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Theoretical studies of the circulation of the Subarctic Pacific Region and the generation of Kelvin type… Thomson, Richard Edward 1971

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THEORETICAL STUDIES OF THE CIRCULATION OF THE SUBARCTIC PACIFIC REGION AND THE GENERATION OF KELVIN TYPE WAVES BY ATMOSPHERIC DISTURBANCES by RICHARD EDWARD THOMSON B.Sc, University of Bri t i s h Columbia, 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Physics and Institute of Oceanography We accept this thesis as conforming to the standard required THE UNIVERSITY OF BRITISH COLUMBIA JANUARY, 1971 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the requirements fo an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia , I agree that the L i b r a r y s h a l l make i t f r e e l y avai l a b l e f o r re ference and study . I f u r t h e r agree t h a t permiss ion fo r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department o r by h i s r e p r e s e n t a t i v e s . It i s understood that copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l not be al1 owed without my w r i t t e n p e r m i s s i o n . Department o f 'Jf $ S ' C ^ The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8 , Canada Date )c<A\ 3 ^ / 7 / Abstract Theoretical studies of two problems concerned with the surface forced, large-scale motions in bounded oceanic regions are presented. In Part I, such motions are considered for a particular area of the North Pacific Ocean known as the Subarctic Pacific Region. Discussion is based on the assumption that the velocity components may be separated into a time-averaged or quasi-steady flow about which fluctuations occur i n the form of transient planetary waves. Some of the characteristics of the latter are b r i e f l y outlined. Several aspects of the time-averaged motions are then considered. A simple circulation, driven by the vertical velocity structure, i s presented for the interior region of the ocean below the upper f r i c t i o n a l layer. Also, using observational data to obtain the depth of the layer between the suface and the main halocline, this upper layer i s found to behave as a geostrophic layer of f l u i d when averaged over many years. Combination of the above observed depths with the mean calculated Ekman divergences permitted calculation of a mean eddy coefficient of diffusivity for density. The results agree very well with those obtained by Veronis for similar oceanic situations. An explanation for the variations in the intrusion of 'warm' water along the top and bottom of the halocline off the coast of British Columbia i s also given. The two f i n a l sections of Part I deal with the overall, quasi-steady circulation of the Subarctic Pacific Region. Here, a theoretical study i s combined with the mean-monthly values of the calculated surface forcing. Curvilinear coordinates are used in order to model the northern boundary formed by the Aleutian-Komandorski island chain. The interior quasi-steady flow, which satisfies a Sverdrup-type balance of vorticity, i s closed to the north by a f r i c t i o n a l boundary layer. Using mean-monthly values for the surface winds over the region, the observed separation of the eastward flowing West Wind Drift into a northern and southern tending flow is found to correspond to the zero of the mean wind-stress curl. In the northern boundary layer, the characteristics of the westward flowing boundary current there, are shown to change down-stream from a Western' to a 'zonal', type boundary current. The s t a b i l i t y of the latter i s dependent upon vortic i t y of appropriate sign being added to the boundary layer flow to balance that generated by f r i c t i o n along the coast. Discussion i s also given for the effect of passes between the Aleutian islands on the zonal boundary current. Through a type of boundary layer 'suction' or, alternately, by mass transport into the boundary layer, the effect of these passes would seem to be to keep the boundary flow attached to the coast. Finally, spectral analysis of the wind-stress curl data, obtained from the mean-monthly surface pressure, i s performed to determine i t s frequency distribution. A demodulation technique i s used to determine the time variations of six of the frequency bands obtained i n the spectral analysis. These results'are then applied to the circulation in the Subarctic Pacific Region in an attempt to relate variations and spatial distribution i n the circulation with the applied winds. The generation by the atmosphere of a type of long, boundary waves, known as Kelvin waves, i s considered i n Part II. In particular, i t i s shown that for a general large-scale distribution of wind and pressure systems that only the longshore component of the wind-stress and pressure can generate such waves. Examples are presented for a semi-infinite wind and moving pressure pattern. Kelvin waves are shown to move away from the force discontinuities at the speed of shallow-water waves. These waves are further found to exhibit a frequency shi f t , typical of non-dispersive waves from a moving source. Using some observed parameters for the atmospheric forcing terms off the Oregon coast of the United States, numerical values for the wave amplitudes for both examples are given. Part II has been published in the form presented here. Reference: J. Fluid Mech. C1970), 42C4), 657-670. TABLE OF CONTENTS ABSTRACT TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES ACKNOWLEDGEMENTS PART I - THE CIRCULATION OF THE SUBARCTIC PACIFIC REGION 1 . INTRODUCTION 1 . 1 General discussion 1 . 2 The Oceanography of the Subarctic Waters 2 . EQUATIONS OF MOTION 2 . 1 Navier-Stokes equations 2 . 2 The vortic i t y equation 3 . THE VORTICITY EQUATION FOR LARGE-SCALE MOTIONS 3 . 1 Derivation ' ' . 3 . 2 Nondimensionalization 3 . 3 Numerical values 4 . PLANETARY WAVES 4 . 1 Introduction 4 . 2 Barotropic Rossby waves 4 . 3 Some properties of Rossby waves QUASI-STEADY MOTIONS 5.1 Numerical values 5.2 A simple thermohafline circulation; absence of vertical shear 5.2.1 Equation of motion 5.2.2 oV/3z = o 5.2.3 8w/8z > o 5.2.4 8w/Sz < o 5.3 The ver t i c a l l y integrated flow 5.3.1 The Ekman and geostrophic transport modes 5.3.2- Mean geostrophic flow i n the upper layer 5.3.3 Warm water intrusion off Br i t i s h Columbia 5.3.4 Mean eddy diffusion coefficient i n the Subarctic Pacific Region 5.3.5 Justification of the neglect of VpxVp-k in section 2.2 A LINEAR, INTEGRATED MODEL FOR THE SUBARCTIC PACIFIC 6.1 Introduction 6.2 The linearized stream function equation 6.3 The interior region 6.4 The eastern boundary region 6.5 Boundary layer region C; arbitrary curvature 6.6 Boundary layer region C; no curvature 6.7 Boundary layer region C; constant curvature 6.8 Passes and their possible effect on the northern boundary layer flow 7. THE GEOSTROPHIC WINDS AND RESULTING CIRCULATION 7.1 Pressure data analysis 108 7.1.1 Introduction 108 . 7.1.2 The surface geostrophic wind-stress 109 7.1.3 Pressure data 112 7.1.4 Computer output of mean monthly quantities 113 7.1.5 Comparison of geostrophic and actual 116 surface winds 7.2 Results 7.2.1 The general distribution of the wind-stress 118 curl 7.2.2 The general distribution of the resulting 127 circulation 7.2.3 Spectral analysis of the wind-stress curl 129 7.2.3.1 Data handling 128. 7.2.3.2 Spectra of the wind-stress curl 130 7.2.3.3 Co-spectra and quad-spectra of the 137 wind-stress curl • 7.2.4 Demodulation of the wind-stress curl 140 8. SUMMARY 147 BIBLIOGRAPHY 153 APPENDICES I. On the validity of separating barotropic Rossby waves 158 and quasi-steady motions II. Glossary of symbols 163 PART I I - ON THE GENERATION OF KELVIN TYPE WAVES BY ATMOSPHERIC DISTURBANCES 1. INTRODUCTION 218 2. EQUATIONS OF MOTION 219 3. RESPONSE TO A GENERAL WIND AND PRESSURE FIELD 220 4. KELVIN WAVE SOLUTIONS 223 5. RESPONSE TO A NON-TRAVELLING WIND PATTERN 225 6. RESPONSE TO A MOVING STORM 229 7. STEADY-STATE SOLUTIONS 234 8. PHYSICAL DISCUSSION OF THE SOLUTIONS C K 234 9. NUMERICAL VALUES FOR THE OREGON COAST 235 10. SUMMARY 239 BIBLIOGRAPHY 241 APPENDIX 242 Glossary of symbols 243 LIST OF TABLES PART I TABLE 1. Values of the mean eddy diffusion coefficient 67 PART II TABLE . 1. Kelvin wave amplitudes A at x = o 240 2. Kelvin wave amplitudes A* at x = o 241 LIST OF Figures ' PART I FIGURE la Geography of the Subarctic Pacific Region. 2 lb Bathymetry (metres). 3 2 Typical s a l i n i t y structures at Station "P". 7 3a Selected density (sigma-t) profiles for the line between , 0 Ocean Station "P" and Swiftsure Bank, October 1966 to March 1967. 3b Selected density profiles for the line between Ocean 10 Station "P" and Swiftsure Bank, April-March 1967. 4a Density (o"t) distribution along long. 175°E Winter 1966. 13 4b Density (a t) distribution along long. 165°W Winter 1966. 14 5 Coordinate systems 16 X, Y Cartesian coordinates ' x, y Curvilinear coordinates 6 A possible form for the mean ver t i c a l velocity structure. 43 7 Mean depth (metres) o f the upper layer, from 1955-1959. 50 8 Mean f/h contours obtained using the depth (h) values 51 o f figure 7. 9 The section of the eastern Subarctic Region to be 54 considered for discussion of warm water intrusion along the coast. 10 Comparative plot of the winter-mean (October-March) 56 meridional wind-stress and the relative area of water whose temperature is greater than 7 C at the top of the halocline in the region of figure 9. 11 Surface temperatures at Ocean Station "P" January (1955)- 58 December (1959). . 12 Comparative plot of the mean (July-June) meridional wind- 60 stress and the relative area of water whose temperature is greater than 6.5 C at the bottom of the halocline i n the region of figure 9. 13 Comparison of theoretical temperature profile for various 65 Peclet numbers to mean summer profile at Station "P". 14 Schematic diagram of the subregions used in the 75 mathematical model. 15 Streamlines for a northern boundary of constant curvature 85 (8 0 = 30 ) and an applied wind-stress curl = T 0 C O S / I T Y. C O S /7T X\ > L = acos9 Q . 2 L 2 1/ 16a Geopotential topography 0/1000 decibars, Summer 1956. 86 16b Geopotential topography 0/1000 decibars, Summer 1958. 87 17a Mean wind-stress curl, x 10-9 dynes cm-3. From June 89 (1955)-May (1956). 17b Mean wind-stress curl, x 10~9 dynes cm-^. From June 90 (1958)-May (1959). 18 Comparison of constant-curvature northern boundary with 92 the 1000 metre depth contour. 19 Temperature (°C) distribution on sigma-t surface 26.90, 101 summer 1959. Shaded region shows approximate extent of the Alaskan Stream. The 300 metre depth contour is also shown. 20 Schematic diagram of the va r i a b i l i t y in location at 102 which southern branches diverge from the main flow as indicated by surface s a l i n i t y . 21 Passes and 1000 metres depth contour i n the vic i n i t y 106 of the Aleutian islands. 22 Schematic diagram of the main features of the circulation 107 in the Subarctic Pacific Region. 23 Balance of forces 111 a. far from the ocean surface b. near the ocean surface 24 Pressure grid for the f i n i t e difference calculations. 114 25 Grid system 115 26 Cumulative vector winds 117 Ocean Station "P" January 1960 27 Cumulative stress vectors 119 Ocean Station "P" January, 1960. 28 Stress vector estimates 120 Ocean Station "P" January, 1960. 29 ; a. Winter mean (October-March) of the wind-stress 122 curl at various stations, b. Summer mean (April-Sept.) of the wind-stress curl at various stations. 30 Mean-monthly wind-stress curl at stations 2, 12 and 23. 123 31a Mean sea level pressure i n winter (millibars). 125 31b Mean sea level pressure in summer (millibars). 125 32 Mean wind-stress curl, January 1945 to Apr i l 1966. 126 33 Distribution of the annual component of the wind-stress 132 curl spectra. 34 Distribution of the semi-annual component of the 134 wind-stress curl spectra. 35 Integral under the wind-stress curl spectrum. 136 36a The demodulation of a 'spike' input signal (width = 142 2 months) for six frequency bands. 36b Demodulation for a sine wave input. 143 37a Summer (1948) mean wind-stress curl. 166 37b Summer (1962) mean wind-stress curl. 167 38a Winter (1948-49) mean wind-stress curl. 169 38b Winter (1962-63) mean wind-stress curl. 169 39a Mean annual wind-stress curl, April 1946 - March 1947. 170 39b Mean annual wind-stress curl, A p r i l 1948 - March 1949. 171 39c Mean annual wind-stress curl, April 1950 - March 1951. 172 39d Mean annual wind-stress curl, April 1952 - March 1953. 173 39e Mean annual wind-stress curl, April 1954 - March 1955. 174 39 f Mean annual wind-stress curl, A p r i l 1956 - March 1957. 175 39 g Mean annual wind-stress curl, A p r i l 1958 - March 1959. 176 39h Mean annual wind-stress curl, April 1960 - March 1961. 177 39 i Mean annual wind-stress curl, A p r i l 1962 - March 1963. 178 39j Mean annual wind-stress curl, A p r i l 1964 - March 1965. 179 40a Station .1. Wind-stress curl spectrum versus frequency. 180 40b Station 2. Wind-stress curl spectrum versus frequency. 181 40 c Station 5. Wind-stress curl spectrum versus frequency.' 182 40d Station 9. Wind-stress curl spectrum versus frequency. 183 40e Station 10. Wind-stress curl spectrum versus frequency. 184 40f Station 11. Wind-stress curl spectrum versus frequency. 185 40 g Station 13. Wind-stress curl spectrum versus frequency. 186 40h Station 17. Wind-stress curl spectrum versus frequency. 187 40i Station 20. Wind-stress curl spectrum versus frequency. 188 40j Station 26. Wind-stress curl spectrum versus frequency. 189 40k Station 43. Wind-stress curl spectrum versus frequency. 190 41a Co-spectrum (month ~1) .• between stations 1 & 2 versus frequency 191 41b Quad-spectrum between stations 1 & 2 versus frequency 3.. (month -1). 192 41c Co-spectrum between stations (month -1). 4 & 6 versus frequency }, 193 4 Id Quad-spectrum between stations 4 & 6 versus frequency '/ (month ~ I ) . 194 41e Co-spectrum between stations (month " I ) . 9 & 10 versus frequency < 195 41f Quad-spectrum between stations 9 & 10 versus frequency 196 Cmonth ~1). 41g Co-spectrum between stations 11 & 12 versus freqeuncy 197 Cmonth ~1). 41h Quad-spectrum between stations 11 & 12 versus frequency 198 Cmonth ~1). 41i Co-spectrum between stations 12 & 13 versus frequency 199 Cmonth - 1 ) . 41j Co-spectrum between stations 18 & 20 versus frequency 200 Cmonth ~1). 41k Quad-spectrum between stations 18 & 20 versus frequency 201 Cmonth ~1). 411 Co-spectrum between stations 40 & 42 versus frequency 202 Cmonth - 1 ) . 42a Coherence of wind-stress curl between the indicated 203 ._ stations, versus frequency. 42b Coherence of the wind-stress curl. 204 43a Phase between wind-stress curl 'signals'. 205 a. Between stations 1 & 2. b. Between stations 1 & 3. 43b Phase between wind-stress curl 'signals'. c. Between stations 13 & 15. d. Between stations 22 & 23. 206 44a Station 1. Demodulation of the wind-stress curl versus 207 time Cmonths, beginning Jan. 1945). 44b Station 2. Demodulation of the wind-stress curl versus time 208 Cmonths, beginning Jan. 1945). 44c Station 5. Demodulation of the wind-stress curl versus time 209 Cmonths, beginning Jan. 1945). 44d Station 9. Demodulation of the wind-stress curl versus time 210 Cmonths, beginning Jan. 1945). 44e Station 11. Demodulation of the wind-stress curl versus time 211 Cmonths, beginning Jan. 1945). 44f Station 16. Demodulation of the wind-stress curl versus time 212 (months, beginning Jan. 1945). 44g Station 20. Demodulation of the wind-stress curl versus time 213 (months, beginning Jan. 1945). 44h Station 23. Demodulation of the wind-stress curl versus time 214 (months, beginning Jan. 1945). 44i Station 26. Demodulation of the wind-stress curl versus time 215 (months, beginning Jan. 1945). 44j Station 43. Demodulation of the wind-stress curl versus time 216 (months, beginning Jan. 1945). PART II FIGURE 1 Path of integration in the s plane, Re s + i Im s. 224 2 . Plots of the amplitudes of the wind generated Kelvin waves 226 [(5.3) + (5.5)] at fixed longshore position, y = y D < 0, for increasing time t* - w(t + y Q / c ) , for four given values of the decay frequency, Oy- yc (co ¥ 0) . -(i) 0" = uc = OJ ( i i ) cr = T(J; yc = 2w ( i i i ) a = -rU); yc = 2OJ 1 (iv) a = 2 OJ; yc = — 0). 3 Schematic diagram of the regions occupied by the pressure 231 generated Kelvin waves of § 6 for a storm moving i n the negative y direction (a) c >V; (b) c <V. 4 Plot of the amplitudes of the pressure generated Kelvin 232 waves with respect to: (a) the leading edge (c > V), and (b) the t r a i l i n g edge (V > c), for various values of a; (i) a = -JQ CO; ( i i ) a = ! a); ( i i i ) a = 2 w. 5 Daily northward wind-stress off the Oregon coast from 238 Aug. 1 to Sept. 31, 1966. ACKNOWLEDGEMENTS Foremost, I would like to acknowledge my supervisor, Dr. Paul LeBlond, for his assistance and advice throughout the development of this thesis, and for his seemingly i n f i n i t e patience. The assistance of Dr. L.A. Mysak i n connection with the Kelvin wave problem is also gratefully acknowledged. Further, the help of Dr. G. McBean, Dr. J. Garrett and Mr. R, Wilson in my computer work and that of Dr. J. Namias, Dr. N.P. Fofonoff, Dr. F. Favorite, Mr. P. Wickett, and Mr. A. Dodimead through the data they supplied, i s much appreciated. My thanks also to Miss F. McGarry for typing this thesis. Finally, the National Research Council of Canada and the Killara Foundation are acknowledged for their financial assistance. PART I THE CIRCULATION OF THE SUBARCTIC PACIFIC REGION 1. Introduction 1.1 General discussion Since the work of Sverdrup (1947) there has been considerable progress towards understanding both the steady-state and the time-dependent circulation of the oceans. Although the ocean circulation i s in t r i n s i c a l l y coupled to that of the atmosphere, the complexity of the problem has led to simplified, decoupled models, i n which surface heating or surface wind-stress acts as the principal energy source for the currents. The latter i s considered to be dominant, except in certain polar regions (the North Pacific excluded) and i n deep (> 1000 metres) parts of the ocean. Because of the importance of the wind as a driving force, i t i s referred to as the Wind-Driven Ocean Circulation. In this thesis I w i l l be mostly concerned with the Wind-Driven Circulation i n a geographically distinguishable region of the North Pacific Ocean known as the Subarctic Pacific Region. Specifically, this region occupies that part of the North Pacific between the land barriers formed by Asia and North America, and the so-called Subarctic Boundary at about 42°N (Figure l a ) . This boundary i s identified primarily by the almost vertical isohaline of 34.0 °/oo which extends from the surface to a depth of about 200 to 400 metres i n an east-west direction across most of the North Pacific (Dodimead, Favorite & Hirano, 1963). To the north of the boundary l i e the Subarctic Waters which have their minimum s a l i n i t i e s at the surface, due to an excess of precipitation over evaporation; i n contrast, the Subtropic Waters to the south have maximum s a l i n i t i e s at the surface as a result of excess evaporation over precipitation (Jacobs, 1951). The Subarctic Boundary also coincides with a significant (by I 6 0 ° I 7 0 ° E 120", I 7 0 ° E I70 °W FIGURE la. Geography of Subarctic Pacific Region FIGURE lb. Bathymetry (metres). an order of magnitude) northward increase i n the phosphate-phosphorus concentration in the upper 100 metres (Favorite, 1969) and a marked gradient in the surface pH (Park, 1966). For reasons not fully understood, this boundary also appears to be the southern limit of the habitat of the Pacific Salmon (Favorite, 1969). To the countries bordering on the North Pacific Ocean, the Subarctic Region is one of the most important oceanic areas in the world. Each year i t yields large commercial catches of salmon, halibut and other species of fis h which contribute significantly to the economy of these countries. In response to the importance of this region, the International North Pacific Fisheries Commission (INPFC) was established by Canada, Japan and the United States in 1953 to promote oceanographic studies that would aid i n fish conservation. The result has been a greatly increased number of observational investigations, which were rather limited prior to 1955. Despite the numerous observational studies now available, there appear; to be few theoretical investigations into the structure of the circulation in the Subarctic Pacific. Two theoretical studies that did include this region were those of Munk (1950) and Munk & Carrier (1950). The pattern they produced for the steady-state circulation was unfortunately devoid of any detail, being eclipsed by the size and structure of the Subtropical Pacific Current systems. Furthermore, the Alaska Current was incorrectly interpreted to be of the-'eastern', rather than of the 'western', boundary type. Subsequent studies have tended to be mostly observational and concerned with particular features of the Subarctic such as the Alaska Gyre (e.g. Bogdanov, 1961; Roden, 1969) and the Alaskan Stream (e.g. Favorite, 1967). Essentially, the aim of this study is to present a theoretical explanation for the large scale quasi-steady circulation in the Subarctic Region. In § 1.2 the relevant oceanography of the region w i l l be discussed. In § 2 the equations of motion w i l l be obtained in curvilinear coordinates which w i l l then be used to derive i n § 3 the vorticity equation for large scale motions. Planetary wave.motion w i l l be discussed in § 4, while the time-averaged or quasi-steady motions w i l l be considered i n § 5. In § 6 the linearized form of the vorticity equation derived i n § 3 w i l l be used to present a quasi-steady, baroclinic model for the circulation of the Subarctic Region. In particular, the relatively strong flow along the more northern coast of the region w i l l be shown to change from a 'western' type boundary current to a 'zonal' type boundary current downstream.: The a b i l i t y of the latter to remain as a boundary current i s dependent upon the downstream influx of water into the boundary. Section 7 w i l l be devoted to the distribution and variation of the wind-stress curl over the region, as obtained from the calculated geostrophic winds. The variations i n the Subarctic circulation resulting from the changes in the wind-stress curl distribution w i l l also be discussed i n § 7. A summary of the work presented, as well as some concluding remarks, w i l l be given in § 8. 1.2 The Oceanography of the Subarctic Waters The purpose of this subsection is to outline some of the more outstanding features of the water structure i n the Subarctic Pacific; a much more detailed account, including a history of observational work, has been presented in an excellent summary by Dodimead, Favorite & Hirano (1963). The sa l i n i t y structure north of the Subarctic Boundary is a unique feature, of the North. Pacific Ocean (Figure 2). Although this structure varies somewhat with time and location, the following permanent characteristics are recognizable (Dodimead et a l , 1963): (i) an upper zone from the surface to about 100 metres; ( i i ) a halocline between 100 to 200 metres, i n which the; s a l i n i t y increases with depth by about 1 °/oo; and, ( i i i ) a lower zone with sa l i n i t y gradually increasing to about 34.4 °/oo at 1000 metres and then very gradually to 34.67 °/oo at 4000 metres. The surface of salinity = 33.8 /oo + 0.1 /oo can be used to define the bottom of the halocline (Doe, 1955; Dodimead, 1958; Fleming, 1958) except near the Subarctic Boundary where sa l i n i t y at that depth increases to 34.0 °/oo. U t i l i z i n g time series data at, Ocean Station 'P' (lat. 50°N, long. 145°W) (Figure la)., Tabata (1961) has shown that the greatest salinity variations occur i n the upper zone and are related directly to the growth and decay of the seasonal thermocline. Commencing after about March of each year, the surface layers of the ocean begin to retain some of the daily solar heat which is mixed down by; .the wind with the eventual formation of the seasonal (or secondary) thermocline by summer. At this time, the characteristic features of the upper zone are a nearly isothermal and isohaline layer to about 30 metres, with a secondary halocline and secondary thermocline extending down to 75 metres. With the advent of the cooling period (October-March), the upper zone becomes mixed and overturned, eventually to an isohaline, isothermal layer by the end of this period. The temperatures at the top of the halocline during summer, therefore, have a similar distribution to the surface waters i n the previous late winter; Tully & Giovando (1963) have given an ill u s t r a t e d description of this process at Station 'P'. In the lower zone, temperatures generally decrease with depth to about 3°C at 1000 metres and to about 1.5°C at FIGURE 2 . Typical salinity structures at Station 4000 metres. South of 50°N, the upper zone and the halocline becomes less distinct, and eventually vanish at the Subarctic Boundary. Here, and to the south, salin i t y minima occur with depth; the water column remains stable due to the temperature decrease with depth and the presence of a permanent thermocline. The water density in the Subarctic Region w i l l have, the same general ve r t i c a l structure as salini t y with detailed variations in the upper zone due mostly to the larger temperature fluctuations. In late winter (February-March), then, the upper zone w i l l be of uniform density, while in summer a secondary pycnocline w i l l coincide with the secondary halocline and thermocline. Figures 3a,b are plots of the density for selected stations occupied by the Canadian weatherships between Ocean Station 'P' and Swiftsure Bank (48°32'N l a t . , 125°00'W long.) (Canadian Oceanographic Data Center, 1967; 1968). The increasing amount of vertical structure in the upper layer, beginning i n late April, is clearly v i s i b l e . The horizontal distributions of salinity and temperature (hence density) are very similar at a l l depths in that they are nearly zonal, except within proximity of the coast. Salinity and temperature generally decrease northwards. From the central part of the Gulf of Alaska to about 175°E long., however, there i s a 'doming' of the water properties which is reflected in the density structure. These features are clearly visible in figures 4a,b obtained from the Boreas Expedition i n the winter of 1966 (Dodimead, 1968). In the Gulf of Alaska, the 'dome' is associated with the Alaska Gyre and Alaska Current (Figure 22), while to the west i t i s associated with the eastward flowing Subarctic Current to the south, and the westward flowing Alaskan Stream to the north. Both sa l i n i t y and DENSITY 26.00 . 27.00 SO 100 200 Z. » o o a. u o 4 O 0 i 800 28.00 L A T . <lV » e ' n L o M a i*** s*' w T o T A i . O C P T H U » M -t-SI0MA-T 20.00 27.00 i ' 28.00 LAT. SO* O*'N T O T M . O t i l T K M UT. So* oo' (4 LONG. "»5 'oo' w lH-S-<>7 (H.S «•"> T O T M . O U T * 1 Lows. i^ A* W ~ T o T M . D S P T H M S O H U V t . »-V o V M 2s-s-cr c>.o HO T o T W - O W T l t V f S 6 H FIGURE 3b. Selected density profiles for the line between Oceon Station " P" and Swiftsure Bank, April-March 1967. DENSITY: SIQMA-T 24.00 26.00 26.00 SO 100 200 • 3 0 0 a 400 8 0 0 LKT. *» n >* us***. u V » V w T O T A L O O P r K I I * X LAT. *«•»«' KJ L O M 6 . l 7 £ * O o ' w 1 4 - S - b 7 CIS.* M ' l TOTAL S c e n t I OS *  L A T . *-B>' » B N L o r t t . ' I t * o <5 - « l H - I O - f e f e ( O I . S M r . ^ T o T A L D W » T * t o q 14 25.O0 SI0MA-T 26.00 25.00 SIOMA-T 26.00 L A T . l o ' o r M Lor*>. ! « * • « ' W " | . - M - . e < U . - f « ' ' . ' > T o T A L D l P T H 1 U O H L M . « o " oo 'rO L O N G , i v v sa' r* H - l I - 6 b l " V O Mr. ) T O T A L » W * » M . O M 27.00 L A T . 