2.0 Figure *r.io: 4.0 24.0 6.0 2 0.75 0.75 0.00 0.75 Maximum Growth Rate 0.56 0.41 0.37 0.07 As in figure f.4 with T = 5 -5.01 1 1 ' 0.0 2.0 4.0 6.0 k U l / u 2 | |w3/w2l 41 53 1.09 0.26 56.68(+) -75.27(-) 0.41 1.17 -11.76(+) 78.50(+) 1.26 0.15 45.28(+) 9.36(+) 1.51 0.26 13.09(+) - 5.2K+) 2.0 r 5.0i 1 1 1 0.0 2.0 4.0 6.0 k Figure *F.LD S 2 Maximum f.rowth Rate -0.75 0.54 -0.75 0.38 0.00 0.37 0.75 0.06 As in I" iRiirr tt(wi th t = 10 k Vllv2\ |u 3/w 2l 61 6 3 1.12 0.15 59.64(+) -64.93(-) 0.28 1.43 -18.33(+) 82.49(+) 1.29 0.10 44.54(+) 13.14(+) 1.56 0.17 11.15(+) - 3.66(+) (a) 2.0 Figure *\'/]_2- s 2 Maximum Growth Rate -0.75 0.52 -0.75 0.26 0.00 0.36 0.75 0.05 As in figure1l*iwith T = 30 |M3/V2I «3 1.14 0.06 62.6K+) -56.64(-) 0.14 2.11 . -26.62(+) -86.32(-) 1.32 0.04 43.93C+) 17.16(+) 1.64 0.07 13.89C+) - 3.27(+) 58 from -30 to +30 the -wave dispersion curve (which for large T i s a topographic Rossby wave modified by the shears) decreases from being a generally positive root to being a negative root (consistent with the shallow water being to the right of the direction of phase propagation; see Languet-Higgins, 1972). In decreasing from positive values at T = -30 to negative values at T = +30, the S^-wave passes through and interacts with the curvature wave. For S^ = 0 and 0.75 the S2~wave root already l i e s below the curvature wave root at T = 0 and hence no inter-action takes place between these waves for T = 0 . For = - 0.75 the S^-wave does not l i e below the curvature wave for the f u l l range of k considered u n t i l T = +30 and hence some interaction between these waves does occur for T > 0 in the presence of large curvature in the mean velocity profile. Thus, for moderate curvature the interaction between the curvature and S^-waves occurs for T < 0 while for large curvature the interaction generally occurs for T > 0. In either case the effect of topography on the direction of phase propagation i s in opposition to advection in the lower layer (relative to the other layers) thus restraining the phase speed of the waves in the lower layer to remain in the range where interaction with the other waves in the system i s possible. It should be noted here that even a very large bottom slope does not cause significant reduction of the growth rates of i n s t a b i l i t i e s due to the S^- and curvature waves interacting. However, i f one concentrates on the case = - 0.75 in figures <h& to 4.ia(in each case the 59 instability at smaller k i s due to such an interaction) i t w i l l be seen that topography does strongly affect both the phase and amplitudes of these waves in the lower layer. Of particular interest i s the decrease in perturbation amplitude in the lower layer with increasing bottom slope. Hence, even in this case,topography has a stabilizing influence, but i t s significance i s restricted to the lower layer where the bottom slope i s strongly f e l t . Finally we refer the reader to Orlansky and Cox (1973) for an alternative explanation of the stabilizing effect of bottom topography based on energy arguments involving the interchange of f l u i d elements within the wedge of ins t a b i l i t y in a continuously strat i f i e d f l u i d . The application to our model is simple. A large cross-channel bottom slope w i l l cause the f l u i d trajectories near the bottom to also have a significant cross-channel slope and thus the release of potential energy from the lower interface i s inhibited. Hence the bottom intensified waves due to the interaction of the S£ and curvature waves are stabilized by such a slope. However, in the region of the upper interface, the f l u i d trajectories of the unstable waves due to the interaction of the S^-and curvature waves are not nearly as strongly affected by the sloping bottom and hence potential energy continues to be released from the upper interface. Further, a slightly sloping bottom in the same sense as (but shallower than) the slope of the lower interface may actually increase the growth rates of the i n s t a b i l i t i e s which extract potential energy from the lower interface by constraining the f l u i d trajectories 60 in this region to l i e within the "wedge of in s t a b i l i t y " . Relative Layer Thicknesses Figures ?.13 to ^.15" i l l u s t r a t e the effect of increasing H^ /H^  while holding H^ /rL^  constant. Two basic effects are seen. The curvature wave and the S^-wave tend to separate as H^ /H^  increases causing the i n s t a b i l i t i e s to shift to lower wavenumbers. The S2~wave dispersion curve i s affected l i t t l e by changes in H^ /H^  but as the other two curves separate, when the curvature and S^-wave dispersion curves meet, an in s t a b i l i t y results. An exactly analogous situation occurs for H^ /H^  increasing (Figures <Mfc-4-.i!) only in this case the curvature and S^-wave dispersion curves come together as H^/rL^ increases. In general we note that as a layer thickens the perturbation velocities in that layer have a tendancy to decrease. This i s due to the fact that as a layer thickens i t requires more energy to maintain the same'velocities in that layer, yet the thickening of a layer does not make more potential energy available for this purpose and thus a decrease in perturbation velocities in this layer w i l l generally occur (this contrasts with the case of barotropic ins t a b i l i t y where the thickening of a layer makes more kinetic energy available). Alternative p o s s i b i l i t i e s are to have smaller perturbation velocities in the other layers or to have a smaller growth rate for the wave. In general the system responds with a combination of these po s s i b i l i t i e s . k k S2 Maximum Growth Rate |u3/w2l S l «3 -0.75 0.78 1.18 0.63 72.73(+) 36.85(+) 0.00 0.54 1.67 0.27 55.24(+) - 5.79(+) 0.00 0.06 0.34 0.63 - 8.52(+) -82.98(-) 0.75 0.22 1.81 0.46 27.47(+) -21.64(+) S t a b i l i t y boundaries (a) for V H 2 " 0.5 , Hj/H2 - l . f - 0 . i - 0 and f i r s t mode (m-1) dispersion curves corresponding to these parameter values with F 2 - 1. and &2 - -0.75(b), 0.00(c), and 0.75(d) . S t a t i s t i c s for positions of maximum growth rate (marked by plus signs) are given i n the table. 0.0 6 . 6 6 7 13 .333 2 0 . 0 (c) rvl ZJ I L. 3 5.0 r 0 . 0 " 5 . 0 s2 = o.oo 0 . 0 2 . 0 4 . 0 6 .0 Figure 4.1V- S 2 0.75 0.00 0.00 0.75 Maximum Growth Rate 0.64 0.38 0.07 0.09 As in figure ftf) with H^/U^ 3 - 5 . 0 1 1 1 ' 0 . 0 2 .0 4 . 0 6 . 0 k-U 1 / p 2 l Iw3/u 1 6 l *3 0.88 0.51 54 87(+) 64 20(+) 1.19 0.30 48 70(+) - 2 42(+) 0.16 0.66 - 9 28(+) 87 72(+) 1.38 0.56 19 79(+) -11 40(+) , H 3 /H 2 = 1 . I 1_ 9 -5.0 ' 1 ' • 0.0 2.0 4.0 6.0 k i 3 -5.01 ' 1 i 0.0 2.0 4.0 6.0 k 61 63 32.43(4-) -69.38(-) -10.90(+) 72.37(+) 16.06(4-) 0.10(4-) - 9.71(4-) 81.61(4-) Figure /r.15": S 2 Maximum Growth Rate ] p^/u21 lu^/p | -0.75 0.15 0.46 0.21 -0.75 0.43 0.12 0.99 0.00 0.03 0.54 0.39 0.00 0.07 0.03 0.68 As in figure V./3Hl/H2 = 5 , H3/H2 = 1 . Values are given in order of increasing k Maximum Growth Rate |w 3/u 2l 61 *3 -0.75 0.65 0.96 0.56 49.09(+) 87.53(+) -0.75 0.55 0.46 1.36 -10.17(+) 70.87(+) 0.00 0.39 1.14 0.38 49.54(+) 1.20(+) 0.00 0.09 0.12 1.21 -17.47(+) 74.82(+) 0.75 0.11 1.26 0.83 ?h.36(+) -13.50(+) As in figure fc/3with H1/H2 = 1 , H3/H2 = 0.5 . Values are given in order of increasing k . 3 -5.0' ' ' ' 0.0 2.0 4.0 6.0 k Figure 4.17: S 2 -0.75 0.00 0.00 0.75 As in -501 1 1 1 0.0 2.0 4.0 6.0 k 6l S 3 0.88 0.51 54.87(+) 64.20(+) 1.19 0.30 48.70(+) - 2.42(+) 0.16 0.66 - 9.28(+) 87.72(+) 1.38 0.56 19.79(+) -11.40(+) - 1 . Maximum Growth Rate | 0.64 0.38 0.07 0.09 figure with H 1 /H 2 - 1 k K S 2 Maximum Growth Rate 61 63 -0.75 0.51 1.07 0.16 67.92(+) 0. 48(+) 0.00 0.37 1.29 0.11 4 5 . 8 1 0 ) -11 30 O ) 0.00 0.02 0.23 0.13 - 1 . 6 0 O ) - 6 1 4 3 ( - ) 0.75 0.06 1.57 0.15 1 0 . 9 5 O ) - 7 . 2 4 0 ) As in figure with H J / H J » 1 , H 3 / H 2 - 5 . ON 67 The point to be noted is that i t is not always true that the case = = is typical. Density Stratification We f i r s t note that for a given channel geometry (i.e. a n <* ^ 2 2 fixed), ( = f Q L /g'rl^) decreases as the effective density stratification increases either through a decrease, in the rate of rotation or through an increase in the actual density stratification (either of these po s s i b i l i t i e s decreases the slope of the isopycnals necessary for the process of baroclinic 2 instability).Thus through i t s presence in K^/T^ in (4.1) i t is clear that 2 an increase in stratification decreases the range of K in which i n s t a b i l i t i e s m occur. This explains why,for the case of large curvature, the range of unsta-ble waves found by Davey is only about half of that found here (after account-ing for the difference in terminology), (see figure 4.3). This is primarily due to the over-estimation of the stabilizing effect of density stratification when considering a f l u i d of three layers, each of uniform density (again see the last paragraph in section 2). One way to visualize the stabilizing effect of density st r a t i f i c a t i o n on these unstable rotational waves is to consider their effect on the isopyc-nal slopes. An increase in density stratification causes a decrease in the slope of the isopycnals in the mean state and hence a smaller "wedge of inst a b i l i t y " (see Pedlosky, 1971; Orlansky and Cox, 1973). This naturally decreases the range of wavelengths for which i n s t a b i l i t i e s may occur and generally causes a corresponding decrease in the growth rates of the most unstable waves in the system. 68 The strong stabilizing influence of density stratification is also seen from i t s effect on the influence of 3 and bottom slope. As seen previously , an increase in either of these parameters is generally accompanied by a decrease in the range of unstable waves as well as a decrease in the maximum growth rate of the unstable waves. Thus the presence of in the denominator of (3 and T (Equn. 4.1) clearly shows further evidence of the stabilizing effect of density stratification. Curvature in the Mean Velocity Profile The curvature of the mean velocity i s indicated by the difference ^^-S^ . (Note that = corresponds to no curvature of the mean velocity in the middle layer but over the f u l l depth of the f l u i d the velocity f i e l d i s of the form . A study of part (a) of figures 41 to 418 shows that the unstable range i s always small near S2= 1 (recall that S 1 = 1 in these figures) and that the growth rate of the most unstable wave generally increases away from = 1. This i s in agreement with the qualitative statement made by Davey (1977) - "... the range of unstable wavelengths increases as the curvature i s increased from zero". When one notes that i t i s the curvature wave which inter-acts with one of the other waves to generate an instability (at least when i s significant - see section 5) and that in the limit = this 'wave tends to become energetically inactive (recall that for = F^ = F^, and T = £ = 0 the curvature wave reduces to the stable root c = l ^ ) , Davey's result i s not surprising. 69 One may also consider the effect of curvature in the mean velocity profile on baroclinic instability by looking at the mean potential vort i c i t y gradients. Setting = S 2 in the expressions given in (3.1) (with 8, h, and the horizontal curvature of the mean velocities set equal to zero to isolate the effects of vertical curvature) we have: q =0 <>3y = " 2 F3 S1 Since the necessary condition for in s t a b i l i t y states only that a change in sign of q^ must occur somewhere in the f l u i d , this criterion i s clearly satisfied. However, i f the thickness of the middle layer i s significant, the regions in which q y has opposite signs are effectively isolated and i n s t a b i l i t i e s are inhibited. [Note, however, that-as the thick-ness of the middle layer i s decreased relative to that of the other layers (so that becomes large relative fco F^.and F^)the upper and lower layers w i l l interact more strongly and thus generate significant i n s t a b i l i t i e s (see section 5).] If on the other hand, S^S^^ a c h a n 8 e o f s i 8 n l n <ly between adjacent layers may occur and i n s t a b i l i t i e s are more probable. Finally, we note that although in our figures we have only plotted cases in which S 2 < S^ so that the unstable long waves are generally due to the curvature and S^-waves interacting, i t i s important to note that for S^> S^, the S2~wave and curvature wave w i l l interact at long wavelengths. Thus although the relatively long unstable 70 waves shown in the figures of this section tend to be surface intensified, for > S , ,the long unstable waves w i l l tend to be botton intensified. 71 5. Two-layer Models It is of interest here to consider the cases in which one of , or vanishes. We f i r s t consider the case H 2 -*• 0 (Figure £1) . In this case F 2 -*• 0 0 and thus from (2.14) we see that n>2 "jC^" 1" 1!^) • This in turn gives «>2 -^-Cd>3> and u"2 •> yC^+U^ . Equations (3.1) thus reduce to (after replacing the subscript 3 by 2 to relabel the layers appropriately): (a) (U^c) [ ^ y y - k 2 * ^ ^ - ^ ) ] + + 1 ^ - 0 H 3q (b) ( U 2 - c ) [ * 2 y y - k 2 * 2 - ( s - ^ ) F 2 ( ^ 2 - ^ 1 ) ] + ^ = 0 where j ± - 0 - V±yy + F ^ D ^ ) W = B " U2yy + HTbh WV + RQ(H2-bh) \ with <j)^  = o)2 = 0 on y = ±1 . Assuming bh << H 2 , these equations reduce to (2.2.13) of Pedlosky (1964). Hence, as expected, the case H 2 -»- 0 simply gives the two-layer model. Taking the limit H 2 -> 0 in (3.9) we see that one root of the dispersion relation for the three layer model i s C = 0(i.e. c = u"2 = (U^+U^)/2) . Thus the roots of the dispersion rela-tion in this case reduce to OJ - [ (U^+U-j)/2]k = 0 plus those appropriate to a two-layer f l u i d . Figure 21 i s an example of how this transition of roots occurs. ho Figure SA : A cross-section of the two layer model obtained by letting H~ -*• 0 73 2.0 4.0 k 6.0 0.0 ( c ) * n 5.0 2.0 4.0 k Z2 0.0 -5.0 "s 2=i.oo ^ > • • -5.0 6.0 0.0 2.0 4.0 k ' S 2 = I.OO — •v. 6.0 0.0 2.0 4.0 6.0 Figure K2o>: S 2 Maximum Growth Rate »l'«2 6 1 S 3 H 2 - 60m 1.25 0.21 0.28 3.36 54 92(+) -50.77(+) F i gure S . t b : 0.75 0.12 0.60 1.54 60 19(+) -18.52(+) H 2 - 6m 1.00 0.06 0.61 1.65 27 42(+) - 9.84(+) 1.25 0.37 0.19 1.43 20 89(+) -55.97(+) F igure S.2i 1.00 0.09 0.57 1.62 42 02(+) -13.63(+) Hj - 1 0 " 5 o D i sper s i on curves fo r the f i r s t mode (m-l) f o r si • fo , „ - 4 • «• 10 rad. s 0, T - -10 , I - 10km, g ' 0.02m s " 2 , Hj. 180m and Hj - (H T - l l 2 ) / 2 74 Only the case 3 = 0 , T = - 10 has been shown but the same general features are seen in other cases. For definiteness we have taken -4 -1 -2 f Q = 10 rad s , L = 10 km , g' = 0.02 ms , = 180 m , and H l = H3 = ( H T _ H 2 ) / 2 f o r H 2 = 60 m , 6 m , and 10~5 m. For S 1 = S 2 , the case H 2 = 10 m is indistinguishable from the two layer model. From figures {S.lo.) and (f.2b) we see that a small change in curva-ture can cause considerable difference in the st a b i l i t y of the flow. For example in each of these cases when S 2 = 1.25 the curvature wave interacts with the S2~wave to create a significant i n s t a b i l i t y »e«.r K • However, as S 2 the curvature wave tends to become energetically inactive and this i n s t a b i l i t y diminishes. In each case we also see that the cases S 2 = 0.75 and S 2 = 1.00 are very similar at least for the choice of parameters considered here. This i s , of course, only one special case and we should not try to draw any general conclusions from i t . However, i t i s clear that the st a b i l i t y of the system is rather sensitive to changes in the curvature of the mean velocity f i e l d and care is needed when attempting to use a two-layer model to approximate a situation in which curvature i s significant. On the other hand, When S 2 = the three cases considered in figures (£2**.) to (^ .Zc) are encouraging. Although when H^ = H 2 = H^ = 60 m (f.la.) the and S2~waves don't interact, by the time H 2 = 6 m (5V2b) the s t a b i l i t y of the 3-layer system.is very near that of the two-layer system shown in figure (f.2c) ( a l l of these, cases would show better agreement, i f the density differences between the layers were reduced 75 appropriately; however, the choice of an appropriate reduction i s not a t r i v i a l matter). It appears that the two-layer system should be used with great care but is useful when - and is relatively small compared to and H^. When H 2 % t^ i e model must be " f i t t e d " by appropriate reduction of the total density difference. The explanation for our result i s that for the two-layer model the curvature wave reduces to an energetically inactive stationary (relative to the velocity in the middle layer) wave and the S— and S 2— waves can now interact (though they probably should be renamed). Thus the two-layer model i s a good approximation when the middle layer i s thin enough for the upper and lower layers to interact and the curvature wave does not interact significantly with the other waves. The problem with the commonly used two-layer model i s that the effect of curvature i s neglected. If instead of letting -*• 0 one lets -> 0 we get the situation illustrated in figure S.3 In figure 5".f we see that in this case the curvature wave remains active and at least for the parameters we have chosen this model i s a better approxi-mation to the three layer model than i s the model consisting of two constant density layers. For H 1 0 we see from (2.15) that ^ -»• (4^2-iJ>3)/3. Hence -»- (4U 2 o -U^)/3 and $ + (43>2~$3)/3 so that (3.1) gives , {after letting i + i - 1 to relable the layers appropriately): ( U l o " c ) t * l y y - k V 8 V V V / 3 ^ ^ = ° Figure 5.3: A cr o s s - s e c t i o n of the two-layer model obtained by l e t t i n g n± 77 0.0 2.0 4.0 k 6.0 0.0 2.0 4.0 (c) 5 0 r S , = 0.50 6.0 0.0 2.0 4.0 k I. 3 OO -5.0 0.0 2.0 4.0 6.0 k F igure S'.f q : S l Maximua Growth Rate | u 3 / u 2 l 4 1 S 3 - 60B 0.25 0 28 0.18 2.16 29.52(+) -62.47(+) 0.50 0 23 0.28 2.91 40.93(+) -74.77(+) 0.75 0 17 0.24 3.29 52.68(+) -53.54(+) F i gure 5AV. 0.25 0 19 0.52 2.45 26.38C+) -78.94(+) H l " ta 0.50 0 15 1.00 2.76 38.94(+) -75.27(+) 0.75 0 05 3.01 3.40 -28.05(+) 23.49(+) 0.75 0 10 1.24 3.23 50.83(+) -59.33(+) F i gure S'.tc • 0.50 0 13 1.51 2.70 36.19(+) -82.63(+) - 1 0 " 5 » D i sper s i on curves fo r the f i r s t node (m-l) f o r S 2 " l< l0 ' 1 0- 6 r a d . s 6 - 0 T - -10 , L - 10km., g' - 0.02m - 180m. and H 2 - H 3 - ( H T - H 1 ) / 2 78 H -bh 2 9q 2 ^ V c ) [ ( Hj" ) ( $2yy" k V + S F ^ - t y / S ] + $2 3y~ = 0 where O U J - U J ) / 8 9 q l = 6 - U x + 3F 1(U 1-U 2) 9y oyy = B - U , + 8F. ( U . -U.)/3 loyy 1 v lo 2 3 q H3-bh H o 2 (Note that these equations are identical to (5.1) with the density difference between the layers reduced by a factor of 3/8 and Ij^ replaced by U 1 q ,i.e.,the appropriately " f i t t e d " two-layer model.) Further, i f one takes the limit Hj •*• 0 i n (3.9) we see that for g = 0 the dispersion relation reduces to C = +S, (the S,-curve Ao 1 whose slope tends to 3/8 (= 3/8 S ) as •+ 0 i n figure 5 » plus the dispersion relation for the two-layer f l u i d illustrated in figure ?-3. 