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Atmospheric and oceanic 40- to 50-day oscillations in the source region of the Somali Current Mertz, Gordon James 1985

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ATMOSPHERIC AND OCEANIC 40- TO 50-DAY OSCILLATIONS IN THE SOURCE REGION OF THE SOMALI CURRENT by GORDON JAMES MERTZ B.Sc, University Of B r i t i s h Columbia, 1975 M.Sc, University Of B r i t i s h Columbia, 1977 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department Of Physics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA e November 1985 % © Gordon James Mertz, 1985 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of PH V-STCS  The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date Qc-/ . 10 , )E-6 (3/81) i i ABSTRACT Current and temperature data were acquired in the source region of. the Somali Current, j o i n t l y by the U n i v e r s i t i e s of K i e l and Miami, as part of the INDEX p i l o t studies. The data were acquired over a six-month period (January-July, 1976) which spans the springtime Monsoon re v e r s a l . The experiment and the data are described in Duing and Schott (1978). This thesis describes the r e s u l t s of the spectrum analysis of fluctuations found in data from the experiment's two southernmost sensor locations. It i s found that, once the annual cycle i s removed, mo"st~of the variance in these current and temperature records resides in s u b i n e r t i a l f l u c t u a t i o n s . The most prominent spectral feature i s a 40- to 50-day peak. This 40- to 50-day period i s coincident with that of the global-scale c i r c u l a t i o n c e l l s found in the t r o p i c a l atmosphere by Madden and J u l i a n (1971 and 1972). The analysis of wind stress and wind stress c u r l data for the years 1976 and 1979 presented in t h i s thesis indicate that the 40-to 50-day o s c i l l a t i o n was present over the Western Indian Ocean during these years. It i s suggested here that wind-forcing excites a long c o a s t a l l y trapped wave. To test t h i s idea, a wind-forced quasi-geostrophic, three-layer model and a reduced-gravity model incorporating l a t e r a l mean current shear are applied to the Somali Current regime. Model r e s u l t s suggest that the wind forcing i s strong enough to excite the observed current and temperature flu c t u a t i o n s . iv TABLE OF CONTENTS ABSTRACT i i TABLE OF CONTENTS iv LIST OF TABLES v i LIST OF FIGURES v i i ACKNOWLEDGEMENTS xv CHAPTER I Introduction 1 CHAPTER II Global Properties of the 40- to 50-day O s c i l l a t i o n : A Review of the Literature 4 11.1 Introduction 4 11.2 Meteorological Background 5 11.3 Properties and Origin of the Atmospheric 40- to 50-day O s c i l l a t i o n 17 11.4 Oceanographic Background 27 CHAPTER III The 40- to 50-day O s c i l l a t i o n in the Source Region of the Somali Current during 1976 and 1979 45 111.1 Introduction 45 111.2 The 40- to 50-day O s c i l l a t i o n over the Western Indian Ocean during 1976 and 1979 47 111.3 Fluctuations of the Western Indian Ocean 59 111 . 4 Summary 85 CHAPTER IV The Oceanic Response to Atmospheric Forcing ... 87 IV. 1 Introduction 87 IV.2 Prelminary Considerations 88 - IV.3 Response of the Equatorial Ocean to the Global-Scale Atmospheric Circulation C e l l s : Generation of Equatorial Kelvin Waves 96 IV. 4 The—Basin—Response 104 IV.5 Upper-Layer Dynamics 117 IV.6 Multi-layer Dynamics 135 V IV.7 Leakage of Shear Waves into the Equatorial Waveguide 186 V SUMMARY AND CONCLUSIONS . 190 BIBLIOGRAPHY 193 APPENDIX I 199 APPENDIX II 203 APPENDIX III 204 vi L i s t of Tables Table 1. Details about Moorings and Instrument Performance ... 61 Table 2. I n i t i a l Temperature Fluctuations at K2 91 v i i L i s t of Figure Captions Figure 11.1 . Figure II.2. Figure II.3. Figure II.4. Figure II.5. Figure 11.6. Figure II.7. Figure II.8. Figure II.9. Figure 11.10 Figure 11.11. Figure 11.12. Meridional plane of a Hadley c e l l . (From Brock, 1984) .... 7 Schematic representation of a summer monsoon c i r c u l a t i o n 9 P r o f i l e s of dynamic height of standard isobaric surfaces along the equator. The Walker c i r c u l a t i o n i s indicated by arrows . Note the reversal of pressure gradient with height responsible for the Walker c i r c u l a t i o n . (From Bjerknes, 1 969) 11 Dispersion curves for equatorial waves (adapted from L i g h t i l l , 1969) 12 Wave period as a function of wavenumber for various wave types under conditions of neutral conditional i n s t a b i l i t y of the second kind. (From Lindzen, 1974) 1 5 Schematic depiction (from top to bottom) of the 40- to 50-day o s c i l l a t i o n modifying the Walker c i r c u l a t i o n . (From Madden and J u l i a n , 1972). 19 Schematic model of the 40- to 50-day period o s c i l l a t i o n of the anamolous Hadley c i r c u l a t i o n . The maximum stage of the anomalous wind dir e c t i o n i s indicated in the c i r c l e s . Relatively warm areas are shaded. (From Yasunari, 1981). .. 22 Phase difference (degrees) for the zonal v e l o c i t y perturbation at 200 mb and 850 mb. (From Murakami, Nakazawa, and He, 1984) 24 Phase difference versus distance along l i n e AB in Fig.II.8 26 Surface winds (from Diiing, 1970) and surface currents (from Diiing and Schott , 1978) of the monsoon regime of the Indian Ocean 29 Power spectra of (a) zonal v e l o c i t y , and (b) meridional v e l o c i t y , measured on the Island of Gan. (From McPhaden, 1982) 35 Power spectra from the Indian Ocean array for (a) zonal v e l o c i t y at 200m, (b) zonal velocity at 750m, (c) meridional v e l o c i t y at 200m. (From Luyten and Roemmich, 1982) 36 v i i i Figure 11.13. The area of Quadfasel and Swallow's observations. Sites 188 and 189 are positions of moored current meters. (Redrawn from Quadfasel and Swallow, 1984) 38 Figure 11.14, Meandering of s a t e l l i t e - t r a c k e d buoys in the South Equatorial Current during 1976 and 1977. The wavelength of the meanders (1000 to 1300 km) corresponds to about a 50-day o s c i l l a t i o n at the average d r i f t speed of the buoys. (From Quadfasel and Swallow, 1 984) 39 Figure 11.15, Figure 11.16, The t h e o r e t i c a l wind stress c u r l s p e c i f i e d by equation(11.3) for (a) y o = l000km and (b) yo=2000km 41 Locations of moored current meters in the t r o p i c a l P a c i f i c Ocean with nearby bathymetry. Contours are in fathoms. (From Hayes, 1979). 43 Figure 1 1 1 . 1 . Figure III.2. Figure III.3. Figure III.4 Figure III.5. Figure III.6. Figure 111.7 . Figure III.8. Sample surface wind map from Fernandez-Partagas and Duing (1977). Isotachs are in knots. .. 