4 4 * W »l L O H G . I X T ' 4 - O ' W T O T A L O t r T H I S O * » LAT. 4 * " C l ' M LoHft . n a - 4o* *W Z 4 - S - 0 7 l f l * H ^ l T O T A L r * r r « » i T » « FIGURE 3a. Selected densi ty (sigmor-t) profiles for the line between Ocean Station"?" and Swiftsure Bonk, October 1966 to March 11967. sigma-t » (density - I ) x I08-temperature decrease coastward of the 'dome'. The annual variations of s a l i n i t y i n the upper zone are small (about 0.2 °/oo), with the salinity minimum occurring during summer and the maximum in late winter, in oceanic regions. Towards the British Columbia and Alaska coasts, surface salin i t y variations are associated with river runoff i n addition to the amount of r a i n f a l l . Annual ranges of surface temperature are f a i r l y large; at Station 'P', temperature maxima of about 13-14°C in summer decrease to about 5-6°C in winter (Tabata ,1961). Also, because of the onset of north to northwest winds in summer, the west coast of North America, particularly off Oregon, experiences lower temperatures than the oceanic regions at the same latitude due to the upwelling of colder water. The current structure i n the Subarctic Region w i l l be considered in detail i n § 6 so that only a brief description, based on the summary of Dodimead et a l (1963), w i l l be given here. On the western side of the North Pa c i f i c Ocean, part of the Kuroshio current, which carries warm saline water northwards, turns eastward off Japan to form the eastward flowing North Pacific Current (see f i g . 22). This turns south near 150°W longitude. The remaining northward flowing branch of the Kuroshio mixes with part of the cold, lower salin i t y Oyashio Current which is flowing southward along the Kuril Is. The waters formed by the confluence of these two currents then proceed eastward as the slow West Wind Dr i f t . This current divides off the Washington-British Columbia coast, one part going north into the Gulf of Alaska, the rest moving south as the California Current. Finally, that part of the Oyashio that doesn't mix with the Kuroshio forms the eastward flowing Subarctic Current which turns north into the Gulf of Alaska region on the eastern side of the Subarctic T a c k l e . The return flow i n this region is westward along the boundary formed by the Alaska peninsula and the Aleutian island chain. LAT 41° 42° 43° 44° 45° 46* 47* 48° 49° 50° 51* 52* 53* F I 6 U R E 4 0 . Density (crt) distribution along long.l75*E. Winter 1966 . Winter 1966. 2. Equations of motion 2.1 Navier-Stokes Equations We consider a nonuniformly rotating ocean of variable depth and density, having so l i d land barriers to the east and west, a p a r t i a l land barrier to the north, and an oceanic boundary to the south (the Subarctic Boundary). In Cartesian coordinates the east and north axes are X and Y, respectively, while z is the vertical coordinate, positive upwards; the corresponding velocity components are U, V and w. The origin i s taken at mean sea level at the intersection of the northern and eastern boundaries in the corner i f the Gulf of Alaska (figure 5). In order to model the curved northern coastline of the Subarctic Pacific Region, i t is convenient to transform to curvilinear coordinates ( s e e e.g. Goldstein, 1965; p. 119). In these coordinates, the local curvature, K, depends only upon the distance (x) along the boundary formed by the Alaska peninsula and Aleutian-Komandorski island chain (henceforth referred to as the Northern Boundary) and not upon the distance (y) perpendicular to the boundary, nor z; figure 5. Increments in x and y are given by dx[l - K(x)y] and dy , respectively, so that lines of constant y are para l l e l curves of curvature K, while lines of constant x are straight lines. Convergence of the y coordinate lines w i l l not be considered important, thereby confining our model to small curvatures. The east coast, north of about 47°N, w i l l be approximated by a straight line rotated by an angle 0 Q to constant X. Since this study w i l l concentrate on the flow structure east of the Kuroshio and Oyashio current systems, there w i l l be no attempt to model the western boundary. Coordinate systems X,Y Cartesian coordinates x,y curvilinear coordinates The Inverse transformations, from x, y to X, Y are given by, X = J^[l - k(?)y] cos0CO d£ - y sin6 p , (2.1.1) o f l l - K(5 ) y ] sin0(Q d£ + y cos0 o , (2.1.2) Y = or in. which 6(x) measures the angle between lines of constant y and Y, 6(0) f= G Q and where, k C x ) . . deoo ^ ' 3 ) dx The corresponding inverse velocity transformations are: U = u [ l - K(x)y] cos0(x) - v sin0(x) , (2.1.4) V = u[l - K(x)y] sin0(x) + v cos0(x) , (2.1.5) in which u and v are velocities i n the x and y direction, respectively. If we then consider the general vector form of the conservation of momentum equation, and apply the rules for transforming operators (see Batchelor, 1967; pp. 598-599), we obtain for the horizontal components in curvilinear coordinates, 1- u'Vu - uv - fv dt ~ 1 - K y 1_ 1 9p_ r 1 3_ , 3 u / 3 x , 1_ . K V , 1 - K y p 3x 1 - K y 3x 1 1 - K y J 1 - K y 1 1 - K y J + 1 + |?u C 2 > 1 < 6 ) 1 - K y 3 x ^ y 3y 3z2 | r + u-Vv + -=-£— u 2 + fu at ~ 1-Ky 1 3p_ , r 3 r 3u/3x •, _ "3 ,/ v -• , 3 2 v 1 , 3 2v 1-tcy -3x 3z2" (2.1.7) in which t is the time, f i s the Coriolis parameter (f = 2fisin<J) where is the angular rotation of the earth and (J> is the latitude) , p is the water density, p i s the pressure, £ i s the vertical component of relative vorticity and V is the kinematic viscosity. If we make the beta-plane approximation, in which the spherical surface of that part of the earth under consideration i s mapped on to a tangent plane (Veronis, 1963), we have further, f(x,y) = f Q + 3 Q Y (x,y) , (2.1.8) where f Q and 3 Q are constants. This approximation is valid here since we are considering a comparatively small portion of the earth's surface.. The conservation of mass for an incompressible f l u i d i n this coordinate system i s , |^ + |^ [ (1 - icy) v ] + C L - icy) f | = 0 , (2.1.9) while the local vertical component of relative vorticity i s given by, 2.2 The vo r t i c i t y equation If we cross-differentiate (2.1.6) and (2.1.7) and subtract, we obtain the general equation for the ver t i c a l component of vorticity, - ' . ^ g ^ , ^ * . (2.2.1, in. which k i s a unit vector i n the positive z-direction, 1 - ¥ + - v - ( 2-2-2 ) is the Lagrangian derivative following the motion, and VH'U = + |^ [ (1 - Ky)v] , (2.2.3) In the analysis of the following sections, we w i l l always consider the term VpxVp'k as negligible, j u s t i f i c a t i o n for which w i l l be given i n | 5.3.5. Using the divergence equation (2.1.9) we may then write (2.2.1) as, 4r (f + 5) - (f + C) IT " + (Vwx|^) -k (2.2.4) dt oz dz We note here, for future reference, that i n curvilinear coordinates, v-v<&> (2.2.5) { £ _ T _A_ 1-Ky 3x 1 1-Ky 3$ -, + k I C 1 " ^ ¥ ^ i - < y ) 0 > f o r any s c a l a r f u n c t i o n <£>. 3'. The vor t i c i t y equation for large scale motions 3.1 Derivation In this study, I w i l l be dealing mainly with oceanic motions having horizontal scales greater than 10 kilometres and periods longer than one week. I therefore choose one week as the period over which to do a time average. Then, considering an ensemble of such averages, in which the ocean response is obtained under a similar set of conditions, i t is to be expected that the variations associated with shorter period motions such as surface waves, tides, internal waves, turbulence, etc., w i l l occur as fluctuations about a 'mean' for the chosen averaging time. The mean velocity i s assumed to represent those motions that change l i t t l e over periods less than a week. Such an averaging process is j u s t i f i e d i f the amount of energy associated with the one week components i s indeed small compared to shorter and longer period motions; the mean and the fluctuating motions are, therefore, separated by a 'gap' of low energy in the energy spectrum. At present, due to a lack of measurements at periods close to a week, one cannot state definetly that this is the case. The above separation, therefore, cannot be considered to be rigorous. The shorter period responses must, of course, be ultimately considered i f one is to understand the detailed processes of energy influx and dissipation in the ocean. We now let the velocity be separated into a mean component (overbar) and a fluctuating component (prime), viz., u = u + u' , (3.1.1) C = X + C where the characteristic period of the primed components is less than a week. Substituting (3.1.1) into (2.2.4) we find, upon taking the time average, denoted by < >, and requiring that the average of the fluctuating component be zero, that (3.1.2) = VV«'V? < { Vx I u'-Vu' J } -k > where "d : 3 , - _ dt = 3T + r v i s the total derivative for the mean motion, and where < { Vx [ u*-Vu* } 'k > = <; u*-VC' + Vw'^'k - > ~ dz dz ,is the nonlinear effect of the fluctuating vorticity on the time-averaged vortic i t y equation. The term -p < u'-Vu' > = R(x,t) is the divergence of the Reynolds stress, which results from the nonzero correlation between the various motions making up the fluctuating component. For the large scale motions being considered here, we w i l l assume that this is a dissipative mechanism; i.e. that < { Vx R }'K> is positive i n (3.1.2). The dissipative effect of turbulence, wave breaking etc. may then be simulated by assuming that the Reynolds stress is proportional to the strain rate of the mean flow, with the proportionality constant defined as the 'eddy viscosity coefficient' (Fofonoff, 1962). As i n the latter reference, we w i l l distinguish between a ver t i c a l and a horizontal eddy viscosity coefficient. Although i t i s further possible to consider these as variable, they are at best crude approximations to turbulent viscosity so that there i s no particular advantage in doing so. Following the above discussion, we assume, for the sake of concise notation, that u'-Vu' = - A k V-Vu where, since the ver t i c a l oceanic scales are much smaller than the horizontal scales, we have A^ = Aft for horizontal components, (3.1.3) and A^ .= Ay for ve r t i c a l components, with A v « Ah • Using the vector identity, W u = V(V-u) - Vx (Vx u) (3.1.4) we find Vx (V-Vu) = - Vx [Vx(Vx u)] = - Vx [Vx 1] where (3.1.5) C = Vx u (3.1.6) is the three dimensional form of the vorticity. If we again apply the identity (3.1.4), we find Vx(V-Vu) = V-VC - VCV-I) = V-V| , using (3.1.6). Equation (3.1.2) can then be written, = (V + A, ) V-V? (3.1.7) which has the same form as the nonaveraged vorti c i t y equation (2.2.4). As a matter of convenience, the overbar w i l l usually be omitted i n the remainder of part I. In the analysis to follow, we w i l l assume further that, p (V + | | = VXT -k , (3.1.8) where T = ( T x , xy, 0) is the horizontal shearing stress. At the ocean surface the right side of (3.1.8) becomes the wind-stress curl, viz., •'(vxT-'-k) = Vxx w -k , (3.1.9) where ri = n(x,y,t) i s the deviation of the ocean surface from i t s mean level, z = 0. . 3.2 Nondimensionalization An insight into the relative importance of the individual terms of (3.1.7) can be obtained by introducing characteristic scales for the dependent and independent variables. Such scales are intended to approximate typical values for the variables involved. As discussed by Veronis (1965), however, one cannot always be certain that the numerical value chosen for a particular variable is representative. The ultimate test of one's choice i s , of course, whether the results of the theory are consistent with observation. We transform to nondimensional (tildes) variables via the following system of equations: (x,y) = LCx,f) , z = Dz , t = Tt K •= K 0K , p = p 0p , f = f j f (3.2.1) CIx~ ' 9 y 5 f = go C 9 j , ) f , . ' Ou,,v) = U0(u,v) , w = WDw , £ = U0/Lc; , x = T 0T , where in the oceanic interior: L i s the horizontal distance of significant change i n the motion having a characteristic time T and horizontal speed U D; D i s the characteristic depth below which the vertical gradient i n horizontal velocity becomes small compared to the vertical velocity gradient i n the upper layer; K 0 -1 i s the radius of curvature of the Northern Boundary; WQ i s the vertical velocity scale; and T Q approximates the surface wind-stress magnitude. The horizontal scales of the current structure are not independent but related by T = L/UD .. • (3.2.2) Substituting (3.2.1) into (3.1.7) (dropping the overbars) and using (3.2.2), we obtain the nondimensional form of the vorticity equation; RG || - 6* Cf + R 0?) H + u-V CB*f + Ro« + R0S* V wx'l^ -'.k ° dz (3.2.3) i n which the parameters RQ, 8*, 6*, •£*, and axe defined by: *° Lf 0 ' P f Q - ' ° D U D ' E K ° L ' _ AH+V _ _ T N ™ " L 2 f Q > ^ " p 0DU 0f 0 ' In the expanded form, the continuity equation (2.1.9) can be written •P + -I- [ (l-£*Ky)v ] + 6* Cl-e*icy) |t = 0. 13.2.5) dx dy dz (3.2.4) Rowever, as we w i l l also be dealing with the vertically integrated equations, we include the integrated yersibnof the continuity equation in our present discussion. Assuming that the horizontal velocity components are Independent of depth., we integrate (2.1.9) from the ocean bottom, z = -H(x,y), to the surface, z = r|(x,y,t), and obtain, in nondimensional form, |~ [ (6 n + H) u ] + |~ [ (l-e*Ky) (6 n + H) v ] (3.2.6) + 6(l-e*icy) |f = 0 where R = H 0 H , n = n 0n , and $ = Bo Ho is a divergence parameter associated with the barotropic (depth independent) motions. The latter parameter is obtained by retaining the time variations in the surface elevation. The limit 6 -*• 0 is associated with nondivergent .motions; the result of imposing a r i g i d top. In the case of the non-integrated continuity equation (3.2.5), one may formally assume that a l l terms are of the same order, whereby 6* ~ 0(1). For nearly steady (quasi-steady) motions, however, we w i l l consider 6* < 1, basing this partly on calculated data and partly on the assumption that a l l Coriolis terms in (3.2.3) w i l l be of the same order of magnitude. There i s no loss of generality in the latter statement since, for 6* ~ 0(1), one simply has WQ = constant=0 to 0(3*) when the bottom boundary condition [W(z= -H) = 0] is applied. The resulting equation i s then the same as that obtained by simply taking 6* < 1 i n i t i a l l y . 3.3 Numerical values Numerical values are readily obtained for some of the parameters introduced in § 3.2. Values for the remaining parameters are determined only after the type of motion one is interested in i s specified. Typical values, common to a l l large scale oceanic responses i n the area under consideration, are taken as: f Q = 1.195 x-10-4 s-l , B 0 = 1.313 x lCT^cm-is- 1 , 107cm < L < 108cm , 108cm < K 0 - l < l()9cm , (3.3.1) p Q = lgm cm - 3 , = 6 x 10 6cm 2s - 1 , V = 1.5 x 10~ 2cm 2s - 1 , T Q = 1 dyne cm - 2 . -The Coriolis parameters (f 0,B 0) are evaluated at 55°N, while A H i s taken from Arons & Stommel (1967) who give typical values of 6-7 x H)6cm2s based on the steady-state balance of advection and lateral diffusion of dissolved oxygen in the North Atlantic. The magnitude of the radius of ..curvature, i | . K 0 _ l | , i s , in general, variable but for the present w i l l be taken as approximately the mean curvature of the Northern Boundary. The latter i s consistent with the neglect of the convergence of the y-coordinate lines. Values for L are considered to be typical for both wavy motions: and for the large-scale wind systems. Also, the upper limit of L of 1000 kilometres i s about the maximum allowable length scale within the beta-plane approximation; studies at larger scales necessarily include the curvature of the earth's surface. The lower l i m i t , 100 kilometres, i s undoubtedly below the minimum horizontal scale for which the results of Arons & Stommel (1967) are applicable. Despite these shortcomings, we use (3.3.1) in (3.2.4) and find: 1.1 x 10-2 < 3* < l . l x 10-1 , lO- 2 < e* < 1 , (3.3.2) 5 x 10 - 4 > E H > 5 x 10 - 6 . These values w i l l be applied to both the periodic and quasi-steady motions to be studied in the sections that follow. 4. Planetary waves 4.1 Introduction Since the motions satisfying (3.1.7) have relatively long periods and large horizontal scales, they are strongly influenced by the earth's rotation. For 'thin,.stratified sheets' of f l u i d , such as the ocean, the ^nost important component of this rotation i s the one locally v e r t i c a l to the surface of the earth, the Coriolis parameter. The motions that occur away from the boundaries at large'scales, then, are mainly geostrophic; that i s , they are a result of a balance between horizontal pressure gradients and the Coriolis force, In general, there w i l l be a departure from geostrophy i n the interior regions because of nonconservative forces (e.g. friction) non-linear effects and time variations. However, these are usually not large and the resulting flow can be said to be quasi-gepstrophic. Even in the case of intense boundary layer currents, such as the Gulf Stream, the downstream component remains quasi-geostrophic, although the cross-stream component is strongly influenced by nonlinearities, bottom topography and turbulent f r i c t i o n . Two types of large scale, nearly geostrophic, oceanic responses w i l l be considered in the sections to follow. Both w i l l occur simultaneously with interactions between them generally assumed negligible, a j u s t i f i a b l e assumption.provided certain conditions are satisfied by their relative scales. A discussion on this w i l l be given i n the Appendix (I). The f i r s t , known as planetary waves, are periodic quasi-geostrophic motions resulting from the application of surface forcing to the ocean. Because of their broad range of time and space scales, they form the basic transient response of the sea to variations in the surface driving forces for periods greater than a half pendulum day. The fact that these motions are nearly geostrophic suggests that they may be regarded as moving current systems. In middle to high latitude ocean basins, the accumulated effect of these transient planetary waves over a long period (> 100 years) is to establish an integrated (mean) circulation. As a result of the repetitiveness of the annual form of the atmospheric forcing, the integrated circulation over each year occurs as a modified version of the climatological mean current patterns, the amount of departure depending upon the accumulated effect of the planetary wave scales in the appropriate year. This i s particularly true of the depth independent velocity mode (barotropic mode) which constitutes the major form of response for periods between one week and about a year; nonlinear interactions with the depth dependent velocity mode (baroclinic mode).are neglected. The circulation patterns, averaged over some period, are 'steady' within the averaging period, although they are really an approximation to continuously varying current systems. They represent the second type of motion to be studied and w i l l be termed 'quast-steady' motions. Planetary waves then occur as relatively rapid transients superimposed upon these longer period, quasi-steady motions. Furthermore, i f the fluctuations of the atmospheric forcing terms are not large within even shorter, periods (months), i t is tentatively possible to discuss the current variations about the mean flow over that period. Planetary waves in a f l u i d of constant thickness (also called Rossby waves) were studied f i r s t by Rossby (1939). Using the beta-plane approximation he was able to account for the wave-like appearance of upper atmosphere isobaric surfaces. Recently, there has also been an increasing amount of interest i n planetary waves to account for the v a r i a b i l i t y of the ocean. Veronis & Stommel (1956) showed, that for a constant depth, unbounded, two layer, beta-plane ocean the response to moving wind systems was in the form of barotropic and baroclinic Rossby waves. Further, they found that at mid-latitudes most of the energy goes into barotropic waves for periods of forcing of one to seven weeks, while for very long periods (of the order of decades) the energy goes into baroclinic waves. For intermediate periods, both modes attain part of the wind energy. A similar study was done by Fofonoff (1962) who also included the effect of a constant steady velocity on the wave motions. More recently, L i g h t h i l l (1969) has shown that a linear Rossby wave theory is applicable near the equator and that, because of the smallness of the Coriolis parameter there, the response is mainly baroclinic for short periods, contrary to mid-latitudes. The result of this is that the response of the Indian Ocean to the Monsoon is rapid (1 month), creating the strong, highly baroclinic northward flowing Somali Current. Planetary waves, for which the depth is varying (topographic waves), are also important i n understanding the time-dependent and steady-state response of the ocean, Veronis (1966), for example, showed that over most of the ocean, topographic effects dominated the beta effect. Studies by Warren (1963), Robinson & N i i l e r (1967), N i i l e r & Robinson (1967), and Warren (1969), for steady motions, show the importance of topography in controlling the Gulf Stream meanders. A recent study by N i i l e r & Mysak (1970), for long period oscillations along the western shelf of the Atlantic Ocean in the presence of the Gulf Stream, shows the possibility of unstable Continental Shelf waves accounting for the observed meanders along the continental rise of this region. Finally, Rhines (1969a,b) has demonstrated how such waves may be trapped around islands, seamounts and along depth, discontinuities. Despite the probable Importance of planetary waves i n determining the time dependent response of the large scale ocean circulation, no direct observation of them has been made. However, various authors have inferred their existence. Lpnguet-Higgins (1965a, p. 62), for example, suggests that certain deep water velocity measurements obtained by Swallow (1961) off Bermuda could be due to internal Rossby waves. By analyzing t i d a l data off the Oregon coast of the United States, Mooers & Smith (1968) have shown there is a dominant period of 0.35 c.p.d. corresponding to topographic waves in;the form of Continental Shelf waves. Further evidence of topographic waves was obtained by Thompson (1969) from the long-term current records taken at Woods Hole Oceanographic Institution site 'D'. The quasi-steady motions that w i l l be considered in;the next chapter are those to which one usually refers when describing the circulation patterns of the ocean. That they exist at a l l i s , of course, due to the fact that the atmospheric forcing i t s e l f can be viewed as fluctuations about, say, an annual mean. The variations i n this annual mean give rise to a trend which i s important i n determining the changes in the mean annual oceanic circulation and vice versa, since the two represent a coupled system. 4.2 Barotropic Rossby waves If we assume that horizontal f r i c t i o n i s negligible, then use (3.1.8) i n (3.1.7) and vertica l l y integrate the resulting equation over the depth of a water column, we find that for barotropic motions the free solutions must satisfy, ~TJ[^ G(P) (4.2.1) where G is an. arbitrary function of the pathline, P, of a given water f + r column. Along a pathline, the potential vorti c i t y , + ^, must be conserved in the absence of nonconservative forces and internal f r i c t i o n . For the case of Rossby-type planetary waves, any northward displacement (f increasing) of an i n i t i a l l y zonal moving water column requires a compensating decrease in the relative vorticity and/or an increase i n the surface elevation, T\. Both give rise to a restoring force which causes the pathline to turn towards smaller f. Since the Rossby number RQ = £/f approaches unity for small horizontal scales, we expect the relative vorticity effect to dominate over the vortex stretching for these scales. It can be seen from (4.2.1) that the effect of a decrease in H , for- topographic waves, w i l l be analogous to an increase i n f i n the case of Rossby waves. This effect has been used by Warren (1969) in an attempt to explain the branching of the Gulf Stream south of the Grand Banks of Newfoundland. The conservation of potential vorticity when f is also allowed to vary has been employed successfully by Warren (1963) and by Niiler'& Robinson (1967) to explain the Gulf Stream meanders east of Cape Eatteras. Both studies use integration of the potential vorticity along bottom topography similar to that followed by the Stream, for various i n i t i a l conditions. It is found that topographic effects dominate over the Coriolis effects i n determining the downstream quasi-geostrophic motions in the Gulf Stream. 4.3 Some properties of Rossby waves We have seen from tbe physical discussion of the previous section that Rossby wave motion can result i f the time rate of change of relative v o r t i c i t y , 8C/9t, is balanced by the planetary vorticity effect. That i s , i f the Rossby number Ro = 3* (4.3.1) in (3.2.3). Furthermore, from (3.3.2), f r i c t i o n may be neglected for the horizontal scales of interest, although for shorter scales i t w i l l undoubtedly be important. Also, since we wish only to examine the properties of already generated Rossby waves we may neglect the forcing term in our present discussion. Therefore, i n the case of nondivergent (6 = 0) barotropic motion, v e r t i c a l integration of (3.2.3) over the total depth, with the use of the above results and the integrated continuity equation (3.2.6), yields (4.3.2) 0 , where for convenience the tilded notation has been dropped, i n C3.2.6) a stream function, may be defined by; uH = -9r/dy , (l-e*Ky-)vH = 3T/9x , l i { f c C H v ) - h 1 (1-£*Ky) Hu.]1 + (H u) + (l-e*Ky) (H v) + (l-e*Ky) Hu-V? = dx dy • ~ . With 6 = 0 (4.3.3) so that (3. 2.6)^is satisfied identically. Equation (4.3.2) becomes, in vector notation, | ^ ( V - W ) + W x V( f + - 0 ( 4 . 3 . 4 ) the standard nonlinear, nondivergent Rossby wave equation. The linearized form of this equation can be seen to have periodic solutions of the form, T = exp I i(k X + 1 Y - cot ] (4.3.5) where k = (k,l,o) is the horizontal wavenumber along (X,Y,z) and co i s the radian frequency. The dispersion relation i s therefore, nondimensionally, -k k2+l2 to = . ( 4 . 3 . 6 ) For the horizontal scales being considered here, this equation is f a i r l y accurate. The error introduced in the zonal phase speed, to/k, by assuming nondivergent motion is about 10% for the maximum wavelength of 1000 kilometres (Clarke, 1970), and the error decreases for decreasing wavelength. For larger scales, however, the 'tidal effect' (the surface displacement, n, due to long waves) is no longer negligible. The linear, divergent dispersion relation, 6 ^ 0 , has been shown by Longuet-Higgins (1965b) to be, to = — . (4.3.7) k2+l2+f 02/ g D As indicated by equations (4.3.6-4.3.7), the zonal phase speed, c = to/k, i s always to the west,and using inverse transformations from (3.2.1) can be seen to have a value, c ~0(3 oL2) . (4.3.8) At 50°N this corresponds to speeds of about l l m s - l for wavelengths of order 1000 kilometres, and 11 cm s _ l for those the order of 100 kilometres, The barotropic response for short scales i s therefore relatively slow. The group velocity, Cg, i s given by k 2 - l 2 2kl Sg = ((k 2+l2)2 • (k2+l2)2 > °> <4-3-9> and i t always makes an angle with the eastern direction of twice that made by the phase speed, c. The angle <}> the group velocity makes with the eastward direction satisfies ^ , '•-301.30) 2kl . i n . t a n ( ) > = 3 1 % " k2Tl2 > (4.3.10) and i f we consider the three possible cases, • i) k 2 » l 2 i i ) k 2 ~ l 2 (4.3.11) i i i ) k 2 « l 2 we see that only in the second case is the direction of energy propagation not restricted. For the two remaining cases ( i , i i i ) " , |(f>| ~ 0. Thus, when one of the horizontal wave scales i s small compared to the other, the energy associated with Rossby wave motion i s confined to nearly zonal propagation. Typically, the characteristic north-south length scales of the surface wind-curl in the North Pacific Ocean (as well as other world oceans) i s much smaller than the corresponding east-west scale, so that we can expect case ( i i i ) rather than case (i) to occur most often. If we then substitute k 2 « l 2 into the equation for the zonal component of the group velocity (4.3.9), we find, k 2 - l 2 CgX = (k2+l^)2 < 0 ; the energy propagates westward. In case (i) the opposite occurs. Thus, for large east-west horizontal scales, Rossby waves carry energy westward, while for short scales, they carry i t eastward. As suggested by Pedlosky (1965), the waves reflected at the 'western' oceanic boundaries w i l l consist of short waves, while those reflected at the eastern boundaries w i l l consist of long waves. This i s an important result since i t can, at least p a r t i a l l y , explain why intense currents such as the Kuroshio Current and Gulf Stream are found along the 'western' boundaries. (We have used i t a l i c s because, by 'western', we w i l l mean any sol i d continental boundary that acts as a barrier to westward propagating motions, and is therefore not necessarily s t r i c t l y north-south). 