79 In the case where the curvature wave interacts with the Sj-wave, the limit 0 (figure 5.5) may be of interest but we shall not consider this case here. Due to the possible qualitative difference between the three-layer model and the two-layer model with constant densities in each layer i t i s suggested that care be taken when either the middle layer has significant thickness or curvature of the mean velocity profile i s present. When the simplicity of a two-layer model i s strongly desirable, perhaps one of the other po s s i b i l i t i e s introduced here should be considered to choose an appropriate density difference between the layers (a case where this idea i s useful w i l l be considered in Chapter IV, Section 3 of this thesis, where we consider a three-layer model with linear density variation through the upper layer, and a density discontinuity between the lower layers). Figure 5.5- A cross-section of the two-layer model obtained by letting H_ -*• 0 8 1 6. Conclusions In general, three vertical modes may exist in the system studied here. They have been classified as shear waves here due to the role of the vertical shear in destabilizing the flow. They could equally well have been classified as v o r t i c i t y waves. The S^-wave which is strongly affected by variations in S as well as B i s clearly linked to 3 q l — — . Similarly the curvature and S2~waves could be classified by 7 9 q2 8 q3 their dependence on and respectively. It i s shown that when the thickness of the middle layer i s significant the S^ and S2~waves cannot generally interact but the curvature wave Is very strongly interactive with either of these waves. Density stratification i s shown to stabilize the flow both by decreasing the range of unstable wave numbers and by increasing the effect of B and topography. B stabilizes both the unstable S^-wave and the unstable S2~wave, while topography only has a significant stabilizing influence on the S2wwave. Increasing either or relative to rl^ tends to stabilize the flow. The three-layer model with uniform density in each layer i s a good approximation to the model developed here in the case of zero curvature i n the velocity profile but for large curvature (i.e.for |S^—S^l large) i t overestimates the density str a t i f i c a t i o n and must be compensated for. The model consisting of two constant density layers i s shown to be a good approximation to the three-layer model only when the curva-ture in the mean velocity profile and middle layer thickness are small. In particular the agreement seems best when the shear i s slightly stronger in the upper layers than in the lower layers. Improved agreement may be found by reducing the density st r a t i f i c a t i o n appropriately. Although 82. i t i s generally not clear how this should be done, for a linear density variation through the upper layer, i t is shown that the density difference should be taken as 3/8 of the density difference through the top layer. 83 Appendix a I have found i t convenient to rewrite (2.25) in the following form, especially for the purpose of finding s t a b i l i t y boundaries: 3 2 aC + BC + yC + 6 = 0 6 4 2 where a = A,K + A.K + AJt 6 m 4 m 2 m A6 = 1 A. = 3F_ + 8F 0 + 3F_ 4 1 2 3 A 2 = 8(F 1F 2+F 2F 3+F 1F 3) 3 = B6Km + B4Km + B2Km + B0 B 6 = S 2 " S l o o 9q 3q 9q B4 = < S 2 o " S l o ) + ^ + W + WL B 2 = 8(S 2 -S ) ( F 1 F 2 + F 2 F 3 + F 1 F 3 ) + ^  (8F +3F ) + -r-^ (3F +3F ) + -g-i (3F +8F ) .Mi 3q2 3q_ ^ 1 3 ^ 1 2 B = 8 ( — — F F H — F F + — — F F ) o °V3y 2*3 + 3y '1*3 9y 1 2} 6 4 2 Y = C,K + C.K + C„K + C 6 m 4 m 2 m o C6 " " S l S2 o o 3q9 9qo 3q, 3q, C. = -S. S 0 (3F.+8F.+3F,) - S, (^— 1 + - r - ^ ) + S„ (-r-^. + ^ - ) 4 lo 2ov 1 2 3' lo 3y 3y 2o 9y 9y ' 84 C2 - - 8 S l o S 2 o ( F l F 2 + F l F 3 + F 2 F 3 ) " S l o ( i T (^+^ + ^0^+8^)) + S 2 o ( i T W + i T ' < 3 F l + 3 F 3 > > 9 q x 8 q 3 ^ 8 q 2 8 q 2 ^ + 9y 9y + 9y 9y + ay 3y 3q 3 3q 2 9q 9q„ Co = - S l o ( 8 F l F 2 3y~ + 8 F 1 F 3 IT* + S 2 o ( 8 F 2 F 3 3y + 8 F 1 F 3 ^ 3 q ? 94o 9q-, 3q, 3q, 9 q 9 1 dy dy 2 3y 3y 3 9y 9y 6 = D.K 4 + D_K 2 + D_ 4 m 2 m 0 9q 9 D 4 - " S l S2 9 T o o J 9q ? 3q 9 9q 9 3q, 3q 9q D, - - S l o S 2 o ( 3 F 3 — + 3F, w ) - 8 ^ ^ + ^ ^ 3q 2 9q 9 3q, 3q, 3q 2 D Q = - S L S 2 ( 8 F l F 3 — ) + S l i-3h ^ f j f ) + S 2 (3F 3 ^ ^ ) o o o o J 3qx 3q 2 3qo 3y 3y 3y The case of an advected topographic Rossby wave i s found by letting S 1 = S 2 = 0 ,6 = 0. Then [ K 4 + (3F.+8F 0)K 2 + 8F F 0]T g _ m 1 2 m 1 2 KmtKm + ( 3 F l + 8 F 2 + 3 F 3 ) K m + B C F ^ + F ^ + F ^ ) ] and i f we take the limit F 2 °° (H2-K)) we obtain the dispersion rela-tion for a topographic planetary wave in a two-layer system 85 (e.g. Helbig and Mysak, 1976). T(K 2+FJ c = m 1  K 2(K 2+F +F ) m m ± J CHAPTER III MIXED BAROCLINIC-BAROTROPIC INSTABILITY IN TWO- AND THREE-LAYER MODELS 86 87 1. Introduction Pedlosky (1964b) has studied mixed baroclinic-barotropic instability in the two-layer model with upper and lower layers of uniform densities p * and p * respectively ( p * < p * ) . To simplify the analysis he has considered the case in which the velocity in the lower layer i s uniform across the channel. The f i r s t main purpose of this chapter i s to extend the work of Pedlosky (1964b) to include the effect of horizontal shear in the lower layer (section 2). The mean velocity profile i s taken to be a cosine jet in each layer with the amplitude of the velocity in the lower layer varying relative to that in the upper layer, i.e. = UQ(1-COS Tr(y+1)), = eU^ with £ a constant whose value i s varied. To concentrate our attention on the effects of horizontal and vertical shears of the mean currents, the effects of 3 and topography are not considered although they w i l l be important in most applications. The f i n a l sections of this chapter are concerned with a model in which the density stratification i s modelled by three layers. Davey (1977) has studied pure baroclinic i n s t a b i l i t y in a three-layer model similar to the two-layer model of Pedlosky. In chapter II, on the other hand, pure baroclinic i n s t a b i l i t y has been studied in a three-layer model derived from the equations of motion relevant to a specialized continuously stratif i e d f l u i d in which the upper and lower layers are of uniform densities p. and p„ respectively ( p <p ) and the density of the middle 88 layer varies linearly from to p^. The second main purpose of this chapter i s to further develope; the model introduced in chapter.II. In section 3 (of this chapter) some general results analogous to those of Pedlpsky (1964a,b) for the two-layer model are presented for the three-layer model. The energy equation i s discussed, bounds on phase speeds and growth rates of unstable waves are found and the condition for marginally stable waves with phase speed within the range of the mean velocity i s presented. Section 4 i s concerned with some specific results on mixed baroclinic-barotropic i n s t a b i l i t y in the three-layer model. The flow in the lower layer i s assumed to be uniform (and thus set equal to zero) while the velocity profiles in the upper layers are chosen as in the study of the two-layer model, 1 . e. Uj= U Q(l-cos TT (y+1)), U 2 = eU^. By varying e, mixed baroclinic-barotropic i n s t a b i l i t y i s examined. Finally some general conclusions are made in section 5. 2. The Two-layer Model The two-layer model introduced by P h i l l i p s (1951) to study the st a b i l i t y of quasi-geostrophic flows has been extensively used in both meteorology and oceanography. The reason for i t s popularity i s the simplifications which result in replacing a singular non-separable p a r t i a l d i f f e r e n t i a l equation with a pair of coupled, singular ordinary d i f f e r e n t i a l equations. The analysis i s often further simplified by making the assumption that the mean flow in the lower layer is uniform (both v e r t i c a l l y and horizontally) and hence without further loss of generality i t i s set equal to zero. The main 89, purpose of this section i s to investigate the effects of non-uniform flow in the lower layer. The analysis w i l l generally be restricted to the case 3 = 0 . Killworth (1978) has pointed out that the effect of 3 i s mainly quantitative rather than qualitative except in cases where 3 i s sufficiently large.to stabilize the flow. This is particularly relevant to the very long waves which may be stabilized by a relatively small value of 3 and hence we w i l l r e s t r i c t our attention to unstable waves at moderate wavelengths. We begin with a brief discussion of the cases in which the mean currents vary over a horizontal length scale which is either small or large compared to the local internal deformation radius in each layer. This i s followed by a detailed discussion of the case in which these length scales are of the same order. The case of a cosine j e t in the upper layer of a two-layer f l u i d with = 0 ( f i r s t studied by Pedlosky, 1964b) i s further. considered followed by a discussion of the case U = U . For these cases ' 2 1 some simple analytical results are discussed. The remainder of this section consists of a numerical study of the effects of horizontal shear i n the lower layer. The effect of taking u"2 f 0 w i l l , of course, depend on the horizontal length scale of the motion. More precisely, i t depends on the ratio of the horizontal length scale of the mean flow to the internal (Rossby) radius of deformation. This fact i s clear from the equations for the non-dimensional (complex) amplitudes of the stream functions for the two layers. ( U r c ) [ * l y y - k 2 ( | , l + F l ( * 2 - * l ) ] + * l [ B - U i y y + F l ( U r U 2 ) ] = 0 (U2-c)t<(>2yy-k2<f2+F2(^1-((.2)] + +2[0-D2 +F 2(U Z-U 1)] = 0 (2.1) 9.0. where the nondimensional q u a n t i t i e s are r e l a t e d to the corresponding dimensional ( s t a r r e d ) q u a n t i t i e s as. i n s e c t i o n 2 of chapter I I . i . e . : * * (x ,y ) = L(x,y) * * U ( c , U 1 > 2 ) (c ,U 1 > 2) = * k = k/L * * i = UL<(>i A , 2 8 = 8U/L (f=f Q+3*y*) (2-2) i n which L and U are t y p i c a l h o r i z o n t a l l e n g t h and v e l o c i t y s c a l e s , f i s the l o c a l value of the C o r i o l i s parameter, g 1 i s the reduced g r a v i t y ( g ' = [ ( P 2 _ P ^ ) / p 2 ] ' g ) , , H 2 are the thicknesses of the upper and lower 2 2 l a y e r s i n the absence of motion r e s p e c t i v e l y . F i n a l l y , F_^  = f^L /g'H_^ i s the square of the r a t i o of the h o r i z o n t a l l e n g t h s c a l e of the mean flow to the i n t e r n a l deformation r a d i u s of the i 1 " * 1 l a y e r . We note < t h a t the d e f i n i t i o n of the i n t e r n a l deformation r a d i u s i n t r o d u c e d here i s not e q u i v a l e n t to the u s u a l d e f i n i t i o n (r. =/—= ) but i t has the same /fo<Hi+V b a s i c p r o p e r t i e s and we w i l l f i n d i t very convenient t o r e f e r to the q u a n t i t y / g ' t i ^ / f ^ as the i n t e r n a l deformation r a d i u s of the i * " * 1 l a y e r . We w i l l b r i e f l y c o n s i d e r the cases F « 1 , ¥^ ^ 1 and F i >> 1 s e p a r a t e l y . A more d e t a i l e d treatment of these l i m i t i n g cases may be found i n K i l l w o r t h (1978). 91 Case I: F « 1 (i=l,2) . , Assuming that F^ and F^ are of the same order, the following expansions are appropriate: <f>. =<!>? + ^ F, + ... ; 1 i I 1 c = c° + c1¥1 + . .. (2.3) Substituting these expansions in (2.1), we find that to leading order in F^ , aS^  , (|>2 and satisfy: ( U r C° ) [ < ! >?yy- k 2 < , >? ] + ^ ^ " V 1 = ° ( U 2 - C ° ) [ * 2 y y - k ^ 2 ] + ^ " ^ v y 1 = ° ' ( 2 ' 4 ) Hence, in this case the equations decouple, each layer yielding a Rayleigh i n s t a b i l i t y problem. If $ u~2 , an eigenvalue of the f i r s t equation w i l l not generally be an eigenvalue of the second equation so that the perturbation motion is generally trapped in one layer where in s t a b i l i t y may occur due to pure barotropic i n s t a b i l i t y . (To next order in F^ there w i l l , of course, generally be motion forced in the other layer). In this case, only the form of the mean velocity i n the layer in which i n s t a b i l i t y occurs i s of any consequence. A typical example of this type of i n s t a b i l i t y i s shown in - 4 - 1 Figure 2.1 where we have taken f^ = 10 s , g = 0, L = 57.735 km, g 1 = 0.66 ms " 2 , H = H 2 = 5 km (F.^F^O.01) , 1^ = U Q(l-cos iT(y+l)) and U 2 = -U^ . We note that with the exception of L these parameter values have been chosen to model atmospheric flows. This choice was made in order to f a c i l i t a t e comparison with the studies of Pedlosky (1964a,b) and Brown (1969a,b), however the results only depend on the values of F^ and F 2 and are 92 immediately applicable to appropriate oceanic flows. For these parameter values, the layers are essentially uncoupled and <f>^  and <j>2 satisfy, to a good approximation: _ [(U 0-c)-U 0cos ir(y+l)][<|> - k 2 ^ ] + ^ [ - A Q C O S TT(V+1)] = 0 [(eU 0-c)-eU 0cos 7T(y+l) ] - k 2 ^ ] + ^[-AlJgCOS IT (y+1)] =0 (2.5) Since we have taken 8 = 0 , the growth rate of any i n s t a b i l i t y w i l l simply be proportional to the maximum velocity in that layer. This i s the only role of E at these very small horizontal length scales. Further details on this type of i n s t a b i l i t y can be found in Lin (1945, parts I, II, III), Pedlosky (1964b), Drazin and Howard (1966),Brown(1969a,b) and Kuo (1973) as well as a multitude of others. Case II; F ± » 1 (i=l,2) . In this case the horizontal length scale i s much greater than the internal deformation radius and we expect any i n s t a b i l i t y to be basically baroclinic in nature. This being the case, the appropriate horizontal length scale for the perturbation i s the internal deformation radius rather than the scale on which the mean flow varies. Hence we introduce the follow-ing transformations y' = y ( F x ) 1 / 2 k» - k ( F 2 ) - 1 / 2 ( 2 . 6 ) and expand <J>_^  and c as: * ± = <(>? + ^ F" 1 + ... c = c + c F 1 + . . . . (2.7) 93 (a) (b) y y Figure 2.1: (a)Mean potential vorticity gradients in the upper layer (solid line) and the lower layer (dashed l i n e ) . (b)Complex amplitude for the stream function in the upper layer (c^^-O). (c)Transfer of Available Potential Energy (T.A.P.E.). (d)Transfer of Kinetic Energy (T.K.E.) in the upper layer. Since ty^-O, the transfer in the lower layer is very small, f =10~ 4s _ 1, B=070, L=57.735 km, g'=0.66 ms - 2, \ m ^ 2 " 5 ( F 1 = F 2 = 0 - 0 1 ° ' Ul=l-cos1I(y+l) , U2--U1 . 94 Making these substitutions in (2.1), we find that to leading order, <f>!j| and cj>2 satisfy: (U1-c°)[,j,Jy,y,-k'2<J)J+(<f»°-<(>J)] + * J [ B + ( U r U 2 ) ] = 0 (U2-c°) [<)>2y,y,-k'2^+H1/H2((l)J-()>°)] + ^ [ & + R 1 / } l 2 ( v 2 - V 1 ) ] = 0 (2.8) These equations constitute the j u s t i f i c a t i o n for considering local values of the mean flow in studies of baroclinic i n s t a b i l i t y and are valid for 2 (r^/L) << 1 . The study of these equations amounts to the study of the effect of weak horizontal shear on baroclinic i n s t a b i l i t y . Further comments relevant to this case w i l l be made in the following section, but here I w i l l just note that for this case, the marginally stable waves marking the short wave cut-off w i l l be baroclinic in nature as opposed to the expected barotropic nature conjectured by Pedlosky (1964b). This i s due to the fact that the short wave cut-off for barotropic i n s t a b i l i t y i s proportional to the horizontal length scale of the mean shear while that for baroclinic i n s t a b i l i t y i s roughly proportional to the internal deformation radius. Thus for 2 (r_^/L) << 1 we expect the short unstable waves to be baroclinic in nature. This prediction, i s , i n fact, verified by numerical computation. It must be noted here that the horizontal shear may influence the cross-2 channel structure of the most unstable waves even for (r./L) << 1 . The l marginally stable waves at the short wave cut-off w i l l have a meridional scale of the order of the internal Rossby radius and hence are unaffected by the horizontal shear. However, Simmons (1974) has shown that the meridional 1/2 length scale appropriate to the most unstable wave i s (Lr^) and hence one must be careful not to neglect the effects of horizontal shear when these waves are to be studied. Of course i f one is interested in the local (on 95 the scale of ) s t a b i l i t y properties of the flow then i t i s reasonable to consider the case of no horizontal shear but th i s w i l l not give any i n f o r -mation on the meridional structure of the waves occurring on the much larger scales. Case I I I : F ^ 1 (i=l,2) This i s the case of greatest interest here. Pedlosky (1964b) has considered the case of a cosine j e t i n the upper layer with u*2 = 0 and has given the s t a b i l i t y boundary corresponding to the short wave cut-off. "In the i n t e rest of completeness, the f u l l s t a b i l i t y boundary i s given i n figure 2.2 which has been found by the methods described i n appendix a. For the case U"2 = 0 , Pedlosky was also able to demonstrate numerically that the minimum value of UQ at which i n s t a b i l i t y occurs i s precisely that value for which the necessary condition for i n s t a b i l i t y i s just s a t i s f i e d ( i . e . precisely the minimum value of IL. for which q changes sign somewhere) . I t turns 0 y out that this r e s u l t i s r e l a t i v e l y easy, to prove a n a l y t i c a l l y for the cosine j e t and that t h i s point on the s t a b i l i t y boundary corresponds to c=0, k = IT (see appendix b) . I t i s int e r e s t i n g to note that t h i s value of k gives precisely the short wave cut-off for pure barotropic i n s t a b i l i t y (the reason for t h i s r e s u l t becomes clear i n appendix b). F i n a l l y i t i s of int e r e s t to look at the case F^ = F 2 , U = U"2 a n a l y t i c a l l y . The s t a b i l i t y boundary for this case i n U Q vs. k space i s given i n figure 2.3 for f = 1 0 _ 4 s - 1 , B = 0.0 , L = 1000 km , g f = 0.66 m s " 2 , H x = H =' 5 km , 0 =» U 2 = l y i - c o s ir(y+l)) . For F l = ^2 ^1 = ^2 ° U r e 9 u a t i ° n s r e c ^ u c e t o ( U r c ) [ * l y y - k 2 * l + F l ( V * l ) ] + V 3 " U l y y ] = ° (U1-c)[(f»2yy-k2<))2+F1(<j)l-<|>2)] + 4> 2[3-U l y y] = 0 (2.9) Figure 2.2: S t a b i l i t y boundary corresponding to the parameter values f - 10~V\ B - 1.5 , L - 1,000 km , g' - 0.66 m.s."2, H - H - 5 km. o (F - F - 3 ) , U, - U 0(l-cos it(y+D). U 2 = 0 . > 1 I i & 0 1 * 0 2 0 1 * 0 2 I N S T A B I L I T I E S I N S T A B I L I T I E S S T A B L E c s » 0 . 0 i i 1 > o . o 1 — 1 h ^ i 0.0 l.0| 2.0 I 3.0 k Figure 2 . 3 : S t a b i l i t y boundary corresponding to the parameter values f Q - l O ^ s " 1 , 6 - 0.0 , L - 1,000 km., g' - 0.66 , - H 2 - 5 km. ( F x - F 2 - 3 ) , U X - U 2 - U Q U - C O S ir(y+l)). 9.7 The phase speed at the short wave cut-off i s given by c = u ^ ( y g ) where 2 9 (y ) = 0 (see Pedlosky 1964b). This gives c = U_ - B/TT which reduces l y s U our equations to: " . * l v y- k\ + + ^ \ = 0 Kiyy *2yy"k2,|,2 + + ^2 = P (2.10) . Substituting «f>± = A ± sin ~-(.y+l) , (i=l,2) we find that the condition for a non-trivial solution i s : k 2 = < 2 TT -2 TT -r \ 2 ^ - 2F UJ ^ run .2j by Clearly only n = 1 gives k > 0 , so the short wave cut-off i s given 2 2 k = —j— or k = —. 2F_ 4 4 1 The significance of these two cutr-off values has been clearly pointed out by Dr. J, Pedlosky (thesis report). His comments are as follows. "When the mean flow is barotropic the perturbations can be resolved on the N v e r t i c a l resting state modes of the system. The s t a b i l i t y problem then reduces to 2 2 2 the classical barotropic s t a b i l i t y problem with k replaced by k + ^ n where ^„ 2 2 i s the ver t i c a l wave number of the mode. Since c = c (k + j* ) , while the growth rate i s kc_^  i t i s apparent that the maximum i n s t a b i l i t y (which i s always barotropic) must correspond to the barotropic mode which has = 0. The short wave cut-K>ff for the V L ^ mode i s at (k^ - where k^ i s the. barotropic cut-off." In the preceding work N = 2. The short wavelength 98 c u t - o f f f o r the b a r o t r o p i c mode, k , i s t3ir/2 and that f o r the f i r s t B 2 h b a r o c l i n i c mode, which has = -^ 2» i s ( 3 T r ^ ~ 2 F ^ • T n e corresponding regions of i n s t a b i l i t y are marked a p p r o p r i a t e l y i n f i g u r e 2.3. To c o n s i d e r the e f f e c t s o f v a r y i n g i n g e n e r a l we consider the - 4 - 1 -f o l l o w i n g parameter v a l u e s : f ^ = 10 s , 3 = 0, L ='2,000 km, - . g' =0.66 m s i = H 2 = 5 km (F = ? 