48 Sample surface wind stress map from Fernandez-Partagas, Samuels, and Schott (1980). Arrows indicate the wind stresses at each point on the 2° g r i d ; the wind stress magnitudes are in dynes/cm 2, the directions (forward) are given in degrees from true north 49 Components of the wind stress f i e l d at point E. Dashed l i n e s represent a s p l i n e - f i t to the mean stress f i e l d 51 Autospectra of the components of the wind stress at point E. The spectral density i s in units of (dynes/cm2 ) 2/cpd 52 Autospectrum of the c u r l of the wind stress calculated about point E. The spectral density i s in units of 10" 1 4 (N/m 3) 2/cpd 53 Zonal wind and station pressure from Canton Island treated with a 47-day band pass f i l t e r . (From Madden and J u l i a n , 1971). . 54 (a) Low-passed wind stress record from point E (b) Low-passed wind stress c u r l record (in units of 10" 8 N/m3) 55 (a) Autospectrum of the zonal component of the ix Figure III.8 Figure III.9, wind stress at point E'. The spectral density i s in units of (dynes/cm 2) 2/cpd. (b) As in (a) , except for the meridional component 56 (c) Autospectrum of the zonal component of the wind stress at point H'. Units as in (a). (d) As in (c) , except for the meridional component 57 Low-passed zonal wind stress H* record from point 58 Figure III.10. Mooring array off Kenya from mid-January to mid-July, 1 976 (from Diiing and Schott, 1 978 ). .. 62 Figure I I I . 1 1 Vector time series of currents at locations K1 and K2 . Vectors point in the d i r e c t i o n of the flow, with north upward. Average depth and standard deviation of individual sensors are indicated. For cases where speed i s missing an asterisk indicates that the values are obtained from regression with speed record at l e v e l below (from Diiing and Schott, 1978) 63 Figure I I I . 12. Schematic d i s t r i b u t i o n of surface currents in the western Indian Ocean during both Monsoon seasons and related nomenclature (from Diiing and Schott, 1978) 64 Figure I I I . 1 3 Figure III.14, (a) The longshore v e l o c i t y record from the top sensor of station K2. The dashed l i n e represents a s p l i n e - f i t to the mean current. (b) As in (a) , except for the middle sensor at K2. (c) As in (a) , except for the bottom sensor at K2 . Note that the v e r t i c a l scale i s one-quarter that in (a) 67 Current p r o f i l e s ( l e f t ) and l a t e r a l d i s t r i b t i o n (right) of the deep flows at s i t e B, a station not far from K2. Each consecutive p r o f i l e has been sh i f t e d by 60cm/s to the right of the previous p r o f i l e . The data were co l l e c t e d during INDEX (1979). From Leetmaa, Rossby, Saunders, and Wilson ( 1980) 68 Figure I I I . 15. The bottom topography at stations K1 and K2 69 Figure I I I . 16. A conceptual model of the mean flow near station K2 during the period of strongest flow (late May) 69 X Figure III.17. Autospectrum of the temperature fluctuations at the top sensor of station K1. The spectral density i s in units of (°C)2/cpd 70 Figure III.18. (a) Low-passed longshore v e l o c i t y from the top sensor at station K1. The time of the high speed burst i s indicated by an arrow. (b) As in (a), except for the middle sensor at K1 . (c) As in (a) , except for the bottom sensor at K1 71 Figure III.18. (d) Low-passed temperature record from the top sensor at station K1. The time of the high speed burst i s indicated by an arrow. (e) As in (d) , except for the middle sensor at K1 . (f) As in (d) , except for the bottom sensor at K1 72 Figure III.19. (a) Autospectrum of the temperature fluctuations at the top sensor of station K2 . The spectral density i s in units of (°C) 2/cpd. (b) As in (a) , except for the middle sensor at K2. (c) As in (a), except for the bottom sensor at K2. Note that the v e r t i c a l scale i s about one-sixth that in (a) 75 Figure 111.20. (a) Low-passed temperature record from the top sensor at station K2. (b) As in (a) , except for the middle sensor at K2. (c) As in (a), except for the bottom sensor at K2 76 Figure III.21. (a) Low-passed longshore ve l o c i t y record from the top sensor at station K2. (b) As in (a) , except for the middle sensor at K2. (c) As in (a) , except for the bottom sensor at K2. Note that the v e r t i c a l scale i s one-fifth that in (a). 77 Figure III.22. (a) Autospectrum of the longshore velocity fluctuations at the top sensor of station K2. The spectral density i s in units of (cm/s) 2/cpd. (b) As in (a), except for the middle sensor at K2. Note that the v e r t i c a l scale i s twice that in (a). The spectrum was calculated over the last one hundred days of the record. (c) As in (a), except for the bottom sensor at K2. Note that the v e r t i c a l scale i s one-thirtieth that in (a). The spectrum was calculated over the last one hundred days of the record 78 X X Figure III.23. Band-passed (12.5 to 25 days) longshore velocity record from the top sensor at station K2. . 79 Figure III.24. Location of measurement s i t e s for Schumman's Natal Experiment. (From Schumann, 1981). .. 80 Figure III.25. (a) Coherency and phase spectra between measurements at moorings off Port Edward and Southbroom (see Fig.III.24). S o l i d l i n e indicates longshore current components, dashed l i n e indicates offshore current components, and dotted l i n e indicates temperature. (b) Coherency and phase spectra between the longshore wind measured at Louis Botha airport and the longshore current ( s o l i d line) and the offshore current (dotted line) measured at Richard's Bay. (From Schumann, 1981) 81 Figure III.26. Amplitude of Hough functions (H^H^) versus latitude and measured amplitude of a 16-day wave at 100 mb. (From Madden , 1978) 82 Figure III.27. Autospectrum of the offshore velocity fluctuations at the top sensor of station K2. The spectral density i s in units of (cm/s) 2/cpd. 83 Figure III.28. Low passed offshore v e l o c i t y record from the top sensor at station K2 83 Figure IV..1. Temperature fluctuations ( s o l i d l ine) at the mid-sensor of station of station K2 compared to the lo c a l wind stress c u r l (dashed l i n e ; in units of 10-8 N/m3) 95 Figure IV.2. The gaussian d i s t r i b u t i o n of wave amplitude spec i f i e d by equation (IV. 28) 100 Figure IV.3. The basin model . The parameters are fixed at L=l000km, B=6000km, H (depth) = 4000m, for purposes of c a l c u l a t i o n 105 Figure IV.4. (a) The basin model streamfunctions for t=0. 