5. Quasi-steady motions 5.1 Numerical values In the case of the second type of motion to be considered in this thesis (i.e. quasi-steady motions; § 4.1), the parameters (3.2.4) which determine the important mechanisms in the vorticity equation can be obtained by reference to observational data. Therefore, i n addition to the numerical values (3.3.1), we find: L = 108cm , D = 2 x 104cm , Wc = 10_4cm s ~ l , (5.1.1) U D = 4 cm s - 1 where the upper bound of the horizontal length scales in (3.3.1) i s the most appropriate to the interior region of the Subarctic, D i s the depth below which vertical variations in the horizontal velocity become small relative to those in the upper layers, WQ is the mean vertical velocity over this depth and U 0 i s the mean d r i f t velocity. The parameters (3.2.4) then become: Ro = 3.4 x 10-4 t g* = o.l , 6* = 0.1 , 0.1 < e* < 1 , (5.1.2) E H = 5.0 x 10-6 = o.1 . As the Rossby number, RQ, measures the relative importance of the nonlinear terms to the Coriolis terms, or alternately, the ratio of the relative v o r t i c i t y to the planetary vorticity, we may safely ignore nonlinearities in the interior regions. Similarily, the smallness of the horizontal Ekman number, E H, indicates the unimportance of the horizontal f r i c t i o n compared to the Coriolis effect in the interior. However, as i n the. case of the nonlinearities, horizontal f r i c t i o n may be important in boundary regions. 5.2 A simple thermohaline circulation 5.2.1 Equations of motion In the interior of the ocean i t i s evident from (3.2.3) and (5.1.2) that the dimensional form of the quasi-steady vorticity equation i s to a good approximation, + u-Vf = - | - ( v x T . k ) . (5.2.1) 3z ~ P oZ ~ If we now define a function 5(x,t) by, I d| _ 3w ( . £ dt " 3z ' V.t.t) and substitute into (5.2.1), we obtain, iF (f> " hh<**Z'b (5.2.3) where d/dt is for quasi-steady motions, and 3/3z (Vxx-k) is the vert i c a l gradient of the curl of the mean horizontal shear. More specifically, in the absence of any shearing forces, as could occur below the upper wind-stress driven Ekman layer, the potential vort i c i t y , f/£, is conserved along a mean pathline (quasi-geostrophic motion); that i s , f/5 = G(P M) ' (5.2.4) where G i s now a functi on of the mean pathline, Pj^ « A qualitative picture of the quasi-steady interior flow, below the depth of direct wind forcing, may be obtained by considering the various forms of 9w/3z. Physically, the process is linked to a simple thermohaline circulation since a vertical flux of relatively cool, high sa l i n i t y water is required to maintain the observed density structure in the Subarctic Pacific (Figures 3a,b). This, i n turn, gives rise to vortex stretching, d£/dt, which i s then balanced by the horizontal advection of planetary vort i c i t y . A similar interpretation was f i r s t given by Stommel (1965). .5.2.2 ow/8z = 0 In this case, the long-time average of the vertical velocity i s maximum at some depth z = Z(x,y,t), say, or constant over some depth interval. In the former case, such a depth could be associated with the permanent halocline where, because of. the divergence of the surface wind-induced Ekman layer, compensating vertical velocities would be large. Roden (1969), for example, using data from a February 1967 oceanographic cruise, calculates values of 0(10-3cms-l) for ve r t i c a l velocities at the top of the halocline in the Gulf of Alaska. If we assume that Z represents a continuously connected surface, then from (5.2.2) we see that, X = constant and the motion must be zonal at this level. The same holds for 'layers' of constant vertical velocity, such as over f l a t regions of the ocean bottom away from the bottom Ekman layer. 5.2.3 3w/3z > 0 .If an element of f l u i d moves into a region i n which the vertical velocity is increasing upwards, then following the motion, £ w i l l increase. Since.f/^ must remain constant, the Coriolis parameter, f, must increase; the flow turns northward. With the maximum in the vertical velocity being at some upper layer depth, Z, the vertical velocity would be expected to decrease towards the bottom. If i t does so monotonically the deeper waters should have a general northward trend. 5.2.4 3w/3z < 0 This case is obviously the opposite of the previous one so that the flow w i l l have a general southward trend. Therefore, i n the absence of f r i c t i o n and nonlinear effects in the oceanic interior, a simple ve r t i c a l velocity structure such as shown in figure 6 should result i n a southward trending upper layer and a thicker, northward trending lower layer. The fact that the inflection point, Z, w i l l occur at different depths over the region due to differential surface heating, evaporation and Ekman divergence means that at any particular level the flow w i l l vary, depending on whether § 5.2.2,.§ 5.2.3 or § 5.2.4 holds. The result would be a meandering type of large-scale pattern, particularly at depths where a l l three cases could occur in distances of the order of 1000 kilometres. Although the previous discussion gives a qualitative picture of a simple thermohaline circulation i n the interior, i t is presently impossible to obtain quantitative results since the detailed mean verti c a l velocity structure is unknown. As a result, one usually resorts to assuming that the horizontal velocities are depth independent, except, 2 0 0 4 0 0 e oo in 03 1_ E 0. Ill o • O O J IOOO 1 2 0 0 1 4 0 0 V E R T I C A L V E L O C I T Y Xio"* era aT (upwards) 6 1 6 0 0 4 FIGURE 6. A possible form for the mean vertical velocity structure. perhaps, i n the thin Ekman layer, or to ver t i c a l integration which yields transport equations. The thermohaline and wind-driven circulations are thus considered to be uncoupled. 5.3 Vertically integrated flow 5.3.1 The Ekman and geostrophic modes Although we have avoided inclusion of the horizontal stresses in the previous discussion by considering motions below the surface Ekman layer, they are, in fact, important in producing vertical velocities in the upper oceanic layers through the process of Ekman layer divergence. Below the base of the main pycnocline, however, thermohaline processes w i l l also be important in determining the vertical velocity. In order to understand how the Ekman layer divergence works to produce v e r t i c a l velocities, we consider the vertically integrated equations in the manner of Stommel (1965). If we integrate (5.2.1) from a depth z = -H(x,y), where the vertical transport, pw, is assumed to be negligible, to the mean ocean surface, we find M-Vf = Vxxw-k , (5.3.1) (where (3.1.9) has been used and where M = (Mx, My, 0) = Jp(u,v,o)dz , (5.3.2) -H is the mass transport. The fact that the fractional, vertical rate of change of density (p - 13p/3z) is much less than the fractional v e r t i c a l rate of change of vertical velocity (w-13w/3z) has also been used i n deriving (5.3.1). Equation (5.3.1) i s the Sverdrup relation, which expresses a balance between the vorticity added to the ocean through the upper surface by the wind, and the advection of planetary vorticity. In Cartesian coordinates this equation can be written as 3M Y = VxTw-k , (5.3.3) so that the wind-stress curl determines ex p l i c i t l y the steady, total north-south transport at any particular locality in the ocean, far from the lateral boundaries. The continuity equation (3.2.5) gives, upon integration, the further equation, VH-M = |fX + | ^ [ Cl-Ky)!!?) ] = 0 . (5.3.4) Following Stommel (1965), we:separate the transport into two parts: the f i r s t M,g., M^ , represent the Ekman wind-drift transport components, while the second, Mg, MY, represent the geostrophic transports. Then using the linearized, horizontally-frictionless, steady forms of (2.1.6-2.1.7) we set, - f (iHCy)Mj « ; fM* = ... (5-3.5) and -f(l-Ky)tf£ = (l-Ky ) T* ; fM| = T £ , (5.3.6) where P = Jp dz, and p(z - -H) is assumed to vanish. Cross differentiation of (5.3.5) shows that (5.3.7) while c r o s s - d i f f e r e n t i a t i o n of (5.3.6) gives, VxTw-k - M E-Vf f (5.3.8) The t o t a l h o r i z o n t a l divergence i n t h i s quasi-stationary process vanishes by (5.3.4) so that the sum of (5.3.7) and (5.3.8) leads back to (5 . 3 . 1 ) . As pointed out by Stommel (1965), however, the separation i n t o two modes shows that t h i s l a t t e r equation i s simply the r e s u l t of v e r t i c a l v e l o c i t i e s i n t o or out of the Ekman layer causing vortex s t r e t c h i n g i n the adjacent water below. This transmits v o r t i c i t y as the 'spin-up' process, with the r e s u l t that a compensating, convergent geostrophic flow occurs, as seen by (5 . 3 . 7 ) . The v e r t i c a l v e l o c i t i e s are upwards f o r a diverging Ekman la y e r and downwards f o r a converging one. Their maximum values occur near the base of the la y e r at about a depth (Ekman, 1905) and using the numerical values ( 3 . 3 . 1 ) . We may obtain the v e r t i c a l v e l o c i t y at the bottom of the Ekman 1/2 ~ 40 metres D i s found D V assuming a balance of C o r i o l i s force and v e r t i c a l f r i c t i o n l ayer, z =-z E' by i n t e g r a t i n g (5.2.1) from the surface to t h i s depth. With the r e s u l t s j u s t given, i t can be shown that, % * J V x T w - k - C M E + M*) - V f J / f p (5.3.9) = VH-ME Mg-Vf p " p f ' where and Wg i s the v e r t i c a l v e l o c i t y at the base of the Ekman l a y e r . I f we assume that the h o r i z o n t a l geostrophic advection i s small i n the Ekman layer, and use the fac t that i n (5.3.8), M ..7f = V f x (5.3.10) we may obtain an estimate of Wg d i r e c t l y from the surface wind data, v i z . , % - ^ # - * S £ £ . (5.3.11) In the i n t e r i o r regions, away from the coas t a l b a r r i e r s , the magnitude of the wind-curl term i s usually much greater than the wind-stress term and so determines the Ekman divergence. I f i t i s assumed that over a number of years, the mean v e r t i c a l v e l o c i t y produced by the mean wind-c u r l gives r i s e to stationary p o s i t i o n s of the isopycnals, then, since the Ekman l a y e r response to the wind i s rapid (~ 1 day), deviations about these l e v e l s should be r e l a t e d to deviations of the wind-curl from i t s mean. Further, as the response i s bar o t r o p i c for periods of weeks to a year, such deviations should produce s h i f t s i n the isopycnals throughout the water column. These s h i f t s w i l l be cumulative within the b a r o c l i n i c response time of the ocean, so that the ver t i c a l velocity times the period over which the wind-curl variations are occurring should be equal to the ver t i c a l migration of isopycnals. Fofonoff & Tabata (1966), using these arguments, have compared the computed migration to the variations i n the dynamic height anomaly (an integrated measure of isopycnal migration). They found that the two were in phase but that the computed values were about ten times smaller than those observed. 5.3.2 Mean geostrophic flow i n the upper layer The effect of the wind-curl over a number of years i s to produce a steady circulation. In such a circulation the isopycnals attain a mean level through a balance of diffusion, vertical motions in and out of the Ekman layer, and the associated horizontal advection. One such surface i s the top of the main pycnocline which corresponds, approximately, to the top of the permanent halocline. As discussed i n § 1.2, this level also corresponds to the thickness of the uniform-density upper zone in winter and remains intact during summer, due to protecting thermoclines. Suppose that we make the assumption that the density" is uniform in the top layer (which i t i s i n late winter) and integrate (5.2.1) from the bottom of the upper zone, z = -h(x,y), to the mean ocean surface, z = 0. If we consider this equation averaged over a long period (years), we could expect that the momentum imparted by the wind to the upper layer i s ..distributed uniformly with depth, on the average. Then Vx x-k can be-considered as the curl of a body force on this layer. If we further assume that the average vertical distribution of u i s uniform, the -integrated version of 5.2.1 yields, or, since the quantities are time averaged, u ' V (. | ) = 0. (5.3.12) Equation 5.3.12 simply states that the long-time averaged velocity in the upper 'homogeneous' layer should be geostrophic, with flow along lines of constant f/h (the streamlines for time averaged depth ?h = h(x,y)). It i s possible to obtain a mean, or climatologically averaged, depth of the bottom of the upper zone from the oceanographic data collected by various agencies since 1955. For the period from 1955 to 1960, the data summary by Dodimead, Favorite, & Hirano (1963) has been used for this purpose, with the mean depth, h, plotted in figure 7. In general, these data were collected over most of the Subarctic Pacific Region in the summers of the years used. In the winters of these years, however, data from the more interior regions were not collected. The calculated f/h contours are plotted i n figure 8. These were then compared to the calculated geostrophic flow at the surface, relative to the assumed depth of no motion (1000 metres), as calculated in the previous reference (see Figure 22, also). The good agree-ment indicates that, averaged over a long enough time, the depth of the upper zone in the Subarctic Pacific Region so adjusts to the applied wind-stress as FIGURE 8. Mean 7h contours obtained using the depth (Ii) values of figure 7. Units: x I0" 8cm" ,s" 1. to give rise to a layer of geostrophic flow i n the interior region. 5.3.3 Warm Water Intrusion off B.C. Near the eastern boundary, the component of the Ekman transport normal to the coast dominates the wind-stress curl in (5.3.8) in determining the divergence of the Ekman layer. The way in which the coastal waters respond to the normal transport component is seen as follows. A northward meridional wind results in a zonal Ekman transport to the right of the wind (facing downwind). Due to the presence of the coastal barrier, the resulting transport tends to accumulate along the coast and by continuity w i l l cause sinking of the isopycnals by downward flow and/or longshore transport. Similarly, a southward meridional wind component w i l l cause offshore transport which w i l l induce upwelling of cold saline waters and/or longshore advection to the depleted area. The annual variation of the wind induced Ekman transport off the southern Br i t i s h Columbia coast has been shown by Fofonoff & Tabata (1966) to be related to the observed rise and f a l l of isopycnals along the coast, and to zonal intrusions of dilute surface waters. In winter, the onshore transport causes sinking of the deeper isopycnals and accumulation of dilute surface waters along the coast. In summer the opposite occurs, with upwelling and westward intrusion of dilute surface water. Another feature of this region that appears related to coastal divergences or convergences is the yearly variation of the northward intrusion of relatively warm water adjacent to the British Columbia coast. This phenomenon has been well documented by Dodimead & Pickard (1967), who considered the relative amount of water whose temperature at the top (bottom) of the halocline was greater than 7°C (6.5°C) in summer, in the region outlined in figure 9. The temperature at the top of the halocline in summer is approximately that which the water had when i t was part of a homogeneous layer the previous winter. Therefore, the horizontal temperature distribution at the top of the halocline in summer is similar to that of the late-winter surface distribution (Dodimead, 1961a). The bottom of the halocline, which may be defined by the surface of salinity = 33.8 °/oo + 0.1 °/oo (see §1.2), i s below the direct influence of surface cooling, heating and evaporation. Thus, any temperature changes there w i l l be due to internal processes, such as advection. From the previous statements, we may conclude that, i f the 'warm' water phenomenon i n the upper zone is related to onshore Ekman transport, i t should be correlated to the meridional winds in winter. Physically, we expect this to occur as follows: during the cooling period i n the Subarctic Pacific (October-March), the surface waters are being cooled and mixed to an ever-thickening isothermal layer; i f the Ekman transport is onshore, relatively warm water, due to accumulation near the coast, i s sinking and mixing with the deeper upper zone waters; thus, over the cooling period, the mean temperature of the upper layer i s warmer than i t would be in the absence of the onshore transport, or i n the presence of an offshore transport. The latter would result in mixing of colder waters with surface waters. In brief then, increased onshore transport gives rise to a warmer than normal upper layer near the coast i n winter, particularly i f the sea surface temperature in the surrounding region i s above normal i n the preceeding summer and f a l l . By March, the temperat at the top of the halocline has been'established and can only be altered by internal processes, such as diffusion and advection. 160* |80 I 4 0 » 150° I 2 0 » FIGURE 9. The section of the eastern Subarctic Pacific to be considered for discussion of warm woter intrusion. Numbers are port of a grid (see fig. 25 ). In figure 10 the winter means (October-March) of the meridional wind-stress (computed using mean-monthly sea-surface pressures; see §7) at various locations within the grid system used by Dodimead & Pickard, are plotted with the relative area of water whose temperature was greater than 7°C at the top of the halocline. The warm summers of 1958 and 1963 were correlated with large increases in the meridional wind-stress during the previous winters. The decrease in the amount of intruded warm water in 1965 was correlated with a large decrease in the meridional wind, while the slight increase in 1966 was associated with an increase i n the meridional wind. Onshore transport during the winter of 1960-61, however, was associated with a slight decrease in the amount of warm water intrusion and the large decrease of onshore transport in 1961-62 with a slight increase i n the amount of intruded warm water. The latter case is also true for the winter of 1956-57. Obviously, 1961 should have been a relatively warm year, with an increased amount of warm water intrusion over the previous year, while 1957 and 1962 should have been relatively cold years, i f the only process determining the warm water intrusion were the Ekman transport convergence or divergence against the coast. The reason the correlation i s not perfect i s that we have neglected the yearly variations of temperatures in the upper zone over the region. It i s well documented (United States Dept. of Commerce, Fishery Information; 1957-) that anomalous pools of relatively warm, or cold, surface waters may persist into the winter over parts of the Subarctic Pacific (as well as other oceans) and have a strong influence on the variations of the atmosphere and ocean from their norms (Namias, 1959, 1963, 1969). For example, the surface temperatures at Station 'P' were well above the mean from August (1957) to July (1958) and from October (1959) into 1960 87 3 9 61 6 3 65 YEAR FIGURE 10. Comparative plot of the winter-mean (October-March) meridional wind-stress and the relative area of water whose temperature is greater than 7°C at the top of the halocline In the region of fig. 9 ( ). CFigure 11). Throughout 1955 and early 1956, they were below normal, and slightly below normal in early 1959. Thus, in the winter of 1957, the surface temperature reinforced the Ekman transport as far as producing warmer- than normal water at the top of the halocline i n the summer of 1958. In the same manner, the slight decrease of surface temperature in late 1958 to early 1959 combined with the slight decrease of onshore transport to give a decrease in warm water intrusion. The effect of the much greater than normal sea surface temperatures i n the winter of 1959-60, in the region, appear to have dominated the effect of the decrease in meridional wind^-stress, while the effect of the cold winter of 1960-61 dominated the effect of the meridional wind increase in the same period. Again, there i s reinforcement of the onshore winter transport effect by the anomalously warm winter of 1962-63. During the winter of —1963-64, the sea surface temperature were s t i l l greater than normal, but less than the previous winter, with a resulting counterbalance to the increased meridional wind component. Therefore, i t appears that a combination of warmer than normal —temperatures.in the upper oceanic layers during late f a l l (which persist - i n t o winter), and strong meridional winds, result in a large intrusion of warm water along the British Columbia coast at the top of the halocline. Small variations in the amount of intruded water from year to year are "then due to these two effects opposing one another; cold summers, with l i t t l e intruded warm water, occur when low meridional winds and colder than normal surface temperatures combine-. A prominent feature of figure 10 is the sharp decrease of the meridional wind-stress in the years previous to the warmest years of 1958 and 1963. Since the winter meridional wind component is the result of Jan. F e b March April May June July Aug. Sept. Oct Nov. Oec. FIGURE II. Surface temperatures at Ocean Station "P", January(1955)-December(1959). the. isobars of the Aleutian Low being 'pressed' against the coastal mountains, these low values must be caused by a weaker-than-normal or westward shifted Aleutian Low, or by 'leakage' across the coastal mountains. The penetration of winter cyclones across the mountains i s small, however, (Fofonoff & Tabata, 1966). Also, any westward sh i f t of the Aleutian Low would need to be large to cause a significant change. Such a sh i f t was not noticed. The remaining choice, a weakened Aleutian Low, is consistent with Namias' work (Namias, 1959, 1963, 1969) since the colder-than-normal sea surface temperatures throughout 1956 and 1961 imply that increased anticyclogenesis occured in the respective winters. Persistent anticyclogensis into the next summer, over the Northeast Pacific, would have led to anomalously warm water and hence, to a strong Aleutian cyclone in the winter that followed. Although the combination of onshore flow and sea surface temperature in the cooling season i s considered here as the main cause of temperature variations at the top of the halocline, one other feature w i l l be important, particularly below the depth of surface influence. This is the relative geostrophic flow set up by the slope of the isopycnals near the coast. As we have stated in the beginning of this section, onshore transport w i l l result i n downward migration of the isopycnals, while offshore transport w i l l lead to upward migration. For downward sloping isopycnal surfaces there w i l l be a northward geostrophic transport. The combinations of variations in the mean northward flow from year to year, with the fact that temperature increases southward, w i l l result in variations i n the temperature at isohaline surfaces. Figure 12 is a plot of the mean meridional wind-stress from July-June (to approximate the summer observation times) with the relative amount of water at the 33.8 °/oo sa l i n i t y / FIGURE 12. Comparative plot of the mean ( July-June) meridional wind-stress and the relative area of water whose temperature is greater than 65°C at the bottom of the halocline in the region of fig.9 ( ). surface whose temperature i s greater than 6.5°C i n the region of figure 9; (There are no s e r i a l oceanographic data after 1962, although during 1965 and 1966 extensive bathythermograph data were taken; Dodimead & Pickard 1967). The correlation appears good except for 1962. It seems that the amount of relatively warm water established during 1961 remained throughout 1962, although no reason for this i s easily apparent. Since the mean sea surface temperatures over the North Pacific are available for each month (U.S. Department of Commerce, National Marine Fisheries Service), as are the monthly mean surface pressures (Extended Forecast Division of the U.S. Weather Bureau), i t should be possible to predict, qualitatively,' the summer to summer variation i n the relative amount of intruding warm water at the top of the halocline. For example, in the winter of 1969-70 the sea surface temperatures in a large area (> 3002km2) off the British Columbia-Washington coasts were above normal. Thus, i f the coastal meridional winds that winter were weaker than normal, l i t t l e relative change i n the warm water intrusion from the previous summer would have been observed in the summer of 1970. However, i f these winds were normal or stronger than normal, the intrusion of warm water would have increased (unfortunately my data only extend to early 1969). On the other hand, the present (November 1970) lower than normal sea surface temperatures off the northwestern coast of North America suggest that the intrusion, i f measured i n the summer of 1971, w i l l be less than, or about'the same as i n 1969, depending upon whether the meridional wind stress i s normal (or above) in strength, or weaker than normal, respectively. 5.3.4 Mean Eddy Diffusion Coefficient in the Subarctic Pacific Region As we have seen i n previous sections, the vertical velocities generated by the Ekman layer divergence play an important role in maintaining the vertical density structure, particularly for waters immediately below this layer. This fact can be used to obtain a mean eddy diffusion coefficient for heat i n the upper layers of the Subarctic Pacific. It i s well known (e.g. Fofonoff, 1962) that the steady-state balance of diffusion and advection of heat and salt i s given by . 3_ . • 3T* , 8 ( 3T*. -3 3T* 3X L K x 3X ; + "9Y BY + !&> C K z 37 } (5.3. 3T* 3T* 3T* where T* = T D* + (T - T Q) - C J / C J (S - S 0) i s the apparent temperature having a surface value T D*, T i s the actual temperature, S i s the sal i n i t y (S Q at the surface), C^ and C2 are constants in the linearly approximated density equation . p = P o [1 - CiCT - T G) + C 2 (S - S 0)] , and i s the anisotropic turbulent eddy coefficient of dif f u s i v i t y . Also, i t has been shown (e.g. Overstreet & Rattray, 1969) that d_ dz (5.3.14) is a very good approximation to (5.3.13) in regions where the temperature dominates in determining the variations in the density structure. A study by Fofonoff & Tabata (1966) using data from stations between Station 'P' and Swiftsure Bank (see § 1.2 and figure la) shows that, away from the coast, s a l i n i t y accounted for only 15% of the seasonal density variation, so that (5.3.14) is valid here. As further mentioned by Overstreet & Rattray (1969), the reason for dropping the horizontal advection terms is not that they are negligible, but that they approximately cancel each other. where T Q and are, respectively, the surface temperature and the temperature at the bottom of the pycnocline; below this T decreases slowly with depth. Introducing dimensionless variables by, The boundary conditions on the vertical temperature profile are: T(o) = T D T(-h) = T h k = K z / k | o z = z/h c w = w/w 0 = T - Th, equation (5.3.14) becomes, (5.3.15) with boundary conditions 0Co) = 1 , 0C-1) = 0 • (5.3.16) Here, k and wQ represent mean values and (5.3.17) is a mean turbulent Peclet number representing the ratio of advective to diffusive mass transfer. Over the northern part of the Subarctic Region we consider the case of a divergent Ekman layer, with upwards vertical velocity. Assuming w and k are constant over the upper layers (i.e. w and K z are equal to their mean values wG and k D, respectively), the solution to (5.3.15) for (5.3.16) i s , ePz _ p-P 0 = . • (5.3.18) 1 - e -" In figure 13, plots of (5.3.18), from Overstreet & Rattray (1969), for various Peclet numbers are compared with the mean profiles obtained at Ocean Station 'Pr (Dodimead et a l , 1963). It i s easily seen that the low numbers give best agreement, those between 1 - 4 most closely. Therefore, we assume, 1 < P < 4 . (5.3.19) Furthermore, Overstreet & Rattray (1969) compared the ve r t i c a l temperature profiles obtained from two dissimilar vertical velocity profiles with that obtained by the above mean value approximation (i.e. in which the ver t i c a l velocity is approximated by i t s mean value over the whole depth). They found that the mean value approximation closely approximates the form of the observed ve r t i c a l temperature structure, although the details are \ \ • o * FIGURE 13. Comparison of theoretical temperature profile for various Peclet numbers to mean summer profile at station "P". not adequately described. When is nondimensionalized by w_h = w_h/wQ , where wo = T; < VH-ME > (5.3.20) Po is the long-time mean of the Ekman divergence, (5.3.17) may be written as P = < VH.ME > . Combining this with (5.3.19) yields, (5.3.21) The observed depths, h Q, from 195 5-1959 may be obtained from figure 7 (from Dodimead et al, 1963), while the mean value of (5.3.20) for the same period may be obtained from (5.3.8) and the geostrophic wind data (§7). Expression (5.3.21) is then found to be 0.5 x 10 - 1 < ko < A.O x 10"1 cm2s-1 , for the range of Peclet numbers (5.3.19). Table 1 gives some calculated values. The larger values agree fairly well with those obtained by Veronis (1969), viz., 0.15 < k < 2 cm2s-l , o ' ko< < VJL-ME > ^ ~ P o . p D = 1.03 gmcm-3 LATITUDE <VH'ME> h Q k^xlO-l-Cm 2 "^dynes cm-3 s) (xlO 4 cm) P = 1 P = 4 (Alaska Gyre) 60°N 5.0 0.8 4.0 1.0 50°N (Central Region) 2.2 1.0 2.2 0.6 50°N CNorthern tip of Vancouver Island) 2.2 0.8 1.8 0.5 using observed thermocline depths in various thermohaline circulation models. This is,particularly- true when one realizes that the smallest -value, 0.15 cm 2s~l, was obtained using the inverse power law solution which i s only applicable to cyclonic regions (i.e. divergent Ekman layers), and the Discovery stn. 3527 at 46°30'N, 33°21'W (Fuglister,1960). 