2 = 12), w i t h 13 = (1-cos Tr(y+1)), \3 = E U^ . Note t h a t i n the absence of both 3 and topography, the v a l u e of U (the h o r i z o n t a l v e l o c i t y s c a l e ) i s i r r e l e v a n t . The choice of these parameter va l u e s g i v e s the r a t i o o f terms i n v o l v i n g the v e r t i c a l shear o f the mean c u r r e n t s to those i n v o l v i n g the h o r i z o n t a l shear of the mean c u r r e n t s to be of order u n i t y ( i . e . F, (U -U„) ^ U ) . F u r t h e r , the choice of a cosine 1 1 2 lyy j e t i s a p p r o p r i a t e s i n c e we wish to c o n s i d e r a case i n which both b a r o c l i n i c and b a r o t r o p i c i n s t a b i l i t i e s a r e p o s s i b l e . Figure 2.4 gives approximate s t a b i l i t y boundaries i n e v s . k space f o r the parameter va l u e s g i v e n above (see appendix a f o r a f u r t h e r d i s c u s s i o n of the s t a b i l i t y b o u n d a r i e s ) . P r o b a b l y the o n l y meaningful c o n c l u s i o n which can be drawn from t h i s f i g u r e i s t h a t , as expected, away from £= 1.0 the short wave c u t - o f f f o r u n s t a b l e waves i s s h i f t e d to h i g h e r wavenumbers due to the presence of v e r t i c a l shear. The u n s t a b l e r e g i o n s beyond the short dashed l i n e are probably not meaningful. We w i l l f i r s t c o n s i d e r a s e r i e s of cases i n which both energy sources are expected to be important (k = 1.5, £ v a r y i n g ) and then we w i l l l o o k b r i e f l y a t l a r g e r v a l u e s of k (k = 3.5). In each case, a t t e n t i o n w i l l be focused on the most u n s t a b l e waves. The most u n s t a b l e wave f o r k = 1.5, E = -1 ( s t r o n g v e r t i c a l shear) Figure 2.4: Approximate stability boundaries in z vs. k space for the two-layer model with parameter values: fg'= 10 s , 8 = 0.0 , L = 2,000 km., g' = 0.66 m.s."2, H = H2 = 5 km. (F 1 = F 2 = 12), = 1-cos n(y+l) , U 2 = eU^ . Also shown here is the stability boundary appropriate to (L/r^) <<1 (short dashed l i n e ) . Note: These boundaries are qualitative. The long dashed line indicates that value of k below which the apparent i n s t a b i l i t i e s -are l i k e l y real. Beyond this line the apparent i n s t a b i l i t i e s have very small growth rates and details of the boundaries shown should not be taken seriously. vo VO 100 has CD = 0.00 + i 1.50 and is illustrated in figure 2.5. It's energy source i s almost purely baroclinic with only a very small contribution from barotropic sources (Transfer of Kinetic Energy [TKE] < 1/50 Transfer of Available Potential Energy [TAPE]). As one might have expected, this wave has zero phase and group velocities making i t a very special case and hence probably only of limited use in modelling. Other apparent i n s t a b i l i t i e s at k = 1.5 , e = -1 have very small growth rates: OJ = 0.37 + i 0.09 -mixed energy source extracting potential energy from the interface and kinetic energy from the upper layer in roughly equal amounts and OJ = -0.37 + i 0.09 with the same properties as above but i t s source of kinetic energy is the lower layer. The most unstable wave at e = -0.5 (k = 1.5) has to = 0.43 + i 1.10 and is illustrated in figure 2.6. In this case the net conversion of kinetic energy in the upper layer is nearly zero while that in the lower layer is again very small compared to the conversion of potential energy. The only other in s t a b i l i t y found at k = 1.5 is a bottom intensified wave (perturbation energy of lower layer - 4x perturbation energy of upper layer) whose energy source i s mainly barotropic in s t a b i l i t y in the lower layer with TAPE < TKE/2 in the lower layer. The TKE in the upper layer is small and negative. At E = 0.0 the most unstable wave has co = 0.84 + i 0.56 and is illustrated in figure 2.7. It is a potential energy converting wave losing a small amount of kinetic energy to the mean flow in the upper layer. (The conversion of kinetic energy in the lower layer vanishes since = 0 ). Also significant at e = 0.0 i s the unstable wave shown in figure 2.8. It has the same basic energy conversion properties as the most unstable wave but the TAPE i s more evenly distributed across the channel and this i s (d) <e) o Figure 2.5: (a)Mean potential vorticity gradients in the upper layer (solid line) and lower layer (dashed lin e ) . (b)Transfer of available potential energy. (c)Transfer of kinetic energy in the upper layer (^transfer in the lower layer which is indicated by a dashed line in this and future figures). (d)Complex amplitude of the stream function in the upper layer, (e)Complex amplitude of the stream function in the lower layer. Parameter values as in figure 2.A; e=-1.00, k=1.5 . Figure 2.6: As in figure 2.5 but c - -0.5 Figure 2.7: As i n figure 2.5 but c - 0.0 Figure 2.8: As in figure 2.7 - a second s i g n i f i c a n t I n s t a b i l i t y at the same position in parameter space. 105 reflected in the form of the eigenfunctions (especially ) . This wave is basically a higher mode insta b i l i t y . The most unstable wave at e = 0.5 (k=1.0) has co = 0.76 + i 0.34 and is illustrated in figure 2.9. Unlike the most unstable waves of previous cases this wave has substantial energy conversion contributions from barotropic ins t a b i l i t y especially in the upper layer. In fact the TAPE - TKE . One significant consequence of the TKE in the upper layer i s that the wave i s considerably intensified in the upper layer. (When considering baroclinic-barotropic instability in the three-layer model we w i l l see that an apparently insignificant conversion of kinetic energy may cause significant i n t e n s i f i -cation in the corresponding layer). A second significant instability also occurs at e = 0.5 . It is illustrated in figure 2.10. This wave extracts most of i t s energy from the t i l t of the interface near the centre of the channel and is i t s e l f somewhat concentrated towards the centre of the channel. The growth rate of this wave i s considerably reduced by a loss of kinetic energy to the upper layer. The perturbation i s , however, not significantly intensified in either layer. At £ = 1.0 , the only source of energy for the perturbations i s the horizontal shear of the mean currents. This case has already been considered in some detail analytically. Since F^ = F 2 - 12 for the case considered here, the wave with $~ = is not present (recall that the short wave 2 2 3ir cut-off with this class of waves is given by k = —^ 2F^ ) and hence the only unstable wave found here has <j>^  = <j>2 (see figure 2.11). This wave' has co = 0.93 + i 0.39 and actually grows somewhat faster than the corresponding wave at e = 0.5 (to = 0.76 + i 0.34). In conclusion we note that at k = 1.5, F n = F, = 12 the most unstable Figure 2.9: As in figure 2.5 but t= 0.5 Figure 2.10: Aa i n f i g u r e 2.9 - a second s i g n i f i c a n t i n a t a b i l i t y at the same p o s i t i o n i n parameter space. o . _.. i fW\ T v v fr A P F =m OI rf, f=(t_1 Parameter *ly v _ H 2 y ; values as in figure 2.4; e = 1.0 , k = 1.5 Figure 2.11: (a) q (=q ) (b) T.K.E. (T.A.P.E. =0) (c) ^ ( s ^ ) Parameter 0 0 109 waves with e•<_0 are essentially baroclinic energy converting waves extracting energy from the t i l t of the interface and losing small amounts of kinetic energy to the mean flow. For this type of in s t a b i l i t y , the most important quantity to estimate is the magnitude of the vertical shear. For e > 0 however, this i s not the case. The most unstable waves have significant contribution from kinetic energy conversions and taking e = 0.0 as a f i r s t approximation may give very misleading results. Finally, l e t us consider the i n s t a b i l i t i e s present at values of k greater than that corresponding to the short wave cut-off for barotropically unstable waves. In particular we have considered k = 3.5 . At values of e where i n s t a b i l i t i e s occur at k = 3.5 the energy source i s almost purely baroclinic, however a significant loss of kinetic energy to the mean flow may occur for e > 0 . Two effects thus serve to stabilize the flow near e = 1 . The f i r s t is that the net conversion of potential energy i s roughly proportional to the mean vertical shear and hence decreases as e = 1 is approached. The second effect is that the loss of kinetic energy to the mean currents increases as e = 1 i s approached. Both of these effects w i l l be noted on comparing figure 2.12 (e=0.0) with figure 2.13 (e=0.5). We thus conclude that even in the case where we are interested in waves considerably shorter than the short wave cut-off for barotropically unstable waves, the neglect of horizontal shear in the lower layer may be significant (especially for e > 0 ) in that we are neglecting a potential stabilizing effect. However, we note again (as in the case of "small" k ) that for e < 0 this effect i s probably not very significant and the study of baroclinically unstable waves is reasonably approximated by setting e = 0.0 . (a) (b) Figure 2.12: Parameter values as in figure 2.U; k « 3.5 , e - 0.0 Figure 2.13: Parameter values (b) in figure 2.4; k - 3.5 , e - 0.5 112 3. The Three-layer Model (Qualitative Results) In chapter II a three-layer model was derived from the equations of motion for a continuously strat i f i e d f l u i d with densities o.f the upper and lower layers given by p * and p^ respectively (p*>p*) and p * varies linearly from p * to p * (FigureZ.i chapter II). There i t was found that for bh/H 3«l the linearized equations expressing the conservation of potential vorticity for the model described above are: at + u i o ax [ V H V F l ( 3 5 r 4 5 2 + 5 3 ) ] + 5lx qly = ° _a_ _a_ 9t + U2o 3x t W 4 F 2 ( V 2 W ] + 52x q2y = 0 3t 3o 3x [¥3- F3 ( ?r 4 ?2 + 3 ?3 ) ] + 53x q3y = ° (3.1) where 1 = 0 - U + 2F. (2U -3U.+U ) ly lyy 1 1 2 3 q2y " e- U2oyy ' q3y = 6 " U3yy + 2 F 3 ( U r 3 U 2 + 2 U 3 ) + ( b / R 0 H 3 ) h Y (3.2) are the northward gradients of the potential vorticity of the mean currents in each layer. Subscripts 1, 2 and 3 refer to quantities defined in the upper, middle and lower layers respectively and the additional subscript o indicates that the quantity i s to be evaluated at the middle of the appropriate layer. Note that since quantities in the upper and lower layers are vertically uniform, TL = XL , U„ = U_ but ( c f . chapter II) lo 1 3o J 113 u 2 o = ( 6 v u r u 3 ) / 4 - ( 3 ' 3 ) The purpose of t h i s sec t ion i s to extend the q u a l i t a t i v e r e s u l t s of Pedlosky (1964a,b) to the three - layer case. (a) The Energy Equation In chapter II i t was found that i f the i 1 "* 1 equation of (3.1) i s m u l t i p l i e d by S ^ ^ i a n d i n t e S r a t e c * across the channel and over a wavelength i n the x - d i r e c t i o n , a reg ion we have r e f e r r e d to as R, the fo l lowing energy equation i s der ived . _9 9t +(!|^_+ ih_h)2 + + L ( h - 2 W 2 ] dxdy [ V l x V F l + U 2 o y ? 2 x ? 2 y / F 2 + ^ ( 3 ' 4 ) R /r^i?2x ( ur u 2o> + ^ix'VV + ^ 2 ^ 2 0 - ^ d x d ^ J R + 'R This equation expresses the fac t that the l o c a l rate of change of t o t a l per turbat ion energy i s due to one (or both) of two d i s t i n c t energy conversion mechanisms. The f i r s t of these mechanisms extracts energy from the mean flow made a v a i l a b l e to the perturbat ions by the h o r i z o n t a l shear of the mean current s . When t h i s i s the dominant energy source for the per turba-t ions the i n s t a b i l i t y i s r e f e r r e d to as baro trop ic i n s t a b i l i t y . I f , on 114 the other hand, the perturbations extract energy principally from the available potential energy of the mean currents (due to the effect of rotation on a vertically sheared, strati f i e d fluid) the ins t a b i l i t y i s known as baroclinic instability. In this chapter we are basically interested in the mixed baroclinic-barotropic i n s t a b i l i t y case where both mechanisms may be significant. (b) Necessary Conditions for Instability The necessary conditions for instability have been given in chapter II (equations 3.5 and 3.6). The f i r s t of these i s : (3.5) which implied that for unstable waves (i.e.for Im(c)^O) the potential vorticity gradient must change sign somewhere, either within a layer or in going from one layer to another. An enlightening interpretation of this condition has been given by Pedlosky (1964a). Basically (3.5/> results from the fact that in the absence of external forcing, momentum can be redistributed between the mean and the perturbations but no net change in momentum may occur. The second necessary condition for instability i s : 115 1 3 k c i f Z!(i*±yi2+k2 U i l 2 ) / F i + 2 U * r * 2 l 2 + iv^i 2 + ^ I ^ ^ V S l ^ d y -1 1 = 1 =kc .13 D ±|* ±| 9q ± ± ( H " " 2 -i 1 = 1 F i l u i o - c ' 9y dy (3.6) It can be shown (see Pedlosky 1964a) that (3.6) i s equivalent to the statement 1 3 U. $ 2 9 ^—[Perturbation Energy] « TTC . dt l r „ i o | * i j .^iy _ Za 2 dy'exp (2kc ±t) (3.7) , i=l F. |U. -cl -1 l ' 1 0 I For an unstable wave, the l e f t hand side of (3.7) is positive and hence we see that the product U. -r— must be somewhere positive for ins t a b i l i t y to 1 0 dy r J occur. This result may also be seen directly from (3.6), but the form (3.7) (and i t s derivation)is more satisfying physically. (c) Bounds on Phase Speeds and Growth Rates In this section bounds w i l l be found for the phase speeds and growth rates of unstable waves in the three-layer model studied here (3 and topographic effects are now included). Since the details are somewhat more complicated than for the two-layer model a f u l l derivation w i l l be given. 116 We begin with Eq. 3.1 from chapter II with bh«H, ( U l o " c ) ^ l y y - k V F 1 ( 3 V 4 W ] + * l « l y = 0 ( U2o- c )f' $2yy- k V 4 F 2 ( V 2 W 1 + $ 2 q 2 y = 0 (U3o-c)[<D3yy-k2<I>3-F3($1-4$2+3$3)] + * 3 q 3 y = 0 . (3.8) Since for unstable waves, lm(c)^0, i t i s permissible to make the transformations *. = (U. -c)9. .(i=l,2,3) 1 io 1 ' (3.9) From (3.8) i t can be shown that the equations satisfied by 9^, 9 2 and 9 3 are: _d dy (l^-c) dQ1 T. dy" -k 2(U,-c) | — e x - ( u r c ) [3 ( u r c ) e r 4 ( u 2 o - c ) e 2 + ( D 3 -c)9, + ( u r c ) —f [6+2F1(2U1-3U2+U3)]91 = 0 117 _d_ dy ( U2o" c ) d9, dy -k ( U2o" c ) F 92 + 4 ( U 2 o " c ) [ ( U 1 - c ) e r 2 ( U 2 o " c ) 6 2 + ( U 3 ~ c ) 9 3 ] ( U2o- c ) + — [9 - 6 F 2 ( u 1 - 2 u 2 + u 3 ) ]e 2 = 0 _d_ dy (U 3-c)" d&3 F 3 dy -kz—|i e 3 _ (u 3-c)[(^-0)9^4(^^0)62+3(^-0)93] (U3-c) + —-j [3+2F3(U1-3U2+2U3) + (b/R()H3)h ]63 = 0 (3.10) ,th If the i equation i s multiplied by 9 i (where a * is used to represent the complex conjugate), integrated from y = -1 to y = +1 , and the three equations added, the imaginary part of the resulting equation gives: c r <VV*3> = + U 2* 2 + U31>3 - f (J 1 +J 2+J 3) - ^  h y J 3 (3.11) where 1 2 . , 2 1 n 1 2 |61 |" + k |9 I . - 2 1 7 g 1 + 2 | 9 ^ 2 | 2 - | | 8 ^ 3 l 2 * 2 " K | 2 + k2|8 | 2 2 2 _ + 2 | 9 i - e 2 | 2 + 2|92-83r 3 l 8 3 y l 2 +k 2l9,] 2 + 2| e 2 - e 3 1 2 - jkj-e J . = 1 F. 1 (1=1,2,3) (3.12) and an overbar denotes integration from y = -1 to y = +1 . (3.11) i s 118 directly analogous to (4.2.5) of Pedlosky (1964a). However, further analysis is complicated by the possibility of one of ^ or T-^  being negative in some region (It seems plausible that £-^ > ^ 2> ^ 3 a r e a l w a Y s positive for an unstable wave but I have been unable to prove this. If this result could be proved, bounds identical to those given by Pedlosky would follow in the same manner as in the two-layer case). To proceed further, we l e t IL = (U,+U„)/2 + 6U and rewrite (3.11) in the form: 2o 1 3 : r ( t 1 + W = V*1+fp2) + U3(?3+fp2) + 6U* 2 - f^+W - 2 R ^ h y J 3 (3.13) Now, using the identity | a | 2 + , b | 2 = I a 4 | l i + J a = b j l ( 3 > l 4 ) it i s easily seen that + ^2 a n (* ^3 + 2^ 2 a r e Pos*t:*-ve definite quantities •1 2 2 1 2 (e.g. £ + ~P2 = I e 1-9 2l + I © 2—© 3 [ ~ 2"l 9i" 9 3l + positive quantities = 2"l e 1" 2 9 2 + 9 3 I + (positive quantities)). Now, since for any reasonably well behaved function g which vanishes on y = ±1 we have '1 2 2 (1 l s y l dy 1 ( f ) J , g [ 2 d y t ( 3 1 5 ) -1 -1 it i s easily shown that T^+t>2+?3 > (k 2+(^) 2) (Jj+JfJj > (k 2+(|) 2) (3.16) 119 and using (3.J4) W * 3 - * 2 ' ( 3 ' 1 7 ) 1 3 By considering the cases c r >^  U ' = maxCU^.U^) (note that this is max ^ 2 not necessarily greater than the maximum of IL, ) and c < U ! = min(U-,U0) Zo r — mm l j and using (3.13), (3.15) - (3.17) the following bounds are readily found for c r (the details are analogous to those of Pedlosky 1964a). U 1! 3 + min(6U . ,0) ^ 9 - V^*"'^  mm mm' 2 ( f c2 + ( | ) 2 ) 2R 0H 3 ( k 2 + ( | ) 2 ) , . min(h ,0) <c <U 1' 3 + max(6U ,0) - ~ h r o TT ( 3 ' 1 8 ) - r - max max 0 3 (k 2+(f) 2) In the case in which U 0 = (U1+U_)/2 , these bounds reduce to the same co i J form as found by Pedlosky (1964a) for the two-layer model. To derive a semi-circle theorem applicable to our model the real part of the equation corresponding to (3.11) is used. This equation can be arranged into the form: + io*2 + U ^ 3 " <cr+cl> W * 3 + 2 l 9 1 ^ 2 | 2 ( U r U 2 o ) 2 + ^ V ^ X " " / - | | e i - e 3| 2(U 1-U 3) 2 + B(U 1J 1+D 2 oJ 2-HJ 3J 3) + U 3(bh y/R 0H 3)J 3 (3.19) Further considerations w i l l be restricted to the case in which U„ = (U-+U„)/2 2o 1 3 (in any case where "P > 0 , i = 1,2,3 , this restriction is not necessary). Now, letting a = U , d = U . and using t,+-^P0 > 0 , > 0 max • mm 1 2 2 — 3 2 2 — we have: 120 0 > (ILj-a) (U^d) C^+f^) + (U3-a) (U3-d) (^3+^2) = U ^ + U 2 / f , 2 + U 3 1 > 3 " (a+d) (Vl"^  t 2 + U 3 1 ' 3 ) + a d ^ 1 + t > 2 + 1 > 3 ) + ( ( U r U 3 } / 2 ) ^ 2 2. 2, ( ^ 2 H * 3 ) + 8 ( V l + D 2 o J 2 + D 3 J 3 ) + U 3 ( b h y / R 0 H 3 ) J 3 . r , r r 3 , 2 , r r 3 / . ^ y j J + a d ( r ^ 3 ) + T ( U f D 3 ) * 2 - (a+d){c (P.+'P„+1'-)+T(J 1 +JO +JQ) +• (3 ,2 ) + i ( u r u 3 ) 2 , e r e 2 l 2 + |(v u 3 ) 2 , 9 2" 9 3l 2 _ i ( u r u 3 ) 2 , e i " 9 3 l 2 where ( 3 . H ) and (3.19) have been used in the second equality. Using ( 3 .14 ) and the definition of ? 2 ( c f . ( 3 . 1 2 ) ) the sum of the last four terms in ( 3 .20 ) is easily seen to be positive. Hence ( 3 .20 ) gives: a-d v. J (t 1+f 2+T 3) + e v 2 , (J1+J2+J3) - (UJ^+UJ J 2+u 3 J 3) \ + a+d — - - U 2 3 R Q H 3 h y J 3 '„ , 0 2 a-d CP1+l'2+£3) + ( J , + J 0 + J 0 ) + 2 j v- 1"-2 ,' J3' R Q H 3 Using ( 3 . 1 6 ) , we fina l l y have: max1 2 3 h y J 3 c } U +U . max min 2 + c 2 < 1 — f \ U -U . max min c — _ r I 2 J I 2 JB ( U -U . ) max mm k2+ ( f , 2 V 3 max i U +U , max min _ 2 U3 -1 k 2+(f) 2 (3.21) Using max U +U . max min 2 3 V <klJ -U . ) h J —2 max mm y max this can be replaced 121 by the convenient form: — U +U . max mm 2 + c 2 < f ~\ U -U . max mm c -r 2 c v 1 — 2 (u -u . ) + (3+(b/R H ) |h | ) m a X m n (3.22) 0 3 1 y'max r „ 0* k2+(f)2 Although (3.22) has only been derived for the case U = (U +U_)/2 , 2o 1 3 comparison with the corresponding relations found for the two-layer and continuous cases (Pedlosky, 1964a) suggests that i t i s probably valid in general. A proof of this result appears to depend on P , P 2 and P^ being independently positive, a proof of which I have been unable to produce. To find a bound on the growth rate which does not increase as 3 increases we again follow the work of Pedlosky (1964a). Multiplying the i ' * 1 equation * of (3. 8) by $j_<^j_ » integrating from y = -1 to y = +1 and taking the imaginary part of the sum of these equations we have: (1 c. i -1 l * l y l 2 + k X ' 2 - , i * 2 y | 2 + k 2 l * 2 l 2 , |4>3y|2+k2l<j,3l2 + 2[|<J>1-((,2|2+|<(>2-<(>3|2+j |<ri-2*2+*3|2]|dy = J ^ i ^ - U ^ K ^ - f ^ ) - ^ (U1-U3)((|.1*3-<|>1*3)+2i(U2o-U3)(*2<fr3-<|.2<r3)|dy •dy (3.23) ^l 2 -1 ^ ( * 1 V * l V ( V2y-*2*2y> + 1^ (*3 V * 3 V Now, using (3.24) and 122 2k|<j).