109 Figure IV.4. (b) The basin model v e l o c i t y magnitudes (U 2+V 2)' a for t = 0 110 Figure IV.4. (c) -The basin model streamfunctions for t= ir/4( 1/o> ) 111 Figure IV.4 (d) The basin model v e l o c i t y magnitudes (U 2+V 2) x i i for t= 7T/4( 1/oo ) 112 Figure IV.4. (e) The basin model streamfunctions for t = 7T/2( 1/OJ ) 113 \, Figure IV.4. (f) The basin model ve l o c i t y magnitudes ( U 2 + V 2 ) z for t= TT/2( 1/OJ ) 114 Figure IV.5. Ocean ridges near s i t e s A,B,C of Hayes' program. (Adapted from King, 1962). 116 Figure IV.6. The upper layer-dynamics model. 122 Figure IV.7. (a) Nondimensional streamfunction , amplitude ( s o l i d l i n e and phase (dashed line) for the forced shear modes for the case k=0.2. The current p r o f i l e i s given in equation(IV.51); the peaks of the amplitude at x=1,2 correspond to the di s c o n t i n u i t i e s in V. (b) Nondimensional longshore v e l o c i t y , amplitude ( s o l i d l i n e ) and phase (dashed line) for the forced shear modes for the case k=0.2. .... 129 Figure IV.7. Figure IV.7. Figure IV.7. (c) As in (a) , except for the case k=0.4. (d) As in (b), except for the case k=0.4. . 130 (e) As in (a), except for the case k=0.6. (f) As in (b), except for the case k=0.6. . 131 (g) As in (a), except for the case k=0.8. (h) As in (b), except for the case k=0.8. . 132 Figure IV.8. Figure IV.9. The N-layer model. 136 Model of the channel flow considered in the response analysis. VI, V2, and V3 are assumed constant. The i n t e r f a c i a l slopes are drawn appropriate for a Southern Hemisphere situation with the mean vel o c i t y decreasing monotonically with depth 150 Figure IV.10, Surface current data c o l l e c t e d during INDEX (1979). Current arrows are centered on the observation point. (From Diiing, Molinari, and Swallow, 1980) 158 Figure IV.11 (a) The temperature record from the top sensor of station K2. The dashed l i n e represents a spline-f i t to the mean temperature. (b) As in (a), except for the middle sensor. Note that the ordinate has been shifted by 5°C. (c) As in (a), except for the bottom sensor. Note that the ordinate has been sh i f t e d by 8°C. . 159 x i i i Figure IV.12, Figure IV.13 Figure IV.14, Figure IV.15, Figure IV.16, Reconstruction of v e r t i c a l temperature d i s t r i b u t i o n by using simultaneous temperature and pressure records from the top sensor at mooring M1 . (From Diiing and Schott, 1978) . 161 Temperature p r o f i l e s : the dashed curve i s from Cox (1976) and represents mean Indian Ocean conditions; the s o l i d curve i s from K2 data and represents l o c a l conditions in late A p r i l . The surface point on the K2 curve i s estimated from data in Brown, Bruce , and Evans (1980). .. 162 Potential temperature (0) p r o f i l e s from Cox (1976). .. and density ( °<9 ) 163 Sigma^ vs. Q for the upper 300 meters of the s t r a t i f i c a t i o n data in Cox (1976) (see Fig.IV.20) 164 (a) Temperature s t r a t i f i c a t i o n at station K2 during the period of lowest temperatures (late May). The point on the surface i s estimated from data in Brown, Bruce and Evans (1980). (b) Temperature s t r a t i f i c a t i o n at station K2 during the period Of maximum temperatures (early March) . The point at the surface i s estimated from data in Brown, Bruce, and Evans (1980). 166 Figure IV.17. Figure IV.18. Figure IV.19, Figure IV.20 Depth-averaged longshore of time v e l o c i t i e s as a funct ion 168 The dispersion r e l a t i o n for the maximum shear case , Vj=(100,25,8) cm/s, with other parameters having the values given in equation(IV.78). The three modes evident here are described in the text 170 (a) The nondimensional amplitude ( s o l i d l ine) and phase (dashed line) of the upper-layer transfer function under conditions of maximum shear, with the other parameters as l i s t e d in equation(lV.78), and C X=10" 5 m/s , C p=3x10- 5 m/s. The phase shown i s that of the longshore v e l o c i t y fluctuations r e l a t i v e to the forcing. (b) As in (a) , except for the second layer. Note that the amplitude scale i s one-tenth that in (a). As in (a) , except for the bottom layer . Note that the amplitude scale i s one hundredth that in (a) 173 Layer amplitudes for the upper-layer shear mode as a function of the i n t e r f a c i a l f r i c t i o n xiv Figure IV.21 parameter Cj.. This figure applies to the maximum shear case , with the other parameters as l i s t e d in equation(IV.78). Note that the horizontal axis i s logarithmic and that separate v e r t i c a l scales are used for the upper-layer and the two lower-layer amplitudes 175 The resonant wavenumber and wavelength for the upper-layer shear mode as a function of time. 177 Figure IV.22 Figure IV.23, Figure IV.24, Figure IV.25, Comparison of longshore velocity f l u c t a t i o n s from the top sensor at station K2 ( s o l i d line) with wind stress c u r l (dashed l i n e ; the c u r l is in units of 1 0"8 N/m3) 178 (a) The amplitude ( s o l i d line) and phase (dashed line) of the wind-induced longshore v e l o c i t y as a function of the resonant wavenumber for the upper layer. The physical s i t u a t i o n i s as described in the text. (b) As in (a) , except for the middle layer. Note that the v e r t i c a l scale i s one-quarter that in (a). (c) As in (a) , except for the bottom layer. Note that the v e r t i c a l scale i s one-fourtieth that in (a) 180 Comparison of longshore velocity f l u c t a t i o n s ( s o l i d line) with wind stress (dashed l i n e ) . 182 (a) The amplitude ( s o l i d line) and phase (dashed line) of the wind-induced longshore v e l o c i t y as a function of the resonant wavenumber for the upper layer in the case of bottom forced motion. The physical situation i s described in the text. (b) As in (a), except for the middle layer. (c) As in (a), except for the bottom layer. 