5.3.5 Justification of the neglect of VpxVp'k i n § 2.2 , We. are. now i n a position to jus t i f y the neglect of the linear Interaction between the horizontal pressure gradient (Vjip) and the horizontal density gradient (Vjjp) i n § 2.2. Fir s t , we nofe that below the Ekman layer the flow i s quasi-geostrophic (§ 5.2.1). The barotropic component, with velocities independent of depth, i s by definition such that isopycnals and isobaric surfaces are pa r a l l e l ; hence VpxVpg'k = 0 for the barotropic pressure PB . In the baroclinic mode, with depth dependent velocities associated with the inhomogeneity of the water density, the pressure and density surfaces are no longer p a r a l l e l . If we separate the density gradient into two components, one lying in the same plane as the pressure gradient and the other perpendicular to i t , we see that only the component perpendicular to Vp gives a nonzero cross product in the direction of k. The magnitude of this cross product w i l l be small, however, since i t depends on the magnitude of the density gradient component in the direction of the quasi-geostrophic flow; i f the flow were purely geostrophic (g), VpxVp^-k would be equal to zero. To demonstrate the previous remarks, we consider motions i n the Ekman layer where the departure from geostrophy i s greatest. Now, i n general, VpxVp.k = -|Vttp|. |VHP| sin9 1 0 (5.3.22) and we expect 0, which measures this departure from geostrophy, to be small. From (3.2.1) we have that the horizontal pressure gradient, VHp ~ f 0 U 0 P 0 » while V R p ~ Ap/Lp , where Ap is the density difference over a horizontal distance L p . Thus, i f we compare the scaled value of (5.3.22) to p^u-Vf in (2.2.1), we find a ratio Y = VpxVp-k f o G3t£> 10 9 c.g.s r.p But since the fractional change of water density anywhere in the oceans is small, v i z . , ^ - 10-3 , P we have, Y ~ (.j^). 10^ c.g.s. units, This implies that, unless density changes of the order of 10 3 take place in distances of 10 kilometres, or less, the effect of the linear interaction between tbe pressure and density- gradients C5.3.22) on the Ekman l a y e r f low, and t h e r e f o r e on the flow throughout the water column, i s n e g l i g i b l e . In the S u b a r c t i c P a c i f i c , distances of many hundreds of k i l o m e t r e s are r e q u i r e d before the observed change of mean de n s i t y i s of the order of 10~ 3. 6. A linear, integrated model for the Subarctic Pacific 6.1 Introduction In § 5.3, the Sverdrup relation (5.3.1) was obtained by integrating the linear, horizontally frictionless vorticity equation from a depth z = -H(x,y), where vertical motions are assumed to be negligible, to the surface (approximately z = o). Basically, this equation gives the total quasi-steady, baroclinic transport (see § 4.1) i n the interior region of the ocean, where the divergence of the Ekman layer is compensated by a convergence of the baroclinic mode and so is not affected by bottom topography. We now define a dimensionless mass transport stream function, by MX = -9t/V9y , (6.1.1) (1 - e*Ky)Mv = +3TJ>/3X. • The continuity equation (5.3.4) in nondimensional form, viz., || X + |^ [ (1 - e*Ky)W ] = 0 , (6.1.2) is then sati s f i e d identically, while (5.3.1) becomes J O M ) = (1 - e*Ky)V x T w • k , (6.1.3) where J is the Jacobian. It i s straightforward to show that (6.1.3) has solution, . rXCx.y) iKx,y) = k - J V x T W IE, Y(x,y)]dE , (6.1.4) glY(x,y).] ~ where g[Y(x,y)] i s a constant of integration. The stream function of the total transport for the oceanic interior i s thus determined once the curve g{Y] is obtained from the boundary conditions. The fact that there are two horizontal boundaries to be considered but only one constant of integration, however, results in the loss of a boundary condition and therefore i n the existence of a boundary layer. That i s , some of the higher order terms in (3.2.3) can no longer be neglected near one of the boundaries. In order to close the circulation in the Northeast Pacific Ocean a boundary current of relatively strong flow must be added to the previously derived interior solutions. The model presented i n this thesis w i l l retain .horizontal turbulent f r i c t i o n , rather than nonlinearities, for this purpose. Although much success has been achieved by i n e r t i a l models in describing the nature of the intense western boundary currents such as the Gulf Stream (e.g. Robinson, 1963; Warren, 1963; Robinson & N i i l e r , 1967; N i i l e r & Robinson, 1967), the general agreement between the time averaged circulation of the theoretical model (§ 6.6) and the observed geopotential flow of the eastern part of the Subarctic Region demonstrates the applicability of a linear theory. We may further j u s t i f y , at least partly, our neglect of the nonlinear effects by f i r s t noting that, south of the Alaska Peninsula and Aleutian island chain, where a boundary layer flow is known to occur (e.g. Favorite, 1967), the depth contours are very regular, paralleling the coast for depths up to 8000 metres (Figure lb). The increase in depth away from the narrow continental shelf i s very rapid, going to depths between 5000-8000 metres in distances less than 200 kilometres; a common feature of such Island Arc - Deep Sea trench systems. The contour regularity and rapid increase to maximum depth associated with the continental slope suggests less control by the bottom contours, when compared to the case of the Gulf Stream. Furthermore, current speeds i n this boundary layer are smaller than those i n the Gulf Stream by about a factor of three, which also implies weaker i n e r t i a l effects. Finally, because of the requirement of zero velocity along the boundary (the 'no-slip' condition), f r i c t i o n is necessarily important near the coast, even for highly nonlinear currents such as the Gulf Stream (Stewart, 1964). 6.2 The'linearized stream function equation In this section we w i l l retain horizontal f r i c t i o n in order to close the circulation, at the same time neglecting the nonlinear terms. Basically, we are considering, RQ . -f- 0 + , E R f i n i t e . (6.2.1) Applying the above to (3.2.3) and using from (5.1.2), we obtain, -f Cl - e*Ky) 3w + (1 - e*Ky) u-Vf 8z + (1 - e* icy ) 1 | ^ (Vxx-k) , where % * = 3 ^ 3 C6.2.3) We then integrate (6.2.2) from the depth z = -H(x,y) to the surface, as in § 5 . 3 , and use the definitions ( 5 . 3 . 2 ) and (6.1.1) to get an equation for the stream function, J ( * , f ) TT * J 9 / 1 3 R ' ' 1 r 3 r ' 1 S i l i ^ ' S r ,-, * ^ i n n . = E H { 87 { IZPZf 37 1 I^ KV { 37 [ 1 ^ ^ ] + 37 [ d - e * K y ) ^ » » + Cl-£*Ky)VxT w.«k . (6.2.4) Solutions to this equation w i l l be obtained using the method of power series expansion in the dependent variables in each of three regions. These regions are (Figure 14): (A) the interior region, away from the boundaries at x = o and y = o; this region has been; discussed at length i n § 5; (B) the eastern boundary region, near x = o north of point 'a', and near LX = L q J X (l-e*Ky) cosl'0 (S)]dE - yL sin 6 0 = a -s.-in.60 , south of point 'a'; (C) the Northern Boundary region near y = o. No attempt to model the Subarctic Pacific west of 165°E is made. 6.3 The interior region In the oceanic interior, away from the f r i c t i o n a l influence of the boundaries, the horizontal viscous terms w i l l be negligible. Therefore, (6.1.3) with solution (6.1.4) applies to order E^*. However, as 150° 170° E I70 °W ISO0 ISO 0 FIGURE 14. Shemafic diagram of the subregions used in the mathematical model: land barrier, approximate position of the subregion boundaries, approximate limit of the boundary layer regions. subsequent analysis w i l l show, the requirement that the northern boundary layer flow match to the interior flow makes i t convenient at this point to divide the latter into two subregions, labelled and A2 i n figure 14. This division i n the interior i s purely for convenience and only serves to make the presentation more coherent; by i t s very definition the interior region cannot be directly dependent upon the form taken by the boundary flow. Furthermore, the higher order corrections to the zeroth order streamlines w i l l be due to the horizontal f r i c t i o n a l effect so that the wind-stress w i l l be considered to be a zeroth order function only. Expanding formally, we write 2 V i = Vol +• E5 y i V n + E ^ l i|, 2 1 + . . . H = lol and h = V 02 + V12 + EH y 2 v22 + • • • i n which and u 2 are constants to be determined by matching to their appropriate boundary regions (§§ 6.6-6.7). Equations (6.1.3) and (6.1.4) now take the forms, JOJ^.f) = (1 - £*Ky)VxT w i • k , and (6.3.3) ^ FX(x,y) -%Cx,y) = k . J VxTWia,Y)d5 , g[Y(c,y)J Sub-} Region (6.3.1) A l Sub-} Region (6.3.2) A 2 respectively, for i = 1,2, and where the expansions are only carried to O(E^). The function g(Y) is determined only after consideration of the eastern boundary region. Since the wind-stress has been taken to be of zeroth order, i t determines the streamlines e x p l i c i t l y to order E*j (i.e. un t i l f r i c t i o n becomes important) in regions away from the Northern Boundary. Thus, i t i s seen that upon substitution of (6.3.1) and (6.3.2) into (6.3.3) and upon comparison of orders of E^ that, *12» *21» fe2» • • • , (6,3.4) up to order E*j, are identically zero i n the oceanic interior region. 6.4 The eastern boundary region Formally, one could perform a boundary layer expansion near the eastern coast by introducing a stretched variable for the coordinate direction perpendicular to the boundary. This corresponds to the usual assumption that cross-stream variations are much greater than downstream variations in a boundary layer. Alternately, one can simply make certain assumptions which by-pass the algebra. This was done by Munk (1950) who showed that i n dimensional form, ||(X) = | { (Vxx srk) x - (VxTs-k)a.-exp[-k(avX)] } (6.4.1) where k = ( 3 / A H ) l / 3 « l , a n d a'= - Y tan0o . The last term in (6.4.1), which distinguishes the eastern boundary solution from the interior solution, is very small. As with most previous authors we choose to neglect i t (see Carrier & Robinson (1962), §§ 2.2-2.4 for a good discussion of this point). The function g(Y) in (6.3.3) i s then determined, e x p l i c i t l y , by requiring that the coast be the zero streamline. Thus, in effect, we have neglected the eastern boundary layer so that the interior solution now satisfies the eastern boundary condition of zero normal transport. The interior region is extended up to the eastern coast. The southward flowing California current, south of point 'a', has been shown by Munk (1950) to be of the 'eastern' boundary type. Such currents depend upon a strong, local wind-stress curl i n the boundary region for their existence (see § 6.7). The absence of any organized boundary flow north of point 'a', except near the northeast corner of the region (Figures 17a,b), i s consistent with the preceeding statements since i t is in the latter region that the maximum in the wind-curl spectra occurs (§7.2.3.1). 6.5 Boundary layer region C : arbitrary curvature In this region higher order terms must be important, since in satisfying the eastern boundary condition we have determined the one constant of integration, g, i n (6.3.3). Observationally this i s supported by the existence of a concentrated westward flow along the Northern Boundary. The terminology to be followed for the boundary current w i l l be that of Favorite (1967) who uses observational data to distinguish between the Alaska Current, east of about 160°W, and the Alaskan Stream, to the west. This distinction w i l l also appear naturally in the analysis, for the case of non-zero curvature, as a result of assuming a balance between horizontal f r i c t i o n and the Coriolis terms i n the boundary layer. Now, since cross-stream Cy) variations i n the flow are expected to be much greater than the downstream (x) variations, we proceed formally by transforming to the inner variable, y, * V * V T ( 6- 5- 1 > where i n particular we assume <j>(EH*) = E * ^ i , (6.5.2) i = 1,2. The constants and y 2 a r e determined by requiring that the largest f r i c t i o n a l term be of the same order as the largest Coriolis term along the boundary. Letting a circumflex represent variables appropriate to the boundary layer, we have, ^i " %± + E H y ± V i i + • • - . (6.5.3) Iwl ~ l o i Substituting (6.5.1) and (6.5.3) into (6.2.4) we find, F l 8x EH F2 9 y-+ y r a-M) ^ l 0 J ] } (6.5.4) + (1 - f i K y ) ( f ^ - E* - W j|k> , i n which. e = e* E*Ui C6.5.5) and the functions F-^  and F2 are: F 1(x;6 0) = cos 0(xL) (6.5.6) F 2(x,y ;0 o ) = [l-§K(xL)y] sin 0(xL). The characteristic breadths, b^, of the boundary layer subregions are found from, (6.5.7) In the sections to follow (§§ 6.6, 6.7), the boundary layer . -equations w i l l be solved for the particular cases K = o and K = constant 4 o, respectively. The latter case can be used to approximate the whole Northern Boundary very closely. 6.6 Boundary layer region C : zero curvature To determine the effect of a non-zonal Northern Boundary on the circulation, we consider the case K = o. Equation (2.1.3) then yields, 0(xL) = constant = 0 o (6.6.1) If we substitute this and K = o into (6.5.4-6.5.6) we obtain, + 2E- + E; 'H *l-4u. 9 4 $ i 1 „ A/. (6,6.2) 9x ^ 9y where now, ?1 = cos e0 , F 2 = sin 6 Q . The terms, F-^  and F 2 , are constant i n the boundary layer so that there is no distinction, as far as the power series expansion i s concerned, between the two subregions C-^  and C 2 ; the i may be dropped. To determine y, we assume that terms of largest order on each side of (6.6.2) balance. Then, i f sin 6 Q ~ 0 (cos 9Q) , we must have, u = 1/3 , (6.6.3) which using (6.5.7) and values from (3.3.1) gives a characteristic boundary layer width of, b ~ 35 km . (6.6.4) From (6.6.2) i t i s seen, that to orders E*o . l ^ a + s l n e l$o = 9Tg . C 6 6 . _ _R*1/3 . 2% Q 9^ 1 o e 9^ o _ 3j£ . (6 6 6) % • 3^4 + s i n °o 3£ " c o s °o 3 X 3 X > t o . 0 . 0 ; plus higher order equations. The appropriate conditions at the boundary are: (i) the coast is the zero streamline, \JJ(X,O) = o ; ( i i ) there is no s l i p along the boundary ; 9$/9y(x,o) = 0 (6.6.7; i-i v ) Ciii) the solution is bounded for large negative y ; (iv) the solution matches to the interior solution; since only ^ o this requires that, to order Eg , $j_ , $2 > . . . o away from the boundary layer. It should be emphasized here that the equations derived so far would be invalid i f sin 6 Q « cos 6 0, that i s , i f there was no significant rotation (< 10°) of the x,y axes relative to the X,Y axes. Since we are modelling the Alaska peninsula and the coast of British Columbia, i t can be seen from figure 14 that 30° < 6 0 < 45°. The solution to the homogeneous (H) form of (6.6.5) is 4 Y$0(x,9) = A Q(x) + I A,(x) exp[ a j(x)y] , (6.6.8) ,j.=l for which the four roots, aj, satisfy the characteristic equation, a j 4 + F 2aj = o , and are a i - o ; a 2 = E 21/3(| + ; ' a 3 - ^'\\ " *4> > (6.6.9) a 4 = - F 2 l / 3 . The nature of (6.6.8) w i l l depend upon the size of F 2 , as can be seen by applying the boundedness condition (6.6.7; i i i ) . For the case F 2 > o ( 0 O > o), the root a^ gives rise to a divergent solution, hence A4 = o. In the case F 2 < o (6 Q < o), the roots a2 and a 3 give divergent solutions, hence A2 = A3 = 0 there. The general solutions for both cases, which satisfy the remaining boundary conditions (6.6.7; i , i i , i v ) are: Vx>?) = f^o(x»y) - p$0(x»y)J ^  1 - exp[F 2 1 / 3y/2] [cosC^| F 2 1/ 3^) - 1 / 3 s i n ( ^ F21./3y) ] } + M*>V { 1 - *l°tZf) exp(F 2l/^/2) [ c o s ( ^ F 2 l / 3 y ) ] } (6.6.10) -yi s l n 4 F 2 1 / 3 y ) i > + vf k1'3 h [ P V x ' o ) ] e xp(V / 3y/2) s i n / | F 2 1 / 3 y ) for 6 0 > o, where p ^ 0 i s a particular solution of (6.6.5) and \\)Q is the interior solution (6.3.3); $o(x>y) = i%t*,y)' " p i o^'y)] I 1 - exp(-F 2 1 / 3y) ] (6.6.11) + p$o(x,y) [ 1 - ex P(-F 2l/ 3y) ] for 8 0 < o provided, Vx>y> = - df [PVx»°>J + PV x>y> - PVx>°> in (6.6.11). These results show that for 8 Q > o i t is possible to have a zeroth order boundary layer flow, ^ 0(x,y), i n the absence of direct wind forcing / \ in the boundary layer; i.e. for p ^ 0 = — ^ = o . Such boundary currents are of the 'western' boundary type, in that they form on the 'western' (see % 4.3) sides of the oceanic regions. For 0 O < o, however, a boundary layer flow can exist only in the presence of local wind f o r c i n g i n the boundary r e g i o n . Such boundary c u r r e n t s , when they form, are known as 'eastern' boundary type currents s i n c e they flow along the eas t e r n boundaries of oceanic regions. The A l a s k a boundary l a y e r s o l u t i o n s (6.6.10) and (6.6.11), are g e n e r a l i z a t i o n s of Fofo n o f f ' s (1962) treatment of f r i c t i o n a l boundary l a y e r s ; (6.6.10) reduces to h i s 'western' boundary s o l u t i o n s when ^ Q = o. That (6.6.10) corresponds to a western boundary current and not to a p u r e l y z o n a l boundary current i s , of course, due to the r o t a t i o n of the A l a s k a coast r e l a t i v e to the X,Y coordinates. This important p o i n t was overlooked by Munk (1950) i n h i s d e t a i l e d study of ocean c i r c u l a t i o n as he c l a s s i f i e s the Al a s k a Current under 'eastern' boundary c u r r e n t s . S i m i l a r l y , Fofonoff (1962) considered only a p u r e l y z o n a l boundary. Stewart (1964) p o i n t e d out that western type boundary currents are only p o s s i b l e i f the boundary generated v o r t i c i t y gained by the concentrated flow near the coast can be balanced by a corresponding gain i n p l a n e t a r y v o r t i c i t y . I f i t can't, the r e q u i r e d v o r t i c i t y must be s u p p l i e d by the l o c a l w i n d - c u r l . For the s i m p l i f i e d model used i n t h i s s e c t i o n t h i s i s ob v i o u s l y the case. Figure 15 shows the streamline p a t t e r n f o r the i n d i c a t e d w i n d - c u r l d i s t r i b u t i o n i n the case 9 Q > o, and f o r which we have assumed n e g l i g i b l e w i n d - c u r l i n the boundary l a y e r . Comparison o f the c a l c u l a t e d streamlines to the mean g e o p o t e n t i a l surfaces (Figures 16a,b), r e f e r r e d to 1000 metres (Dodimead, F a v o r i t e & Hirano, 1963), i n d i c a t e s that i n the eastern h a l f of the Northern Boundary, where zero curvature i s a p p l i c a b l e , both the w i n d - c u r l d i s t r i b u t i o n and the presence of a western type boundary l a y e r are fundamental i n the formation of the Al a s k a Gyre as w e l l as the general S u b a r c t i c c i r c u l a t i o n ; the western boundary closes the flow i n i t i a t e d by the winds i n the i n t e r i o r r e g i o n . FIGURE 15. Streamlines for a northern boundary of constant curvature (0o = 3O°)and an applied wind-stress curl = TO cos cos (^f), L = acos0 o . (The w i n d - c u r l d i s t r i b u t i o n used here i s an approximation to the mean annual w i n d - c u r l obtained by me from the mean-monthly pressure data over the North P a c i f i c ; two examples are given i n f i g u r e s 17a,b; see a l s o § 7). We may now compare the mean t r a n s p o r t , IJJ(< 10 x 1012 gm s - l ) , and the mean sur f a c e speed, c(< 20 cm s --*-), of the Alaska Current, as obtained from the g e o p o t e n t i a l topography ( F a v o r i t e , 1967), w i t h the zeroth order t h e o r e t i c a l values. The t r a n s p o r t s c a l e , TJJ , f o r the t h e o r e t i c a l values i s obtained from the d e f i n i t i o n s (5.3.2) and (6.1.1). Together w i t h the numerical values from (3.3.1) and (5.1.1) they y i e l d V* - P 0 uoLD ~ 8 x 1 0 1 2 gm s - 1 , i n agreement w i t h the geostrophic c a l c u l a t i o n s . In e f f e c t , t h i s a l s o lends support to our choice of U Q, L and D. To o b t a i n the sur f a c e current s c a l e , c*, we assume a l i n e a r decrease of h o r i z o n t a l speed to zero at 1000 metres and consider the boundary l a y e r to have a t o t a l width of 3b = 3 ( A J J / B 0 ) a s suggested by the r e s u l t s of t h i s s e c t i o n . With the cross s e c t i o n a l area (s) = 3b(10^) cm 2, the speed cft _ — i b * „ 18.5 cm s - 1 , ps r ' which again agrees w i t h o b s e r v a t i o n . 6.7 Boundary l a y e r r e g i o n C : constant curvature I f f o r now we neg l e c t the passes connecting the North P a c i f i c Ocean to the Be r i n g Sea, we can c l o s e l y model the Northern Boundary (approximately represented by the 1000 metre depth contour) by a curve of constant r a d i u s , K ~ l (Figure 18). In t h i s case, the forms taken by the ordered d i f f e r e n t i a l equations w i l l change as 6(x) v a r i e s from 8 D FIGURE 16a. Geopotential topography u/IOOO decibars, summer 1956. —Velocity (sea miles/day). FIGURE 16 b. Geopotential topography u/IOOO decibars, summer 1958. —z*Velocity (sea miles/day). "s. N, • >» • • *^  1 ' * LP^ / I 1 A \ \ 7.5^ 12 I0 .0 V •s X * \ \ o \ -\ \ \ > \ 1 V \ - 8 . 0 . t / 4 \ \ > _ , 2.! - '2 .5 -0 - - - " * ""* «•» \ .... „ o " • > - 2 . 5 • ^ *"" "™ - — / / • -2.5 -2.5 y • * - - 8 . 0 -\ \ \ » > \ \ t i / * * -2.S •X I 1 ' I \ > 4 / ^" FIGURE 17a. Mean wind-stress curl, x I0-9 dynes cm- 3. From June(l955)- May (1956) . — V " — % 1 o *» . \ \ \ N \ \ \ s * \ - 0 , s * 'AIJ, > ' 1 1' / / s' i \ x to 4 s f H / / • / y / 1 / / / - S . O -2 . 6 -— -/ / - 0 \ f \ \ \ / s - - ' I \ N / •%* -0 _ - ' - ' ' ' > . - 2 . S . ~ ~ A - — "' 4* • *"* -2.ST "s. FIGURE 17 b. Mean wind-stress curl, x I0~9 dynes c m - 3 . From June ( 1958) - May (1959). through, zero, to negative v a l u e s , n e c e s s i t a t i n g the two p r e v i o u s l y o u t l i n e d subregions (see §§ 6.3 and 6.5). Subregion 1: The subregion f o r i = 1 i s that i n which sin0(xL) and cos8(xL) are of the same order i n the boundary l a y e r . With the assumption that the l a r g e s t f r i c t i o n a l and C o r i o l i s terms balance i n t h i s l a y e r , we have, • P l =? 1/3 , (6.7.1) which a l s o was obtained i n § 6.6. Using (6.7.1) and t a k i n g the maximum value of e* (~1), (6.5.4) y i e l d s to 0 ( E * 1 / 3 ) : EH° : ^ + 3in0(xL) l | l = * f|l (6.7.2) E * l / 3 . 3 % 1 _ K 3_ r l ^ o l + v i ^ o l i * 9y4 o 3y 9y2 • 3y3 J (6.7.3) + s i n 6 ( x L ) [|$1X -K D y#*]. " c o s & | ^ o l dy dy dx 3T£I _ >. 3TQ1 3x" K ° y 3? i n which 6(xL) = 9(x) = (1 + x ) 9 Q when e* ~ 0 ( 1 ) . -The general s o l u t i o n s to (6.7.2) f o r 9(x) > o and f o r G(x) < o are given by (6.6.10) and (6.6.11) r e s p e c t i v e l y , but where F 2 i - s n o w x-dependent, v i z . , F 2 ( x ) = s i n 9 ( x ) 140° 150° 160° 170° W 180» 170° W 160° 1 5 0 ° ' I 4 0 a I 8 0 » 1 2 0 ° I 1 1 1 1 I I I I I J 140° 1 5 0 ° 160° 1 7 0 ° E 1 8 0 * I 7 0 » W 160* IB0» I 4 0 » I S O * 120* FIGURE 18. Comparison of constant curvature northern boundary M with the 1000 metre depth contour ( ). So far, then, the analysis shows that for subregion 1 , where Eg ' < tan0(x) < 0 ( 1 ) , the concentrated flow south of the Northern Boundary behaves as a western boundary type current provided•9(x) > o. For 0(x) < o i t behaves as an eastern boundary type current which can only exist in the presence of a local wind-curl. Subregion 2 . As the current of subregion^1 progresses westward, i t begins to flow along a nearly zonal boundary. In this subregion, Eg-W si neCx) ~ 0 [ cos9(x) ] ~ ' 0 ( 1 ) . A balance of horizontal friction and Coriolis terms in the vorticity equation (6.5.4) now yields, u 2 '= 1/4 for which the zeroth and first order equations are: E*° = | t ° 2 + E*-1M s i „ e ( X ) - § f ° 2 - :coseCx) - f f ° * ' ( 6 . 7 . 4 ) (6.7.5) E R 1 A s ^ U + ^ , i n 9 ( x ) | | l 2 - _ ; c o s e ( x ) | | l 2 _ K [ 2 - ^ i ° 2 + V^5° 2 1 - K V - ^ ° 2 + ^ ° 2 Ko 1 ^ 9^ 1 + y 9yt K° y:9x + dy Now, equation (6.7.4) may be -approximated ito'order 9(x) by | f o 2 + E * - l M 6 ( x ) | o 2 . . , | f e 2 | ? 2 ( < « . , . « , since the angles, 0(x), for which (6.7,4) lis valid are less than IT/ 8 radians (10°). - ' Solutions to (6.7.6) are not simple, but we may obtain an insight into their behaviour by replacing 9(x) with an average value, < 6 > , over a range (x + m, x - m) where o < m << 1. A solution, HV02> t o the homogeneous form of (6.7.6) is found by separation of variables. Taking A H$ o 2(x,y;A) = exp (Ax) E exp(d^) , (6.7.8) j=l where A is the separation constant, dj and Dj are constants, and d.4 + E J 1 / 4 < G > d, - A = o . (6.7.9) 3 • *J ' With 0 = ( E H ) - 1 / 4 < 6 > the four roots of (6.7.9) are; d l j 2 = | 1 / 2 •{. i + i ( 1 +.||/2 ) 1/2 } f (6.7.10) 1/2 d 3 , 4 = \ '{ "I + i ( 1 " f f/2 > 1 / 2 > » (6.7.11) where 2 I 1 + p Q A + { | 2 [ 1 - ( 1 + 33 | 4 (6.7.12) The general solution to the simplified form of (6.7.6) is therefore, %2M;V = expCXx) QCy) + p$ o 2(x,y) , (6.7.13) where p $ o 2 i s a particular solution of (6 .7 .6) . At the Northern Boundary we find, using the boundary conditions (5 .5 .7 ; i - i i ) , that exp(Ax) Q(o) = - p$ o 2(x,o) , (6.7.14) and , x , 3Q(o) 3 r 7 / \ i exp(Ax) ,^ = - [ p\J;p2(x,o) ] . Therefore, i n order to satisfy the boundary conditions, we require that the particular solution be proportional to exp(Ax) in the boundary layer; the general solutions w i l l thus be f a i r l y restrictive. Equation (6.7.13) now becomes $ o 2(x,y;X) = exp(Ax) [ Q(y) + p $ * 2 (y) ] , (6.7.15) where * p\po2(y) = exp(-Ax) p^ o 2(x,y) , and p ? ; 2 c o ) = -Q(O) by (6 .7 .14) . In order to understand the physical significance of the parameter A i n subregion 2, where A = a l k l - M o 2 / i £ (6.7.16) q(x) tp02' i t i s necessary to f i r s t consider subregion 1. The reason for this is not obvious u n t i l i t i s realized that the flow is observed to move westward along the Northern Boundary. The wind-induced northward transport, v 1 M1 = p- Vxxw-k (5.3.3), is greater than zero (see § 7.2.1) over the interior po ~ and boundary regions of subregion 1, so that in curvilinear coordinates ^ ^ /\ /\ MY-S- o also. Then, since W = 8^0^/3x, we have S^^/Sx > o. However, ip02 = o at the east coast (x = o) by boundary condition (6.6.7; i ) , whereby < o i n subregion 1. Following the flow westward, we see that there are two p o s s i b i l i t i e s : f i r s t , i f ipQ-^ changes sign somewhere along the boundary i t must go to zero, which implies separation of the current from the coast; there would be no continuous boundary layer flow; second, i f no sign change occurs in ip0]_ then IJJ02 < o in subregion 2. The latter corresponds to a continuous boundary layer flow from subregion 1 to subregion 2. Furthermore, i t is now clear that the sign of A in (6.7.16) depends only upon the sign of 3^02/9x, since ip02 < o. The following cases arise: for 3^Q2/3x < o we have X > o, which corresponds to the case of decelerating downstream transport i n the boundary layer subregion 2; for 3iJ;02/3x > o, which corresponds to increasing downstream transport into the boundary layer and an accelerating current, we have A < o. The case A = o corresponds to no downstream change in the boundary layer transport (SiJ^/Sx = o) . Using the unapproximated equation (6.7.4), i t is seen that a solution i n this case is possible only i f SiJ^/Sy = ° also. The latter condition, however, does not permit a boundary layer solution to exist. This implies that, i f the transport into the boundary layer decreases to zero somewhere in subregion 2, due to a change in the wind-stress curl from positive to negative values over this region, the flow becomes unstable. The physical interpretation of this i s that the constraint of no downstream increase in transport prevents the f r i c t i o n a l vorticity generated at the boundary from balancing the decrease i n planetary v o r t i c i t y . To obtain a more complete understanding of the significance of the parameter A, we consider the small area of the boundary subregion 2 subtended by the angle 6(x) ~ o. Using (6.7.10-6.7.12) or (6.7.9), the four roots, dj, are: d'l,2 = ± i A l / 4 > d3,4 = ± A l / 4> for A > o, and d l , 2 = + |A/4| 1 / 4 (1-i) ; d 3 j 4 =.