<f>. | < k2|<f> |2 + |<f). |2  1 1 iy' — 1 i 1 1 ly 1 (3.25) we obtain the following inequality |kc.| £ 2(F 1+F 2)k 1 2o | max k 2 + ( f ) 2 , ( F l + F 3 ) k l U r U 3 f m a x , ^ k 2 + ( f ) 2 2(F 2+F 3)k U2o- U3 max k 2-Kf> 2] max U + 2X max + U 3y_ max (3.26) This method of bounding the growth rate has been interpreted by Pedlosky (1964a) as simply the bound obtained by majorizing the energy conversion terms due to vertical and horizontal shears. A f i n a l bound on the growth rate can be found from (3.8 ). Multiplying the i * " * 1 equation by <J>./F.(U. -c) , integrating from y = -1 to y = +1 , 1 1 10 summing, extracting the real part and using (3.5) we have: -1 A i |2 3 y F. i=l U. -c y i 1 io 1 dy = f 3 |*. | 2 + k 2 j < U 2 -1 i=l F. l + 2[U1-<f.2|2+U2-<03|2+|k1-2<l)2+<j)3|2]}dy (3.27) 3q. Since we have shown that, for this case, (U. -5—) „ > 0 for ins t a b i l i t y 10 <Jy max (cf (3.6)) the following bound is easily derived from (3.2 7) (kc,) 2 < (k 2/k 2 + ( ^ ) 2 ) ( T 4 J ) , 3y max (3.28) Finally, i f |U— c | >_ 6 (i.e. i f c r l i e s outside the range of U ) 123 this bound can be replaced by: (kc,) 2 < <k 2 /k 2 +(y ) 2 ) (uU> " < k 6 ) 2 ( 3 - 2 9 ) 1 — i. dy max (d) Marginally Stable Waves The condition for marginally stable waves whose phase speed l i e s in the range of the mean velocity of one or more of the layers is derived in exactly the same manner as for the two-layer case (cf Pedlosky, 1964b) and hence only the fi n a l result shall be given. It is 3 IO2 3q ± J, .t. i . r irwri^r--0 (3-30> i=l c r i t i c a l points I 1 ioy in the i f ch layer th A c r i t i c a l point in the i layer is defined as the point y c where U. (y ) = c . The result (3.30) i s identical in form to that corresponding io c r to a two-layer f l u i d . With the results of this section in hand we now proceed to consider the effects of mixed baroclinic-barotropic instability in the three-layer model. i 124 4. Baroclinic-barotropic i n s t a b i l i t y in the three-layer model (A numerical study) In this section we consider the three-layer model discussed above with = 0 and U,, = eU^ where = UQ(1-COS Tr(y+1)) . In many respects the work done here i s similar to that of section 2 on the two-layer model but one must bear in mind that the vertical shear between the lower layers is an important energy source for baroclinic i n s t a b i l i t y (especially near e = 1.0 ) . As in section 2 we are interested in the effect of varying e^l^/U^) and we look at this effect in the three cases, F.<<1,F.>>1 and F. ^  1 . 1 1 1 Cases I. II: F i « 1 and F ± >>1 (i=l,2,3) These cases may be studied in exactly the same manner as was done for the two-layer model. For F^ << 1 , 1=1,2,3 (horizontal length scale of the mean currents much smaller than the internal deformation radii) the three equations decouple to leading order in F^ , and for each layer there i s the possibility of barotropic ins t a b i l i t y . For F^ >> 1 , i=l,2,3 the main energy source for unstable waves i s the t i l t of the two interfaces, and the most unstable waves are essentially the same as the corresponding waves in the absence of horizontal shear (this has been verified by direct numerical studies). This case has been studied in some detail in chapter II. 125 Case III: F ± <\, 1 (1=1,2,3) This is the case of greatest interest here for i t is in this case that both baroclinic and barotropic energy sources are li k e l y to be important. To f a c i l i t a t e comparison with the work on the two-layer model in section 2 -4 -1 -2 the following parameter values were chosen. fQ = 10 s , 0 = 0.0 , g' =1.00ms (Note that this i s somewhat larger than for the two-layer case), H^=H2=H3= 3333.33m (Note that 11^ = 10 1cm as in the two-layer case) and L = 2000 km. The choice of g' is such that F^ (i=l,2,3) are a l l equal to the corresponding parameters in the two-layer model (i.e. F^=F2=F3=12). Since away from e = 0.5 i t was found that baroclinic energy sources strongly dominated the barotropic sources for these parameter values, L = 1,000 km was also considered in some detail. This i s of considerable importance in i t s e l f since i t indicates the fact that the baroclinic energy source i s apparently considerably stronger for the three-layer model than for the two-layer model. The explanation for this appears to l i e in the fact that the term involving U in the equation expressing the conservation of potential vorticity i s poorly estimated in a two-layer model but somewhat better estimated using the three-layer model. In fact i t was shown in chapter II that the two-layer model corresponds to the limit of the three-layer model as B^ 0 with U2=(U +U3)/2 (i.e. with the f i n i t e difference approx-imation to U in the middle layer set identically equal to zero), zz Approximate s t a b i l i t y boundaries for these two cases are given in figure 4.1. Several general features of these boundaries deserve some explanation. Possibly the most striking feature is the cusp at e = 0.5 . This feature has been discussed above and is due to the fact that the vertical - I.0L Figure A. l : Approximate stability boundaries in e vs. k space for the " -4 -1 three-layer model. f Q - 10 s , 8 - 0.0 , g' - 1.0 , H - H 2 - Hj - 3333.33 m.; L - 57.735ki,,- approximately vertical line; L • 1,000 km - dashed line; L - 2,000 km - solid l i n e . 127 curvature of the mean flow i s small here. The branches extending to large values of k near e = 0.0 and e = 1.0 have been explained by Davey (1977) in a study of pure baroclinic i n s t a b i l i t y (no horizontal shear) in a similar three-layer model. At these short wavelengths, the layers are only weakly coupled and a t i l t of one interface acts as effective topography for the two layers above or below as the case may be. The branches may thus be related to similar features of a two-layer model with topography. We also note that i n s t a b i l i t i e s extend to much shorter wavelengths in a three-layer model than they do in the corresponding two-layer model (compare figure 4.1 with figure 2.4). This feature i s expected since in the three-layer model, the minimum vertical scale of the motion i s two thirds that of the corresponding two-layer case, and we know that for baroclinic i n s t a b i l i t y to occur the vertical and horizontal length scales must be related such that, on the average, the slope of the flu i d trajectories in the y-z plane l i e within the"wedge of instability*(see Pedlosky, 1971; Hide and Mason, 1974). Finally, we note that the sta b i l i t y boundaries for L = 1,000 km and L = 2,000 km are nearly identical outside of a factor of two due to scaling with respect to the half-channel width. If the horizontal length scale had been chosen as the internal deformation radius for the system the two stab i l i t y boundaries would nearly overlap. This is due to the fact that by L = 1000 km, the short wave cut-off is dictated almost entirely by baroclinic instability with the short wave cut-off for barotropically unstable waves occurring at much smaller values of k . In order to study the effects of varying E we shall consider the three values e = 0.5, 0.75 and 1.0 for different values of k . The main emphasis w i l l be on the case L = 1,000 km but any significant 128 differences between L = 2,000 km and L = 1,000 km w i l l be pointed out. At k = 1.0, E = 0.5 the most unstable wave has changed from a layer-limited barotropic i n s t a b i l i t y at L<<r^ to a baroclinically unstable wave with substantial energy input from barotropic i n s t a b i l i t y at L = 1,000 km 2 ((L/r^) ^3) (figure 4.2). Due to the influence of barotropic i n s t a b i l i t y in the upper layer this wave i s substantially intensified in the upper 2 layer. Even at L = 2,000 km ((L/r^) vL2) where the net transfer of available potential energy is of order six times the net transfer of kinetic energy, the eigenfunctions of the most unstable wave are very similar to those in figure 4.2. Also present when L = 2,000 km is a baroclinically unstable wave which loses energy to the mean flow through the horizontal Reynolds stresses. This wave i s shown in figure 4.3. It is significant that this wave i s equally as strong in the upper and lower layers and considerably diminished in the middle layer. Also considered for E = 0.5 was the case k = 2.5 . At L = 1,000 km the most unstable wave had approximately equal contributions from baroclinic and barotropic energy sources with the major source of potential energy being the t i l t of the lower interface and the major source of kinetic energy being the shear of the mean currents in the upper layer as expected (figure 4.4). By L = 2,000 km the situation i s quite different (figure 4.5). The transfer of kinetic energy represents a loss from the perturbations in both of the upper layers and the energy source is baroclinic. The most unstable wave at k=1.0, E=0.75 has ^ =0.33:+ i 0.26 and i s shown in figure (4.6). Comparing this with figure 4.2 (e=0.5) we see that the phase speeds, growth rates and eigenfunctions for these two cases are very similar. The relative magnitudes of the net TAPE and TKE are also very similar, but their distribution i s quite different. At e = 0.5 (a) (b) (c) Figure 4 . 2 : (a)The mean potential vorticity gradients in the upper layer (solid l i n e ) ; the middle layer (long dashes); the lower layer (short dashes). (b)Transfer of available potential energy due to shear between the upper layers (solid line) and that due to shear between the lower layers (dashed li n e ) . (c)The transfer of kinetic energy in the upper layer (solid line) and in the middle layer (dashed l i n e ) . The transfer in the bott layer is zero since U-j-0. (d)The complex amplitude for the stream function in the upper layer. (e)The complex amplitude for the stream function in the middle^layer. (f)The complex amplitude for the stream function in the bottom layer. fo=10 s ~ l , 8=0.0, L= 1,000km, g'=1.00ms-2, ^ =^=^=3333.33m (F^F^F.^3) , U^l-cosfl (y+1) , U 2 = e U i » £ = 0 * 5 * Fip.urc 4 .3 : As i n f i g u r e 4.2 but L = 2,000 km. Figure 4.4: As in figure 4.2 - the i n s t a b i l i t y at larger k Figure 4.5: As i n figure 4.3 - the i n s t a b i l i t y at l a r g e r k ho 134 the source of potential energy i s almost equally distributed between the upper and lower interfaces whereas at e = 0.75 the t i l t of the lower interface i s the main source of potential energy. Also, with the increase in U2 the TKE in the middle layer i s substantially increased. Unlike the case e = 0.5 the most unstable wave at L = 1,000 km s t i l l remains at L = 2,000 km with relatively minor changes in phase speed, growth rate and eigenfunctions. However, as expected the relative magnitude of the TKE is considerably reduced compared to the TAPE (compare figures 4.6 and 4.7). One point of considerable interest i s clear from these figures (especially figure 4.7). Although the major source of energy for the perturbations is the t i l t of the lower interface, the wave i s intensified in the upper layers. In the absence of horizontal shear, the unstable wave here would be intensified in the lower layers. It appears that a_ relatively small TKE i s capable  of a_ very significant vertical redistribution of energy in the growing per- turbation. This behaviour has frequently been observed in this study and appears to be of major importance whenever the TKE represents a source of energy for the perturbations. This behaviour w i l l be observed again for £ = 1.0 in figure 4.10. Although the most unstable wave at L = 1,000 km is s t i l l present at L = 2,000 km i t no longer has the largest growth rate. The wave with the largest growth rate i s due to baroclinic i n s t a b i l i t y and loses very small amounts of kinetic energy (relative to the TAPE ) in both of the upper layers (figure 4.8). Now, i f one considers k = 3.5 (with e = 0.75 s t i l l ) , things are much as expected. We are well beyond the short wave cut-off for barotropic instability and the waves are not substantially influenced by the presence Figure 4.7: As in figure 4.2 but L - 2,000 km., t - 0.75 Figure 4.8: As in figure 4.7. 137 of the horizontal shear. As shown in figure 4.9 the most unstable wave for L = 1,000 km i s a bottom intensified baroclinically unstable wave extracting i t s energy primarily from the .t i l t of the lower interface. There is a small gain through the Reynolds stresses i n the upper layer, and a loss in the lower layer but neither of these appear to have any significant effect. The only major difference at L = 2,000 km, i s that both the phase speed and the growth rates are substantially increased (co = 3.06+i 1.94 for the most un-stable wave at L = 2,000 km compared to co = 2.35+i 0.72 at L = 1,000 km ). The energy transfer properties and eigenfunctions are not substantially altered although the eigenfunction i s sl i g h t l y less intensified i n the lower layers. The explanation for the seemingly large discrepancy in phase speed (we expect the growth rate to be increased) l i e s in the fact that we are comparing waves with the same non-dimensional wavelength when scaled with the horizontal length scale of the mean currents when the appropriate length scale is the internal deformation radius. In fact the phase speed of the most unstable wave at k =7.0, L = 2,000km i s very near that for k = 3.5, L = 1,000 km as i t should be. The eigenfunctions and energy transfer properties of these two waves are also in excellent agreement. Finally, we consider e = 1.0 . This case is in fact very similar to e = 0.75 and hence w i l l only be considered b r i e f l y . The figure corresponding to figure 4.7 (e = 0.75) i s given in figure 4.10. We see that the-only differences that occur are quantitative and due mainly to the increase in TKE in the middle layer. The figures corresponding to figures 4.6, 4.8 and'4.9 show similar agreement and w i l l not be reproduced. In figure 4.10 we see that the effect of the small TKE in redistributing the energy of the perturbation i s again surprisingly strong. In fact, the effect i s so strong that in spite of the fact that (a) (b) 13.22 UJ % 6.59 f--0.04 1.0 0.0 y 1.0 0.23 UJ ^-0.44 -1.10 / J J i J i - i .o 0.0 y i.o (c) (d) (O y y y Figure 4.9: As in figure 4.6. Figure 4.10: Aa In figure 4.# but I - 2,000 km., e - 1.0 . 140 TAPE a- 10 x TKE i t is s t i l l reasonable to refer to these waves as baro-tropically unstable waves modified by baroclinic effects. The f i n a l case of interest is the i n s t a b i l i t i e s associated with the branch extending to the right near e = 0.0 and e = 1.0 in figure 4.1. As mentioned earlier this branch has been explained by Davey (1977) as a result of the layers effectively decoupling and the t i l t of one or the other of the interfaces acting similar to bottom topography. No attempt was made to study these waves in detail as the growth rates of these waves were very small and i t would take a large number of cross-channel modes to properly resolve them. Studies with 10 symmetric cross-channel modes did however indicate that Davey's conclusions are correct. 141 5. Conclusions We have considered baroclinic-barotropic in s t a b i l i t y in two- and three-layer models. For the case of a mean flow with a short horizontal length 2 scale ((L/r/) << 1 ) i t i s shown that the study of barotropic i n s t a b i l i t y i s appropriate with the thickness of the layer in which the large horizontal shear occurs used as the fl u i d depth. From the proof given in section 2 i t is clear that this result holds for any region (layer) in which the condition 2 ( L / r . K « l holds. This result may be immediately extended to a continuously 2 str a t i f i e d f l u i d . The condition (L/r/) «1 simply states that the vertical s t a b i l i t y of the f l u i d is so large that the process of baroclinic in s t a b i l i t y i s inhibited (i.e. that the rotation rate is so small or the density stratification so strong that the isopycnal slopes are insufficient for significant amounts of potential energy to be released by the process of baroclinic i n s t a b i l i t y ) , however the process of barotropic in s t a b i l i t y involves only horizontal exchanges of fl u i d (and- vorticity) and hence i s affected l i t t l e by the vertical s t a b i l i t y . Thus in a region of high vertical s t a b i l i t y , "layer limited" barotropic in s t a b i l i t y i s possible. This fact i s very significant since averaging over the depth of the fl u i d may eliminate or greatly diminish the instability i f the large horizontal shears occur over a limited depth of the f l u i d , thus such averaging is not recommended in the study of cases where the process of barotropic instability is believed to be dominant. A better approximation 2 would be to consider the region in which (L/r/) « 1 separately. 2 Now, as (L/r/) increases the mechanism of baroclinic in s t a b i l i t y becomes increasingly important. Since baroclinic in s t a b i l i t y occurs on the scale of the internal deformation radius, and barotropic instability occurs on the scale of the horizontal shear of the mean currents, for 142 2 (L/r.) small, baroclinic i n s t a b i l i t y w i l l be limited to very large length A • scales (i.e. very small k = k L ). Further, the growth rates of such baroclinically unstable waves w i l l be small relative to those of barotropically 2 unstable waves in the system. However, for (L/r^) ^ 1 substantial baroclinic and barotropic i n s t a b i l i t i e s may occur at the same wavelengths and the interaction can be strong. The (basically) baroclinically unstable waves generally lose kinetic energy to the mean flow through the horizontal of Reynold's stresses while the barotropically unstable waves extract additional energy from the mean state through the process of baroclinic i n s t a b i l i t y . As expected, the main effect of the losses due to the Reynold's stress on the baroclinically unstable waves (both above and below the long wave cut-off for barotropic instability) i s to reduce the growth rates of the waves. It appears that the study of these waves may be reasonably undertaken by taking the shear between the layers to be an appropriate mean value across.the channel. However this approximation , eliminates the second class of i n s t a b i l i t i e s (the basically barotropically unstable waves) which we have seen may have 2 significant growth rates even for (L/r^) large due to their a b i l i t y to extract potential energy from the mean flow. A very interesting property of these waves is that their vertical energy distribution appears to be dictated primarily by the transfer of kinetic energy even when their dominant energy source i s the transfer of available potential energy. 2 For (L/r.) >>1 the barotropically unstable waves are limited to very long wavelengths (relative to those of the most unstable baroclinic -waves) where the growth rates are much smaller than for the shorter baroclinically 2 unstable waves. Thus for (L/r ) >> 1 the flow w i l l be essentially baro-c l i n i c a l l y unstable and may be studied by considering uniform shears across the 143 2 channel. It i s , however, important to note that the condition (L/r/) » 1 must be strongly satisfied before such a study i s j u s t i f i e d (the studies with the cosine jet suggest that this condition i s more accurately stated 2 2 as (L/r.) >> TT ) Finally, we mention the work of section 3 in which the three layer model of chapter II is further developed. The energy equation is discussed, necessary conditions for in s t a b i l i t y derived, bounds on phase speeds and growth rates are found and the condition for marginally stable waves with phase speed within the range of the mean currents is discussed. A l l of these results are essentially identical to the corresponding work done on the two-layer model (Pedlosky, 1964a,b) although the bounds on the phase speeds are somewhat weaker unless = (Uj+U^)/2 . The work done here suggests that in most cases of real oceanic flows the study of i n s t a b i l i t i e s should include the effects of both horizontal and vertical shears (as well as the effects of topography and 6 which have not been considered here). A l l of these aspects are considered in chapter IV in which the three-layer.model i s applied to real ocean situations. 