183 Acknowledgments I would l i k e to express my gratitude to a l l the persons involved in the completion of t h i s thesis. I p a r t i c u l a r l y thank Dr. L. A. Mysak for his guidance and exceptional patience. I also thank Drs. R. W. Burling, P. H. LeBlond, and K. Hamilton for reading t h i s thesis and for many helpful suggestions. The oceanographic and meteorological data were supplied by Dr. F. Schott at the University of Miami. I am grateful to the University of B r i t i s h Columbia for support through Graduate Fellowships and to the Natural Sciences and Engineering Research Council of Canada for Post Graduate Scholarships. I am also grateful to the U.S. O f f i c e of Naval Research (Code 422PO) for support as a Research Assistant at the la t e r stages of this work. 1 I. Introduction It i s now widely appreciated that the t r o p i c a l atmosphere and ocean exert an important influence on weather and climate worldwide. Kerr (1984) notes that "It i s an axiom of meteorology that no part of the atmosphere stands alone; everything i s interconnected. It i s a f r u s t r a t i o n of meteorology that major, sometimes catastrophic, atmospheric e f f e c t s arise from the complex interconnection of numerous, subtle, and obscure causes. E l Nino has recently become a prominent example of how elusive changes in the ocean and atmosphere can sometimes lead to dramatic s h i f t s in the behavior of the atmosphere, sending abnormal weather through the entire system. It now seems that another prominent phenomenon, whose existence long eluded meteorologists and whose significance is only now becoming apparent, sets the c i r c u l a t i o n of the tropics pulsating at a far faster pace than the irregular 3- to 8-year recurrence time' of E l Nino. From a s a t e l l i t e ' s perch, the new phenomenon appears as a wave of cloudiness that f i r s t develops every 40 to 50 days in the Indian Ocean..." This 40- to 50-day o s c i l l a t i o n i s the subject of this thesis. Recently, this o s c i l l a t i o n has become the object of intense meteorological interest since i t spreads from i t s o r i g i n 2 in the Indian Ocean throughout the P a c i f i c Ocean with meridionally propagating branches spreading as far off as the poles. Perhaps most s i g n i f i c a n t l y , t h i s 40- to 50-day o s c i l l a t i o n may play a role in triggering the onset and withdrawal of the Indian Monsoon, causing monsoon rains to pause in midseason. The 40- to 50-day o s c i l l a t i o n has even been implicated in reshaping the jet stream (via modification of the monsoon) that plays a central role in North American weather. The strength of this o s c i l l a t i o n in the atmosphere implies the l i k e l i h o o d of finding i t in the ocean. However, i t is only r e l a t i v e l y recently that t h i s o s c i l l a t i o n has been d e f i n i t i v e l y i d e n t i f i e d there (Luther, 1980). Even less work has been done on the r e l a t i o n between the atmospheric and oceanic 40- to 50-day o s c i l l a t i o n s . In t h i s thesis I w i l l examine the 40- to 50-day o s c i l l a t i o n , in both the atmosphere and the ocean, in a very interesting area, namely, the source region of the Somali Current. This area forms part of the western boundary, of the Indian Ocean, and i s thought to be the'source of the 40- to 50-day o s c i l l a t i o n . The area i s one of marked land-sea contrast, known to be important in generating monsoons and other convective phenomena. Nearby l i e the East African Highlands which profoundly a f f e c t the c i r c u l a t i o n of t h i s region. The annual reversal of the major Somali Current system must have an important e f f e c t on any o s c i l l a t i o n s present. I w i l l show that the properties of the 40- to 50-day o s c i l l a t i o n are indeed unique in t h i s area; most importantly, I i d e n t i f y a 40- to 50-3 day signal propagating as a coastal wave that appears to be driven by fluctuations in the wind stress c u r l (this i s also the f i r s t i d e n t i f i c a t i o n of a 40- to 50-day fluctuation in the wind stress c u r l ) . The plan of the thesis i s as follows: In Chapter II the observations of the 40- to 50-day o s c i l l a t i o n reported in the l i t e r a t u r e to date are reviewed. Relevant meteorological and oceanographic background information i s provided. Chapter III presents an analysis of oceanographic and meteorological data co l l e c t e d near the source region of the Somali Current, and establishes the existence and properties of the 40- to 50-day o s c i l l a t i o n in t h i s area. Chapter IV i s concerned with r e l a t i n g the atmospheric and oceanic 40- to 50-day flucuations. Various models are presented to elucidate t h i s relationship. The results are summarized in Chapter V. 4 II. Global Properties of 40- to 50-day O s c i l l a t i o n ;  A Review of the Literature II. 1 Introduction In t h i s chapter I w i l l focus on the global-scale properties of the 40- to 50-day o s c i l l a t i o n in the ocean and atmosphere as established in the l i t e r a t u r e to date. (This contrasts with Chapter III where I w i l l discuss the properties of the 40- to 50-day o s c i l l a t i o n primarily in the l o c a l i t y of the source region of the Somali Current, presenting an o r i g i n a l analysis of atmospheric and oceanic data acquired in this region). Since Madden and Julian (1971 and 1972a) announced the discovery of the 40- to 50-day o s c i l l a t i o n in the atmosphere, great progress has been made in elucidating the properties of this o s c i l l a t i o n . Much less progress has been made toward the understanding of the origi n of the 40- to 50-day o s c i l l a t i o n . It i s only r e l a t i v e l y recently (Luther, 1980) that the 40- to 50-day o s c i l l a t i o n has been p o s i t i v e l y i d e n t i f i e d in oceanic data, and l i t t l e work has been done on re l a t i n g t h i s oceanic o s c i l l a t i o n to i t s atmospheric counterpart. 5 II.2 Meteorological Background In the introduction of thi s thesis I emphasized the importance of the t r o p i c a l atmosphere to global meteorology. As Holton (1979) states: " . . . i t i s the tropics [where] the majority of the solar energy which drives the atmospheric heat engine i s absorbed by the earth and transferred to the atmosphere. Therefore, an understanding of the general c i r c u l a t i o n of the tropics must be regarded as a fundamental goal of dynamic meteorology." This absorbed radiation becomes the energy source for various i n s t a b i l i t y mechanisms in the tro p i c s . The primary energy source for these i n s t a b i l i t i e s i s latent heat release. This i s in contrast to mid-latitude meteorological systems where the potent i a l energy available to make disturbances grow i s provided by the substantial existing horizontal temperature gradients. The significance of t r o p i c a l disturbances i s i l l u s t r a t e d by the case of Easterly (or African) waves centered at around 16°N. These waves are generated by barotropic i n s t a b i l i t y of an easterly thermal wind created by the strong horizontal temperature gradient over the Sahara in Northern Hemisphere summer. Propagating westward from the zone of o r i g i n , these waves are occasionally the progenitors of t r o p i c a l storms and hurricanes in the Western A t l a n t i c . In thi s section I intend to focus on these t r o p i c a l disturbances, and