+ IX/4.1'1/* (1+i), for X < o. The general solution to (6.7.6) satisfying (6.6.7;i-iii) i s , for A < o, ^ o 2(x,y;A) = - p^o 2(o) exp(my + Ax) [cos(my) - sin(my)] - ^ sin(my) exp(my + Ax) |p [ p$o2(°) ] (6.7.17) + p^o2 ( y^ exp (Ax) , - in which » - l|l 1 / 4 For A > o, the general solution near B(x) ~ o i s , Vo 2Cx,y;X) = exp(Ax) { D { exp(v5my) - [sin(%/2my) + cos(V2my)] } - cos^Zmy) p$* 2(o) - ^ ^ S l ^ [ p } * 2 ( 0 ) ] (6.7.18) where D is a constant. This solution contains f r i c t i o n a l terms which do not decrease in amplitude away from the boundary. As this implies that horizontal f r i c t i o n would be important in the interior, contrary to assumption and observation, such values of A lead to flow i n s t a b i l i t y and perhaps boundary layer separation. Any flow that does separate would necessarily merge with the eastward flowing Subarctic Current to the south (see Figure 22). In summary, then, we see that for the range - 0 0 < A < o (accelerating flow), boundary layer transport entering subregion 2 from the eastern part of subregion 1 can exist westward into the western part of subregion 2 where 8(x) becomes negative. This flow can continue as a boundary layer into the western part of subregion 1 only i f supported by the local winds. For o < A < 0 0, however, boundary layer i n s t a b i l i t y can occur. Theoretically, the concentrated boundary current entering the area near 8(x) = o should break-up and separate from the boundary. In reality, i n e r t i a l effects, and perhaps 'leakage' through the passes (see § 6.8), w i l l tend to hold the main current, and/or the branches formed from i t , together a short distance westward of this area. The solution for a Northern Boundary of zero curvature can be expected to be valid for the eastern part of the Subarctic Pacific Region. To ful l y understand the circulation over the whole region, however, i t is necessary to consider the results of this section for the case of a curved Northern Boundary. With these results, and the vorticity arguments used by Stewart (1964) in describing oceanic circulation, we can now give a physical description of the Subarctic circulation. The wind-stress curl north of about 45°N adds net positive (counterclockwise) vorticity to the oceanic interior. The quasi-steady, Sverdrup-type flow that results from this accumulated wind-stress curl effect consists of a net northward transport to latitudes of more positive planetary vorticity. That part which concentrates into a western boundary current (6.6.10) in the vici n i t y of the eastern corner of the Northern Boundary i s known as the Alaska Current; the characteristic width of this boundary layer being 0[(AR/3O) ]• Since the boundary current i s bounded by a line of zero vorticity, i t cannot contain or transport any relative vorticity (Stewart, 1964). Therefore, considered over the width of the current, relative vorticity is not a determining factor in the overall vorticity-balance. Also, as a western boundary current, the Alaska Current can exist in the absence of a local wind-curl. Therefore, the gain of more negative planetary "vorticity by the current as i t moves southward i s balanced by the diffusion of negative vorticity from the coast. As the flow proceeds westward along the ever northward turning boundary, however, i t s gain of negative planetary vorticity decreases. This results in the formation of a zonal boundary current known as the Alaskan Stream and an increase in the characteristic boundary width to 0[(A I 1L/B O) 1/ 4 ] due to greater southward diffusion of vorticity generated at the coast. Eventually, the planetary vorticity variation can no longer balance the diffusion of clockwise vorti c i t y from the boundary. If this accumulating v o r t i c i t y i s not balanced by other terms i n the v o r t i c i t y equation (6.5.4), the boundary l a y e r becomes uns t a b l e ; the st e a d y - s t a t e s o l u t i o n s are no longer v a l i d . As the a n a l y s i s of t h i s s e c t i o n shows, the req u i r e d balance can be obtained by a t r a n s p o r t of water i n t o the boundary l a y e r or, i f one p r e f e r s , by a p o s i t i v e wind-stress c u r l over the boundary l a y e r . Therefore, i f the w i n d - c u r l , which i s p o s i t i v e over the c e n t r a l and e a s t e r n regions of the S u b a r c t i c (Figures 16a-,b) , changes s i g n i n t h i s p a r t of the boundary, i n s t a b i l i t y and perhaps boundary l a y e r s e p a r a t i o n w i l l occur. Figure 19 shows the western and southern extent of the Alaskan Stream, as i n d i c a t e d by the temperature d i s t r i b u t i o n on the s i g m a - t ^ t ) s u r f a c e 26.90, i n summer 1959 ( F a v o r i t e , 1967). One branch i s seen to separate from the boundary w h i l e the r e s t flows n o r t h through Near Pass (see Figure 21). As f u r t h e r i n d i c a t e d by the sur f a c e s a l i n i t y d i s t r i b u t i o n j u s t south of the A l e u t i a n i s l a n d s , there i s a l a r g e amount of v a r i a b i l i t y i n the l o c a t i o n at which southern branches diverge from the main flow ( F a v o r i t e , 1969); see f i g u r e 20. As s o c i a t e d w i t h the changing character of the boundary l a y e r equations i s the stream f u n c t i o n d i f f e r e n c e A = E^-/** - E^-l^, obtained by comparing h i g h e r order c o r r e c t i o n s from each subregion. These higher order c o r r e c t i o n s apply only to the boundary l a y e r , as the w i n d - c u r l , ^together w i t h the eas t e r n boundary c o n d i t i o n determine the i n t e r i o r s o l u t i o n - independently of h o r i z o n t a l f r i c t i o n . For EJJ = 5 x 10 A ~ 5 x 10~ 2 or 5% of the ze r o t h order flow. B a s i c a l l y , t h i s d i f f e r e n c e i s a r e s u l t of r e q u i r i n g that the C o r i o l i s and f r i c t i o n a l terms balance i n the boundary _l I I 1 I I I I 6 6 » 1 7 0 ° 1750 E |80o I 7 5 ° W I 7 0 » I 6 S » FIGURE 19. Temperature (°C) distribution on sigma-t surface 26.90, summer 1959. Shaded region shows approximate extent of the Alaskan Stream. The 300 metre depth contour ( ) is also shown. FIGURE 2 0 . Schematic diagram of the variability in location at which southern branches diverge from the main flow, as indicated by surface salinity. layer. Since ip0y_ and \p0^  are less than zero, we have the following situation: for 1JJ12 < o an additional f i r s t order correction transport of 0(A) must enter the boundary current from the interior region (or through the passes) in the transition region; for > °» t n e f i r s t order correction transport of 0(A) must leave the boundary layer in the transition region. The increase of the characteristic width of the boundary layer within the transition region, however, w i l l l i k e l y A lead to > °« This is consistent with the fact that transport through the eastern Aleutian passes is northward; through the more central passes i t is southward (Favorite, 1969). In this section the curvature parameter, £*, was assumed to take i t s maximum value as obtained from (3.2.4) and (3.3.1); i.e. e* = K DL ~ 0(1) . (6.9.1) This meant that §, where from (6.5.5) e = e* EgUi , (6.9.2) was of order E ^ u i and could be neglected i n the zeroth order boundary layer equations. To determine i f this is reasonable, we suppose that curvature effects are directly important in the zeroth order flow; i.e., that, e ~ o ( i ) . Then, from (6.9.1) and (6.9.2), the radius of curvature r. = i = L E * ^ . (6.9.3) o / * AH In the A l a s k a Current, w i t h = 1/3 and ER = 0 ^ - 3 , the radius of curvature Po 35 k i l o m e t r e s w h i l e i n the Alaskan Stream, where y 2 = 1/4, r 2 , (f±)W 80 k i l o m e t r e s I n both cases, the curvature enters d i r e c t l y i n t o the z e r o t h order s o l u t i o n only i f the e f f e c t of r e l a t i v e l y s m a l l i s l a n d s i s i n c l u d e d . Since the s o l u t i o n s already obtained have neglected the convergence of the c u r v i l i n e a r l i n e s and bottom topography, i t h a r d l y seems reasonable to suppose that they may be a p p l i e d t o such s m a l l topographic f e a t u r e s . One c o n c l u s i o n that may be drawn, perhaps, i s that the Alaskan Stream i s more s e n s i t i v e to the curvature e f f e c t of r e l a t i v e l y s m a l l c o a s t a l features than i s the A l a s k a Current. 6.8 Passes and t h e i r p o s s i b l e e f f e c t on the Northern Boundary Layer flow I n the previous d i s c u s s i o n s , the assumption that the Northern Boundary i s s o l i d appears reasonable up to 173°E s i n c e the only passes having s i g n i f i c a n t area between there and 165°W are, the Amukta (19.3 km 2), Amchitka (45.7 km 2) and B u l d i r (28.0 km 2) passes ( F a v o r i t e , 1967). Of these, only Amchitka pass at 180° has a s i l l depth exceeding 1000 metres. To the e a s t , however, Near and Kamchatka passes have a combined area of more than. 570 km 2 and s i l l depths over 2000 metres, rendering a s o l i d boundary approximation i n v a l i d (Figure 21). One e f f e c t the passes east of 173°E could have on the westward boundary l a y e r f l o w i s the p o s s i b i l i t y of northward f l u x i n t o the B e r i n g Sea a c t i n g as a s i n k f o r the negative v o r t i c i t y generated at an upstream boundary, i n analogy w i t h laminar boundary l a y e r s u c t i o n . A l s o , southward flow through.the passes could supply the r e q u i s i t e t r a n s p o r t f o r boundary l a y e r s t a b i l i t y i n the absence of t r a n s p o r t from the i n t e r i o r r e g i o n . The boundary current would be more l i k e l y to remain attached to the coast than i t would i n the absence of the passes. This i d e a i s f u r t h e r encouraged by the e x i s t e n c e of northward t r a n s p o r t through the eastern passes ( F a v o r i t e , 1969), and by the f a c t that passes only e x i s t west of about 165°W where the t r a n s i t i o n from a western boundary current (Alaska Current) to a z o n a l boundary current (Alaskan Stream) occurs. - Figure 22 gives a schematic o u t l i n e of the main fea t u r e s of the quasi-steady c i r c u l a t i o n . FIGURE 21. Passes and 1000 metres depth contour() in the vicinity of the Aleutian Islands. FIGURE 22. Schematic diagram of the main features of the circulation in the Subarctic Pacific Region. 7. The geostrophic winds and resulting circulation 7.1 Data analysis 7.1.1 Introduction In this section, the mean-monthly pressure data over the North P a c i f i c Ocean w i l l be used to obtain the distribution of the mean-monthly, ver t i c a l component of the wind-stress curl. A spectral analysis and demodulation of this wind-stress curl distribution w i l l be performed i n order to determine the dominant Fourier components and how they vary with time. These, together with the general wind-curl distribution, w i l l then be used in an attempt to understand some of the main features of the circulation i n the Subarctic Pacific. The distribution of wind-stress over the sea determines, through momentum transfer across the sea-air interface, the wind-driven ocean circulation. According to Ph i l l i p s (1966, p. 145), this momentum transfer i s accomplished by direct viscous shear at the surface and by momentum flux, from a thin viscous layer at the surface, to short waves. An empirical relationship that appears to represent the situation i s , 131 - PacDU,2o (7.1.1) . / . . . . where t i s the wind-stress on the ocean, p a is the density of ai r , Cn i s the drag coefficient and U 1 0 is the speed of the wind at 10 metres above the surface (an international convention). Measurements of the surface wind-stress show that Cp increases with wind speed, although the exact reasons are not completely understood (Phillips, 1966). Equation (7.1.1) w i l l be the basis of the ensuing analysis with the drag coefficient assumed constant; v i z . , C D = 1.3 x 10-3 (7.1.2) i n agreement with the r e s u l t s of various experimenters p l o t t e d by P h i l l i p s (1966, p. 144). Obviously i t i s not possible to obtain the v e l o c i t y , U 1 0 , for a l l times over the ocean and assumptions must be made as to i t s value based on measurable q u a n t i t i e s . To do t h i s , we w i l l follow Fofonoff (1960) and assume that the surface geostrophic winds, with corrections made f o r the e f f e c t of surface f r i c t i o n , can be used to approximate U 1 0 • The v a l i d i t y of t h i s assumption w i l l be considered s h o r t l y . 7.1.2 Surface wind-stress To obtain the geostrophic winds away from the f r i c t i o n a l influence of the earth*s surface, one assumes that the pressure gradient balances w i t h the C o r i o l i s force, i n the same manner as for geostrophic ocean currents, i . e . , P a f u = » tf-1-3* P a f v - F^bs^fc ' ( 7'1-4 ) where s p h e r i c a l coordinates are r e a d i l y useable and where u = (u,v,o) i s the geostrophic wind v e l o c i t y , f = 29, sin<j> (fi i s the angular speed of the earth's r o t a t i o n ) , <j> i s the l a t i t u d e , A i s the longitude and R i s the radius of curvature of the earth's surface. In order to obtain a more r e a l i s t i c approximation to the surface winds, we apply a transformation of the form " S = r 1 K 1 ! v s I a 2 b 2 v C7.1.5) where u s = ( u s,v s,o) i s now the s u r f a c e wind v e c t o r and a±, ..., b 2 are constants which c o n t a i n both a r o t a t i o n and a magnitude r e d u c t i o n to account f o r the f r i c t i o n a l b r a k i n g of the sea s u r f a c e ; i . e . , u s = r exp ( i y ) u > (7.1.6) where r i s the r e d u c t i o n f a c t o r (0.7, here) and Y i s the r o t a t i o n angle which i s u s u a l l y l e s s than 10° over the sea (Flohn, 1969). This i s I l l u s t r a t e d i n f i g u r e 23 which shows how surface f r i c t i o n causes the s u r f a c e winds to blow towards lower pressure at the angle Y to the upper l a y e r i s o b a r s (500 m i l l i b a r l e v e l ) . The components of s u r f a c e wind-stress are then given by T w = p a C D u s ( u s 2 + v g 2 ) 1 / 2 , (7.1.7) and 4 = Pa CD v s ( u s 2 + v s 2 ) 1 / 2 , (7.1.8) from which we o b t a i n the v e r t i c a l component of the wind-stress c u r l , VxT w-k = " 1 , { |T- (T X COS<$>) ~ | Y 4 > • (7.1.9) R coscp dcp w 3A w The beta-plane components of the above s u r f a c e wind-stress are, of course, w l w and (7.1.10) T w T w * LOW PREssune C o r i o l i s f o r c i HIOH PRESSURE a. Geostrophic winds. LOW PRESSURE HIOH PRESSURE b. Friction modified winds. FIGURE 23. Balance of forces o. far from the ocean surface. b. near the ocean surface. 7.1.3 The pressure data The data used in the computations of (7.1.9) are the mean-monthly surface atmospheric pressures compiled i n card-image format by the Extended Forecast Division of the United States Weather Bureau, Washington D.C. These consist of surface pressure less 900 millibars at discrete points between 15°N and 90°N for every five degrees of longitude, such 1 that nonzero values are given when either, both the longitude and latitude are even multiples of 5 or are both odd multiples of 5 (e.g. 55°N, 135°W; 45°N, 125°W; 60°N, 130°W; ,...). For each month there are 72 card image records (corresponding to 360° of longitude) consisting of: 1) an identification, giving the year and month; 2) the longitude, west to 355°W (5°E) ; • . 3) the surface pressure less 900 millibars beginning with the higher latitudes, 90° or 85°, depending whether the longitude i s even or odd, respectively. The oceanic grid for the North Pacific i s chosen between 65°N and 15°N with different longitudinal extent for each latitude to approximate the eastern and western coasts. Wind-stresses calculated for 20°N w i l l be in error at certain longitudes because of a lack of pressure data at 15°N. Any d i f f i c u l t y with this i s avoided by simply considering 25°N as the southern limit of interest. The f i n i t e difference calculations used to obtain the pressure gradient, as well as the second derivatives (32P/3A2, 32p/3A3(J> , S^P/3(p^), at an interior point (I,J)- of the oceanic grid have the forms (Froese, 1963) 3P/3A = I2CP 5-P6) + Pl. + P4 " P2 - P 3 J / 6 1 . 3P/3$ = (? x + P 2 - Pj - P4>/21, 32P/3A2 = (P 5 - 2P D + P 6 ) / l 2 , 32P/9(J)3X = CPi - P 2 + P 3 - P A)/1 2, and 32P/3c}>2 = I2CP! + P 2 + P 3 + P 4) - P 5 - P 6 - 6PD)/12, where 1 = (10°) TT/180° i s the separation of the data points, and where PD,..., Pg are as shown i n f i g u r e 24. The computer program to which these approximations are applied i s b a s i c a l l y that w r i t t e n by- Fofonoff & Froese (1960). This was reprogramed by Froese & Fowler (1964) i n Fortran IV for the old IBM 7040. system at the computing center of the U n i v e r s i t y of B.C. I t has been furth e r adapted (Thomson, 1969) f o r the IBM 360/67 i n current use at the computing center. 7.1.4 The computer output of monthly-mean q u a n t i t i e s The monthly-mean qua n t i t i e s f o r which computer output i s obtained are; ( i ) the zonal and meridional wind-stress components, and T$ , r e s p e c t i v e l y ; ( i i ) the v e r t i c a l component of the c u r l of the surface wind-stress, using (7.1.9). These are p r i n t e d out, together with the surface pressure (less 1000 m i l l i b a r s ) i n tenths of m i l l i b a r s , as a l o n g i t u d e - l a t i t u d e format f o r each 5° between 65°N and 15°N l a t i t u d e and 115°E and 100°W longitude. Also, since we are mainly i n t e r e s t e d i n the Subarctic P a c i f i c , i t i s convenient to construct a numbered g r i d of 'stations' for t h i s region. This.eliminates tedious s p e c i f i c a t i o n of the longitude and l a t i t u d e when considering a given oceanic p o s i t i o n (Figure 25). FIGURE 24. Pressure grid for the finite difference calculations. L s (10°) 7T/I80° radians. We note here, that, i f (1) is used in (5.3.6) and ( i i ) in (5.3.3), the total Ekman wind d r i f t components and the total integrated northward transport may be obtained. Such computations for a l l months up to the present, beginning in January 1945, have been performed by members of the Pacific Oceanographic Group, Nanaimo, B.C. The original program begun by Fofonoff (1960) has been extended by Wickett (1966) and by Wickett, Thomson & Dienaar (196?) with the surface transports now being plotted as vectors for each 5° grid point. 7.1.5 Geostrophic and actual surface winds The conditions under which geostrophic wind velocities are a reliable estimate of the actual surface wind velocities i s s t i l l a subject of investigation. As far as the computations performed here are concerned, the use of mean-monthly pressure data and large grid spacings (5°) are deemed responsible for most of the difference between the geostrophic and the actual winds. This w i l l be particularly true i f the magnitude and direction of the winds change often within a month. Peak winds w i l l be underestimated because of the coarseness of the grid. For January 1960, Fofonoff (personnal communication) compared the twelve-hourly observed and geostrophically calculated winds at Station 'P' to determine the effect of v a r i a b i l i t y on the estimated winds. The resulting wind vectors are shown in a cumulative vector diagram (Figure 26), i n which the changing strength and direction of the wind at a given point i s represented by joining the vectors to one another in chronological order. There is f a i r l y good agreement between the two, although, as Fofonoff points out, there can be large errors i n the stress estimate because in the square law (7.1.1) strong winds have higher weighting. * \ *30 @ OBSERVED WINDS » — A GEOSTROPHIC WINDS - 9 ' 9 7 9 10 10 - A ^ , J 0 29 \ A28 \ \ 27* 21 24 9, 22. 2^25 22 a * AI3 ~3>.ir-24 K • m I /2I 72 14 CUMULATIVE VECTOR WINDS Ocean Station"p" January I960. A-^9-20 20 »I9 A l 8 >o/8 4* I6 A - A — r ; * /5 N 500 jOOO Kilometres. FIGURE 26. F i g u r e 27 i s a l s o from the same p e r i o d . In i t , the three-hourly measured winds and the t w i c e - d a i l y geostrophic winds have been used to o b t a i n the s u r f a c e s t r e s s . The shape of the two curves i s s i m i l a r , suggesting that the geostrophic values c o r r e c t l y i n d i c a t e the changes i n d i r e c t i o n , although the magnitude of the changes i s c l e a r l y underestimated. Figure 28 compares the geostrophic wind-rstress obtained using the mean-monthly pressure w i t h that obtained u s i n g the t w i c e - d a i l y pressure, f o r January 1960 at s t a t i o n 'P'. The l a t t e r i s d e f i n i t e l y underestimated compared to the f a c t u a l ' wind-stress f o r t h i s month. I f the angle of d e v i a t i o n between the estimated mean-monthly wind v e c t o r and the r e s u l t a n t observed v e c t o r i s r e l a t i v e l y c o n s i s t e n t , there w i l l be l i t t l e e f f e c t on the v e r t i c a l component of the surface w i n d - s t r e s s c u r l estimate, although i t s magnitude w i l l be somewhat low. The one 1 month of comparison presented here i s , of course, not n e c e s s a r i l y r e p r e s e n t a t i v e of other times or regions. A l s o , s i n c e wind v a r i a b i l i t y i n January of each year i s much greater than average, the agreement between geostrophic and a c t u a l s u r f a c e winds i s probably at i t s worst. Over longer p e r i o d s , the r e s u l t s are more encouraging s i n c e there i s reasonably good agreement w i t h averages obtained from wind roses, such as computed by Hidaka f o r the P a c i f i c (N.P. F o f o n o f f , personnal communication). F u r t h e r , as we have j u s t seen, the geostrophic estimates may be g r e a t l y improved by use of pressures averaged over much s h o r t e r p e r i o d s ; d a i l y pressure charts over the P a c i f i c could perhaps be employed f o r t h i s purpose. 7.2 R e s u l t s 7.2.1 General d i s t r i b u t i o n s of the wind-stress c u r l As one would expect, the w i n d - c u r l at a p a r t i c u l a r s t a t i o n i s FIGURE 27. 1.0 N I E u </> <w c >. Q >% 0.5 STRESS VECTOR ESTIMATES Ocean Station "p" January, I960. Estimated from : 1. Monthly mean pressure 2. Twice-daily pressure. 0 5 r*, Dynes c m " z Fl G U R E 28. 1.0 1.5 dependent upon the time and the location. In general, i t appears that during the period of heating (April-September) the wind-curl distribution is of small magnitude with a clockwise tendency (i.e. tending to be more negative than in the cooling period), while in the cooling period (.October-March) i t tends to become counterclockwise (positive), with an increase in magnitude associated with a l l but the more southern stations (Figure 29). In figure 30 a plot of the wind-curl for the individual months between September 1954 and April 1962 i s given. This shows that the larger (more positive) values do occur in the cooling period in most cases and minimum values in the warming period. Also, the sharp decrease in the cooling period means after 1959, in the northeastern part of the region, is seen to be associated with a decrease in the maximum winter wind-curl which appears to occur always between December and February in that region. Spatially, the wind-curl distribution over the heating and cooling periods is approximately zonal, far enough from the continental boundaries, going from positive values in the northern part of the region to negative values to the south (about 40°N). This zonal distribution i s most pronounced during the cooling period, while i n the warming period i t tends to be 'patchy'; cf. figures 37 and 38 (pages 168-171). The large positive values in winter are associated with the strong cyclone centered off Alaska (The Aleutian Low; figure 31a) that establishes i t s e l f over most of the North Pacific at that time. The weaker values i n summer are associated with the North Pacific High anticyclone (Figure 31b) which centers somewhere in the eastern part of the region, at mid-latitudes. The a b i l i t y of these two systems to establish themselves in their respective seasons determines the v a r i a b i l i t y an'd magnitude of the general wind-curl distribution. Thus, the striking decrease of the cooling period FIGURE 31b. Mean sea level pressure in summer (millibars). means after 1959. are due to the decreased pressure gradients in the Aleutian Low in winter and a south-western shift of the. area of maximum depression. The distribution of wind-curl remains uniform throughout the summer for the same years, suggesting l i t t l e change i n the North Pacific High patterns compared to those of the mean Aleutian Low patterns i n winter. The western part of the region, enclosing the Aleutian islands, does not reflect the changes i n the northeastern corner. Instead, i t appears to,remain f a i r l y uniform from year to year. In figures 39a,j, the spatial plots of the mean annual (April-March) wind-stress curl are given for every second year to show the variation i n their accumulated effect over the f u l l period being considered. The basic features are: the near zonality of the small magnitudes i n the central region; the relatively large values over the western Aleutians; and.the change of the strong values in the Gulf of Alaska, from a maximum in 1958 to much reduced values after 1959. The southward shift of the maximum in the Gulf of Alaska, after 1958, is also apparent. For a l l years shown, excluding 1964, a minimum in the wind-stress ,etij?l distribution occurs over the south-western part of the Alaskan Peninsula. It reached i t s lowest values i n 1958 when the wind-curl over the Gulf of Alaska was at i t s maximum. Finally, the mean wind-stress curl from January 1945 to April 1966 is shown i n figure 32, The most obvious features are: zonal distribution, except i n the northeast corner where a maximum occurs; and a relative minimum, centered near station 3, over the eastern part of the Aleutian island chain. Also, the wind-stress curl at stations enclosing the Aleutian-Komandorski island chain has a relatively- large and uniform, climatological mean. 7,2.2 General distribution of the oceanic circulation Since the wind-stress curl determines the quasi-steady transport streamlines through the Sverdrup balance (5.3.1), except i n the boundary layer regions, the general features of the quasi-steady and steady total transport fields can be described. The eastward flowing Subarctic Current and West Wind Drift (see figure 22), which form as eastern extensions of the waters originating from the Oyashio and Kuroshio currents, move into a region of nearly zonal annual wind-curl distribution. The mean axis of this general eastern flow w i l l then be the wind-curl zero, with a northern increasing northern component of transport to the l e f t of the flow (positive wind-curl) and a southern increasing southern component (negative wind-curl) to the right. Near the coast of North America, the current w i l l be forced to turn, with flow north of the wind-curl zero going north and that to the south going south. Therefore, the observed current s p l i t t i n g off North America appears directly related to the wind-stress curl distribution; variations in the mean annual distribution w i l l result in variations in the position of the sp l i t t i n g . There w i l l also be shorter period effects, which tend to be more local, associated with the monthly fluctuations. The maximum values in the northeastern part of the Gulf of Alaska cause a large transport into that region with the formation of the strong Alaska Current as a result. If we consider the climatological mean of the wind-stress curl, in this corner region, to have a value of 7 x 10-9 dynes/cm as suggested by figure 32, then using (5.3.3), the total northward transport per unit of latitudinal distance moving into the region i s , < JJY > = < V x | w . k > ~ 7 x 10 6 gm m - 1 s " 1 i n rough agreement w i t h g e o s t r o p h i c a l l y c a l c u l a t e d values f o r the Alaska Current ( F a v o r i t e , 1967; p. 14). As the wind-stress c u r l over the northern p a r t o f the r e g i o n , p a r t i c u l a r l y the northeastern p a r t , becomes much more p o s i t i v e i n w i n t e r than i n summer, there w i l l be a corresponding i n c r e a s e i n the t o t a l t r a n s p o r t of the Alaska Current-Alaskan Stream system. This w i l l manifest i t s e l f i n an in c r e a s e i n current v e l o c i t y and/or widening of the boundary l a y e r , as indeed has been observed (Reed, 1968). A l s o , the t r a n s p o r t v a r i a t i o n s from w i n t e r t o w i n t e r could r e s u l t i n v a r i a t i o n s i n the maximum western extent of the Stream through some s o r t of ' i n e r t i a l overshoot', a l l other f a c t o r s being equal. The s i g n i f i c a n t values of the w i n d - c u r l , over the A l e u t i a n -Komandorski i s l a n d c h a i n , west of about 175°W a l s o support the ex i s t e n c e of a westward extended boundary flow w i t h subsequent northward t r a n s p o r t through the l a r g e western passes (see § 6.8). The reduced northward t r a n s p o r t i n the v i n c i n i t y of the A l e u t i a n i s l a n d s i n summer, and the observed annual r e c i r c u l a t i o n of some Alaska Stream water i n t o the S u b a r c t i c Current at t h i s time (e.g. Dodimead, 1958), i s c o n s i s t e n t w i t h the r e s u l t s of the t h e o r e t i c a l model (§ 6.7). F i n a l l y , the s i g n i f i c a n t decreases o f the w i n t e r means a f t e r the w i n t e r of 1958-59, i n the nor t h e a s t e r n corner, i n d i c a t e t h a t l e s s t r a n s p o r t moved i n t o t h i s r e g i o n than i n previous w i n t e r s . The mean s t r e n g t h of the currents i n the re g i o n bounded by the most s i g n i f i c a n t l y effected stations, namely 1, 2 and 9, w i l l therefore be reduced. This i s not true of the summer averaged transport, which as seen from figure 29, would have remained relatively constant over the f u l l period. The maximum values of the wind-stress curl, occurring north of 50°N and into the Gulf of Alaska region in 1958, suggest that this would have caused large anomalous transports and currents to be recorded over the region, as indeed were found by Dodimead (1961b) from geostrophic calculations. In comparison, 1964 would have been a year of smaller annual transports and weaker currents over the same period. 