144 Appendix a - Method of Solution We have already seen in chapter II how a simple analytical solution of (3.40) (with ((K = 0 on y = ±1 , i = 1,2,3) may be found for the case U. = constant io h = constant y Under these restrictions solutions are found in the form <|>^  = vu sin -^(y+l) with the P^'s being determined from the differential equations. In order to extend this method to more general cases the following expansions are made. U.(y) = u. + I u. cos -^(y+l) 1 x0 k=l \ U. (y) = uc. + I uc. cos ^(y+l) i y y X0 k-1 \ 2 lit bh /H. = hs. + Y hs cos -^ -(y+1) y 3 0 1=1 I 2 • i = I y i m s i n T ( y + 1 ) ^ m=l The second of these expansions permits the study of cases in which the differentiated series for the f i r s t case may be slowly or non-convergent. Substitution of (6.1) in (3.10) results in an i n f i n i t e dimensional matrix eigenvalue problem of the form: (4 - c S ) $ = 0 (A.2) where c i s the (complex) phase speed, A. and B_ are square matrices, and $ i s a column matrix containing the coefficients for the eigenfunction 145 expansions. The components of these matrices are given below. a 3 j - 2 , 3 m - 2 = - i ( < - F l ) ( u - u ) - S F ^ u -u ) + F ^ u -U3 ) |m-j I m+j |m-j I m+j |m-j | m+j 1 2 - T ( u c . -uc.. ) + (B-K u . -ua. +F_ ( u . - 6 u ~ + 2 u , ) )6 . 2 1'|m-j| 1 ' m + j m l 0 X0 1 ^ 20 30 m J a„. . . = 2 F ( u - u ) + 4 F u 6 3 j - 2 , 3 m - l 1 l | m _ . | 1 ^ . 1 1 Q mj a = -F,/2(u _ u ) _ F l u 6 . 3J-2,3m 1 1 ^ 1 1 Q mj a 3 j - l , 3 m - 2 " 2 F 2 ( u 2 o , . , - U 2 o > + 4 F 2 U 2 o n 6 m j J ' I m-j I m+j 0 a 3 j - l , 3 m - l - - I ( K m + 8 F 2 ) ( u 2 o , . f U 2 o > " I .|~UC2o > J |m-j| m+j |m-j| m+j - 3 F 2 ( U ; L - u - 2 u 2 + 2 u 2 + u 3 - u 3 ) |m-j I m+j I m - j I m+j [m-j | m+j + (B - ( K 2 + 8 F ) u „ -uc - 6 F ( u - 2 u +u ) ) 6 m 2 2 o Q 2 o Q 2 1 Q 2 Q 3 Q mj a.. = 2F_(u. -u_ ) + 4 F u 6 . 3 j - l , 3 m 2 2 o i .1 2 o , . 2 2 o _ mj J |m- j I m+j 0 aT.l C m O = - F Q U 0 6 m 4 - F /2(u, -u, ) 3 j , 3 m - 2 3 3 Q mj 3 3 j m _ . | 3 ^ . a.. « = 4F„u, 6 + 2F (u -u ) 3j,3m-l 3 3 Q mj 3 3 , ^ 3^. = -K 2/2(u 0 -u 3 ) - j ( u c 3 -uc 3 ) 3 j , 3 m m' v 3. . 1 3 . . ' 2 3i .1 3 . . J ' [m-j I m+3 |m-j [ m+j 1 1 3 11 m-j I V l 2|m-j| 2m+j 1 3|m-j| 3m+j + (B-K2u_ -uc, +2F„(u +3u +^u )+hsQ/R ) 6 + (^-) (h -h ) m 30 30 3 X0 20 2 30 ° ° m J 2 R o S|m-j| Sm+j 146 b,. „ , . = -(K2+3F.)6' . 3j-2,3m-2 m 1 mj b-. 0 „ = 4F.5 . 3j-2,3m-l 1 mj b«. 0 „ = -F..6 . 3j-2,3m 1 mj b _. - „ 0 = 4F„6 . 3j-l,3m-2 2 mj b T i o i = -(K2+8F0)<S , 3j-l,3m-l m 2 mj b-. n Q = 4F_6 . 3j-l,3m 2 mj b3j,3m-2 = ~ F3 6mj b~. \ , = 4F0<5 . 3j , 3m-l 3 mj b_. _ = -(3F_+K2)6 . 3j , 3m 3 m mj $3m-2 = u l m $3m-l = y2m $3m = y3m where K = k + (-—) m L U2o. = ( 6 u 2 j - U l j - U 3 j ) / 4 U C2o. = (6uc 2.-uc 1.-uc 3.)/4 This system was approximately solved by truncating the eigenfunction expansions at a point where the eigenfunction expansions appeared to have settled down. For the most unstable waves, only a few modes were generally required although the number varied with the form of the mean currents. In this paper, only symmetric velocity profiles are considered and we have 147 taken h = 0 . Under these restrictions, the solutions of (3.10) are either symmetric or anti-symmetric i n y (since the coefficients i n 3.10 are then even functions of y ) and these two classes of solutions can be considered separately (at a considerably reduced expense). Since the symmetric modes were generally the most unstable, only these modes are discussed. In practice, ten symmetric modes were used in a l l the calculations presented in 10 (2m-l)ir this paper (i.e. $. was approximated by <j>. = \ u. sin ^ (y+1)) • m=l 2m-1 This i s considerably more than is generally required. Finally, we note that for cases such as those presented in figures 2.2 and 2.3, the value of c at the short wave cut-off i s known exactly from the work on marginally stable waves ( 3(d)). In this case i t is preferable to f i x 2 c in (3.10) and solve the resulting eigenvalue problem with k as the eigenvalue. This eigenvalue problem is simply expressed in terms of the definitions given by (A.3). It is given by: (I - k 2B)£ = 0 2 IUTT 2 where A and B" are given by the following. F i r s t , set K = (—) = = m z TT 2 2 instead of (^r-) + k in (A.3). Then A = A -c B and ¥ is given by 2 — — s— — b ± , = 0 ( i * j) 3j-2,3m-2 2 3-1m_j | V j 1 Q s mj b3j-l,3m-l = I(u2o, .rU2o^? + ( u2o " Cs ) 6mj J |m-j I m+j 0 3j,3m 2 3|m_.| 3^. 3 Q s m3 Finally the approximate s t a b i l i t y boundaries presented in figures (2.4) 148 and (4.1) need some further discussion. The method used in this paper to study unstable waves is not well suited to the calculation of s t a b i l i t y boundaries due to the singular nature of the governing differential equations (and hence the resulting eigenvalue problem) for c^=0 (or SO). For this reason we stress that while we may use the method employed here to study the most unstable waves, the s t a b i l i t y boundaries calculated by this method are qualitative in nature and small details should not be taken seriously. In fact the unstable wave associated with the branches extending to the right in both figures 2.4 and 4.1 have very small growth rates and are not very well resolved. Due to the inherent qualitative nature of these boundaries, figure 4.1 was calculated using only 3 symmetric modes and checked at several points with 10 modes for qualitative accuracy (the results agree quite well). In spite of the qualitative nature of these figures, they do contain much useful information. 149 Appendix b In this appendix i t is shown that the minimum value of U q at which ins t a b i l i t y occurs in a two-layer model with = U^Cl-cos ir(y+l)) , TJ^  = 0 is precisely the minimum value for which q y changes sign somewhere. Since the phase speed at the high wavenumber cut-off i s given by c = U^(yg) where y is given by q 1 (y ) = 0 and the low wavenumber cut-off S -L S y consists of retrograde waves the minimum value of U q for which in s t a b i l i t y occurs must have c = 0 . The equation for the second layer then gives <f>2 = 0 (i.e. we are essentially considering a case of barotropic instability) and the upper layer equation reduces to U l [ ( f > l y y - k 2 < t , r F l * l I + * l [ 6 - U l y y + F l U l 1 = 0 2 It is now easily verified that for U = 0/TT (the minimum value for o which q = 0 somewhere and also precisely that value which makes q^ <* ) ^ 2 2 ^ <$>^  = A sin^y- (y+1) i s a solution for k = IT - (—-) . Clearly only the f i r s t mode can be unstable and the value of k corresponding to the minimum value of U_ on the sta b i l i t y boundary i s k = (/3/2)TT . CHAPTER IV OCEANIC APPLICATIONS 150 151 1. Introduc t i on Energetic current fluctuations with periods longer than a day have been observed i n many regions of the ocean during the past decade. Probably the most accepted explanation f o r the presence of these fl u c t u a t i o n s i s that p o t e n t i a l energy (available from the t i l t of the isopycnals i n a r o t a t i n g s t r a t i f i e d f l u i d i n the presence of v e r t i c a l shear) may be released to perturbations at these frequencies by the process of b a r o c l i n i c i n s t a b i l i t y (Charney (1947) , Eady (1949), Green (1960), Pedlosky (1964)). A second possible energy source f o r these fl u c t u a t i o n s i s the k i n e t i c energy of the mean currents made ava i l a b l e by h o r i z o n t a l shear i n these currents. When t h i s i s the p r i n c i p l e source of energy f o r the perturbations the mechanism by which energy i s extracted i s known as barotropic i n s t a b i l i t y (Lin (1945), Kuo (1949, 1973), Drazin and Howard (1966)). Using the model developed i n chapter I I , both of these energy sources are included i n a study of the s t a b i l i t y of the current systems i n Juan de Fuca S t r a i t and the northern region of the C a l i f o r n i a Undercurrent (off Washington and Vancouver Island). In each case, i t i s found that the major source of energy i s the p o t e n t i a l energy due to sloping isopycnals and that the main influence of h o r i z o n t a l shear i n the mean currents i s to l i m i t the region i n which the v e r t i c a l shear i s such that energy i s released to the perturbations. Mysak (1977) has studied pure b a r o c l i n i c i n s t a b i l i t y i n a two-layer model of the C a l i f o r n i a Undercurrent and found r e s u l t s consistent with observations. Through a study of the s t a b i l i t y of a three-layer system in c l u d i n g h o r i z o n t a l shear and non-constant bottom slope, the r e s u l t s of Mysak's study are extended and several questions r a i s e d i n h i s paper are examined. In p a r t i c u l a r we f i n d that replacing the upper layer of strongly s t r a t i f i e d 152 flu i d with a ri g i d l i d i s valid in the study of the waves examined by Mysak. However, this approximation f i l t e r s out a very important class o f i n s t a b i l i t i e s which extract potential energy from the t i l t of the upper interface, and i t appears likely that this class of in s t a b i l i t y may be responsible for the observed wave-like perturbations in the region of the California Undercurrent. 153 2. Juan de Fuca Strait Juan de Fuca Strait consists of two basins separated by an effective s i l l extending southward from Victoria at a depth of about 100 m (see figure 2.1). Although we w i l l principally concentrate on the western basin, we w i l l find that the most unstable waves in the s t r a i t are strongly surface intensified and hence the s i l l probably does not act as a barrier to these waves. The western basin i s approximately 20 km across and 90 km long with a relatively uniform rectangular cross-section throughout i t s length. The most complete measurements of the currents i n this basin have been presented by Fissel and Huggett (1976), Fissel (1976), Cannon and Laird (1978) and Holbrook and Halpern (1978) . The reports by Fissel and Huggett, and Fissel are concerned with the same set of measurements collected in the period from late May to mid-July 1975. The positions at which these measurements were taken are shown in figures 2.1 (circles) and 2.2. Figure 2.2 also includes the values of the mean long-st r a i t currents during the measurement period. The average density profile across this cross-section, measured during a cruise on June 14-15, 1975 is indicated in figure 2.3 together with the approximate density str a t i f i c a t i o n corresponding to the model developed i n chapters II and III of this thesis. The approximation i s clearly quite reasonable. A more complete set of measurements of the mean currents than those given in figure 2.2 (appropriate to the nearby cross-section indicated by the x's in figure 2.1), measured during the same season (June-August 1977), has been presented by Cannon and Laird (1978) and is reproduced in figure 2.4. These two sets of mean current measurements appear to be 154 Figure 2.1: (top) Juan de Fuca S t r a i t and nearby geographical features, (bottom) Juan de Fuca S t r a i t with p o s i t i o n s at which F i s s e l ' s data (•) and one section of Cannon and Laird's data (x) were c o l l e c t e d . (from Cannon and L a i r d , 1978) SHER1NGHAM PI PILLAR PT. CANADA USA. DISTANCE IN KMS. Figure 2.2; The cross-section of Juan de Fuca S t r a i t at which F i s s e l ' s data were collected. Numbers above each station are used to indicate the station considered i n figure 2.5. Along-channel v e l o c i t i e s are given i n cm s . The dashed l i n e indicates the topography used i n H e model of the s t r a i t , (from F i s s e l , 1976) Figure 2.3; The mean density profile on June 14-15, 1975 across the section at which Fissel's data was collected (from Fissel and Huggett, 1976) 157 Figure 2.4; Along-channel t o t a l - r e c o r d average currents (cm/sec.) through the cross-section marked by x's i n figure 2.1 (from Cannon and L a i r d , 1978) 158 reasonably consistent and hence i n our studies we s h a l l use the more complete set of values presented i n figure 2.4. F i s s e l (1977) defines the r e s i d u a l currents as the flow remaining a f t e r currents with frequencies greater than 0.8 cycles per day have been removed. His analysis shows that the r e s i d u a l currents i n Juan de Fuca S t r a i t between Sheringham Point and P i l l a r Point have the following properties: (1) 80% of the t o t a l variance of the r e s i d u a l currents i s concentrated at frequencies less than 0.25 cycles per day, (2) the amplitude f l u c t u a t i o n s i n the u p - s t r a i t component of the r e s i d u a l currents are larger by a factor of two than those of the cross-s t r a i t currents, (3) the amplitude f l u c t u a t i o n s of the r e s i d u a l currents at 20m depth are considerably larger (by a factor of two or three) than the fl u c t u a t i o n s at 120 m depth, (4) the r e s i d u a l u p - s t r a i t currents are poorly correlated at the lowest frequencies where most of t h e i r a c t i v i t y occurs. Even f l u c t u a t i o n s of the current at p a i r s of stations that are adjacent to one another (with a t y p i c a l separation of 4km) are not con s i s t e n t l y correlated at the 90% si g n -i f i c a n c e l e v e l . V e r i f i c a t i o n of the f i r s t three statements above can be found i n figure 2.5 which gives the s p e c t r a l d e n s i t i e s of the r e s i d u a l currents measured at the various stations shown i n fig u r e 2.2. The fourth statement i s discussed further at the end of t h i s s e c t i o n . The p r i n c i p a l subject of t h i s section i s the study of the s t a b i l i t y of the mean currents to low-frequency quasi-geostrophic wave perturbations. The model used, to study the s t a b i l i t y of the mean currents 159 Figure 2.5; The power spectral density of the residual currents for the current meter stations of F i s s e l (1976). Stations 130-136 are shown in figure 2.2 and station 137 i s the additional station a few kilometers to the west of Sheringham Point seen i n figure 2.1 (bottom), (from F i s s e l , 1976). F i g u r e 2 .6: (a) Approximation t o the mean c u r r e n t s used t o model Juan de Fuca S t r a i t ; (b) the mean p o t e n t i a l v o r t i c i t y g r a d i e n t s corresponding to the three l a y e r s of our model: upper l a y e r - s o l i d l i n e , middle l a y e r -long dashes, lower l a y e r - s h o r t dashes; (c) the e i g e n f u n c t i o n corresponding to the most unstable wave i n t h e model; (d) the t r a n s f e r of a v a i l a b l e p o t e n t i a l energy corresponding to t h i s wave: the s o l i d l i n e corresponds to the t r a n s f e r of energy due to the shear between the upper l a y e r s and the dashed l i n e to t h a t due to the shear between the lower l a y e r s ; (e) the t r a n s f e r of k i n e t i c energy f o r t h i s wave: s o l i d l i n e corresponds to the upper l a y e r , long dashed l i n e t o the middle l a y e r and the s h o r t dashed l i n e t o the bottom l a y e r . Table 2.1: Properties of the most unstable wave found i n the three-layer model of Juan de Fuca S t r a i t . This wave i s i l l u s t r a t e d i n figure 2*6. (A p o s i t i v e value of c or c g corresponds to an eastward v e l o c i t y . ) Model U ,U ,U Period Wavelength e-folding Phase Group 1 2 3 (days) (km) time v e l o c i t v v e l o c i t y (x 16 cm sec" 1) (days) (km day"*) (km day - 1) Figure 2.6 figu r e 2.7a 13.7 76 6.6 -5.5 -6.8 162 in Juan de Fuca Strait to propagating quasi-geostrophic perturbations has been derived i n Chapter II, section 2. A three-layer model was derived from the equations of motion for a continuously strat i f i e d f l u i d with densities of the upper and lower layers given by and respectively and p * varies linearly from p*^  to p"3 over the middle layer in the absence of motion (the basic state density approximation is shown in figure2.1.chapter II). There i t was found that the linearized equations^ expressing the conservation of potential vorticity for the model described above are: ( U l o _ c ) [hyy ~ k \ " ^ ( 3 ^ - 4 ^ + ^ ) ] + ^ = 0 ( U2o- c ) [*2yy " ^ 2 + 4 P 2 ( * r 2 * 2 + * 3 ) 1 + *2q2y = ° ( 2 ' 1 } H -bh 2  ( U3o- 3 ) [ (-H7- ) (*3yy- k *3} ' F 3 ( * r 4 * 2 + 3 * 3 } 1 + * 3 q 3 y = ° where q, = 3 - U. + 2F ( 2 U . - 3U_ + U ) ^l y lyy 1 1 2 3 q^ = 8 " U, " 6F„(U -2U.+U ) 2y 2yy 2 1 2 3 H -bh q 3 y = ( _ L _ ) ( g - u ^ ) + 2F 3(U 1-3U 2 +2U 3) + (b/R 0H 3)hy 163 In the study made here and i n that made i n the next s ec t ion the e f f e c t of 3 i s very small and hence i t i s neg lec ted . I n . f a c t , as pointed out by D r . Roland de Szoeke (pr ivate communication), t h i s i s probably genera l ly true for the study of b a r o c l i n i c i n s t a b i l i t y i n the oceans s ince the e f f ec t of 3 i s not l i k e l y to be s trong ly f e l t on the scale of the i n t e r n a l deformation rad ius of the oceans. Th i s i s , of course , not true for atmospheric flows where the i n t e r n a l deformation radius i s much l a r g e r . The bottom topography and dens i ty s t r a t i f i c a t i o n used i n our model are shown i n f igures (2. l) and (2. 3). The s t ruc ture of the mean currents (averaged over each l ayer ) are modelled by the approximation shown i n f i g u r e (2.fca) . A l so shown i n t h i s f i gure (part (b)) are the mean p o t e n t i a l v o r t i c i t y g r a d i e n t s . The form of the p o t e n t i a l v o r t i c i t y gradients i s of great i n t e r e s t i n the study of e i t h e r b a r o t r o p i c or b a r o c l i n i c i n s t a b i l i t y . I f the p o t e n t i a l v o r t i c i t y grad ient changes s i g n w i th in a given layer then the p o s s i b i l i t y of b a r o t r o p i c i n s t a b i l i t y e x t r a c t i n g i t s energy from the k i n e t i c energy of the mean currents i s in troduced . A change of s ign i n going from one layer to another introduces the p o s s i b i l i t y o f b a r o c l i n i c i n s t a b i l i t y i n which energy i s extracted from the a v a i l a b l e p o t e n t i a l energy due to the t i l t of an i n t e r f a c e . As can be seen from the f igures showing the t r a n s f e r of a v a i l a b l e p o t e n t i a l energy ( T . A . P . E . ) and the t r a n s f e r of k i n e t i c energy ( T . K . E . ) f or the most unstable waves found ( f igure 2.6) the major source of energy i s p o t e n t i a l energy and i t s re lease i s centred on the reg ion i n which q n ^ * q i s l arge and negative (- .7 < y < . 3 ) . 164 The T.K.E. i s very s m a l l but i t i s i n t e r e s t i n g t h a t i t s l a r g e s t e f f e c t i s a l o s s o f energy i n the lower l a y e r due to the s t a b i l i z i n g e f f e c t o f topography. The e i g e n f u n c t i o n s corresponding t o the most unstable wave i n t h i s system are shown i n f i g u r e 2.6(c). I t i s i n t e r e s t i n g t o note t h a t the t r a n s f e r of p o t e n t i a l energy i s s t r o n g l y i n t e n s i f i e d i n the r e g i o n o f maximum v e r t i c a l shear (see F i g . 