especially those of 6 40- to 50-day time scale. It i s helpful to f i r s t discuss the general c i r c u l a t i o n of the tr o p i c s . Lau and Lim (1982) note that "...at least to a f i r s t approximation, the t r o p i c a l large-scale c i r c u l a t i o n , including the monsoons, can be regarded as being driven by l o c a l heat sinks and sources..." The heat sources create large-scale v e r t i c a l motion, giving r i s e to meridional c i r c u l a t i o n c e l l s (Hadley c e l l s ) and east-west oriented c i r c u l a t i o n c e l l s (Walker c e l l s ) . The heat release i s generally accomplished by conditional i n s t a b i l i t y . That i s , a moist parcel of a i r r i s i n g a d i a b a t i c a l l y becomes saturated at some l e v e l (cloud base) and gives r i s e to condensation and the release of latent heat. This heat warms the parcel, and i f the rate of warming i s greater than the lapse rate (rate of change of ambient temperature), then the parcel w i l l continue to r i s e (pseudoadiabatic ascent) and release latent heat, thus experiencing conditional - i n s t a b i l i t y . At the largest scale, the entire tropics are a heat source, giving r i s e to the Hadley c i r c u l a t i o n i l l u s t r a t e d in Fig.II.1. The area of r i s i n g a i r between 5° to 10°N constitutes the In t e r t r o p i c a l Convergence Zone (ITCZ). Here, large cumulonimbus clouds ("heat-towers") carry heat from the low l e v e l of convergence to the upper troposphere in the updrafts of the cloud core. [This transported heat is then dispersed poleward, 7 North Pote Equator F i g . II.1. Meridional plane of a Hadley c e l l (from Brock,1984). thus s a t i s f y i n g the global heat balance]. The ITCZ appears in s a t e l l i t e photographs as a series of cloud clusters with scales of the order of a few hundred kilometers. These cloud clu s t e r s are associated with weak wave disturbances propagating along the ITCZ. The aforementioned l o c a l heat sources are associated with monsoon a c t i v i t y . The term "monsoon" i s , according to Holton 8 (1979), "commonly used in a rather general sense to designate any seasonally reversing c i r c u l a t i o n system". Further, "Most t r o p i c a l regions are influenced to some extent by monsoons. However,the most extensive monsoon c i r c u l a t i o n by far i s the complex c i r c u l a t i o n associated with the Indian subcontinent, producing warm wet summers and cool dry winters." Monsoon c i r c u l a t i o n i s the result of the contrast of the thermal properties of land and sea surfaces. The penetration of solar heating on land i s limited (by conduction) to a depth of the order of one meter. Thus land surfaces have small e f f e c t i v e heat capa c i t i e s . The ocean, in contrast, i s mixed by wind to depths of the order of 100 meters and can thus absorb a comparatively large amount of heat. In summer conditions the land becomes warmer giving r i s e to v e r t i c a l advection over the land with consequent enhanced cumulus convection and, in turn, the release of latent heat which causes warming throughout the troposphere. The si t u a t i o n i s i l l u s t r a t e d in Fig.II.2. The monsoon i s thus driven by diabatic (latent and radiative) heating. In the winter season the thermal contrast is opposite to that in summer, and the continents become cool and dry with p r e c i p i t a t i o n found over the r e l a t i v e l y warm oceans. Major monsoonal heat sources include the three continents straddling the equator, i . e . , South America, equatorial A f r i c a and the "maritime continent" of Borneo and Indonesia (Lau and 9 Pr-essu /-<_, cH~ Ssct /L«vt ( Fig . II.2. Schematic representation of a summer monsoon - c i r c u l a t i o n . Lim, 1982). The f i r s t two sources are important in Northern Hemisphere summer, while in winter the Borneo-Indonesia region 10 is by far the strongest of the three heat sources. It i s in fact the large thermal contrast between the Borneo-Indonesia heat source and the immense cold source over northern China and Siberia which drives the East Asia winter monsoon. These sources contribute to the Walker c i r c u l a t i o n , i l l u s t r a t e d in Fig.II.3. Superimposed on the large scale features of the t r o p i c a l c i r c u l a t i o n are certain d i s t i n c t classes of disturbances. Before discussing the properties of the 40- to 50-day o s c i l l a t i o n , I w i l l now b r i e f l y review the information accumulated on other t r o p i c a l disturbances. Boyd (1977), in reviewing equatorial waves in the atmosphere, considers the existence of 4 modes1 to be well established: (i) the 40- to 50-day o s c i l l a t i o n we are interested in here. This wave i s i d e n t i f i e d as a Kelvin mode of planetary wavenumber s=1 (one wavelength around the circumference o.f the earth = 40,000 km). ( i i ) tropospheric "5-day" waves2 propagating as n=1 (lowest l a t i t u d i n a l l y symmetric) mode Rossby waves. ( i i i ) A stratospheric Kelvin wave with planetary wavenumber one to two and a period of 10-15 days. (iv) A stratospheric mixed Rossby-gravity (Yanai) wave 1 Refer to F i g . II.4 for i d e n t i f i c a t i o n of the equatorial modes in terms of their dispersive c h a r a c t e r i s t i c s . 2 The "5-day" waves discussed here are s t r i c t l y t r o p i c a l in nature and are not to be confused with the global-wide 5-day waves discussed by Madden and J u l i a n , 1972b. The l a t t e r are westward propagating in the tropics with planetary wavenumber (as defined in (i) above) s=1, while the former are easterly waves with a planetary wavenumber of about s=12. 