7.2.3 Spectral analysis 7.2.3.1 Data handling In order to obtain some idea of the dominant time scales for the observed variations of the mean-monthly wind-curl, a spectral analysis was performed on this data from January 1945 to April 1966. The transformation from the time domain to the frequency domain was done through use of the Institute of Oceanography's FTOR program, while the spectra etc., where obtained using the complementary SCOR program. These are part of a program library written by J.R. Wilson andJ.F. Garrett (.Garrett, 1970) of this Institute, and modified for card input by G.A. HcBean. Briefly, the data are transformed using a fast-Fourier-transform algorithm which requires that there be 2 m (m is a positive integer) discrete data points. The resulting 2 m - l 4- 1 complex Fourier coefficients are printed out on tape, from which, the spectra, cospectra and quadrature spectra are computed using the real and imaginary parts of the Fourier components. 7.2.3.2 The wind-curl magnitude spectra The spectral plots are shown in figures 40a,k (page 182) . At a l l stations, from the Gulf of Alaska south to 50°N (except station 12), the annual harmonic, resulting from the winter build-up and summer break-down of the Aleutian Low (figure 31a), is the most dominant. This i s particularly true of stations 1, 2 and 9 where the magnitude of the annual harmonic greatly exceeds that of the next largest, the semi-annual harmonic. These large coastal values are due to the deflection of winter cyclones by the mountains with a resulting 'squashed' appearance to the isobars and large curvatures. Similar spectra occur at stations 40-44 for the annual variation, but only for station 43 is the semiannual component large compared to other frequencies. In this case, the relatively large annual values are due to the westward movement of the Aleutian Low from the Gulf of Alaska, in winter, to the western Aleutian islands, as summer approaches. As the pressure system is weakening during i t s motion, the resulting wind-curl values are much less. The annual component i s again dominant for stations 15-19 and 23-25 although greatly reduced, both in absolute value and i n comparison to other frequency bands. Magnitudes for stations 20-22 are small at a l l frequency bands. These lower, southern wind-curl magnitudes are due to the North Pacific High anticyclone which dominates the regions during summer, and which eventually occupies the northeastern portion of the Pacific (figure 31b). The pressure gradients associated with the High are much less than those associated with the Aleutian Low in winter. Very low magnitudes at station 20 result from the almost nonexistent pressure gradient over this station and the lack of curvature of the Isobars to the west of i t . Figure 33 is a plot of the spatial distribution of the intensity of the twelve-month harmonic. It essentially repeats, in a more concise manner, what was revealed by the summer and winter spatial distribution plots of §"'7.2.2. The extremely large values in the Gulf of Alaska again Indicate that the summer to winter differences in wind-induced northern transport into this region w i l l be large. Even though the direct response i s mostly barotropic for periods shorter than six months (see § 4), there w i l l be observable changes in the baroclinic mode due to interactions between these two modes. In the very northeastern corner, where the summer to winter difference in the wind-curl is most pronounced, a marked change in current velocities and/or complexity would be expected. Dodimead (1958) finds that between August 1956 and February 1957 the baroclinic transport into the Gulf of Alaska increased noticably, but that from February 1957' to August 1957 there was no corresponding decrease in transport. Bogdanov (1961), using data from 1957 to 1959 obtained from the International Geophysical Cooperation, does find an increase in the winter baroclinic transport over the summer baroclinic transport in the Gulf of Alaska, although i t is much smaller than expected. That there does not appear to be a consistent winter to summer change in the observed transport, compared to the consistent change in the wind-curl by factor of 3-10 in the same period, suggests that other effects besides that of the wind-curl are important in causing variations about the steady flow. Two such effects could be, an increased amount of northward transport in some summers due to a southward shift of the line of zero wind-curl, and/or bottom topography i n the shallowing northeast corner of the region. The latter would cause a departure from geostrophy and complicated interactions FIGURE 33. Distribution of curl spectra the annual component of the wind-stress ( id 1 5 dynes2 cm9). between the b a r o t r o p i c and b a r o c l i n i c modes. A l s o , the dynamic c a l c u l a t i o n s on which the 'observed' t r a n s p o r t s are based represent a time and space averaged mean flow and could vary g r e a t l y from the a c t u a l t r a n s p o r t f i e l d , p a r t i c u l a r l y i n such a complex region as the Gulf of A l a s k a . The only other region of s i g n i f i c a n t w ind-stress c u r l magnitude i s that west of 180° over the Aleutian-Komandorski i s l a n d chain. As p r e v i o u s l y mentioned (§ 7.2.2), t h i s i s c o n s i s t e n t w i t h the requirement of t r a n s p o r t i n t o the boundary l a y e r i n order to have the observed westward extent and s t a b i l i t y of the Alaskan Stream. Figure 34 i s a p l o t of the six-month component f o r comparison to f i g u r e 33. The s t r u c t u r e i s approximately the same as that of the annual c y c l e but much reduced i n magnitude i n the nor t h e a s t e r n corner. Magnitudes at a l l other s t a t i o n s are w i t h i n a f a c t o r of t h r e e , or so, of the annual values. The comparatively l a r g e magnitudes at s t a t i o n s 12 and 13 are due t o tbe w i n d - c u r l reaching a maximum, both: i n w i n t e r and i n summer; a r e s u l t of being i n a p o s i t i o n to experience moderately s t r o n g winds (~ 1 dyne/cm 2) both from the A l e u t i a n Low i n w i n t e r and the North P a c i f i c High i n summer. Near the North American coast, north of 50°N, the A l e u t i a n Low completely dominates the annual wind p a t t e r n s . This i s a l s o t r u e of the A l e u t i a n i s l a n d r e g i o n s i n c e the summer a n t i c y c l o n e i s not l a r g e enough to have an i n f l u e n c e over great d i s t a n c e s . A fea t u r e of the more c e n t r a l s t a t i o n s (6, 13-19) i s the r e l a t i v e s i z e of the frequency band about 1 c y c l e / 3 months compared to other bands at the same s t a t i o n . This i s probably a r e s u l t of the i n t e r p l a y between the developing (or weakening) North P a c i f i c High and the westward moving (or developing) A l e u t i a n Low. This harmonic i s s t i l l , however, weaker than the twelve-month c y c l e in most cases and below the magnitude FIGURE 34. Distribution of the semi-onnuol component of the . . . -is 2 -e wind-stress curl spectra (10 dynes cm ). of the three-month cycle at more northern stations. The strongest occurrence of any 'high' frequency ( i . e . > 0.4 month occurs at station 10 near the northern tip of Vancouver Island. Here, the frequency of 4.59 x 10 -1 month--'- has a value comparable to the semi-annual component and is apparently an anomalous feature for the north-eastern region where the spectrum for this frequency is usually very low compared to others. Because of the strength of this component, there exists the possibility of directly forced Rossby waves occurring. If we assume these waves would have the same frequency, F, as the wind-curl forcing, then use of (3.2.1) and (4.3.6) gives a wavelength L = 4TT2F/3 500 kilometres. This is consistent with the observed distance between countercurrents along the line joining Station 'P' and Swiftsure Bank (Fofonoff & Tabata, 1966). The possibility of the wavelike pr o f i l e , presented by the countercurrent-current pattern, representing stationary Rossby waves was considered unlikely by Fofonoff & Tabata since Rossby-type waves near the eastern coast would tend to move northwards in a region where the mean flow was also northwards. Further discussion is not warranted, however, since theoretical studies of Rossby waves at boundaries in general, and observations i n time and space in this region i n particular, are insufficient. An estimate of the 'energy' due to a l l frequencies may be obtained by integrating the spectral curves over frequency. A plot of the results is given in figure 35. The spatial distribution of tbe wind-curl magnitude, squared, due to a l l frequencies is similar to that for the twelve-month component simply because that is- the usual dominant frequency, However, the plot does emphasize the importance of other frequencies at stations outside the station 1 in figure 33 but is 1/10 i n figure 35. Thus, the other frequencies tend to make up for the difference resulting from the peak of the 1/12 month"! component, or, i n other xrords, the 'energy' is more evenly distributed when a l l frequencies are considered. Furthermore, the homogeneity' of the integrated spectrum outside the Gulf of Alaska indicates that a nearly constant amount of water is transported about by the wind-curl effect (in the Sverdrup sense) over the whole period 1945-1966; i.e., that the root-mean-square value over this period is within a factor of two to three for the entire Subarctic Pacific, excluding the Gulf of Alaska. curl The cospectrum, C A B ( F ) , is obtained from the real part of the product of the complex Fourier coefficients of two spatially separated wind-stress curl signals ( A and B) over the same frequency band. It t e l l s how well the wind-stress curl at the two stations, for a given frequency band, is i n phase. The quadrature spectrum, Q A B ( F ) , which is the imaginary part of the above product, gives the out of phase component. From these we obtain the coherence of the two signals, northeastern region. For example, station 6 has about 1/100 the value of 7.2.3.3 The cospectra and quad-spectra for the wind-stress SA(J).SBCF) FIGURE 35. Integral under the wind-stress curl spectrum -is 2 . ( 10 dynes cm ). where S^(F) and Sg(F) are the respective spectra. The phase difference is given by The maximum co-spectra for stations 1, 2 and 8-12, in the central-northeastern part of the Subarctic, are at the annual cycle. As the cospectra between stations 9 and 10 and between 11 and 12 show, there is also a large amount of in-phase variation at the six-month period for this latter region; between stations 12 and 13 the co-spectrum is maximum at this frequency. In the eastern section of the Bering Sea (stations 4-6), the co-spectra are not dominated by any one particular frequency as i s also the case for the co-spectra between station 3 and those surrounding i t . To the south of 50°N, except for the low spectral region near station 20, the annual cycle again emerges as the dominant frequency. This i s so over the Aleutian^Komandorski island chain also, with no other peaks i n the spectrum. The quad-spectra are usually less than the co-spectra except for regions of low spectral values. Again the twelve-month component emerges as the dominant frequency, although less so than for the co-spectrum. Figures 41a,1 are given as examples (see pages 193 to 204 ) . Coherences between various stations are plotted i n figures 42a,b (pages 205to 206). Those between 6-7, 16-17, 18-19 and 22-23 show a trend of decreasing coherence for increasing frequency. This suggests a greater amount of independence, and hence smaller characteristic wind-curl patterns, with increasing frequency. This is unlike the situation between stations 1 and 2 where coherences of nearly 1.0 occur at the annual and semi-annual p e r i o d s , d r o p - o f f , then again become l a r g e (> 0.8) beyond frequencies of 0.30 month -!. Combining t h i s w i t h the low coherences between s t a t i o n s 2 and 3 i n d i c a t e s that the w i n d - c u r l systems g i v i n g r i s e to these coherences, at r e l a t i v e l y b i g h frequency, have l o n g i t u d i n a l s c a l e s of about 10° i n the no r t h e r n p o r t i o n of the Gu l f of A l a s k a . The phase of the frequency components between two s t a t i o n s v a r i e s g r e a t l y w i t h frequency; f i g u r e s 43a,b (pages 207to 208). I f we consider the twelve-month c y c l e , we f i n d a d e f i n i t e p a t t e r n emerging, however. The developing w i n d - c u r l d i s t r i b u t i o n spreads out from c e n t r a l s t a t i o n s C3, 6 and 14) w i t h s t a t i o n s 10, 20 and 43 apparently being among the l a s t to r e c e i v e the s i g n a l . Since the minimum coherences are between the c e n t r a l s t a t i o n s (3, 6 and 14) and surrounding ones, the spreading w i n d - c u r l p a t t e r n a t t a i n s i t s maximum st r e n g t h away from the c e n t r a l s t a t i o n s . Thus, although the p a t t e r n begins to develop about the c e n t r a l S t a t i o n s i t does not a t t a i n s t r o n g values there. The phases between s t a t i o n s near 45°N are approximately zero, so that the annual w i n d - c u r l p a t t e r n i s changing simultaneously at each s t a t i o n (on the average) over t h i s r e g i o n . The exception here i s s t a t i o n 20 which i s about 180° out of phase w i t h s t a t i o n s 18 and 19. Near 40°N the p a t t e r n appears to develop i n the opposite sense, i n that s t a t i o n s west of 21 i n c r e a s i n g l y l a g one another. The above s i t u a t i o n s are e x p l a i n a b l e i n terms of the developing A l e u t i a n Low which s i t s approximately over the region centered by s t a t i o n s 3, 6 and 14 i n w i n t e r . Near i t s c e n t r e , the deepening pressure trough remains r e l a t i v e l y - uniform, w h i l e as i t s dimensions i n c r e a s e w i t h w i n t e r the more p e r i p h e r a l regions take on l a r g e pressure g r a d i e n t s . The curvature of the f u l l y developed cyclone i s maximum against the mountainous coast of the.Gulf of A l a s k a i n mid-winter, w i t h r e l a t i v e l y l a r g e value over the Aleutian-Komandorski i s l a n d chain a l s o . To the south, the developing North P a c i f i c High i n summer, centered about 20° west of C a l i f o r n i a , gives r i s e to a westward moving disturbance, i f the annual frequency i s considered. By the time i t reaches A l a s k a , the system i s very weak. Therefore, considered oyer a year, the summer a n t i c y c l o n e has l i t t l e e f f e c t on the d i s t r i b u t i o n i n the Gulf of Alaska. I t does, however, a f f e c t s t a t i o n s l i k e 11 and 12, f o r example, i n the oceanic r e g i o n j u s t south of the Gulf of Al a s k a by i n t e r p l a y w i t h the A l e u t i a n Low. This gives the s i g n i f i c a n t six-month component there. The low magnitudes at s t a t i o n 20 are a s s o c i a t e d w i t h the s m a l l curvature of the i s o b a r s there during summer. Since these i s o b a r s have a s l i g h t l y c y c l o n i c curvature i n summer, w h i l e those over other surrounding oceanic s t a t i o n s , such as s t a t i o n 18, l i e near the center of the North P a c i f i c High, the phase d i f f e r e n c e between these and s t a t i o n 20 should be about 180°; as observed. < 7.2.4 Demodulation Through the use of a demodulation technique, i t i s p o s s i b l e to estimate the time v a r i a t i o n of the magnitude of the w i r i d - c u r l f o r a p a r t i c u l a r frequency band. B a s i c a l l y , i t c o n s i s t s of s h i f t i n g the frequency band of i n t e r e s t to zero frequency, and then a p p l y i n g a low pass f i l t e r . Here, the hanning f u n c t i o n (Blackman & Tukey, 1959) 1/2 { 1 - cos [ J } i s a p p l i e d , where 1/2 -y8/3 i s the n o r m a l i z a t i o n constant and j = 1, 24 i s the time l a g . The f i r s t nonzero value occurs at January 1946 since this is taken as the position of the estimated demodulation values for the f i r s t twenty-four month period, January 1945 - January 1947. Demodulated values are then given for the frequency bands centered at frequencies, K T.-1 F = Y2 month x ; K = 1, 6. As a reference, figures 36a,b show the frequency response for a spike*' and a sine wave, respectively. The spreading of the energy to twelve months on either side of the spike-input is simply a manifestation of the weighting. In the sine wave case, the demodulation works as follows; there is again the spreading over months outside the actual data interval for the nominal frequency Cl cycle/12 months in this case), but the maximum values are recorded within this interval. When the ends of the data record are not within the weighting interval, a constant; maximum value for the magnitude of the wind-curl over the interval i s obtained; that i s , the magnitude at 1 cycle/12 months is a constant. At higher frequencies, when the weighting includes the ends, 'energy' from the side lobes of the spectral-window leaks into the magnitude estimate from the 1/12 cycles/month frequency. When no ends are present in the weighting at these higher frequencies, there is no such contribution and the spectral estimate is constant. The amount of leakage is greatest from the next lowest and highest frequency for any given frequency, so that, in the case of the 1 cycle/6 months component, leakage from the annual cycle could be significant. Some representative data and corresponding demodulated values are FI GURE 36 a. The demodulation of a 'spike' input signal ( width - 2 months) for six frequency bands. plotted in figures 44a,j (pages 209to 218). The scale of the ordinate axis for the smallest frequency. (the annual cycle) i s twice that of the others in order to prevent overlapping of the graphs. The most striking feature of the demodulation appears at stations 1, 2 and 9 where the large magnitudes at the annual and semi-annual periods, between January 1945 and January- 1959, begin to rapidly decay. By January 1960 they have decreased by factors of 3-5. This i s particularly true at station 1, where no recovery, whatever, appears to occur. Up to January 1959, and after January 1960, the magnitude of the twelve-month component at stations 1 and 2 i s only slowly varying, with main peaks i n January 1956 and January 1959. At station 9, however, the variations are much more rapid with important peaks occurring in January of 1952, 1954, 1956 and 1959. The semi-annual cycle tends to be similar to the annual one but with greater fluctuation. Of the six frequency bands at stations 1, 2 and 9, the 1 cycle/4 months and 1 cycle/3 months appear to have the least trend; values prior to January 1959 are comparable to those in following years. At higher frequencies, which are approaching the sampling rate of one sample per month, the period between January 1945 and January 1959 again produces the most sign i f i c a n t variations, particularly in January of 1950, 1951, 1957 and 1959. At other stations, the magnitudes of the six frequency components ate usually much smaller than those of stations 1, 2 and 9 prior to 1960, and exhibit a large amount of variation. Trends, when they appear in the modulation, as for example in the magnitude increase after 1958 for frequencies 1/12, 1/6, 1/4, and 1/3 month - 1 at station 16, tend to be small. S t a t i o n s which, e x h i b i t the s m a l l e s t magnitudes and v a r i a t i o n s at a l l frequency bands appear c l o s e to s t a t i o n 20, which d i s p l a y s the smoothest data r e c o r d of any other i n the p a r t of the S u b a r c t i c Region considered. I n g e n e r a l , the v a r i a t i o n s o f the magnitude of the w i n d - c u r l frequency components over the whole region show no apparent, o v e r a l l p a t t e r n , except i n the northeast corner. The twelve-month component g e n e r a l l y presents the most constant or s l o w l y v a r y i n g values simply because i t i s r e l a t e d to the annual c y c l e of s o l a r h e a t i n g . One could, t h e r e f o r e , be f a i r l y confident i n choosing a s c a l e f o r t h i s component although the occurrences at s t a t i o n s 1, 2 and 9 should warn of complete f a i t h i n the value chosen. The magnitudes i n the s p e c t r a l estimates f o r the twelve-month component, as given i n § 7.2.3, are t h e r e f o r e a reasonable r e p r e s e n t a t i o n of the magnitude of tth'e w i n d - c u r l , except i n the n o r t h e a s t e r n corner. For h i g h e r f r e q u e n c i e s , however, the time v a r i a t i o n s are g e n e r a l l y more e r r a t i c , w i t h periods of n e a r l y constant magnitude becoming i n c r e a s i n g l y r a r e r w i t h i n c r e a s i n g frequency. This i s to be expected s i n c e monthly v a r i a t i o n s i n the mean-monthly w i n d - c u r l are o f t e n l a r g e even i n summer, so that the response approaches that from a s e r i e s of broad p u l s e s . The h i g h e r the frequency then, the l e s s l i k e l y that a s i n g l e number can be used to p r o p e r l y s c a l e the w i n d - c u r l . Therefore, the s p e c t r a l estimates of the previous s e c t i o n have meaning only as a s o r t of average. One e x c e p t i o n to the l a t t e r i s the 5° region centered at s t a t i o n 20, where s m a l l magnitudes w i t h l i t t l e v a r i a t i o n occur over the analyzed p e r i o d . The r e l a t i v e l y s m a l l v a r i a t i o n s in. the amplitude of the twelve-month component means that the general y e a r l y d i s t r i b u t i o n of t o t a l t r a n s p o r t v a r i e s l i t t l e throughout the sampling p e r i o d (1945-1966). Thus, i n the northern p a r t of the S u b a r c t i c r e g i o n , the l a r g e p o s i t i v e w i n t e r values decay to s m a l l e r values i n summer. A f t e r 1959, at s t a t i o n s 1, 2 and 9, t h i s was s t i l l the case but now the w i n t e r and summer values had become more comparable. At more southern s t a t i o n s (e.g. 23), negative w i n d - c u r l and, t h e r e f o r e , southward t o t a l t r a n s p o r t o f t e n occur i n w i n t e r w i t h near zero values i n the summer. Since the magnitudes of the twelve-month components are u s u a l l y l a r g e r than those of other frequencies i n the Gulf of A l a s k a , but only of comparable s i z e i n other r e g i o n , the w i n t e r to summer d i f f e r e n c e s i n t r a n s p o r t are expected to be l a r g e r i n the former case. The six-month component i s g e n e r a l l y q u i t e s i m i l a r to the twelve-month component i n shape but u s u a l l y much more v a r i a b l e . The reason f o r i t s s i g n i f i c a n t magnitude i s the change from A l e u t i a n Low domination i n w i n t e r to P a c i f i c High domination i n summer over a l a r g e p o r t i o n of the S u b a r c t i c Region. In the A l e u t i a n i s l a n d s region the e f f e c t of the summer High i s not f e l t and any six-month c y c l e i s r e l a t e d to the westward s h i f t of the A l e u t i a n Low as summer approaches. 