2.6(a) and (d)) i n s p i t e of the f a c t t h a t the channel i s only about two i n t e r n a l deformation r a d i i i n wid t h . Hence averaging the mean c u r r e n t s over a couple of i n t e r n a l (Rossby) r a d i i of deformation i s not j u s t i f i e d i f l a r g e h o r i z o n t a l shears are present. F u r t h e r , i t i s i n t e r e s t i n g t h a t w i t h h o r i z o n t a l shear i n c l u d e d , the s t a b i l i t y of the system was found t o be r e l a t i v e l y i n s e n s i t i v e to small changes i n the mean c u r r e n t s . A case i n which the s t r a t i f i c a t i o n and mean c u r r e n t s of the flow are r e t a i n e d as i n f i g u r e s 2.3 and 2.6 but topography i s t o t a l l y n e glected ( i . e . the bottom i s assumed f l a t ) was a l s o s t u d i e d . The r e s u l t s f o r the most unstable wave were not s i g n i f i c a n t l y d i f f e r e n t from those given i n f i g u r e 2.6 and t a b l e 2.1, which g i v e s the prop-e r t i e s of the most unstable wave i l l u s t r a t e d i n f i g u r e 2.6. These r e s u l t s are encouraging as they i n d i c a t e that the changes i n mean cu r r e n t s and bottom topography which occur over the l e n g t h of the channel may be reasonably neglected. Note, however, that although t h i s i s the case f o r the study made here, i n the general s i t u a t i o n these r e s u l t s w i l l not always be t r u e . (eg. i f the main source of energy was the t i l t of the lower i n t e r f a c e , e f f e c t s of bottom topography would be s i g n i f i c a n t ) . Table 2.1 gives the p r o p e r t i e s of the most unstable wave found i n Juan de Fuca S t r a i t . Though the observations are s e r i o u s l y l i m i t e d , we see 165 that the predictions of our model are at least consistent with what is known. The period i s in the right general range, and the wavelength is such that the ratio of long-strait to cross-strait energy i s of order four, as observed The e-folding time and group velocity are such that the perturbation v e l o c i t i e-fold in the time that the group travels about 45 km (i.e. about two channel widths). Hence the growth rate i s certainly sufficiently large to have significant influence on the low-frequency motions in the s t r a i t . Finally, from figure 2.6(c), we see that the perturbation i s predicted to be surface intensified as observed. We thus conclude that the low-frequency motions observed in the strait may be at least partially due to baroclinic i n s t a b i l -ity of the mean currents, however, we have not yet explained the observed lack of coherence between the stations. It seems li k e l y that i t i s due to several processes occurring simultaneously. Some po s s i b i l i t i e s are: (1) influence from low-frequency motions in the Strait of Georgia, (2) influence of varying outflow from Puget Sound and Frazer River, (3) effects due to the proximity of boundaries for the near-shore stations, (4) wind forcing over the ocean causing a "piling up" of water along the coast creating an adverse pressure gradient in the upper layer, (5) the diffraction of shelf waves or other waves propagating up the coast (which may be generated many kilometers to the south) into Juan de Fuca Strait, (6) the effects of non-linearity and geostrophic turbulence, (7) the effects of strong "high-frequency" (u) ~ f) tides. The most significant of these are probably the last four. Cannon et a l (1978) have found that, during the winter, deceleration of the along-channel 166 currents generally occurred during strong southerly winds o f f the coast and during i n c r e a s i n g sea surface height at Neah Bay. They found that s i g n i f i c a n t c o r r e l a t i o n s e x i s t e d between: (1) the along-channel 4-m currents at S i t e A (Fig. 2.1) and the north-south P a c i f i c winds with currents lagging winds by 42 hours (process (8)); (2) the along channel 4-m currents at S i t e A and the sea surface height at Neah Bay with currents lagging sea-surface height by 6 hours (process (9)); and (3) the north-south P a c i f i c winds and sea surface height at Neah Bay with sea-surface height lagging winds by 24 hours (the winds are apparently e i t h e r generating or r e i n f o r c i n g wave motions on the s h e l f ) . These observations c e r t a i n l y support the assumption that motions on the s h e l f and winds over the open ocean have s i g n i f i c a n t influence on the currents i n the S t r a i t . Without doubt the e f f e c t s of n o n - l i n e a r i t y are also strong. The study made here assumes that the perturbations are much smaller than the mean currents and p r e d i c t s that these small perturbations w i l l grow at the expense of the p o t e n t i a l energy of the t i l t i n g isopycnals. However, when the perturbation v e l o c i t i e s become comparable with the mean v e l o c i t i e s (observations i n Juan de Fuca S t r a i t show that perturbation v e l o c i t i e s ~ mean v e l o c i t i e s ~ 20 cm s ^ ) the governing equations f o r the "perturbations" become highly non-linear and the e f f e c t s of geostrophic turbulence probably dominate the flow. Though the e f f e c t of strong n o n - l i n e a r i t y on a r e a l i s t i c channel flow i s not w e l l understood i t i s c e r t a i n that some or a l l of the low- (and high-) frequency waves w i l l i n t e r a c t and tend to i n i t i a t e an energy cascade to-167 ward larger scales (Charney, 1971). Due to the proximity of boundaries, the studies of geostrophic-turbulence in the open ocean are not directly applicable since they rely strongly on the assumption of isotropy. However, the basic dynamics are probably similar and interactions are undoubtedly strong. Such strong non-linear interactions almost certainly act to decrease the coherence between stations. Finally we must consider the possible effects of strong, "high-frequency" (diurnal and semi-diurnal) t i d a l currents. The inclusion of ti d a l currents i n our model introduces some interesting effects. The Rossby number corresponding to the tides in the channel i s not small and the scaling used in our model must be revised. The result i s that we must consider the effect of an imposed high-frequency ageostrophic wave on the low-frequency geostrophic waves studied in this thesis. To study this problem, the f u l l ageostrophic equations must be considered and no simple solutions appear possible. Rao and Simmons (1970) have shown that i n s t a b i l i t y can occur as a result of a coupling between an internal gravitational mode (an ageostrophic wave) and a rotational wave (the high-frequency analogue of the geostrophic waves studied here). However, this i n s t a b i l i t y occurs at much higher frequencies than those considered in this thesis and gives l i t t l e insight into the above mentioned problem other than to demonstrate that interaction i s possible. Since the t i d a l frequencies are much greater than those of the low-frequency quasi-geostrophic i n s t a b i l i t i e s studied here, i t has been assumed that their effect on these i n s t a b i l i t i e s i s negligible. However, this i s by no means obvious and a more complete study including ageostrophic effects would certainly be enlightening. 168 We thus conclude that although baroclinic i n s t a b i l i t y i s probably significant in Juan de Fuca Strait, i t i s not l i k e l y the dominant energy source for the observed low-frequency motions. It seems quite l i k e l y that the effects of motions on the continental shelf are of at least equal importance and that the effects of tides are also strong. Finally, due to the large "perturbation" velocities in the s t r a i t , the effects of non-linearity are certainly important and must be included before any firm conclusions can be made. 169 3. The California Undercurrent off Vancouver Island The California current is a broad eastern boundary current off the west coast of Canada and the United States formed at mid-latitudes where the eastward flowing Subarctic Current (west wind drift) separates into northern and southern components. The northern component forms the Alaska Current while the southern component forms the California current. At intermediate depths beneath this weak (~ 5-10 cm s \ , Tabata (1975), Reed and Halpern (1976), Bernstein, Breaker and Whritner, 0.976), .Halpern, Smith and Reed (1978)) current, a strong (mean speed of order 10 cm s with maximum speeds as high as 100cm s * ; Tabata, 1975), narrow (~ 50 km ) poleward flowing jet known as the California Undercurrent occurs. The major features of this current system have been discussed by Mysak (1977) and w i l l be summarized here. The undercurrent extends a l l the way from California up to Vancouver Island and i t s water properties suggest that i t has i t s origin in the North Equatorial Counter Current (Tabata, 1975). Off California i t consists of approximately an equal mixture of Pacific Equatorial water and subartic water, but further north the percentage of equatorial water is significantly reduced. Off California where extensive measurements have been made, meso-scale eddy-like formations have frequently been observed through the analysis of hydrographic data (eg. Wooster and Jones, 1970; Wickham, 1975). Further, the formation of these eddies has been observed through the use of satellite-borne infrared scanners (Bernstein, Breaker and Whritner, 1977). Though our primary interest i s with the flow off Vancouver Island, the analysis produced below may also be used to explain the presence of these eddies. 170 Off Vancouver Island, the data base i s f a r more l i m i t e d than o f f C a l i f o r n i a ; however, as seen i n figure 3.1 such mesoscale eddies as those observed further south are again observed here. We s h a l l show that, as f i r s t suggested by Mysak (1977), the presence of these eddies may be explained i n terms of b a r o c l i n i c i n s t a b i l i t y of the undercurrent. The presence of the undercurrent i n these northern regions i s now f a i r l y w e l l established. I t has been observed over the slope and s h e l f o f f Oregon by Huyer (1976), who suggests i t may be a part of the wind-induced coastal upwelling regime. (It had e a r l i e r been shown (Pedlosky, 1974) that a deep topographically c o n t r o l l e d poleward undercurrent i s part of the steady-state response of a wind-driven flow i n a r o t a t i n g s t r a t i f i e d f l u i d i n a channel with a sloping bottom.) Halpern et a l (1978) have also noted the existence of a northward flow over the depth range of 200-500m over the slope o f f Oregon i n the f i r s t h a l f of July. Dodimead et a l (1963) found a northward flow during winter i n the same depth range, and Ingraham (1967) observed that within about 200 km of the shore there was a net northward volume transport r e l a t i v e to 1500 m. Over the depth range of 200-500m the maximum current speeds were estimated to be 10-20 cm s F i n a l l y , Reed and Halpern (1976) and S. Tabata (see Mysak 1977) have found a northward flow during early f a l l at intermediate depths i n the l a t i t u d e range 46-50°N. Maps of geopotential topography at four l e v e l s o f f Washington and the southern part of Vancouver Island as computed by Reed and Halpern (1976) are shown i n figure 3.2. The 10/1000 db map shows a weak (~ 5cm s - 1 ) southward flow pver the continental slope which i s apparently considerably reduced as one moves further from the coast. The remaining maps a l l show 171 Figure 3.1: Enhanced infrared image of sea surface showing s p a t i a l structure of surface temperature on 10 September, 1975 off the west coast, of B r i t i s h Columbia and Washington. The dark areas are warm water and the grey-white, cold water (after Gower and Tabata, 1976.) Figure 3.2: Geopotential topography (dyn m) of the 10, 150, 300 and 500 db surfaces (referred to 1000 db), 7-20 Septebmer, 1973. Open c i r c l e s refer to time-series stations. The 100 and 500 fathom (1 fathom = 1.829m.) isobaths are also shown ( from Reed and Halpern, 1976). 173 a northward flow over the slope region with a southward flow further west. The 150/1000 and 300/1000 db maps are nearly i d e n t i c a l while the 500/1000 db map shows a considerably reduced flow. From Reed and Halpern's estimates of the volume transports of the 150/1000 db layer o f f Vancouver Island, Mysak (1977) has estimated the maximum speed of the undercurrent i n t h i s region to be of the order of 15-20 cm s The mean flow over the slope region o f f Washington and Vancouver Island can thus be described (at l e a s t to a f i r s t approximation) as a weak southward flow of order 5 cm s ^ i n the upper 200m (decreasing slowly to the west) with a stronger narrow poleward j e t with speed of order 10 cm i n the approximate depth range of 200-600m. Beneath t h i s depth, the flow i s assumed to be r e l a t i v e l y quiescent and i n our model we s h a l l take the mean flow to be i d e n t i c a l l y zero at such depths. Mysak (1977) has noted that the low frequency eddy-like motions discussed above are l i k e l y due to an i n s t a b i l i t y of the mean flow i t s e l f to quasi-geostrophic perturbations at the observed length and time scales. Since, i n the upper layer, the mean currents are small and the density s t r a t i f i c a t i o n strong (see figure 3.4), Mysak assumed that t h i s layer acts e f f e c t i v e l y as a r i g i d l i d on the flow below and modelled the undercurrent by the two layer system shown i n figure 3.3. The r e s u l t s of h i s analysis do indeed i n d i c a t e the p o s s i b i l i t y that the undercurrent i s b a r o c l i n i c a l l y unstable. However, as Mysak noted, the v e r t i c a l shear i n t h i s upper layer could be s i g n i f i c a n t and thus the neglect of t h i s surface current could be a serious omission. We are thus led to consider a three-layer model. Mysak has also noted the possible errors i n introducing an a r t i f i c i a l outer v e r t i c a l wall and neglecting l a t e r a l shear i n the mean currents. I s h a l l show that both of these approximations are j u s t i f i e d i f the 174 Z . t 500 m I A 2000 m. Strongly Stratified Region Approximated by Rigid Lid / ' / / / / / / / / / / / / / Pt •» 1.027 gma. cm: 3 U, ="10 cms. »: 1 P 2 — 1.0275 gma. cm."3 u. = 0 cms. ». slope = I.34 X 10 75 km. Figure 3.3: The two-layer model studied by Mysak (1977) 175 position of the outer wall i s chosen correctly. In figure 3.4 a typical set of temperature, salinity and a profile in the slope region off Vancouver Island i s shown. Clearly the three-layer model of the density stratification considered in chapters II and III of this thesis i s not appropriate and we are led to consider the approximation indicated by the dashed line in figure 3.4. Following the method described in chapter II we find that the linearized equations expressing the conservation of potential vorticity in the three-layer model corresponding to this system are: ( U l o - c ) [ ? l o y y - k 2 5 l o + 8 F l ( 5 2 - 5 l o ) / 3 ] + ^ l o x ^ l o y y ^ V V ^ ^ = 0  ( V c ) [ ? 2 y y " kV 8 F2 ( 5lo^2 ) / 3 + *2RlW] + 52x [ ^ 2 ^ 2 ( U l o " U 2 ) / 3 + F 2R(U 2-U 3)] = 0 H -bh H -bh (U 3 - c ) [ ( - | — ) ( 5 3 y y - k 2 ? 3 ) - R F 3 ( 5 3 - C 2 ) ] + 5 3 x [ ( — ) ( B - U 3 y y ) + F 3 R ( U 3 - U 2 ) • b hy] = 0 R H_ o 3 (3.1) with a l l definitions and non-dimensionalizations exactly as in Chapter II. section 2. R is defined by: R = (P 2-P 1)/(P 3-P 2) O - 2 ) and U = (9U -U )/8 . £, is the (non-dimensional) perturbation stream lo 1 2 1 function in the. i * " * 1 layer and the subscript o indicates that the quantity is evaluated at the middle of 176 31.00 32.00 SALINITY 33.00 34.00 35.00 36.00 i 1 1 1 1 TEMPERATURE 200 4.00 6.00 8.00 10.00 12.00 14.00 0' & 1000 22.00 23.00 24.00 25.00 26.00 2700 28.00 SIGMA-T Figure 3.4: Temperature, s a l i n i t y and o f c p r o f i l e s , 10 September, 1973, at 49°N, 127° 19'W (right-hand open c i r c l e i n top l i n e of time series stations shown i n F i g . 3.2). (from Holbrook, 197S) 177 the layer. Note that the three-layer model described by 3.1 i s equivalent to the three-layer model of Davey (1977) with the density diff e r e n c e between the upper layers given by 3/8 of the density difference between the top and bottom of the upper layer i n f i g u r e 3.7 and replaced by U^c We must now make an appropriate choice f o r the bottom topography. Figure 3.5 shows isobaths o f f Vancouver Island and figure 3.6 shows topographic cross-sections corresponding to l i n e s A - E i n figure 3.5. Since the undercurrent l i e s below the s h e l f break, Mysak (1977) has assumed that the motion on the s h e l f has l i t t l e influence on the s t a b i l i t y of the undercurrent and hence models the steeply sloping region near the shelf-break by a v e r t i c a l w a l l . Further, he has estimated that the more gently sloping region has a mean slope of approximately 0.0134. We s h a l l follow Mysak i n both these choices. F i n a l l y , Mysak considers uniform flow i n each layer and considers a channel of width 75 km with constant sloping bottom. We s h a l l consider t h i s model (extended to include the e f f e c t s of the upper layer ) as w e l l as the model incl u d i n g the e f f e c t s of h o r i z o n t a l shear and a f l a t ocean f l o o r of the same width as the sloping bottom. The three-layer models studied here are shown i n figures 3.7 and 3.8. To compare our three-layer model with the two-layer model of Mysak (1977) we begin by considering the system depicted i n fig u r e 3.7, with bh (bottom topography) neglected with respect to H 3 . For t h i s system the cross-channel modes decouple and a simple a n a l y t i c s o l u t i o n i s possible (see chapter II) The dispersion curves corresponding to the f i r s t two cross-channel modes are shown i n figure 3.9. The regions of i n s t a b i l i t y corresponding to those studied by Mysak are marked as such. These waves are e a s i l y i d e n t i f i e d as they e x t r a c t most of t h e i r I30°W I25°W I20°W !30°W I25°W I20°W Figure 3.5: Isobaths (m) o f f Vancouver Is land and Washington. Topographic c ros s - sec t ions a t l i ne s A - E are p lo t ted i n f i gure 3.6. (from Mysak, 1977) 300 DISTANCE FROM COAST (KM) 200 100 H500 A\ooo DEPTH (M) HI500 H2000 J 2500 Figure 3.6: Topographic cross-sections at l i n e s A- E shown i n figure 3.f. (from Mysak, 1977). 180 Figure 3.7; The three-layer, channel model analogous to the two-layer •odel of Mysak, 1977). Table 3.1: Comparison of the i n s t a b i l i t i e s found by Mysak (1977) and the corresponding i n s t a b i l i t i e s found using the analogous 3-layer model. 6^  and 6^  are the phases of the wave in the upper and lower layers with respect to the middle layer. (A positive value of c or c corresponds to a southward velocity.) ^ Model U l ' U 2 ' U 3 Period Wavelength e-folding Phase Group <J>2 6^ , 6 3 -1. (days) (km) time velocity velocity I-—I, • I-—I x 10 cm sec ) J ,, , ., . J . „ , -\. '4>J' '<fr ' . (days) (km day -1) (km day A) T2 2 (degrees) figure 3.3 -, -1.0, 0.0 9.2 65 13.1 -7.0 -3.3 -, 0.9 -, 78 figure 3.7 0.5, -1.0, 0.0 11.3 54 13.0 -4.7 -3.3 0.4, 0.6 0, 80 Table 3.2: Properties of the most unstable wave corresponding to the system shown in figure 3.7. 6^  and 6^  are the phases of the wave in the upper and lower layers relative to the middle layer. (A positive value of c or c corresponds to a southward velocity.) Model U1' U2' U3 (x 10 cm sec "S figure 3.7 0.5, -1.0, 0.0 Period (days) 55.7 Wavelength (km) 107 e-folding time (days) 6.9 Phase velocity (km day~l) -1.9 Group velocity (km day~i) -1.0 • l *3 *2 9 *2 1.7, 0.06 V * 3 S I—1 (degrees) 63, 114 / ft == 1.024 , , , / / / , / / / / / / / / / / / / / / / < / \ ^ U, = 0.375 - 0 . 