11 F i g . II.3. P r o f i l e s of dynamic height of standard isobaric surfaces along the equator. The Walker c i r c u l a t i o n i s indicated by arrows. Note the reversal of pressure gradient with height responsible for the Walker c i r c u l a t i o n . (From Bjerknes, 1974). of planetary wavenumber three to four and a period of about 5 days. Boyd does not mention the observation of 3,4, and 5.5 day o s c i l l a t i o n s in sea-level records from the t r o p i c a l P a c i f i c , which correspond to the zeroes of the group velocity of the gravity modes for n=1,2,4. Also, Boyd omits reference to the 1 2 F i g . II.4. Dispersion curves for equatorial waves (adapted from Lighthill,1969). easterly waves (discussed in the introduction to t h i s chapter), presumably because they are not true equatorial waves, being associated with the ITCZ. The two stratospheric waves have been i d e n t i f i e d in the output of a general c i r c u l a t i o n model (Hayashi, 1974) and i t is found that they are important in the general c i r c u l a t i o n : the Kelvin waves transport westerly momentum upward into the stratosphere, resulting in a westerly 1 3 acceleration of the zonal flow; the mixed Rossby-gravity waves transport sensible heat away from the equator in the stratosphere, implying an easterly acceleration of the zonal flow by the induced mean meridional c i r c u l a t i o n . Given that the waves exist and may be important, what i s thei r origin? Boyd (1977) suggests that the stratospheric waves are not generated by in s i t u i n s t a b i l i t y , but rather are generated as by-products of the often vigorous weather of the t r o p i c a l troposphere. Thus, understanding tropospheric dynamics should y i e l d an accounting for a l l the modes discussed in th i s section. Holton (1979), discussing t h i s problem, notes: "Some t r o p i c a l disturbances probably originate as midlatitude b a r o c l i n i c waves which move equatorward and gradually assume t r o p i c a l c h a r a c t e r i s t i c s . However, there can be l i t t l e doubt that most synoptic scale disturbances in the equatorial zone originate in s i t u . Baroclinic i n s t a b i l i t y cannot account for the bulk of these disturbances because, except in a few isolated regions such as North A f r i c a and India, the meridional temperature gradient .is very small. Thus, there i s an i n s u f f i c i e n t source of available potential energy to account for the development and maintenance of equatorial disturbances." Holton then goes on to discuss the role of barotropic i n s t a b i l i t y in generating easterly waves. However, Holton notes that 14 "Since t r o p i c a l disturbances are observed to exist in the absence of strong l a t e r a l shear, i t i s unlikely that barotropic i n s t a b i l i t y i s the primary energy source for the maintenance of the waves over most of the oceanic equatorial zone." This leaves as the only l i k e l y candidate for generating t r o p i c a l waves the Conditional I n s t a b i l i t y of the Second Kind (CISK) mechanism. The notion of CISK was introduced by Charney (1964) to explain the growth of hurricanes. Charney noted that conditional i n s t a b i l i t y is generally (in Holton's (1979) terminology) "released" on the scale of individual cumulus clouds, which i s too small to account d i r e c t l y for the growth of large t r o p i c a l storms. Thus, Charney suggested that a cooperative process takes place; the cumulus c e l l supplies heat energy to drive a (hurricane) depression, and the depression provides low l e v e l convergence of moisture to the c e l l in a surface Ekman layer. This positive feedback cycle apparently accounts for hurricane growth. However, a v a r i a t i o n of t h i s mechanism i s needed to account for the existence of t r o p i c a l waves , since they can exist at the equator while hurricanes cannot since as Linzden (1974) notes "the effectiveness of Ekman pumping close to the equator seems dubious." Accordingly, Lindzen (1974) has formulated a theory of wave-CISK, wherein the low-level convergence f u e l l i n g cumulus convection i s provided by synoptic scale (^1000km) waves. Results of his c a l c u l a t i o n are shown in Fig.II.5. The dispersion curves c o r r e c t l y predict the 15 F i g . II.5. Wave period as a function of wavenumber for various wave types under conditions of neutral conditional i n s t a b i l i t y of the second kind. (From Lindzen, 1974) 16 period of 50 days for a zonal wavenumber 1 Kelvin wave. However, Lindzen showed that the growth rate curves do not single out either this wavenumber or period, leaving the question of scale selection open. 17 II.3 Properties and Origin of the Atmospheric 40- to 50-day O s c i l l a t i o n Kerr (1984) describes the discovery of the atmospheric 40-to 50-day o s c i l l a t i o n as follows: "The 40- to 50-day o s c i l l a t i o n was the serendipitous discovery of Roland Madden and Paul Julian (1971 and 1972a) of the National Center for Atmospheric Research (NCAR) in Boulder, Colorado. Not that i t is a t e r r i b l y subtle, d i f f i c u l t to detect phenomenon that might be swamped by a l l the other v a r i a b i l i t y of the atmosphere. Fifteen kilometers over Canton Island in the central P a c i f i c where the east-'" west or zonal wind may vary, by 30 kilometers per hour from season to season, the 40- to 50-day o s c i l l a t i o n can vary the wind over a range of 90 kilometers per hour, reversing the wind's direction in the process. In terms of wind changes, the o s c i l l a t i o n i s one-third to one-half as powerful as the changes accompanying a good-sized E l Nino event. The problem with finding the o s c i l l a t i o n was that whenever meteorologists studied a record long enough to contain a s u f f i c i e n t number of cycles, they usually worked with monthly averages that obliterated any sign of the o s c i l l a t i o n . Madden and Julian stumbled on — i t in 1971 while broadening e a r l i e r studies of t r o p i c a l atmospheric waves having periods of less than 14 days. 18 They had no trouble tracing i t s variations in pressure and zonal wind from the Indian Ocean to the eastern P a c i f i c and, at least at higher a l t i t u d e s , a l l the way around the globe. They could not find any evidence of the o s c i l l a t i o n extending much beyond 10° north or south of the equator, although they presumed such a strong o s c i l l a t i o n must exist beyond the t r o p i c s . " Madden and Julian's (1971 and 1972a) analysis of wind and pressure variations showed the disturbance acts as a center of intensifying updraft (with corresponding clouds and prec i p i t a t i o n ) which propagates eastward from the Indian Ocean u n t i l i t eventually dissipates over the eastern P a c i f i c . This r i s i n g a i r would level off at the troposphere-stratosphere boundary and eventually f a l l back toward the lower troposphere, forming a closed c i r c u i t or convection c e l l (see Fig.II.6). The updraft i s driven by surface heating; Madden and J u l i a n (1972a) suggest the process is "associated with some sort of a feedback mechanism as that of sea-surface temperatures and atmospheric c i r c u l a t i o n " . They were not able to d e f i n i t i v e l y i d e n t i f y the mechanism of the 40- to 50-day o s c i l l a t i o n . After t h i s pioneering work, the subject was neglected by the meteorological community for 8 years. Kerr (1984) notes that: "...Explanations for the hiatus vary. Madden and Ju