8. Summary In this part of the thesis, several aspects of the wind-driven ocean circulation in the Subarctic Pacific Region have, been studied. The large-scale motions characterizing the circulation were assumed to have periods much longer than a half pendulum day and to be composed of a quasi-steady (time averaged) flow about which the planetary wave motions occur as fluctuations. In regard to the latter form of motion, i t was pointed out that Rossby waves carry energy to the west i f their zonal scale, resulting from a nearly zonal pattern of the curl of the wind-stress, is much greater than their longitudinal scale. As analysis of the mean-monthly wind-stress curl data showed, this type of situation occurs over most of the Subarctic Pacific Region. In the Gulf of Alaska, the larger latitudinal gradient of the wind-stress curl in winter than in summer could, therefore, give rise to a mote rapid winter than summer response of the water in this region. A f u l l understanding of the oceanic response time i n the Gulf of Alaska would, however, necessarily include the nonlinear interactions between the quasi-steady and planetary wave motions as well as the topographic effects in the shallowing northeast corner of the region. A much more dramatic example of rapid oceanic response to an intense wind-stress curl pattern occurs in the Indian Ocean. There, the fast response (~ 1 month) to the Monsoon, with subsequent formation of the intense Somali Current, is postulated by L i g h t h i l l (1969) to be a result of the strong latitudinal wind pattern developed, as well as- to the f a c t that the C o r i o l i s parameter i s very s m a l l . A n a l y s i s of the quasi-steady motions was c a r r i e d out using the v e r t i c a l component of the v o r t i c i t y equation i n c u r v i l i n e a r c oordinates. In these coordinates, which were used i n order to model the boundary formed by the Aleutian-Komandorski i s l a n d c h a i n , the curvature depends only upon the di s t a n c e along l i n e s p a r a l l e l to the above boundary. Relevant terms i n the equations' were based upon the r e s u l t s of a non-d i m e n s i o n a l i z a t i o n procedure. Since the sur f a c e Ekman l a y e r over most of the S u b a r c t i c P a c i f i c i s a divergent one, a simple c i r c u l a t i o n based on the mean v e r t i c a l v e l o c i t y s t r u c t u r e has been used to describe the general features of the quas i - g e o s t r o p h i c motions below the s u r f a c e f r i c t i o n l a y e r . In p a r t i c u l a r , a v e r t i c a l v e l o c i t y s t r u c t u r e having maximum near the top of the permanent h a l o c l i n e (pycnocline) w i l l r e s u l t i n a southward tending upper l a y e r and a northward tending lower l a y e r , away from the boundaries. A l s o , the f a c t that t h i s upward motion, averaged over a number of years, maintains the observed mean d e n s i t y p r o f i l e has been used to o b t a i n an eddy c o e f f i c i e n t of d i f f u s i v i t y f o r 'density' i n the S u b a r c t i c P a c i f i c Region. The c a l c u l a t e d v a l u e s , based on the divergence of the surface Ekman l a y e r and v e r t i c a l d i f f u s i o n , give mean values between 0.5 x l O - 1 and 4.0 x 10 -1 cm 2 s ~ l which agree very w e l l w i t h those c a l c u l a t e d by Veronis (1969) f o r comparable l a t i t u d e s i n the North A t l a n t i c . Furthermore, the mean (time averaged) thi c k n e s s o f the upper zone away from the coast i s so adjusted to the mean c u r l of the wind-stress that i t behaves as a l a y e r of constant p o t e n t i a l v o r t i c i t y ; the time averaged motion of t h i s l a y e r i s th e r e f o r e geostrophic. The appearance of 'warm* water i n t r u s i o n s at the top of the h a l o c l i n e , i n summer, along the southern coast of B r i t i s h Columbia, has been discussed here. I t has been shown that t h i s phenomenon i s r e l a t e d to the winter-mean m e r i d i o n a l w i n d - s t r e s s , and the sea-surface temperature anomalies o f f the coast i n w i n t e r . Summers i n which a r e l a t i v e l y l a r g e amount of i n t r u d e d 'warm' water has been observed are seen to be preceeded by w i n t e r s i n which the sea-surface temperatures i n the v i c i n i t y of the coast were above normal, as were, a l s o , the n o r t h -wards merdional wind-stresses over the c o a s t a l r e g i o n . A r e l a t i v e l y s m a l l amount of 'warm1 water i n t r u s i o n was shown to occur i n a summer f o r which, i n the preceeding w i n t e r , the m e r i d i o n a l winds were weak (or southward) and the sea-surface temperatures were below normal. On the b a s i s of c o l d e r than normal sea-surface temperatures i n the w i n t e r of 1970-19.71, i t i s p r e d i c t e d that the observed r e l a t i v e amount of 'warm' water i n t r u s i o n along the B r i t i s h Columbian coast i n the summer of 1971 w i l l be r e l a t i v e l y s m a l l , provided the w i n t e r m e r i d i o n a l wind-stresses along the coast are weak or southward. At the bottom of the h a l o c l i n e , the l e s s pronounced 'warm' water i n t r u s i o n appears to be mostly r e l a t e d to v a r i a t i o n s i n the north-south geostrophic currents formed as a r e s u l t of the slope of the i s o p y c n a l s near the coast. The v a r i a t i o n s i n these currents can be f u r t h e r r e l a t e d to v a r i a t i o n s i n the c o a s t a l , m e r i d i o n a l wind-stresses from t h e i r means. In what I consider to be the most s i g n i f i c a n t p a r t of t h i s t h e s i s , a steady, f r i c t i o n a l model was used to e x p l a i n the observed c i r c u l a t i o n i n the S u b a r c t i c P a c i f i c Region. The eastward flow i n the i n t e r i o r r e g ion was balanced by r e q u i r i n g that a westward boundary l a y e r flow, w i t h a modified Ekman number (AJJ/3 0L3) as the i n n e r expansion v a r i a b l e , c l o s e the c i r c u l a t i o n along the northern boundary. In t h i s boundary l a y e r , i t was assumed that the f r i c t i o n a l l y generated v o r t i c i t y at the coast must be balanced by the p l a n e t a r y v o r t i c i t y e f f e c t and/or the l o c a l w i n d - s t r e s s c u r l ; the e f f e c t s of bottom topography and non-l i n e a r i t i e s were neglected. The r e s u l t s of the theory show that there i s a downstream change i n the northern boundary l a y e r c u r r e n t , from a 'western 1 type boundary current (Alaska Current) east of about 165°W. long, to a ' z o n a l ' type boundary current (Alaskan Stream) to the west of t h i s l o n g i t u d e . I n a 'western' type boundary c u r r e n t , the p l a n e t a r y v o r t i c i t y e f f e c t i s capable of b a l a n c i n g both the v o r t i c i t y generated along the boundary and that s u p p l i e d by the l o c a l c u r l of the w i n d - s t r e s s . This balance of forces permits a ' s t a b l e ' boundary l a y e r flow. As the westward f l o w i n g boundary current changes to a 'zonal' type, however, the r a p i d l y decreasing importance of the p l a n e t a r y v o r t i c i t y e f f e c t means that flow s t a b i l i t y becomes dependent upon the a b i l i t y of the wind-stress c u r l to supply the r e q u i s i t e b a l a n c i n g v o r t i c i t y ( counterclockwise, h e r e ) . I f a balance of forces i s not achieved, a steady flow cannot occur and the boundary l a y e r must somehow break down. In t h i s case, the l i n e a r i z e d , steady equations used to model the boundary l a y e r cannot be used to o b t a i n a d e t a i l e d d e s c r i p t i o n of the motion. Although the e f f e c t of the passes between the A l e u t i a n i s l a n d s on the boundary flow was not given a r i g o r o u s treatment, i t appears p l a u s i b l e that they may a s s i s t i n keeping the Alaskan Stream attached to the coast. This would be done through e i t h e r a s o r t of boundary l a y e r ' s u c t i o n ' which would remove v o r t i c i t y generated at the coast or by a t r a n s p o r t of mass i n t o the boundary l a y e r which would have the same e f f e c t as a p o s i t i v e w i n d - c u r l . The Alaskan Stream would, t h e r e f o r e , be found f u r t h e r westward as a boundary current than i t would in the absence of the passes. The effect of ' i n e r t i a l overshoot' and bottom topography were not included i n the study, however, and may yet prove to be important factors i n determining the western extent of this current. In order to obtain some insight into the distribution and variation of the surface wind forces, and their resulting effects on the Subarctic circulation, the mean-monthly sea-surface pressures were used to calculate the mean-monthly values of the wind-stress curl and the two horizontal components of the wind-stress. Plots of the mean annual (April-March) wind-stress curl distribution for every second year, beginning i n 1946, were then given as was also a plot of the mean distribution for the period 1945 to 1966. One basic result of these plots i s , that, the zero of the mean wind-stress curl roughly corresponds to the zonal region separating the southward turning flow from the northward turning flow in the interior region of the Subarctic Pacific. Also, the intense boundary layer current along the northern coast originates in the area of most intense couterclockwise wind-stress curl, the Gulf of Alaska. These results would be expected for a wind-driven ocean whose mean circulation was determined by a Sverdrup type of vorticity balance. Furthermore, the existence of a positive, mean wind-stress curl distribution over the Aleutian islands i s consistent with the theory presented in this thesis concerning the observed s t a b i l i t y of the Alaskan Stream to as far west as 170°E. For the period of January 1945 to April 1966, i t was shown that for most locations in the Subarctic Pacific Region the frequency, spectra were dominated by the annual Fourier component. The six-month component became very important i n the region just south of the Gulf of Alaska. The maximum values of most F o u r i e r components occurred i n the Gulf of A l a s k a w i t h r e l a t i v e maxima o c c u r r i n g over the western p a r t of the A l e u t i a n i s l a n d s . Minimum values occurred over the Alaskan p e n i n s u l a and near the northern coast of C a l i f o r n i a . These features- were expl a i n e d In. terms of the mean pressure d i s t r i b u t i o n s ; the dominant A l e u t i a n Low i n w i n t e r and the more s o u t h e r l y North P a c i f i c High i n summer. The much l a r g e r values of the mean wind-stress c u r l i n w i n t e r than i n summer are a t t r i b u t e d to the f a c t t h a t the A l e u t i a n Low i s much more int e n s e than the North P a c i f i c High. These l a r g e w i n t e r values suggest w i n t e r a c c e l e r a t i o n of the flow i n the Gulf of Alaska w i t h increased t r a n s p o r t and/or widths of the northern boundary c u r r e n t s , as indeed has been observed to occur. F i n a l l y , time v a r i a t i o n s i n the magnitudes of s i x frequency components i n the wind-stress s p e c t r a were determined through use of a demodulation technique. The most s t r i k i n g r e s u l t was the marked decrease i n magnitude of the annual and semi-annual components f o r s t a t i o n s i n the northeast c o mer of the Gulf of A l a s k a a f t e r 1959. Up to the end of my data (January, 1969) there was no tendency of these components to r e t u r n t o t h e i r pre-1959 l e v e l s . I t appears that t h i s phenomenon was r e l a t e d to a weakening and southward s h i f t of the A l e u t i a n Low p a t t e r n i n w i n t e r . The f a c t that the wind-stresses near the n o r t h e a s t e r n corner of the Gulf of A l a s k a became n e a r l y equal a f t e r 1959 i s f u r t h e r i n d i c a t i o n of the l a c k of curvature of the pressure i s o b a r s i n t h i s r e g ion a f t e r t h i s time. B i b l i o g r a p h y Arons, A.B. and Stommel, H. (1967). "On the a b y s s a l c i r c u l a t i o n of the World Ocean - I I I . An advection - l a t e r a l mixing model of the d i s t r i b u t i o n o f a t r a c e r property i n an ocean b a s i n " . Deep-Sea Res., 14, 441. Bat c h e l o r , G.K. (1967). An i n t r o d u c t i o n to f l u i d dynamics. Cambridge U n i v e r s i t y P r e s s , 615 pp. Blackman, R.B. and Tukey, J.W. (1959). Measurement of power s p e c t r a . - New York: Dover, 190 pp. Bogdanov, K.T. (1961). 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On the validity of separating Barotropic Rossby waves and quasi-steady motions The dependent variables in the general vorticity equation (3.1.13) are assumed to be composed of two distinct motions: Cl) the planetary wave motions, which occur as transient responses to the applied wind-stress, and (2) the time averaged, quasi-steady motions resulting from the accumulated effect of the wind-stress over a period of time greater than a week. By considering separate scales for each of these motions i t is possible to determine the conditions for which the two motions are separable; i . e . , when the nonlinear interactions between the motions are small. It is assumed that, £Cx,t) = ' ^Ot . t ) + C2tx,t) , uCx,t) = u-iCx,t) + u2Cx,t) , CA.l) and i Vxx-k = CVXT-10-L + CVxT-k)2 , where subscript (1) refers to the planetary wave motions and subscript C2) to the quasi-steady motions. The dependent and independent variables in C3.1.13) are then scaled for i = 1, 2 by: Cx,y) = I ^ C x ^ y ^ ) , z = Dz*, t = T^t 1, K = KQK* , p = P o p * . f = fof<t C^iy f - B 0 C | ^ , ' - f ^ f * , CA.2) Cu,v) = UiCu^vV), w = w ^ , C = Ui/Li.q, T = T0T<\ where the prime denotes nondimensional v a r i a b l e s and where: i s the wavelength of the p l a n e t a r y waves having p e r i o d T\ and speed U^; L 2 i s the h o r i z o n t a l d i s t a n c e over which a s i g n i f i c a n t change i n the mean current speed U2 occurs i n a time T 2. The remaining parameters are defined i n § 3.2. I f we s u b s t i t u t e (A.2) i n t o (3.1.13), use T ± = L±/V± (3.2.2), and def i n e the f o l l o w i n g parameters; R. 1 o f=± , q = B 0 L i / f o (A. 3) and f u r t h e r assume that i n the c o n t i n u i t y equation (3.2.5) that w ± ~ 0 (DL^/Li) , we f i n d R l 1 3 t i + T 2 3t 2 3 + R 2 1 3ti + T 2 3t 2 J + B? [ uJ + 7I2- u 2 ] • V ' . f + *2 [ ^ "2 * ^ + £ V p c- + %L u i • V 2 C 2 + u 2 • V- q ] , R SWJL TJL 3w_2 1 1 1 3z' + T 2 3z' J = Au { . ^ [ l i n e a r terms + ( n o n l i n e a r terms) + (r^-)"^ n o n l i n e a r terms) L l u l L 2 . -L2 Ro T 1 To n o n l i n e a r terms + T - ^ ( l i n e a r terms) + -— ( n o n l i n e a r terms) } L 2 U i L 2 L X + ^ ^ [ (V! x T • k ) i + ^ CV.'x.T • k ) 9 ] PD 1 ~ x L 2 £ ~ / (A. 4) In the i n t e r i o r regions of the ocean we may ne g l e c t the h o r i z o n t a l f r i c t i o n . A l s o , as we are concerned w i t h the i n t e r a c t i o n s between the two modes, we consider only the f o r c e f r e e equation by dropping the l a s t term of (A.4). Now, s i n c e R2/R1 = T i / T 2 , i t can be seen from (A.4) t h a t , i f the p e r i o d of the p l a n e t a r y wave motions i s much s h o r t e r than that of the long-time (> 1 year) averaged (or quasi-steady) motions, i . e . . T - j . / ^ « 1 , (A. 5) then terms having c o e f f i c i e n t R2 may be neglected. Furthermore, i n the remaining terms, i f we only r e q u i r e that L!/L 2 > OCIO'2) then (A.5) i m p l i e s t h a t , at most, U 2 / U i < 0(1) , w h i l e f o r I^/L^ > 0 ( 1 0 _ 1 ) , U2/U]_ « 0(1) . Using these above r a t i o s , we see that terms i n v o l v i n g n o n - l i n e a r U? Ti i n t e r a c t i o n s between the two modes are s m a l l , to 0[Maximum (yjr- , -^) ]. Therefore, provided the p e r i o d of the p l a n e t a r y motions i s l e s s than that of the quasi-steady motions, and the speed of the former i s gr e a t e r than that o f the l a t t e r , we may neglect the e f f e c t of quasi-steady motions when d i s c u s s i n g p l a n e t a r y waves. I f we now7 average (A.4) over a long p e r i o d of time (1 year, s a y ) , we may consider p l a n e t a r y waves as f l u c t u a t i o n s about the mean c i r c u l a t i o n i n the S u b a r c t i c P a c i f i c . T h e o r e t i c a l l y , these p l a n e t a r y waves may have any p e r i o d g r e a t e r than one pendulum day. However, i t f o l l o w s from T1 = I ^ / U - L (3.2.2) and U1 = 0 ( B O L ^ ) (4.3.8), f o r the s p e c i f i c case of m i d - l a t i t u d e b a r o t r o p i c waves, th a t L-L ~ 0(l/3 oT!) so that waves having periods g r e a t e r than s i x months w i l l have wavelengths l e s s than ten k i l o m e t r e s and w i l l , t h e r e f o r e , be subject to st r o n g damping. Taking < u[ > = 0 , where < > represents the long-time averaging o p e r a t i o n , we f i n d that the steady form of (A.4) over the averaging p e r i o d s a t i s f i e s , + * 2 f A' "I + £ ' i> ? 2 " t R l C 1 9 z ' R2 T 2 S2 3z» + % < (VJ^  + ^ v«) w l x |JL • £ > = 0 The n o n l i n e a r i n t e r a c t i o n s between various modes v a n i s h , as we wanted. There i s , however, a nonzero c o n t r i b u t i o n from the n o n l i n e a r i n t e r a c t i o n s of the p l a n e t a r y waves on themselves, i n much the same manner that f l u c t u a t i o n s s h o r t e r than one-half a pendulum day give r i s e to the eddy c o e f f i c i e n t i n § 3.1. Since the quasi-steady motions are l i n e a r i z e d i n the a n a l y s i s , the s p e c i f i c way i n which these c o r r e l a t i o n s e f f e c t the flow i s of no concern i n t h i s t h e s i s . In the oceanic i n t e r i o r region t h e i r e f f e c t w i l l be n e g l i g i b l e anyway. F i n a l l y , as the r a t i o t ^ / U i i s about of order u n i t y , f o r Ui^ = speed of b a r o c l i n i c Rossby waves, we see from (A.4) that the i n t e r a c t i o n s between t h i s mode and the mean flow i s not n e c e s s a r i l y n e g l i g i b l e as they were f o r b a r o t r o p i c Rossby waves. Appendix II Glossary of symbols, part I X,Y Cartesian coordinates (east, north). i x,y Curvilinear coordinates (along the northern boundary, perpendicular to the boundary). z Vertical distance upwards. K Curvature of the northern boundary. 6 Angle between lines of constant y and Y, except in : § 5.3.4. 0 A mean value for the angle 8 (§ 6). 6 In §5.3.4,a nondimensional temperature. u,v,w, Velocity components in curvilinear coordinates, along x,y,w. t Time. f The Coriolis parameter, p Pressure, p Density. £ Vertical component of the relative vorticity. v Kinematic molecular viscosity. 3 Q The 'beta' parameter; f = fo + 3o Y £ unit vector, vertically upwards. A^ Eddy coefficient of viscosity. T T T Surface wind-stress. ~w T\Q Characteristic deviation of sea surface from i t s mean level. L;D Horizontal length scale; v e r t i c a l depth scale for baroclinic motions. H Q Vertical scale for the ocean depth,(H). h Depth of the h a l o c l i n e ( p y c n o c l i n e ) . hg Depth of the bottom of the Ekman l a y e r . UQ;W0 H o r i z o n t a l v e l o c i t y s c a l e ; v e r t i c a l v e l o c i t y s c a l e . T The time s c a l e , except i n § 5.3.4. T In § 5.3.4, the a c t u a l temperature. R Q , 3*, £*, Nondimensional parameters; see eqn. 3.2.4. EH, E W . 6 = n 0/H 0, a divergence parameter. Mx,My; MX,MY I n t e g r a t e d t r a n s p o r t components along x,y; X,Y. ¥ Integrated stream f u n c t i o n . k,1 Wavenumbers along x,y i n § 4. k - ( 3 0 M n) 1 /'3 > C o r i o l i s - f r i c t i o n wave number i n • § 6. (jj Radian frequency of pl a n e t a r y waves § 4. g A c c e l e r a t i o n of g r a v i t y (4.3.7). g Constant of i n t e g r a t i o n i n § 6. £ A 'depth' f u n c t i o n ; eqn. 5.2.2. k ^ j k y j k g Components of the a n i s o t r o p i c t u r b u l e n t eddy c o e f f i c i e n t of d i f f u s i v i t y . * 3 EJJ = A J J / 3 0 L > a n Ekman number. y Expansion constant; § 6. F^,F2 Components of Vf along y,x r e s p e c t i v e l y . X In § 6, the downstream 'decay' parameter. X In § 7, the l o n g i t u d e . (J) In § 7, the l a t i t u d e . q(x) The x-dependent term of the zeroth order stream f u n c t i o n i n subregion C 2 (see § 6). The y-dependent term of the above (§ 6). The radius of the e a r t h . The p o i n t where the eastern boundary changes from a north-south d i r e c t i o n to an angle 9 0 to t h i s d i r e c t i o n (see f i g . 14). FIGURE 39 i. Mean annual wind-stress curl, April 1962- March 1963: 10"' dynes cnf3 . r -%?• to r -rn -Z ) a. C O o.o STATION 1 ~1 1— 0.2 0.4 FREQUENCY 0.6 1 o.a FIGURE 4 0 a. Wind-stress curl spectrum versus frequency( month"1). CVJ S " to r-0". 3:8-— X Q _ C O a r™ 1 1 1 0.0 0.2 0.4 0.6 0.8 FREQUENCY STATION 2 FIGURE 40b. Wind-stress curl spectrum versus frequency (month-1 ). CO £8' 2: ZD CO a . rvi 0.0 ~1 1— 0.2 0 . 4 F R E Q U E N C Y -r— 0.6 1 0.B STATION 5 FIGURE 40c. Wind-stress curl spectrum versus frequency (month"' ) o cn 3g -a.- CD ^8" a X I — a as-a. co a a a 0.0 STATION 9 -I 1 — 0.2 0.4 FREQUENCY "~1 0.6 "1 0.8 FIGURE 40 d. Wind-stress curl spectrum versus frequency (month"1) r -rvi r ^ ' CO P o -sies-X z: ZD a ' C O O.D -l -r— 0.2 0.4 F R E Q U E N C Y 0.6 o.a STATION 10 F l GURE 40 e. Wind-stress curl spectrum versus frequency ( month"1). Z i ° _ . o. O J —• CO £81 2 ° CL. C O a rvj 0.0 0.2 0.4 FREQUENCY 0.6 0.B STATION 11 FIGURE 40 f. Wind-stress curl spectrum versus frequency (month-1). o •a a . C O r-ru o X 3 a*-to a . ru a o 0.0 STATION 13 -| T— 0.2 0.4 FREQUENCY 0.6 0.B FIGURE 40g. Wind-stress curl spectrum versus frequency (month-1 ). :ZD I O m «^ co O J CO H Z o d o 1 r— 0.2 0.4 FREQUENCY 0 .6 n 0.8 STATION 17 FIGURE 40 h. Wind-stress curl spectrum versus frequency (month-1). ID o Oo .01. CC 0 r— -U J U 3 Q_ CO 0.0 -1 r— 0.2 0.4 FREQUENCY -T— 0.6 ~1 o.a STATION 2 0 FIGURE 40i. Wind-stress curl spectrum versus frequency (month-1). OJ CO a 1 — 0 us-U J Q_ CO o o. 0 ,0 STATION 2 6 ~l 1— 0 . 2 0 . 4 FREQUENCY 0 . 6 0 . 8 FIGURE 40j. Wind-stress curl spectrum versus frequency (month-1). r-z o O J S-Z 3 . o . > CO CM O X 21 ZD CL CO CM 0.0 i r~ 0.2 0.4 FREQUENCY ~T~ 0.6 1 o.e STATION 43 FIGURE 40 k. Wind-stress curl spectrum versus frequency ( month"1). - r — i — r o 1 1 1 0.0 0.2 0.4 0.6 0. FREQUENCY 1 2 FIGURE 41 a. Co-spectrum between stations I & 2 versus frequency (month"1). X I-co H ( J CO r -• rvi >o. ru O o" s: C£ r— O o LU • CO i I Q CX ZD C 3 Q 1 "I 1 1 0.0 0.2 0.4 0.6 0. FREQUENCY 1 2 FIGURE 41 b. Quad-spectrum between stations I & 2 versus frequency (month"1 ). I CO I— Z> CO d o xpj. h-o a s . CL—1 CO I o o o. OJ s 1 1 1 —I O.D 0.2 0.4 0.6 0.B FREQUENCY 4 6 FIGURE 41c. Co-spectrum between stations 4 & 6 .versus frequency ( month-1 ). l CO 2 3 (J to 3 O P £ 3 H . -o x: Z D en r— O o as . CO i i C D C X 1 1 1 1 0.0 0.2 0 . 4 0.6 o.a FREQUENCY 4 6 FIGURE 41 d. Quad-spectrum between stations 4 8 6 versus frequency (month -1 ). X r-CQ z Mo o o o. ru OJ —«o Xif>_ DC o • U j g . D _ -CO I ED (_) in i 1 ~*—T r 1 0.0 0.2 0.4 0.6 0.B FREQUENCY 9 1 0 FIGURE 41 e. Co-spectrum between stations 9 & 10 versus frequency (month -1). i 1 1 1 1 O.D 0.2 0 . 4 0.6 0, FREQUENCY 9 1 0 F l G U R E 41 f. Quad-spectrum between stations 9 8i versus frequency ( m o n t h - 1 ). W. C. UNITS * I 0 " 9 dynes cm" 3 co h-3 6 CO I-z> OJ \— O R UJ CD. Q OJ CO 1 6D O CD a' a i 1 1 1 1 O.D 0.2 0.4 0.6 o.a FREQUENCY 11 1 2 FIGURE 41 g. Co-spectrum between stations II 8s 12 versus frequency (month-1 ). W.C. U N I T S = I 0 " 9 dynes cm" 3 CO h -3 (J £° CO 2: ><o o " ZD Q± I— O o CO 1 I a cx ZD d 00. 1 o CM 1 r 1 1 O.D 0.2 0.4 0.6 0.B FREQUENCY 11 1 2 FIGURE 41 h. Quad-spectrum between stations II 8 12 versus frequency ( month-1 ).. 2 ° ct i— o° LiJd. n r\j CO I €D CJ o a a OJ. i , . 1 1 1 0.0 0.2 0.4 0.6 0.8 FREQUENCY 12 13 FIGURE 41 i. Co-spectrum between stations 12 & 13 versus frequency (month-1 ). X r-CO Z> 6 CO I— 5:8-C L D C O 1 C J o — g_l 1 I 1 1 O.D 0.2 0.4 0.6 0.8 FREQUENCY 18 20 FIGURE 4lj. Co-spectrum between stations 18 8i 20 versus frequency (month-1). W.C. UNITS = I0~9 d y n e s c m ' 3 Z> d o CO Z> O ZD CH h-o COo I a ex ZD C 2 U 9_ o o 03. I r— ! 1 1 O.D 0.2 0.4 0.6 0.8 FREQUENCY 18 2 0 FIGURE 41k. Quad-spectrum between stations 18 8 20 versus frequency ( month-1 ). 1 1 1 r 0.0 0.2 0.4 0.6 0 FREQUENCY 40 42 FIGURE 41 I. Co-spectrum between stations 40 6 versus frequency (month-1 ). C O H E R E N C E C O H E R E N C E o c •JO m ro o —*1 •"I CD Q. C O ro Q 3 O fD ve Q. II V) • « • • O 3" O — • - i o 3 3 o Ui 3 O < 3 -\ O in c t/> ro cn —•» •s CD A .Q C CD 3 O CD 3 O CD O Q. t —« CD CA (/) o cr CD s s* CD CD 3 3 " CD \ O z o &• C O H E R E N C E C O H E R E H C E a. Between stations 1 8 2 • 0 -100 4 / 6 \ 8 / \l A / A 2 21 24 tt6 f 3 0 312 - H A R M O N I C N O . iod% d. Between stations 22 8 23 »o° < X a. 2 \ 4 6 A / / 8 \ l.Q_. 12 /x/\ A 14\ 1 6 / 18 \ / 2 2 \ / 2 4 A / V 2 6 28 3 0 ' 3 2 H A R M O N I C NO. -100 I-FIGURE 43b. Phase between wind-stress curl 'signals'. ( If > 0, 1st leads 2nd ). Frequency = 4(harmonic no.)/256 month"'. T\,^stA A A,A A A ft A, A A A 0.0 20.0 40.0 60.0 80.0 10D.0 120.0 140.0 160.0 " 180.D 200.0 220.0 240.0 260.0 TIME (months after Jon. 1945) Station I, Wind-stress Curl (10 dynes cm" 3). 1/2 C/M T i 1 1 1 1 1 1 1 1 1 r 3.0 20.0 40.D 60.0 80.0 100.0 120.D 140.0 160.0 180.0 200.D 220.0 240.D TJME 260.0 " * ^ ' * " r J * >/i u - " ^ ^ i 0.0 20.0 40.0 60.0 SO.O 100.0 120.0 140.0 160.0 180.0 200.0 220.0 240.0 260.0 TIME (months after Jon. 1945). Station 2; Wind-stress curl (10 dynes em" 3). r~i rrn r\ /yw>„pi i <r-i • j i s ^ — i . - ^ p . r s ^ „ rAn» j~vA. - - A Y ^ f ^ 1 • - W 1 0.0 20.0 40.D 60.0 80.0 100.0 120.0 1 40.0 160.0 180.0 200.0 220.0 240.0 260.0 T IME (months after Jan. 1945). Station5; Wind-stress curl ( IO~° dynes c m - 3 ) . 0.0 20.0 1/2 C/M 260.0 1 240.0 260.0 2 6 0 . 0 260.0 2 6 0 . 0 1 1 1 1 1 1 ^ - ^ T 1 J 100.0 120.0 140.0 160.0 1B0.D 200.0 220.0 240.0 260.0 TIME 40.0 60.0 80.0 0.0 20.0 40.0 EO.O BO.O 100.0 120.0 140.0 160.0 180.0 200.0 220.0 240.0 260.0 TIME Station 9 ; Wind-stress curl (ICT* dynes c m " 3 ). 1/2 C/M t - o O . O 20.0 40.0 GO.O 80.3 100.0 120.0 1 40.0 160.0 180.0 200.0 220.0 240.0 260.0 260. D a .3 20.0 40.0 GO.O fiO.D 100.0 120.0 140.0 160.0 1B0.0 200.0 220.0 240.0 260.0 TIME 0.0 20.0 4 0 . 0 Station 60.0 BO.O 100.0 120.0 TIME Wind-st ress curl (10 dynes c m - 3 ). 140.0 160.0 iso.D 200.0 220.0 240.0 1 260.0 1/2 C/M 2 6 0 . 0 1 I I I I I I I I 1 1 1—' 1 0.0 20.0 40.0 60.0 BO.O 100.0 120.0 140.0 160.0 1B0.0 200.0 220.0 240.0 260.0 T IME l O - i 20.0 1 1 60.0 T 80.0 100.0 120.0 140.0 160.0 180.0 200.0 220.0 240.0 260.0 Stat ion 16; Wind-stress curl (ICT* dynes c m " 3 ) . 0.0 40.0 j j . 0 d - , 0.0 ZD.O -i r 100.0 120.0 TIME 140.0 180 0 200.0 220.0 40.0 T so.o T RO.O 1/2 C/M -1 1 240.0 260.0 1/12 C/M 1 1 1 1 1 : 1 1 1 100.0 12D.0 140.0 160.0 1BO.0 ZDD.O 220.0 240.0 260.0 TIME 20.0 V40.0 60.0 80.0 100.0 120.0 140.0 160.0 180.0 200.0 220.0 w TIME Station 23; Wind-stress curl ( 1 0 " ° dynes c m " 3 ) H ^ - v 1 160.0 1B0.0 200.0 220.0 v 240.0 260.0 1/2 C/M T T 100.0 120.0 TIME i o-i eo.o 100.0 " T - I V * — ° — i 240.0 260.0 0.0 20.0 40.0 60.0 120.0 TIME Stot ion 2 6 ; W i n d - s t r e s s curl (10" dynes c m " 3 ) . 140.0 160.0 IBO.O 200.0 220.0 140.0 160.0 160.0 200.0 220.0 1/2 C/M T 1 240.0 260.0 T— 1 240.0 260.0 250.0 260.0 T 1 240.0 2B0.0 240.0 260.0 1 r 100.0 120.0 TIME * — £ L 4 V 60.0 •^1 ' eo.o 4- ^ 220.0 D.O T V " -2 0 . 0 40.0 100.0 120.0 TIME 1 4 0 . 0 Station 4 3 ; Wind-str ss curl O C T 9 dynes cm"3 ) 160.0. 180.0 r=—-200.0 240.0 260.0 1/2 C/M 1 1 1 1 1 ~~—I 1 1 1 1 0.3 20.3 40.3 60.0 80.0 100.0 120.0 140.0 160.0 180.0 200.0 220.0 240.0 263.0 TIME PART I I ON THE GENERATION OF KELVIN-TYPE WAVES BY ATMOSPHERIC DISTURBANCES 1. I n t r o d u c t i o n The g e n e r a t i o n of l o n g waves by i n i t i a l l y a p p l i e d f o r c e s on the s u r f a c e of a u n i f o r m l y r o t a t i n g i n f i n i t e ocean has been t h e o r e t i c a l l y i n v e s t i g a t e d by such authors as Crease (1956) and Mysak (1969). S i m i l a r i n v e s t i g a t i o n s f o r a non-uniformly r o t a t i n g ocean have been c a r r i e d out by V e r o n i s & Stommel (1956) and by Longuet-Higgins (1965) u s i n g the 3-plane approximation. For the case of a s e m i - i n f i n i t e ocean K a j i u r a (1962) considered the p o s s i b i l i t y of K e l v i n type waves being generated at the boundary. Using a Green's f u n c t i o n approach he i n v e s t i g a t e d the s u r f a c e response to an atmospheric d i s t u r b a n c e of r e c t a n g u l a r -h o r i z o n t a l e x t e n t and was able to show t h a t the s o l u t i o n f o r l a r g e d i s t a n c e s from the source, but c l o s e to the w a l l d i d behave as a K e l v i n wave. I n t h i s paper we a l s o c o n s i d e r the response of the sea s u r f a c e t o atmospheric f o r c e s i n the presence of an i n f i n i t e boundary, but i n s t e a d of a Green's f u n c t i o n approach we s o l v e the i n i t i a l - b o u n d a r y value problem u s i n g t r a n s f o r m methods. The r e s u l t i s t h a t the response behaves as a f o r c e d K e l v i n wave and t h a t t h i s wave appears as p a r t of the exact s o l u t i o n . . A l s o , we f i n d t h a t only the longshore component of the w i n d - s t r e s s and pressure g r a d i e n t can generate these waves, i n agreement w i t h K a j i u r a (Thomson, 1970). In %2 a p a r t i a l d i f f e r e n t i a l e q uation f o r the f o r c e d s u r f a c e e l e v a t i o n i s d e r i v e d from the l i n e a r i z e d shallow-water equations. Transform methods are then used i n § 3 t o determine the sea s u r f a c e response t o a g e n e r a l space-time w i n d - s t r e s s and pressure i n the pressence of an i n f i n i t e boundary. S e c t i o n 4 i s devoted t o a d i s c u s s i o n of t h a t p a r t of the s o l u t i o n which give s r i s e to K e l v i n type waves. These r e s u l t s are used i n %§5 and 6 f o r s p e c i f i c examples: ( i ) a non-moving d i v e r g e n t longshore wind and ( i i ) a moving pressure source, r e s p e c t i v e l y . Through the a p p l i c a t i o n of a Tauberian theorem, the s t e a d y - s t a t e response f o r l a r g e times i s shown i n % 7 to be zero i n each case. I n § 8 a p h y s i c a l d i s c u s s i o n of the r e s u l t s found i n ^ 4 and 5 i s g i v e n . F i n a l l y , i n ^ 9 some num e r i c a l v a l u e s f o r the amplitudes and v e l o c i t i e s are presented. 