1 2 5 eo. "/z ( y + 0_ ________ 0 2 7 U, = " 0 . 4 3 7 5 + 0 . 5 6 2 5 eo. % ^ + •) K H2=500m. U 3 = 0 0 2 7 5 H,= 2 0 0 m. / / / H3= 1900 m. 7///////// 75 km. >k •iop« - 1.34 x 10 75 km. Figure 3.8: The extension of the channel model to include e f f e c t s of h o r i z o n t a l shear and reduce the infl u e n c e of the a r t i f i c i a l l y imposed western boundary. 0.70 p 0.51 -k Figure 3.9; Graph of c^ , (non-dimensional phase speed) i n the wave-number range k » 0 - 5 (wavelength = « - 47.1 km) for the f i r s t two cross-channel modes. The unstable regions corresponding to those studied by Mysak (1977) are labelled as such. Regions of instability are indicated by dashed lines and the position at which the largest growth rate is found i s indicated by a plus sign. 1 8 4 energy from the t i l t of the lower i n t e r f a c e (this i s the only energy source i n Mysak's model). The trans f e r of available p o t e n t i a l energy corresponding to the most unstable wave of t h i s type is\shown i n figure 3.10 and the properties of t h i s wave are given i n table 3.1 together with the properties of the corresponding wave discussed by Mysak (1977). The r e s u l t s of the two models are c l e a r l y i n quite good agreement. We note however that the wavelength of these i n s t a b i l i t i e s i s only about h a l f of that observed (Mysak has considered other cases which show better agreement with observations but the parameter values used i n these studies do not appear to be as reasonable as those used i n the above model). This class of i n s t a b i l i t i e s (those ex t r a c t i n g t h e i r energy from the t i l t of the lower interface) does not however have the l a r g e s t growth rates found i n studying the above three-layer model. The most unstable waves found extract energy p r i n c i p a l l y from the upper i n t e r f a c e (see f igure 3.11). Table 3.2 gives the properties of the most unstable waves found i n the study of the model depicted i n figure 3.7. The properties of t h i s wave are s i g n i f i c a n t l y d i f f e r e n t from those of the corresponding wave at shorter wavelengths (compare tables 3.1 and 3.2). In p a r t i c u l a r we note that t h i s wave i s strongly i n t e n s i f i e d i n the upper layers and has an along-channel wavelength approximately double that of the previously discussed i n s t a b i l i t y (this wavelength i s i n good agreement with the observed wavelength of about 100 km.). The e- f o l d i n g time, phase and group v e l o c i t i e s are also s i g n i f i c a n t l y reduced which, with the surface i n t e n s i f i c a t i o n , makes t h i s the more l i k e l y wave to be observed. Thus the predictions of the three-layer model suggest that although Mysak's 185 CL < \ 1.0 0 . 0 y Figure 3.10: Transfer of available p o t e n t i a l energy (in a r b i t r a r y units) corresponding to the most unstable 1st mode i n s t a b i l i t y analogous to that considered by Mysak (1977). The p o s i t i o n of this wave i n figure 3.9 i s marked by a c i r c l e . (The s o l i d l i n e corresponds to the transfer of energy due to the shear between the upper layers and the dashed l i n e to that due to the shear between the lover layers) Figure 3.11: Transfer of available potential energy corresponding to t •ost unstable wave found using the three-layer model (plus sign in Fig. 3.9) (The solid line corresponds to the energy transfer due to the shea between the upper layers and the dashed line to that due to the shear between the lower layers.) 186 two-layer model gives reasonable results for one type of i n s t a b i l i t y , i t does not exhibit the in s t a b i l i t y which i s most likely observed. The other values of the mean velocities considered by Mysak have also been considered in the above fashion (results are not reproduced here) . In each case the results are similar to those for the case discussed above, i.e. the two-layer model of Mysak (1977) reveals the in s t a b i l i t y which extracts energy from the lower interface (which in some cases gives reasonable agreement with observations) but the most unstable wave i s due to the shear between the upper layers. The model considered above gives the smallest growth rates of the cases considered and i s believed to give a conservative estimate of the i n s t a b i l i t i e s present i n the region of the California Undercurrent. Hence from the above work i t i s clear that this current system is very unstable to quasi-geostrophic perturbations at the observed length and time scales. We now consider the effects of the a r t i f i c a l l y introduced wall at the western boundary of the undercurrent. To do this we consider the more r e a l i s t i c system depicted in figure 3.8. When the western boundary is moved further out, horizontal shear must be included. The choice considered here i s shown in figure 3.12(a), but the form of the mean currents can be varied considerably without significantly changing the results. The potential vorticity gradients in the three layers are given i n figure 3.12b. Clearly there are two possible types of baro-c l i n i c i n s t a b i l i t y in this system; one due to the change in sign of q^ between the f i r s t and second layers and the other due to the change in sign between the second and third layers. In each case the sign change occurs i n the region of large vertical shear and we expect the amplitude of the unstable wave to be largest there. This i s indeed 187 1.0 Figure 3.12: (a) Approximation to the aean currents used to aodel the California undercurrent off Vancouver Island (see figure 3.8); (b) The aean potential v o r t i c i t y gradients corresponding to the three layers of our model: upper layer-solid li n e , middle layer-long dashes, lower layer-short dashes; (c) The Eigenfunctions corresponding to the most unstable wave i n our model; (d) the transfer of available potential energy corresponding to the above wave; (a) The transfer of kinetic energy i n the three layers, (see figure 2.7 for the meaning of the different lines i n (d) and (e).) Table 3.3: Properties of the most unstable wave corresponding to the system shown i n f i g u r e 3.8. This wave i s i l l u s t r a t e d i n f i g u r e 3.12. and 6 3 are the approximate phases of the wave i n the upper and lower layers r e l a t i v e to the middle layer i n the region where the perturbations are l a r g e s t . (A p o s i t i v e value of c or c corresponds to a southward ve l o c i t y . ) Model U1' U2' U3 (x 10 cm sec 'S Period (days) Wavelength e-folding Phase Group (km) time v e l o c i t y v e l o c i t y (days) (km day" 1) (km day - 1) V 6 3 (degrees) figure 3.8 figure 3.12a 57.3 100 8.3 -1.7 -0.3 ~2, -0.03 ~60, ~115 oo 00 189 verified in figure 3.12(c). The in s t a b i l i t y due to the change in sign of between the middle and bottom layers corresponds to the wave studied by Mysak (1977). Its properties are very similar to those of the correspond-ing wave for the system illustrated in figure 3.7 and are not further discussed. Finally we note that the energy source for the unstable perturbations which owe their existence to the change in sign of between the upper two layers i s almost purely potential energy from the t i l t of the upper interface, and the transfer of energy occurs in the region of large vertical shear (figure 3.12(c)). The properties of the most unstable wave existing i n this system (illustrated in figure 3.12(b)) are given in Table 3.3. Clearly the properties of this wave are very similar to the corresponding wave for the system illustrated in figure 3.7 (see table 3.2). The period, wavelength, e-folding time, phase velocity, vertical distribution of energy and phase shifts between the layers a l l show good agreement. The group velocity i s considerably smaller, however both values are quite small and the difference may not be significant. A f i n a l three-layer model was considered in which the topography in figure 3.8 was neglected but everything else was kept the same. The results were very similar to those corresponding to figure 3.8 and w i l l not be presented here. The insensitivity of this class of in s t a b i l i t y (extracting energy from the upper interface) to topography i s expected from the work done in chapter II. Finally we note that just as Mysak (1977) studied the class of i n s t a b i l i t i e s which extract energy from the lower interface i t may be possible to use a two-layer model to study the class of i n s t a b i l i t i e s which extract potential energy from the t i l t of the upper interface. 190 The agreement between the results for the most unstable waves corresponding to figures (3.7) and (3.8) suggest that a simple channel model analogous to that shown in figure 3.7 should suffice to investigate this p o s s i b i l i t y . Since our study of the three-layer model shows that the wave which extracts i t s energy from the upper interface extracts very l i t t l e energy from the lower interface, i t i s probably reasonable to assume that the t i l t of the lower interface acts in a manner similar to a sloping bottom. The effect w i l l , of course, not be as strong as i f the lower interface were replaced by a solid bottom and hence we consider two different two-layer models. The f i r s t has zero bottom slope while the second has a bottom slope equal to the slope of the lower interface due to the vertical shear of the mean currents. It is to be expected that the true situation would be best modelled by something between these cases. Since the mean state is i n hydrostatics and geostrophic balance i t i s easily found that the mean position of the lower interface in the three-layer channel model depicted in figure 3.7 i s given by fo n 3 = g ( p * ^ 2 , ^ 2 U 2 - P 3 U 3 ) Y (3-3) and hence: f n „ / H 0 = - T T T T T T r ( p - U o " • ( 3 ' 4 ) '3y' 2 g(p*-p*) ^2"2 "3-3 2 We thus consider the two-layer models analogous to figure 3.7 with no lower layer and bottom topography given by 191 f * * bh /H = , *" * l (P,U - P*U )L/H y 2 g(p 3-p 2) 2 2 3 3 2 or bh /H. = 0 Y 2 (3.5) The properties of the most unstable waves in these two models are given i n Table 3.4 together with the average of these two cases. The average i s clearly in good agreement with the predictions made using the three-layer model (except for the group velocity, which although s t i l l quite small, is increased significantly) corresponding to figure 3.7, however, the. strong variation of these statistics with changing bottom topography indicates the d i f f i c u l t i e s i n attempting to use such a two-layer model. It appears that at least three layers are required to study a system such as the California Current system. Further, using more than three-layers would probably create unnecessary confusion as the data-base does not permit the mean currents to be sufficiently well defined to warrant the investigation of more complex models. My investigations with a three-layer model do certainly support the conjecture of Mysak (1977) that the eddies observed over the continental slope in figure 3.1 may be due to the inherent in s t a b i l i t y of the current system to perturbations at the observed length and time scales. Unlike Mysak, however, we find that the fastest growing waves in this system probably extract their energy principally from the t i l t of the upper interface. Table 3.4: Properties of the most unstable waves found using two-layer models to study the stability of the upper layers of the California current system. 6 is the phase of the wave in the upper layer relative to the lower layer. (A positive value of c or c corresponds to a southward velocity.) Model U l ' U 2 (x 10 cm sec )^ Period (days) Wavelength (km) e-folding time (days) Phase velocity (km day-!) Group velocity (km day--*-) (degrees) 0 bottom slope 0.5, -1.0 14.4 124.0 5.92 -0.73 -3.3 1.4 56 constant bottom slope 0.5, -1.0 111.1 91.2 5.86 -3.6 -3.0 1.0 75 Average of above cases 0.5, -1.0 62.8 102.6 5.89 -2.2 -3.2 1.2 66 193 4. Conclusions Three-layer models have been used to study the st a b i l i t y of the current systems in Juan de Fuca Strait and the Calfiornia Undercurrent off Vancouver Island to quasi-geostrophic wave perturbations. The model developed in chapter II is applied to Juan de Fuca Strait and i t i s found that the most unstable wave has properties consistent with observations made by Fissel (1976). The lack of coherence between stations separated by as l i t t l e as four kilometers i s not expected from the predictions of our model and i t i s suggested that i t may be due to several processes occurring at the same time at similar length and time scales. It should also be noted that the data analysed by Fissel (1976) only contains about three periods at the time scales under consideration and are not sufficient to make firm conclusions. The observations off Vancouver Island allow us to determine an approximate wave length (~ 100km ) for the eddies and show that the eddies are surface intensified. The results of the three-layer model applied to this area (a modification of that developed in chapter II) are in good agreement with the limited observations available. Further, the model predicts an e-folding time of about 8 days with a group velocity of about 0.3 km day 1 (to the north) making i t very li k e l y that these waves are significantly amplified before propagating out of the undercurrent region which extends a l l the way from California to Vancouver Island. Note how-ever that this large growth rate applied to the i n i t i a l growth of the perturbation and is significantly reduced as the eddies reach f i n i t e amplitude (Pedlosky (1970). Since the magnitudes of the mean currents used are conservative estimates i t appears very li k e l y that the undercurrent 194 system i s highly unstable to perturbations at the observed length and time scales. Both a simple channel flow with constant bottom slope and no horizontal shear, and a more r e a l i s t i c model including a f l a t ocean floor and horizontal shear are considered in the study of the California Undercurrent. The agreement between the two models i s excellent. This agreement i s , of course, strongly dependent on choosing the correct channel width i n the channel model. This width must be chosen to include the region i n which the majority of the energy i s released to the perturbations but must not be much wider. A channel which i s much to wide w i l l predict a perturbation which i s also too wide and too narrow a channel w i l l predict too narrow an eddy as well as not including the f u l l energy source for the perturbations. If the channel width i s chosen correctly though the agreement i s good. For the models considered here i t appears that the choice may be made simply by considering only the region in which the vertical shear i s largest but such a choice may not always be valid. An example of such a case would be furnished by any situation in which the transfer of kinetic energy due to barotropic in s t a b i l i t y i s significant. Finally a two-layer model is used to study the stability of the upper two layers of the three-layer model. It i s found that the system is very sensitive to the choice of bottom slope but with a correct choice, results are in good agreement with those of the corresponding ins t a b i l i t y i n the three-layer model. Due to the sensitivity of the in s t a b i l i t i e s to the bottom slope in such a model, i t appears that to model a current system such as that considered here at least three layers must be used. C H A P T E R V C O N C L U D I N G D I S C U S S I O N 195 196 Concluding Discussion This study was i n i t i a l l y undertaken in an attempt to determine whether or not baroclinic instability could make a significant contribution to the low-frequency motions observed by Fissel (1977) in Juan de Fuca Strait. Since the transition of density from i t s value near the surface (o = 24.7) to i t s value at depth (o^ - 26.7) occurred gradually over a depth of more than 80 m (nearly half the depth of the strai t ; see figure2.3 }chapter IV), the usual layered models did not appear appropriate. Although Davey's three-layer model (with each layer of constant density; Davey (1977)) could have been applied to this situation with appropriate reduction of density differences between the layers, i t i s not clear exactly how much these differences should be reduced. Therefore, i t was decided to derive a three-layer model which would overcome (or at least minimize) this problem. This was done by modelling the observed continuous density stratification by a simpler continuous model (figure 2.3,chapter IV) and then finding an approximate solution to the new problem with three degrees of freedom in the vertical. The resulting equations are very similar in form to those derived by Davey(1977) but the differences are also significant. DAVEY'S EQUATIONS ( extended to include horizontal shear, variable layer depths and weak bottom slopes; the density difference between successive layers i s equal) : 197 3? <!F + U l b[^l + 2 F 1 (W ] + 1 ^ [ B " U l y y + 2Fl(VU2^ = 0 U2 + 2 F 2 ( ? 1 - 2 5 2 + ? 3 ) ] + ^  [ 6 - U 2 y y + 2F 1(U 1-U 2)] = 0 (lt- + U3 b[V\ + 2 F3 ( ?2-^3 ) ] + [ g - U3yy + ^ ^ " V + FK~V = 0 ( l . D (Davey's equations have been rewritten in our notation for convenience). These equations are to be compared with equations 3.2, Chapter II (with bh « H3) : (lr + u i b[*2h - V 3 5r 4W ] + ^ r ^ - V + V 3 ur 4 u 2 o + u3 ) ] - 0 ( l t - + U2o ! r ) [ V % + 4 F 2 ( 5 l - 2 ? 2 + ? 3 ) ] + ^ [B. - U - 4F2(U1-2U2+D3)] = 0 ( f r T + U3 b^h ~ W 4?2 + 3 53 ) ] + ^ t e " U3yy + F 3 ( U r 4 U 2 0 + 3 U 3 ) + FHT\ ] = 0 d.2) o 3, J The f i r s t thing to note is that these two sets of equations can not be made equivalent by any choice of density difference between the layers in Davey's model. This i s due to the presence of 5 3 (and U3) in the f i r s t equation and ^  (and U^) in the third equation of (1.2). The modelling of vortex stretching i s different; in the model developed here, the upper and lower layers feel each others influence much more strongly (and directly) than in Davey's model. One expects this kind of difference since the model 198 developed here takes account of the fact that the vortex tubes are continuous whereas the usual layered models do not. The second major difference i s seen by considering the second layer only. Where we have 4 F ^ , Davey has 27^. Decreasing the density difference between the layers by a factor of two in Davey's model w i l l eliminate this difference (note, however that the equations for the upper and lower layers are s t i l l not in agreement). This explains why, for a given density difference between the layers, the extent of the 2 unstable range (plotted against K^ /F,,) is only about half as wide for Davey's model as i t is for ours in the presence of large vertical curvature. Finally there is the difference between IL (the value of U at the middle of the 2o second layer) and u*2 (the vertical mean value of U in the second layer) in the two models. Although this difference i s relatively small for S^-S 2 > in the presence of large vertical curvature, i t i s significant. It is important to notice here that both Davey's model and the one developed here are essentially f i n i t e difference approximations to the equat-ions for a continuously strat i f i e d f l u i d . This i s implicit in Davey's model and is introduced a p r i o r i by considering three layers of uniform density. The model developed here introduces the f i n i t e difference approximation rather more explicitely through a truncated power series expansion about the middle of the second layer. Such an expansion i s , after a l l , the basis of f i n i t e difference models. By introducing the f i n i t e difference approximation in this manner, the model is automatically " f i t t e d " to the actual density strat-if i c a t i o n and mean currents. Of course, i f many layers are to be used, the_ two models w i l l be essentially equivalent. However, the difference appears very significant i f as few as three layers are being employed. 