l i a n had exhausted the available data. It was only a t r o p i c a l phenomenon, at a time when many considered 19 EAST LONGITUDE WEST LONGITUDE 2 0 ' 6 0 ' 100* 110* ISO" lit' 100' 6 0 ' 2 0 ' i—i—m—i—i—i—i—i—i—i—i—i—i—i—i—i—i— I D ' H X mmm AFRICA INDONESIA Increasing large scale convection over the Indian Ocean. Eastern c i r c u l a t i o n c e l l spreading eastward. Center of large scale convection over Indonesia. The two c i r c u l a t i o n c e l l s are now nearly symmetric Western c e l l i s shrinking, convection weakening. Increasingly weakened' convection. No enhanced convection remains. C e l l s again symmetric. F i g . II.6. Schematic evolution (from top to bottom) of the 40-to 50-day o s c i l l a t i o n modifying the Walker c i r c u l a t i o n . (From Madden and J u l i a n , 1972a). the weather patterns of the mid-latitudes (where most meteorological research was done) and the tropics separate and independent. And fluctuations that were so much slower than day-to-day v a r i a b i l i t y were not a hot topic among meteorologists. In any case, things picked up around 1980. 20 Tetsuzo Yasunari (1981) of Kyoto University reported 40- to 50-day fluctuations in cloudiness over the bay of Bengal and adjacent India, prompting Ju l i a n and Madden (1981) to publish some old equatorial cloudiness data supporting their model of enhanced convection. Then Klaus Weickmann (1983) of the University of Wisconsin at Madison pointed to s a t e l l i t e observations of 30- to 60-day fluctuations in t r o p i c a l cloudiness that tended to propagate eastward into the central P a c i f i c . Weickmann also found during some winters the cloudiness fluctuations were accompanied by fluctuations in the breadth of jet streams far to the north over East Asia, the North P a c i f i c , and North America. John Anderson of the University of I l l i n o i s and Richard Rosen (1983) of Atmospheric and Environmental Research Inc., of Cambridge, Massachusetts, also found the association of t r o p i c a l and mid-latitude o s c i l l a t i o n s by demonstrating that they work together through their variations in the zonal wind to slow and speed up the rotation of the earth by a few tenths of a millisecond." Thus, a picture emerges of a t r o p i c a l o s c i l l a t i o n (equatorial c i r c u l a t i o n c e l l s ) with poleward propagating branches (as evidenced by Yasunari in the cloudiness fluctuations) which disturb mid-latitude weather systems ( l i k e the jet streams). The 21 tr o p i c a l component with the poleward propagation has been modelled by Chang (1977) as a damped equatorial Kelvin wave in which the phase speed c depends on the damping factor D. This model explains the poleward phase propagation in the tr o p i c s . This i s because the Gaussian trapping factor for equatorial Kelvin waves becomes complex when damping i s included. Equation (18) of Chang (1977) takes the form The phase propagation is quantitatively in agreement with the observations of Madden and Julian (1972a) (see F i g . 7 in Chang (1977)). For the phase speed of the 40- to 50-day waves (^10 m/s), one finds y c = l000 km, so that 20° from the equator (^2000 km) the wave amplitude i s reduced by a factor e-"=50. Thus the tr o p i c a l component c l e a r l y cannot account for the large amplitudes observed by Weickmann far north of the equator. Various explanations have been advanced for the o r i g i n of the 40- to 50-day o s c i l l a t i o n . Yasunari (1981) has explored the idea of a tr o p i c a l - s u b t r o p i c a l connection at a 40-day period. Yasunari's observations centered on the Monsoon active-break cycle. Active periods (heavy rain over India) alternate with break periods (less rain) at periods of about 40 days (as noted e a r l i e r , t h i s pattern i s p a r t i c u l a r l y clear in cloud ( C = C--+ i C i ) where ' 'o=o 22 observations). Yasunari notes that cloud variations over the equator occur in phase with cloud variations in the Southern Hemisphere (where a near-standing o s c i l l a t i o n i s dominant from the equator to 30°S). He suggests that the formation of major equatorial cloud disturbances over the Indian Ocean i s clos e l y linked with mid-latitude westerly disturbances and/or fronts in the Southern Hemisphere. The equatorial disturbances are in turn linked to the Indian Monsoon active-break cycles (see Fig.II.7). 30 N " 20 " \a" EO \S 20 MS JOH 20 10 EO 10 70 aos 30N 30 10 EO 10 • ao 90S Cold a i r outbreak over Southern Hemisphere y i e l d s equatorial disturbances formed at the head of the cold a i r mass. India i s dominated by a break condition. Southern Hemisphere c e l l approaching India, equatorial disturbance moving eastward. Maximum r a i n f a l l over India, Southern Hemisphere c i r c u l a t i o n changed. Break condition over India once more- cycle s t a r t s over with outbreak from the Southern Hemisphere. F i g . II.7. Schematic model of the 40- to 50-day period o s c i l l a t i o n of the anomalous Hadley c i r c u l a t i o n . The maximum stage of the anomalous wind directions are indicated in the c i r c l e s . R elatively warm areas are shaded. (From Yasunari, 1981). 23 An explanation emphasizing air-sea interaction has been offered by Goswami and Shukla (1984). Their model studies of the global atmosphere indicate that the Hadley c e l l may develop o s c i l l a t i o n s via i t s interaction with the sea. Air in the low-l e v e l , return flow toward the equator accumulates moisture from the ocean, which condenses and helps to drive the c e l l ' s c i r c u l a t i o n . The faster the a i r returns, the more moisture i t w i l l accumulate, further enhancing convection - but only up to a point. The returning a i r eventually moves through the return leg too quickly to pick up a f u l l load of moisture, with a consequent diminution of convection. The model of thi s feedback process predicted the Hadley c e l l to o s c i l l a t e with a period of between 20 and 40 days. Anderson (1984) also believes the Hadley c e l l to be the source of this o s c i l l a t i o n , but provides a model, predicting o s c i l l a t i o n s of t h i s c e l l , which does not require the inclusion of moisture and condensation. In this model, the c e l l has a natural period between 30 and 60 days determined by the time i t takes the c e l l ' s winds to carry the momentum added by a disturbance around the c i r c u i t . Neither Hadley c e l l mechanism explains why the disturbance propagates to the east. It i s possible that Lindzen's CISK model (presented in the preceeding section; see