2. Equations of motion We consider a homogeneous, uniformily rotating ocean in which nonlinearities and bottom friction are neglected. Further, we assume that the waves are sufficiently long for the hydrostatic equation to be valid. Then the vertically integrated equations of momentum and mass conservation are: 3u - fv : = - g 3? - 1 3Pa + 3t 3x p 3x ph 3v + f u = - g 3 _ £ - _ l 3Pa + 3t . 3y p 3y ph (2.1) jKhu) + 3_(hv) = - H , . (2.2) 3x 3y 3t in which the hydrostatic equation P = Pa + pg(C - z) has been used, and where x , y are horizontal cartesian co-ordinates z is measured vertically upwards t is the time £ is the sea surface elevation u , v are the velocity components along x , y h is the mean depth, assumed constant g is the acceleration of gravity f is the coriolis parameter p is the water density Pa is the air pressure on the sea surface T x , are the components of the wind-stress in the x , y directions. Then from (2.1) we obtain Nu = - / 3J; + f d \ f g ? + P a U ^ / a f + f T v ) (2.3) V 3x3t 3y / \ p J ph [dt J ' Nv = /-S 2 + f 3 \ (s C + P a W 3^ /ST* - f t * \ (2.4) V 3y3t 3x / V p i ph Ut' ' J * where N - f 2 + 3 2 Using equations (2.3) and (2.4) i n (2.2) we obtain I 1 " gh J at where F = a[ (VX T) Z + 1\ • 3L - h V 23Pa ] , a = _ f _ f 3t f 3t ' pgh ' T = ( x x,xy,0) and V 2 = 3 2/3x 2 + 3 2/3y 2 . (2.6) In the general a n a l y s i s of ^§ 3 and 4 we w i l l only require that T and Pa be bounded and i n p a r t i c u l a r that they both are zero at |y| = 0 0 . Our model consists of a s e m i - i n f i n i t e ocean of constant depth h, bounded at x = 0 by a v e r t i c a l w a l l . In t h i s paper the a n a l y s i s i s c a r r i e d through f o r an ocean i n the h a l f - p l a n e x> 0; f o r an ocean i n the h a l f - p l a n e x< 0 i t i s simply a matter of s u b s t i t u t i n g x = -x whenever i t appears. Also we have assumed f > 0 so that the a n a l y s i s i s f o r the northern hemisphere; i n the southern hemisphere we take f < 0. At the w a l l we require that the normal v e l o c i t y component (u) vanish. 3. Response to a general wind and pressure f i e l d We assume that at t = 0 a pressure (Pa) and a h o r i z o n t a l wind-stress d i s t r i b u t i o n begin to q u i c k l y b u i l d up over the surface, which i s i n i t i a l l y at r e s t ; i . e . £ =0 f o r t < 0. We now define the Fourier-Laplace transform of $(x,y,t) by 0(x,X,s) = J exp(-st) dt \ exp(iAy) $(x,y,t) dy (3.1) and specify the following conditions: i) 3£/3t = C = 0 < Mi at t< 0 as t •> + 100 i i ) 3c/3y, C < Mj, as |y |-> + 0 0 (3.2) i i i ) Pa, 3Pa/3y, |T| •> 0 as t, |y|-> + 00 iv) 3£/3x, C< M3 as x + 0 0 where the M-j_ are f i n i t e . These conditions w i l l be used to obtain the general transform of £ as a function of Pa and _T . Since (2.5) i s a third order d i f f e r e n t i a l equation in. time, the Laplace i n i t i a l accelerations w i l l be zero since atmospheric disturbances w i l l be assumed continuous in time. Physically this is the most r e a l i s t i c situation, although mathematically, there is no d i f f i c u l t y in allowing i n i t i a l discontinuities for specific cases since they simply require the inclusion of the i n i t i a l acceleration and forces. transform method requires a knowledge of 3 2£/3t 2 at t = 0. In this paper such With the above condition, and [ 3 . 2 ( i , . . . , i i i ) ] equation (2.5) transforms to (3.3) (3.4) F*(x,X,s) = a [d/dx(ry(x,X,s) + sf*T*(X,X,S)) + iX (TX(X,X,S) - sf \y(x,A,s)) - h f l (d 2/dx 2 -X2) s Fa(x,X,s)] . The most general solution to (3.3) under condition [3.2(iv)] i s then C(x,X,s) = c(X,s) exp(-kx) - 1 ( G , (x|x') F*(x',X,s) dx' , (3.5) 2ks J R o where Gj_(x|x') = G(x|x'|k) is the Green's function Gk(x|x') = exp[-k|x-x'|] . (3.6) To determine c(X,s) we use the boundary condition, u = 0 at x = 0. • Taking the transform of (2.3), using conditions [ 3 . 2 ( i , . . . , i i i ) ] and substituting from (3.5) and (3.6) we obtain, CO (iXf + ks) c(X,s) = iXf-ks \ exp(-kx') F*(x',X,s) dx' 2ks { _ _ _ ( 3 ' 7 ) + ah[( s_ 8_ - iX) Pa(0,X,s)] - a[ s_ T X(0,X,s) + T Y(0,X,s)] f Sx f wherebye (3.5) becomes, after some manipulation, • 2ksf C(x,X,s) = iXf - ks [ Ti(x,X,s) - h f 2+ s 2 Pi(x,X,s) ] a iXf + ks c + iX T 2(x,X,s) + k T 3(x,X,s) (3.8) + h f 2+ s 2 P 2(x,X,s) - 2khs Pa(x,X,s) , c z i n which Ti(x,X,s) = ^exp[-k(x+x')][(fk - iXs) x y(x',X,s) + (iXf + ks) x X(x',X,s)] dx' T 2(x,X,s) = J G (x|x')[ sT y(x',X,s) - f T X ( x ' , X , s ) ] dx' T 3(x,X,s) =JX°sgn(x-x') exp [-k | x-x1 | ] [ f T Y(x',X,s) + s T X(x',X,s)] dx' Pi(x,X,s) = s^exp[-k(x+x') Pa(x' ,X,s) dx' a P 2(x,X,s) = sjG k(x|x') Pa(x',X,s) dx'• (3.9) o and s g n ( q ) = + l q > 0 = - 1. q < 0 . The surface response i s then found by taking the inverse, v i z . , C(x,y,t) = 1 \ exp(-iXy) dX 1 exp(st) C(x,X,s) ds . (3.10) ^ i L vl;.. In the s-plane y i s chosen so that any singularities l i e to the l e f t of the inversion path (see figure 1), while i n the X-plane the inversion path must be suitably indented above or below, any singularities on the real X-axis; the appropriate indentations being determined by the Sommerfield radiation condition. 4. Kelvin wave solutions As seen from expression (3.8), only the f i r s t two terms can possibly contribute poles in the s and X planes, other than at s = 0, which are not totally attributable to the force. Since we are inverting our transform with respect to s f i r s t , these poles are at s = -iXc and s = -if|X|/X (since k > 0) as derived from iXf + ks = 0, and thus arise entirely from the wall boundary condition. Similar i l y , when we f i r s t evaluate for s-plane poles which arise from the e x p l i c i t form of the force we w i l l obtain poles in the X-plane for the zeroes of iXf + k .s^; where k q^s. i s evaluated at the s-plane poles, s-i , of the force. It i s these terms which give rise to Kelvin waves. To partly demonstrate the above remarks, without specifying a force, we consider the poles at the zeros of iXf + ks i n the s-plane. Let £ 4(x,y,t) represent their contribution to the surface response. Now because the force i s assumed bounded for a l l ' t , the poles and/or branch points i t contributes i n the s-plane w i l l also l i e on or to the l e f t of the imaginary s-axis. Thus in this plane we closer.'the '. inversion contour, F, to the right for t < 0 and to the l e f t for t > 0. Hence C1= 0 for t < 0. For t > 0 the contribution from the pole at s = -iXc for the f i r s t two terms of (3.8) i s , using (3.10) and Cauchy's residue theorem, C 4(x,y,t) = dx" exp[- I (x + x')] dX exp[-iX(y + ct)][ a^(x',X,s) i - i _ (4.i) - a^X Ta(x' >^ >S)<L-iXc Im s T Branch cut * - : R e "s FIGURE I. Path of integration in the s plane, s= Res + ilms. ( , branch cuts; x, singularities). Singularities due to the transform of a particular force are not shown. in which a t = - 1_ a and a a = + ih a , (4.2) 27T 277 while that from the pole at s = -if|X|/X, is zero. It is easily seen that (4.1) represents a Kelvin wave in which the wavenumber X w i l l be determined by the e x p l i c i t form of the force. 5. Response to a non-travelling wind pattern . In order to demonstrate the remarks of §4 we take as an example a disturbance that is stationary in space but transient i n time. One such form satisfying boundary condition [3 . 2 ( i i i ) ] i s T Y = T 0H(t) sinoot exp (-at) H(y) exp(-yy) , (5.1) T X = Pa = 0 , where T«, = constant, y and a are real ( > 0), OJ is real and H(q) is the unit step function. This forcing f i e l d could be regarded as a crude approximation to that caused by successive, large onshore moving weather systems in which the pressure difference between each storm's center and edge i s small. Then, applying (3.1) we find xY = -T„ 0) 1_ . (5.2) (s+a)z+ooz iX-y Substituting this into (4.1) we obtain poles at Xc= - i y and X+ = ±U)-icr . c Evaluation is then a straightforward application of Cauchy's residue theorem provided we conform to the following condition: the contour is closed above for y -i- ct < 0 and below for y + ct > 0. This ensures that integration along the large semi-circle of the inversion path vanishes in each case, or, which, amounts to the same thing, that any solution be bounded. In the analysis to follow O > 0 to avoid having poles on the imaginary s-axis, which, depending on whether f ^ OJ, could necessitate an inversion contour other than that i n figure 1. Equation (4.1) then becomes C l v(x,y,t) ,= A {{cos[K(y+ct)] +'a sin[K(y+ct)]} exp[-m(y+ct)] - exp[-y(y+ct)]} H(y+ct) (5.3) (iii) FIGURE 2 Plots of the amplitudes of the wind-generated Kelvin waves [(5.3) (5.5)] at fixed longshore position, y = y o<0, for increasing time t* = a>(t + yo/c), for four given volues of the decay frequency, o--/iC (w*0 ). U) o* = /Jc= W; (ii) <r = lw, / X C = 2O J ; (iii) <r=iw,/ic= w; (iv) C T=2O J , jxz - c^u. where (« , m) = _1 (u) , a) are horizontal wave numbers, a = a-yc and c 00 A = T o . to exp(- — x) . (5.4) pc co2 +(a-yc) 2 Following §4 we next evaluate the f i r s t term of (3.8) at the poles i n the s-plane due to the e x p l i c i t form of the force, i.e. at s + = ±iw - o* from (5.2). Further iiiversion in the X-plane for the poles at the zeroes of iXf + k e± gives a contribution C2\> such that C (x,y,t) = - C 4 V(x-,y,t) H(y) - A exp [-y (y+ct) ] H(y+ct) . (5.5) ~ V . H(y+ct) It i s easily seen that the combination of (5.3) and (5.5) represents a wind forced Kelvin wave plus a surge-like term moving away from the forcing region (y > 0) with the long wave speed, c-(see figure 2). Although i t is not the purpose of this paper to investigate a l l the possible long waves generated by a particular disturbance, we w i l l b r i e f l y outline their general behaviour. If we include a l l terms of (3.8) we find the following: (i) no contribution for the poles at s = 0; ( i i ) for the poles at s = s+ a l l terms, excluding the f i r s t (which gave (5.5)), give a response £ which, using the Method of Steepest Descent, can be shown to behave as £w(x,y,t) ^ A [exp(-at)] sin(wt){ exp[-(f 2-a) 2)' r/c] 1 • lim y,a 0+ lim o 0+ (f 2-oo* )y* (r/c)> cosG ( f -wr r/c -> oo y v - ( 2 7 T ) 1 e x p [ - ( f 2 - 0 J 2 ) 1 y/c]} H(y) (5.6) where A = 1_ Tg_ 1 , (x,y) = r(cos9,sin9) and \Q\ < rr/2; (8TT)* pc (f 2-u) z) y* ( i i i ) for the poles at s+ the f i r s t term has, besides the Kelvin, waveopoles, ..a :pole at X = - i y where for g,y ->0+ the resulting displacement £ , behaves as C y(x,y,t) = ~ la. 1 1 t [exp(-at).] cos (cat) lim y,a -> 0+ pc co ( f 2 - O J z ) V i lim a -> 0 + X exp[-(f 2-w 2j / lx/c] H(y) , (5.7) and f i n a l l y (iv) the branch cut integration for s = ± i ( X 2 c 2 + f 2 ) (when k = 0) gives the following double integral, Cbc» at x = 0; TOO ?bc(°>y>t) = E [\ exp(-iXy) (OJ2+ a 2 - q 2) sin(qt) - 2aqcos(qt) q dqdX (5.8) - J q Z-A zc z • . (u)z+ a z - q2T + 4q 2a 2 (q z - g z ) ^ where E = T 0ifu)c and $ 2 =X 2c 2 + f 2 . For a -»• 0 + i t is straightforward to show that (5.8) consists of standing type waves of the form, sin(ut) sin(u)y/c), cos(u)y/c) sin(wt), ... while the use of Laplace's Method [see Carrier, Krook and Pearson (1966), page 256 for a description of this method] for the branch points of (5.8) y i e l d terms l i k e Sbc(°»y»t) exp (-at) cos(tot) exp[-(f 2- w 2)\/c] + 0(£Vi) . (5.9) lim t -H-oo 6. Response to a moving storm As a f i n a l example we determine the Kelvin wave response to a moving pressure source, which for simplicity w i l l be represented by a delta function. Later in this section we w i l l give some j u s t i f i c a t i o n for such a pressure distribution. We consider the form Tx = -y = 0 • Pa = P Y 2 5(y+Vt) 6(x-xQ) sin(oot) exp (-at) t >, 0, x«> 0 (6.1) where Pa i s now the pressure difference, at sea le v e l , between the center of the storm and the undisturbed pressure far from the disturbance, and i n which P = constant, V i s the speed of the storm, x„ i s the distance from the coast, oo , O ( > 0) are real and y i s a measure of the storm's width [we again retain a >0 so that the inversion contour i n the s-plane has the form of figure 1]. Then using (3.1) we find Pa = P Y 2 <$(x-xj w (6.2) (s+a+iXV)i:+coz which substituted into (4.1) yields poles at A+ = 1 (±u> + i a ) . From residue V-c theory and the requirement that the solution,C^p, obtained from these poles be bounded everywhere, we find C i p(x,y,t) = ± C p(x,y,t) H[±(y+ct)] (6.3) where £p(x.,y,t) = P y 2 foo exp[- l(x+x 0)] exp [-|m| (y+ct) ] PS (6.4) {cos[K(y+ct)] + a sin[K(y+ct)]} a = a/co, (K,m) = (oo,a) 1 and where we use + for c > V V-c - for c < V . As in the previous example we next evaluate the f i r s t term of (3.8) at the poles i n the s-plane due to the force, (6.2). These poles, at s+ = -i(AV±w ) - a, contribute a response^^p, i n which C 2 p(x,y,t) =± (-) £p(x,y,t) H[±(y+Vt)] (6.5) with ± having the same meaning as i n (6.4). The K e l v i n wave response, £ , formed from the sum of (6.3) and (6.5) i s gi v e n by C k ( x , y , t ) = C p ( x , y , t ) { H[±(y+ct)] - H[±(y+Vt)]} sgn(c-V) . (6.6) As seen from f i g u r e s 3(a) and 3(b), £ k as gi v e n by (6.6) occupies two d i s t i n c t longshore regions w i t h r e s p e c t to the storm, depending on the s i g n of c-V. For c > V the K e l v i n waves move away from the storm r e g i o n at group v e l o c i t y c, and at wavenumber K = lo . For c < V there i s a K e l v i n wave 'wake' t r a v e l l i n g c-V i n the storm d i r e c t i o n . I f we tr a n s f o r m V •> -V, to g i v e a storm moving i n the p o s i t i v e y - d i r e c t i o n , we f i n d a K e l v i n wave i n the r e g i o n --ct < y < Vt . F i g u r e s 4(a) and 4(b).are p l o t s of these K e l v i n wave s o l u t i o n s f o r v a r i o u s v a l u e s of ff/oo f o r the cases V < c and V > c r e f l e c t i v e l y . The c o - o r d i n a t e s are f i x e d at y = - c t w i t h the edge y = -Vt not shown, i n order to show the f u l l wave development. I f we choose the p o s i t i o n of t h i s edge, then y*, at y = - V t , becomes y* = -wt so th a t t i s determined f o r each O J. T h i s a l s o f i x e s the r e g i o n occupied by the wave whose l e a d i n g or t r a i l i n g edge (V > c or V < c) decays at the same r a t e as the amplitude of the storm. In the present theory we i n t e r p r e t : ( i ) the d e l t a f u n c t i o n as the l i m i t of a gaussian p r e s s u r e d i s t r i b u t i o n , , v i z . , 6(z) = l i r a 1 e x p ( - z 2 / e 2 ) (6.7) , e -> 0 + £7T^ and ( i i ) y, = y(e) to be much l a r g e r than the wavelength of g r a v i t y waves but s t i l l much s m a l l e r than the c h a r a c t e r i s t i c h o r i z o n t a l wave s c a l e s ( c / f . K 1 ) . For example, i f i n (6.1) we had s t a r t e d w i t h a gaussian d i s t r i b u t i o n i n the x - d i r e c t i o n , centered about x 0 , we would have obtained an x-dependence f o r Cp of (6.4) as £p(x) o: exp[- I ( x + x c ) ] e x p [ ( f e / c ) 2 ] ^ [1 + e r f ( b / e ) ] (6.8) where e r f ( q ) i s the e r r o r - f u n c t i o n and b = ^ (2x Q - f e 2 / c ) . In the l i m i t + f e -»- 0 , L, (x) -»- exp[ -(x + x 0 ) ] , x B> 0, as r e q u i r e d . p c FIGURE 3. Schematic diagram of the regions occupied by the pressure-generated Kelvin waves of § 6 for a storm moving in the negative y direction: (a) c>V; (b) c<V. -1 i , — i i Amplitude £jA* y* ^ -6 - 4 -3 1 (ii)N>-,. (iii) -(b) - - 1 0 FIGURE 4., Plot of the amplitudes pf the pressure-generated Kelvin waves with respect to: (a) the leading edge ( C T - V ) , and (b ) the trailing edge { V > c ), for various values of <r : (i) <r = ^  (ii) <r= Leo ; (iii)S"= 2c«i. The edge for y = -Vt is not shown, fixing it on the figure determines «*t. The important point here, however, is that since f/c is small ( 'V 10~3 km - 1) e (or y) need not be too small before (6.8) is closely approximated by the f i r s t term only, hence justifying the use of the delta function to represent a storm of f i n i t e width. To demonstrate these remarks we note that i f b/e = 2 then the erf(2) = 0.995..., which for e *v 0(10 2 km) also implies that exp[(fe/c) 2] ^ 1.22... . We note further that b/e ^ x 0/e for the above' r value of £ , which in turn implies that x Q = 2e represents a storm of gaussian thickness £ centered offshore at a distance of about 200 km. Of some interest i s the case V ->• c for a storm moving in the negative y-direction. This limit must be applied with care however, since in modelling the storm we have assumed y - 1 much larger than the horizontal wavenumbers (K ,m ). The result of applying V c for f i n i t e m i s that there w i l l be a large surface discontinuity at the leading (trailing) edge y = -ct for c > V (c < V). Immediately behind (in front), y+ct > 0 (y+ct < 0), the surface w i l l decay rapidly to zero. Finally in the l i m i t V = c there w i l l be no waves, as expected, since this also implies OJ = 0. We omit any discussion of the remaining pressure terms of (3.8) except to point out that the last term corresponds to the direct surface response to the force. Also, i t should be noted that sincot was used in the preceeding examples to determine the effect of a time-variable force amplitude. If we consider that the atmospheric force, and resulting long-wave sea surface response, can be Fourier analysed into a sum of sine and cosine components, we could regard the chosen forcing fields as representing one component of the Fourier expansions. In the absence of an oscillatory forcing f i e l d , the response becomes simply a moving Kelvin type 'surge'.. 7. Steady-state solutions As a result of the awkward integrals that arise when attempting to evaluate the fu l l surface response i t is desirable to know the response in the limit t -* +eo (the steady-state solution) without having to approximate the integrals individually. To this end we use the final value theorem, lim G(x,y,t) = G(x,y) = lim s X [C(x,y,t)] " (7.1) t - +<» S - 0 (where X is the Laplace transform) provided C(x,y,t) is a piecewise continuous function for t ^ 0 and has a limit as t -* + » , s approaching 0 + through real values. Application of the above limit.to XCC(x,y,t)] = ^CCCx^s)] (7.2) in which is the inverse Fourier transform, for the examples choj(sen in ^ § 5 and 6, shows that in each case lim C(x,y,t) = 0 ; t — +co the surface returns to its original state of rest. 8. Physical discussion of the solutions Ck In this section we wi l l give a physical interpretation to the Kelvin wave solutions obtained in the previous sections. When a wind is suddenly applied to one-half of the bounded sea, there is immediate motion in the direction of the stress. The discontinuity at the boundary of the generating area wi l l then begin to move away at group velocity '- i_ (gh)2 and to be effected by the rotational forces. Such forces wi l l tend to turn motions to the right (left) of their direction of propagation in the northern (southern) hemisphere. The requirement of zero normal velocity at the wall, however, restricts the formation of a transverse motion outside the forcing region. As a result, for motions having the boundary to the right of their direction of propagation, a surface slope must exist to balance the c o r i o l i s f o r c e . No K e l v i n waves can e x i s t i n the generating region f o r the case considered; forced waves i n t h i s region however have t h e i r wavenumbcrs modified by the r o t a t i o n a l e f f e c t s [ ( 5 . 6 ) - ( 5 . 9 ) ] . The o r i g i n a l y-dependencc of the s t r e s s i s f e l t through the propagating surge given by the l a s t term . of (5.5). For the case of pressure generated K e l v i n waves, we base the d i s c u s s i o n on the r e s u l t obtained i n $ 6 i n which we considered the x-dependence of the atmospheric pressure as a l i m i t of a gaussian d i s t r i b u t i o n . Since the storm's in f l u e n c e would extend to the boundary, any long wave motion s et up by the fo r c e w i l l have i t s onshore v e l o c i t y component r e s t r i c t e d by the w a l l . The c o r i o l i s f o r c e w i l l again be balanced by a sl o p i n g sea surface as i n the f i r s t example. But now, because of the smallness of the atmospheric pressure • gradient at the coast f o r storms f a r offshore ( i . e . l a r g e x e) as compared to those near the coast, the slope of the sea surface need not be as large to give a balance. T h i s i s c l e a r l y manifested i n (6.4) by the weighting f u n c t i o n exp (- -£- x 0 ) . The d i r e c t dependence of the amplitude of (6.4) on f shows tha t these waves are a d i r e c t r e s u l t of the r o t a t i o n and are not a m o d i f i c a t i o n of a surge as i n the wind-stress example. F i n a l l y , the change of the n a t u r a l ' y • frequencies of the storm ( f f , c u ) t o ( a , u ) ) - ~ — i n the K e l v i n waves i s V + c the common 'Doppler' e f f e c t ' f o r waves generated by a moving source. 9. Numerical Values f o r the Oregon coast One appropriate region f o r a numerical d i s c u s s i o n of the amplitudes and v e l o c i t i e s of the wind and pressure generated K e l v i n waves i s the Oregon coast of the United States. T h i s s t r a i g h t , north-south running coast approximates the w a l l of the mathematical model, with the north-south wind component being dominant and the offshore movement of r e l a t i v e l y small storms being common, e s p e c i a l l y i n the winter. Also, numerous t i d e gauges a f f o r d the p o s s i b i l i t y of detecting these waves through the analysis of similtaneous sea level records. An important feature of any real ocean, however, as opposed to the constant depth model, i s the existence of a continental shelf and slope between the coast and the ocean basin. The added complications due to such rapid topographic change might be enough to obscure the Kelvin waves generated. In table 1 the maximum aplitudes, A, of the wind generated Kelvin waves of §5 are presented for the various values of (a - yc)/u) used in the plots of figure 2; A i s given by (5.4). The calculations are for T Q = 1 dyne/cm^ and OJ = 0.12(10"^) sec--'- (day--'-) only, since A is directly proportional to T 0 and inversely proportional to to ( 4 0) . As an example of the magnitude and va r i a b i l i t y of the northward wind-stress off Oregon, data obtained by Oregon State University for the period Aug. 1 to Sept. 31, 1966 are presented i n figure 5; Mooers et a l 1968. These data seem to indicate the presence of a dominant period in the longshore wind-stress component during the time of observation, thus partly justifying the use of the model chosen in (5.1). Also, the period appears to be roughly 2TT days, corresponding to the value of OJ used above. The familiar experimental law T = c'p aU^, where c' = 2.5(10~3) i s the drag coefficient and p a = 1.27(10~3) gm/cm.3 is the density of air, has been used to obtain the stress from the mean wind speed U. The maximum coastal amplitude, A*, of the pressure generated ..Kelvin y2 f w f waves of §6 are given i n table 2; A* = P — r ~ o — exp( x n ) . We (V-c)2 p g ^ o ° consider V « c only, since for typical storms V is usually less than 40 km/hr, while for an ocean depth of 2000 m, c = (gh)l/2 = 504 km/hr. Also, we w i l l take P = 50 mb = 50(10^) dynes/cm^ as representative of an i n t e n s e storm having a gaussian t h i c k n e s s y. As f o r t a b l e 1 only OJ = 0.12(10"^) sec--'- w i l l be considered s i n c e A* i s d i r e c t l y p r o p o r t i o n a l to to. As might be expected, the amplitudes of the pressure generated waves are much l e s s than those generated by the wind. 1 6 12 18 24 30 1 6 12 18 24 30 Aug. Sept. Time (days) FIGURE 5. Daily northward wind-stress off the Oregon coast from I Aug. to 31 Sept. 1966. 239 j 10. Summary In t h i s part of the t h e s i s , the. response of a s e m i - i n f i n i t e , uniformly r o t a t i n g , constant depth, homogeneous ocean to a v a r i a b l e atmospheric force has been considered. I t has been shown that, f o r a general wind and pressure system, forced Kelvin-type waves can be generated and that only the longshore wind component and the pressure can generate them. In p a r t i c u l a r , a s e m i - i n f i n i t e wind and a moving pressure pattern were shown to generate K e l v i n waves that t r a v e l away from the force d i s c o n t i n u i t i e s at the speed of shallow-water waves. The waves, i n the l a t t e r case e x h i b i t a freqeuncy s h i f t t y p i c a l of. non-dispersive waves from a moving source. Using the Oregon coast of the United States to approximate the boundary of the mathematical model, together with some representative data f o r t h i s region, numerical values f o r K e l v i n wave amplitudes have been given. T y p i c a l l y , wave amplitudes for wind-stress generated K e l v i n waves (a few centimetres) are much greater than those generated by pressure d i f f e r e n c e s . The p o s s i b i l i t y of detecting such waves through use of the coastal t i d a l records has been suggested. Table 1. Kelv.in wave amplitude A at x = 0. T 0= 1 dyne/cm3 ; P = 1 gm/cm3 ; g = 103 cm/sec8 ; h = 2000 m • w = 0.12 x 10 sec ( o" , p.c ) / U J A (cm) ( 1 '•, 1 ) 6.0 ( 1 , 1/2) 4 . 7 (1/2 , 1) ( 2 , 1/2). 1.8 (1/2 , 2 ) P « 50 x 103 dynes/cm2 ; f =-.1.03 x lO'V'sec"1; p « 1 gm/cm3 h - 2000 m ; u = 0.12 x lO'^sec'1 ; Y(km) V(lcm/hr) x 0(km) A* (cm) 100 40 100 0.4 (10-1) 200 • 100 .100 0.2 100 40 1000 • . 0.K10"1) 1000 • 40 1 : 1000 2.0 B i b l i o g r a p h y C a r r i e r , G.F., Krook, M. and Pearson, C.E. (1966). Functions of a Complex  V a r i a b l e . New York: McGraw-Hill, 438 pp. Crease, J . (1956). "Propagation of long waves due to atmospheric disturbances on a r o t a t i n g sea". Proc. Roy. Soc. Lond. A 223, 556. K a j i u r a , K. (1962). "A note on the generation of boundary waves of K e l v i n type". J . Oceanographic Soc. Japan, JL8(2), 51. Longuet-Higgins, M.S. (1965). "The response of a s t r a t i f i e d ocean to s t a t i o n a r y or moving wind systems". Deep-Sea Res. 12, 923. Mooers, C.N.K., Bogart, R.L., Smith, R.L. and P a t t u l l o , J.G. (1968). "A c o m p i l a t i o n of observations from moored current meters and thermographs (and of complementary oceanographic and atmospheric d a t a ) " . Department of Oceanography, Oregon State U n i v e r s i t y , V o l . 2. Mysak, L.A. (1969). "On the generation of double K e l v i n , waves". J . F l u i d Mech. 37, 417. Thomson, R.E. (1970). "On the generation of K e l v i n - t y p e waves by atmospheric disturbances". J . F l u i d Mech. 4^(4), 657. V e r o n i s , G. and Stommel, H. (1956). "The a c t i o n of v a r i a b l e wind s t r e s s e s on a s t r a t i f i e d ocean". J . Marine Res. 15, 45. Appendix Glossary of symbols, part II x,y,z Cartesian coordinates; east, north, upwards respectively. z Dummy variable i n § 6. u,v,w, Velocity components along x,y,z. h depth of the ocean (constant). P a . Surface atmospheric pressure. £ Surface elevation about the mean level. g Acceleration of gravity. T x , T v Components of the surface wind-stress. f The Coriolis parameter. p The water density. A Fourier transform variable. s Laplace transform variable. y Path of integration in the A plane in § 3. A measure of the width of the moving pressure system in § 6. c(A,s) Constant of integration in transform space. c =v /gh, the phase speed of shallow water waves. k = (A 2 + [ f 2 + s j / C 2 ) 1 / 2 > 0, a wave number; (see § 3) . G^ The Green's function; (see eqn. 3.6). 0) Frequency of the surface forcing functions (always real). O The time decay constant for the surface forcing functions. U The spatial decay constant along the boundary (y). The wavenumbers o f t h e K e l v i n waves a l o n g x,y r e s p e c t i v e l y . = 1 q > 0 = 0 q < 0 t h e u n i t s t e p f u n c t i o n . A n g l e i n p o l a r c o o r d i n a t e s f o r d i s c u s s i o n o f s t e e p d e s c e n t t e r m (eqn. 5.6). The D i r a c d e l t a f u n c t i o n . Speed o f t h e moving s t o r m . = a/oo • A m p l i t u d e o f t h e K e l v i n waves g e n e r a t e d by t h e w i n d - s t r e s s . A m p l i t u d e o f t h e K e l v i n waves g e n e r a t e d by the p r e s s u r e (moving s t o r m ) . D i s t a n c e f r o m s h o r e o f t h e c e n t e r o f t h e moving s t o r m . L i m i t i n g v a r i a b l e i n t h e d e f i n i t i o n o f t h e D i r a c d e l t a f u n c t i o n ( s e e § 6). 

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