199 Due to the unusual form of our f i n i t e difference equations in the upper and lower layers, we now give a brief derivation of our equation which makes their f i n i t e difference nature more explicitely revealed. We begin with equations (2.10)-(2.12), Chapter II. As in the previous chapters, a subscript o w i l l be used to indicate a quantity eval-uated at the middle of a given layer, and we w i l l use a A to indicate the is z=z m f i n i t e difference approximation of a given quantity. Thus ^P2 Z Z H the f i n i t e difference approximation to p 0 at z=zm ( = - (rL+_JL) /H) . We w i l l zzz J- H need the following expressions: Ap Ap (0) 2z (0) 2zz = (H/H 2)(p^-p(°>) z=z„ 2 1 J z=zm ^(H/H 2) 2(pf)-2p^) +p(°>) (1.3) Using the second of these in (2.11), Chapter II, we get (2.15b), Chapter II, immediately, and no further discussion of this equation is needed. The corr-esponding equations for the upper and lower layers are equally as simply de-rived. In (2.10) we use: Ap (0) 2z z=- 1 2z H z=z + A P m (0) 2zz (H./2H) + 0((H 0/2Hr) z=z m 2 2 = (H/H2)(p{0)-p^0)) + 2(H/H2)(p^0)-2p^0)-p^0)) + 0((H2/2H)?) = (H/H2)(3p^°)-4p^)+p^0)) + 0((H2/2H)2) and in (2.11) we use: 200 A.P (0) 2z z=-H. 1 AP 2z z=z (0) + AP, 2zz|z=z (0) m (-H2/2H) + 0((H 2/2H) 2) H (H/H 2)(pf >- Pf b - 2 ( H / H 2 ) ( p ^ - 2 p ^ + p ( ° ) ) + 0((H 2/2H) 2) (H/H 2)(p{ 0 )-4p 2 (° )+3p^° )) + 0((H 2/2H) 2) where we have used (1.3) in each case. Substituting these expressions in (2.10) and (2.12) we get (2.15a,c). the f i n i t e difference nature of our equations, but i t shows the ease with which they can be derived (although this method of derivation was not obvious at the outset). The important thing to notice i s that, although both this model and Davey's are essentially f i n i t e difference approximations, by introducing step discontinuities in the density profile a p r i o r i , Davey's model has altered the dynamics of the flow more than is necessary and i t i s not clear exactly how the model should be " f i t t e d " to the actual situation. We suggest that i f a f i n i t e difference approximation i s to be used, i t must be introduced in a consistent manner. The method employed in this thesis to find such an approximation i s to f i r s t approximate the actual density profile by a simpler profile which closely resembles that observed and then find the appropriate f i n i t e difference equations corresponding to this approximate model. By finding the equations appropriate to a density profile which closely approximates the real situation this method results in a set of equations which can be used to model r e a l i s t i c oceanic circumstances more accurately than the conventional three-layer model without introducing more layers (and thus greater complexity). There w i l l , This alternative derivation of our equations not only demonstrates 201 of course, s t i l l be an error introduced in making the f i n i t e difference approximation and the study of the precise nature of this error i s indeed a worthy topic for future research. . It must be mentioned here that Davey's model does have some advan-tages over the one developed here. The main advantage i s that in the hands of an experienced modeller (who chooses the density differences between the layers correctly, uses the correct values for U^, U^, and [this includes replacing U by U„ when the ve r t i c a l curvature of the mean currents i s s i g -^ ^o nificant] and interprets results carefully), Davey's model may be immediately applied to a large variety of cases without re-deriving the governing equations. Although i t i s believed the model derived here w i l l yield more accurate results, different density and velocity profiles w i l l require a re-derivation of the appropriate equations. Because of this fact our method w i l l be slightly less convenient, especially i f one only desire a crude approximation before going on to do a more detailed analysis. Finally we mention that the two models may be combined as was done in chapter IV, section 3 to study the California Undercurrent system. In this manner, the region in which the large density transition occurs may be modelled quite accurately using the methods described in this thesis, and'more degrees of freedom in the vertical may be added with a minimal effort. Such a model would be useful in many situations. In particular, i t might be useful in studies of the open ocean where a linear density variation would model the thermocline quite well. In chaper II, I have mentioned that a two-layer model i s probably most useful when the ve r t i c a l curvature of the mean currents and the region of density transition are small, I now wish to stress a point made in chapter III involving the study of pure barotropic i n s t a b i l i t y in which a "one-layer" model i s employed. If over some depth of the f l u i d , the horizon-t a l length scale over which the mean currents vary is sufficiently strong 202 that (L/r. ) << 1 in this region, then barotropic i n s t a b i l i t y may occur int over this depth with l i t t l e effect from the flu i d above and below. Thus, for such a case, averaging the mean currents over the depth of the f l u i d results in a rather poor approximation. This w i l l significantly reduce and may even eliminate any i n s t a b i l i t i e s present, A better approach would be 2 to consider the region i n which C^/r^ ) « . 1 separately. Another interesting result found in chapter III was the presence of a "barotropic" i n s t a b i l i t y which can draw energy from the potential energy of t i l t i n g isopycnals due to the mean flow. This wave i s f i l t e r e d out under the assumption of a horizontally uniform mean flow, but even 2 for (L/r^.) moderately large i t can have a significant growth rate and 2 may, in some cases, be important. Thus, even for (L/r_^) » 1 i t may not be j u s t i f i e d to consider the case of no horizontal shear. (Note, however, that such a study i s valid for the study of baroclinic i n s t a b i l i t i e s ) . An interesting discussion of momentum transports for the case of mixed baroclinic-barotropic i n s t a b i l i t y has been presented by Held (1975). Of particular insterest i s his proof that in an arbitrary zonal flow, linear theory predicts that unstable waves cause a net transport of momentum out of the region of f l u i d in which the generalized Rayleigh criterion for in s t a b i l i t y (the change in sign of ^q/<Jy; see Pedlosky, 1964a) i s satisfied l o c a l l y . Although the author was unaware of this paper u n t i l after the writing of this thesis was completed, i t would certainly be enlightening to re-examing the results of chapter III in the light of Held's work. Work in this direction has now begun but w i l l be reported elsewhere. In each of the case studies (Juan de Fuca Strait and the California Undercurrent) made in Chapter IV i t was found that the significant i n s t a b i l i t i e s 203 are basically baroclinic i n s t a b i l i t i e s extracting their energy from the inter-face between the upper layers. The role of the horizontal' shear was simply to limit the region in which baroclinic i n s t a b i l i t y was possible. The isolation of these i n s t a b i l i t i e s from the bottom made the influence of topography relatively weak. This result w i l l probably be true for many oceanic flows where the large ve r t i c a l shear occurs near the surface. However, one must remember that bottom intensified waves may also exist due to bottom slopes (Rhines, 1970), so i f one chooses to neglect bottom topography on the grounds that only surface intensified motions are being studied, i t must be borne in mind that these waves are fi l t e r e d out. Finally we mention that many effects have not been considered in the model presented here. Some of these have been mentioned in Chaper IV, section 2 in the study of Juan de Fuca Strait. Of these effects the author finds the modifications of these low-frequency waves due to the tides to be a largely overlooked problem. The inclusion of this effect as well as the effects due to motions on the continental shelf off Washington and Vancouver Island seem to be necessary before any definite conclusions can be made about the low-frequency motions in Juan de Fuca Strait. For now, we can only say that the predictions of our model are consistent with observations. The model of the California Undercurrent appears to be- consistent with the very limited observations off Washington and Vancouver Island, however the extremely sparse observations again make definite conclusions d i f f i c u l t . It would be convenient to make detailed measurements here, but the relatively small width of the undercurrent in this region and the extreme-ly long periods predicted by our model (which, by the way, make our assumption of a non-diffusive f l u i d rather questionable), make i t unlikely that an 2Q4 adequate study w i l l be made in the near future. Perhaps a more feasible approach would be to apply the model developed here to the region off California where much more detailed observations have been made,_ If the model predictions in this region show good agreement with observations, one could be more certain that the same would be true further north, 205 REFERENCES CITED Bernstein, R.L., Breaker, L., and R. Whritner, 1977. California Current Eddy Formation: Ship, Air and Satellite Results. Science, 353-359. Bretherton, F.P., 1966a: C r i t i c a l layer i n s t a b i l i t y in baroclinic flows. Quart. J.R. Met. Soc., 92, 325-334. Bretherton, F.P., 1966b: Baroclinic i n s t a b i l i t y and the short wave-length cut-off in terms of potential vor t i c i t y . Quart. J.R.  Met. Soc, 92, 335r345. Brown, J.A., 1969a,b: A numerical investigation of hydrodynamic in s t a b i l i t y and energy conversions in the quasi-geostrophic atmosphere: Part I and II. J. Atmos. Sci., 26 352-375. Brown, R.A., 1972: On the physical mechanism of the inflection point i n s t a b i l i t y . J. Atmos. Sci., 29, 984-986. Cannon, G.A., and N.P. Laird, 1978: Circulation in the Strait of Juan de Fuca, 1976-1977. PMEL, ERL, NOAA Technical Report, (in preparation). Charney, G.A., and N.P. Laird, 1978: Circulation in the Strait of Juan de Fuca, 1976-1977. PMEL, ERL, NOAA Technical Report, (in preparation). 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Huggett, 1976: Observations of currents, bottom pressures and densities-through a cross-section of Juan de Fuca Strait. Pacific Marine Science Report 76-6, 68pp. Institute of Ocean Sciences, Patricia Bay, Victoria, B.C. (Unpublished manuscript). F j ^ r t o f t , R. , 1951: Stability properties of large-scale atmospheric disturbances. In: Compendium of Meteorology, American Meteorology Society, Boston (Reprinted in: Theory of Thermal Convection, ed. by B. Saltzman, Dover Publications, New York, 1962). Gent, P.R., 1974: Baroclinic i n s t a b i l i t y of a slowly varying zonal flow. J. Atmos. Sci., 31, 1983-1994. Gent, P.R., 1975: Baroclinic i n s t a b i l i t y of a slowly varying zonal flow, Part II. J J . Atmos. Sci., 32, 2094-2102. G i l l , A.E., Green, J.S.A. and A.J. Simmons, 1973: Energy partition i n the large scale ocean circulation and the production of mid-ocean eddies. Deep Sea Research, 21, 499-528. Green, J.S.A., 1960: A problem in baroclinic i n s t a b i l i t y . Quart. J.R.  Met. Soc, 86, 237-251. Halpern, D., Reed, R.K. and R. L. Smith, 1977: On the California Undercurrent over the continental slope off Oregon. J. Geophys. Res., 83, 1366-1372. Helbig, J.A. and L.A. Mysak, 1976: Strait of Georgia oscillations. Low-frequency currents and topographic planetary waves. J. Fish. Res. Can., 33, 2329-2339. Held, I.M., 1975: Momentum Transport by Quasi-Geostrophic Eddies. J . Atmos Sci., 32, 1494-1497. Hide, R. and P.J. Mason, 1975: Sloping Convection in a rotating f l u i d : a review. Advances in Physics, 34, 47-100. Holbrook, J.R. and D. Halpern, 1978: Vari a b i l i t y of near-surface currents and winds in the western Strait of Juan de Fuca. PMEL, ERL, NOAA Technical Report, (in preparation). Holmboe, J., 1968: Instability of baroclinic three-layer models of the atmosphere. Geofysiske Publikasjoner, 28, 1-27. Huyer, A., 1976: A comparison of upwelling efforts in two locations: Oregon and North West Africa. J. Mar. Res., 34, 531-546. 207 Ingraham, W.J., 1967: The geostrophic circulation and distribution of water properties off the coasts of Vancouver Island and Washington, spring and f a l l , 1963. Fish Bull., 66-223-250. Killworth, P.D., 1978:. Barotropic and baroclinic i n s t a b i l i t y in rotating s t r a t i f i e d fluids, with application to geophysical systems, Dyn.. Atmos. Ocean, (submitted). Kuo, H.L., 1949: Dynamic i n s t a b i l i t y of two-dimensional non-divergent flow in a barotropic atmosphere. J. Meteorol., 6, 105-122. Kuo, H.L., 1973: Dynamics of quasigeostrophic flows and i n s t a b i l i t y theory. Advances in Applied Mechanics, 13, 247-330. LeBlond, P.H. and L.A. Mysak, 1978: Waves in the Ocean. Elsevier, Amsterdam, 602 pp. Lin, C.C., 1945: On the s t a b i l i t y of two-dimensional p a r a l l e l flows, Parts I, II, III. Quart. Appl. Math., 3, 117-142,..218-234, 277-301. Longuet-Higgins, M.S., 1972: Topographic Rossby waves. Memoires Societe Royale des Science de Liege, 6 e serie, tome II, 11-16 Mclntyre, M.E., 1970: On the non-separable baroclinic pa r a l l e l flow i n s t a b i l i t y problem. Journal of Fluid Mechanics, 40, 273-306 Miles, J.W., 1964a: Baroclinic i n s t a b i l i t y of zonal wind. Rev, of  Geophys., 2, 155-176. Miles, J.W., 1964b,c: Baroclinic i n s t a b i l i t y of the zonal wind: Parts II and III. J. Atmos. Sci., 21, 500-506, 603-609. Mysak, L.A., 1977: On the s t a b i l i t y of the California Undercurrent off Vancouver Island. J. Phys. Oceanogr., 7, 904-917. Narayanan, S., 1973: Quasi-geostrophic waves in the open ocean (Ph.D. thesis), Harvard University, Cambridge, Massachusetts. Needier, G.T. and P.H. LeBlond, 1973: On the influence of the horizontal component of the earth's rotation on long-period waves. Geophysical Fluid Dynamics, 5, 23-46. Orlanski, Land M.D. Cox, 1973: Baroclinic i n s t a b i l i t y in ocean currents. Geophysical Fluid Dynamics, 4, 297-332. Pedlosky, J., 1963: Baroclinic i n s t a b i l i t y in two-layer systems. Tellus, 15, 22-25. 208 Pedlosky, J., 1964a,b: The s t a b i l i t y of currents in the atmosphere and ocean, Parts I and II. J. Atmos. Sci., 21, 201-219, 342-353. Pedlosky, J., 1964c: An i n i t i a l value problem in the theory of baroclinic i n s t a b i l i t y . Tellus, 16, 12-17. Pedlosky, J., 1970: Finite amplitude baroclinic waves. J. Atmos. Sci., 27, 15-30. • . • -• Pedlosky, J., 1971a: Finite-amplitude baroclinic waves with small dissipation. J. Atmos. Sci., 28, 587-597. Pedlosky, J., 1971b: Geophysical Fluid Dynamics. Mathematical Problems  in the Geophysical Sciences, W.H. Reid (editor), V. 13, American Mathematical Society, Providence, Rhode Island, pp. 1-60. Pedlosky, J., 1972: Finite-amplitude baroclinic wave packets, J. Atmos.  Sci., 29, 680-686. Pedlosky, J., 1974: Long shore currents, upwelling and bottom topography. J. Phys. Oceanogr., 4, 214-226. Pedlosky, J., 1975: On secondary baroclinic i n s t a b i l i t y and the meridional scale of motion in the ocean. J. Phys. Oceanogr., 5, 603-607. Pedlosky, J., 1976: Finite-amplitude baroclinic disturbances in downstream varying currents. Journal of Physical Oceanography, 6, 335-344. Ph i l l i p s , N.A., 1951: A simple three-dimensional model for the study of large scale extra tropical flow patterns. J. Meteorol. % 8, 381-394. P h i l l i p s , O.M., 1966: The Dynamics of the Upper Ocean. Cambridge University Press, 261 pp. Pond, S. and G.L. Pickard, 1978: Introductory Dynamic Oceanography, Pergamon Press, 241 pp. Rao, D.B. and T.J. Simons, 1969: Stability of a sloping interface in a rotating two-fluid system. Atmospheric Science Paper No. 151, Colorado State Univ.,;- Fort Collins, Colo., 32 pp. Reed, R.K. and D. Halpern, 1976: Observations of the California Undercurrent of Washington and Vancouver Island. Limnol. Oceanogr., 21, 389-398. 209 Rhines, P.B., 1970: Edge-, bottom-, and Rossby waves in a rotating s t r a t i f i e d f l u i d . Geophysical Fluid Dynamics, 1, 273-302. Rossby, C.G., 1949: On a mechanism for the release of potential energy in the atmosphere. J. Meteorol., 6, 163-180. Simmons, A.J., 1974: The meridional scale of baroclinic waves. J. Atmos. Sci., 31, 1515-1525. Stone, P.H., 1969: The meridional structure of baroclinic waves. J. Atmos. Sci., 26, 376-389. Tabata, S., 1975: The general circulation of the Pacific Ocean and a brief account of the oceanographic structure of the North Pacific Ocean. Atmosphere, 13, 133-168. Turner, J.S., 1973: Bouyancy Effects in Fluids, Cambridge University Press. 367 pp. Wickham, J.B., 1975: Observations of the California counter current, J. Mar. Res., 33, 325-340. Wooster, W.S., and J.H. Jones, 1970: California Undercurrent off northern Baja California, J. Mar. Res., 28, 235-250. 210 Appendix Glossary of symbols^ b - amplitude of topographic variations ~:: c*,c- phase speed C = c - ^ 2o~ Soppier shifted phase speed f - c o r i o l i s parameter f^ - local value of the c o r i o l i s parameter 2 2 F^ = f Q L /g'H - the Burger number for the i t h layer g - acceleration due to gravity g' - reduced gravity h - cross-channel structure of topographic variations (bh gives the height of the bottom above z* = - rLj,) H - thickness of the i * " * 1 layer H^ , = + H2 + H - ve r t i c a l length scale * k,k- long-channel wavenumber L - horizontal length scale A th p_^-dimensional pressure in the i layer p^-non-dimensional perturbation pressure in the 1^ layer q. - potential v o r t i c i t y in the i * " * 1 layer (non-dimensional) r^- internal deformation radius for the i layer R = U/f L - the Rossby number 0 0 T — J°„ h - the topographic parameter R QH 3 y r W^hen a variable appears both with and without a star on i t s shoulder, the starred variable i s dimensional and the other i s non-dimensional. 211 T • T / F 2 u£,iu - eastward component of the velocity in the i layer U*,U\ - mean value of the velocity in the i * " ^ layer. U* U. - mean value of the velocity evaluated at the middle of the i t h laver 30 l O . v*,v_^ - northward component of the velocity in the i ^ layer. t i l W i ' W i ~ V B V t^- c a^- component of the velocity in the i layer. x*,x - coordinate measured positive eastwards y*>y - coordinate measured positive northwards z*,z - coordinate measured positive v e r t i c a l l y B* - the variation of the co r i o l i s parameter with latitude (f = fg+3*y*) 3 = 3*L 2/U 3 = 3/F 2 6 = (p*-p*)/p* - relative density difference between the upper and lower layers 6^ ,6^  ~ the phases of the upper and lower layers with respect to the middle layer. th H * , T U - elevation of the surface of the i layer - amplitude factor for the mean stream function for the i * " * 1 layer * th - density of the i layer to - radian frequency - ve r t i c a l l y averaged stream for the i * " * 1 layer ijj. - time average of ii. ( i . e . . in the absence of perturbations) x x x £. - perturbation of Ui'. x x <|>^  - complex amplitude of £^ 

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