Fig.II.5) might account for the eastward propagation of 40- to 50-day fluctuations i n i t i a t e d by "ringing" of the Hadley c e l l at i t s natural frequency. I now conclude t h i s section with a discussion of Murakami, Nakasawa, and He's (1984) analysis of the 40- to 50-day 24 o s c i l l a t i o n , which w i l l prove to be very important to the considerations of Chapter IV. In Fig.II.8 one sees the results of their analysis of phase propagation of the 40- to 50-day Fi g . II.8. Constant phase contours (in degrees) for the zonal v e l o c i t y perturbation at 200 mb and 850 mb in the Indian and western P a c i f i c Oceans (From Murakami, Nakazawa, and He, 1984). The l i n e AB has been drawn for purposes of wavelength estimation (see Fig.II.9). zonal v e l o c i t y anomaly over the . Indian Ocean. The important result for the work here is that the 850 mb perturbations propagate -roughly p a r a l l e l to the African coast in the Western Indian Ocean, and that the (northward) propagation starts well south of the equator. In Fig.II.9 I have plotted the phases 25 shown in Fig.II.8 along l i n e AB, as a function of distance along l i n e AB, to establish the wavelengths of the perturbation. The estimated slope of the l i n e gives a value of A =3200 km; I w i l l present an o r i g i n a l analysis of data from the Western Indian Ocean in Chapter III which supports th i s wavelength estimate. 26 PHASE DIFFERENCE Fig.II.9. Phase difference versus distance along the l i n e AB in Fig.II.8. 27 II.4 Oceanographic Background I noted in Section II.2 that much of the t r o p i c a l atmospheric c i r c u l a t i o n i s driven by diabatic heating provided by land-sea temperature contrasts. These contrasts reverse seasonally and give r i s e to monsoon v a r i a b i l i t y . The Indian Ocean, being completely bounded to the west and north by land, exhibits perhaps the most dramatic seasonal variations of any ocean. Accordingly, I w i l l concentrate in thi s section on the Indian Ocean [the material of t h i s section also provides background for the discussion of the Somali Current region given in chapter III ]. Regarding these monsoonal changes, Diiing (1970) has noted: " Since c l a s s i c a l times i t has been recognized that the winds over the northern Indian Ocean reverse semiannually. It i s a well known fact that the monsoons were used to great advantage by the Greek seafarers in the f i r s t centuries A.D. in carrying out their extensive Arabian Sea trade with India. From Arabic documents of the medieval period, however, we know that the semiannual reversal of the surface currents was discovered only in the ninth or tenth century... ...The pe r i o d i c a l reversal of the wind and of the surface c i r c u l a t i o n over such an extended area is outstanding when compared to that of the A t l a n t i c Ocean or the P a c i f i c Ocean. The large-scale 28 c i r c u l a t i o n in the northern part of the Indian Ocean (north of 20°S) has an e s s e n t i a l l y nonstationary character, whereas the large-scale c i r c u l a t i o n in the great A t l a n t i c and Pa c i f i c gyres shows an e s s e n t i a l l y stationary behavior." The large-scale, seasonal changes are i l l u s t r a t e d in Fig.II.10. The maps marked February t y p i f y the Northeast Monsoon. During t h i s period (Northern Hemisphere winter) the Northeast Monsoon winds traverse the equator to about 3°S, followed by a belt of doldrums to about 10°S, Further south, the Southeast Trade winds blow quite steadily throughout the entire year. This wind system causes a zonal current system, consisting of the North East Monsoon Current north of about 3°S, an Equatorial Counter Current between 3°S and 10°S and the South Equatorial Current south of 10°S. Water carr i e d westward by the NE Monsoon Current flows southward along Somalia, and subsequently into the Equatorial Countercurrent located in the doldrum region. West of the northern t i p of Madagascar, the South Equatorial Current branches into the southward-flowing Mozambique Current and into the northward flowing East African Coast Current. Near 2°30'S the East African Coast Current converges with the southward flowing (weak) Somali Current. This convergence zone constitutes the root of the equatorial Counter-current. This zone s h i f t s from year to year; southward flow has been seen as far south as Zanzibar. The maps marked August typify the Southwest Monsoon. Under these (Northern Hemisphere) summer conditions the belt of 29 A « - AgJhas Current So * Somali Current MM • Northeast Monsoon Current SE • South Equatorial Current Mo * Mozambique Current SM • Souffjwesr Monsoon Ctri-ronr CC • Equatorial Counter Current CA • £asf African Coast Current CONSTANCY very constant »• fe/rty conalanf — • variable SPEED »• %-Mfcnors *• 1 - 1 % ->2 Fig.II.10. Surface winds (from Diiing, 1970) and surface currents (from Duing and Schott, 1978) of the monsoon regime of the Indian Ocean. doldrums s h i f t s to the north, and north of the equator the winds have reversed against the winter si t u a t i o n toward the northeast. 30 The Southeast Trades have strengthened and penetrated into eastern A f r i c a as a strong narrow jet which, blocked by high ground at i t s western boundary (the East African Highlands), curves into a southerly d i r e c t i o n , across the equator, and then to a southerly jet which crosses the Arabian Sea flowing toward India. This i s the Somali jet (or the East African j e t ) . The oceanic current system now consists of the South West Monsoon Current down to about 8°S, and south of i t again the South Equatorial Current. There i s no Equatorial Counter Current during t h i s season. The South Equatorial Current i s stronger owing to the strengthening of the Southeast Trade winds. During both, seasonal t r a n s i t i o n s (May and October) a narrow eastward equatorial jet develops. This jet i s generated by westerly winds blowing in the equatorial region during the t r a n s i t i o n periods. This i s the only surface current in any ocean which flows eastward at the equator. The reversal (onset) of the Somali Current south of the equator occurs about one month before the onset of the SW Monsoon over the i n t e r i o r of the northern Indian Ocean. Leetmaa (1972,1973) argues that this southerly onset i s caused by l o c a l winds and a switching action of the East African Coast Current. Diiing and Schott (1978) show that the Somali Current south of the equator sets in as a rather shallow, intense, northeastward flow extending only gradually to below the thermocline. It appears that northward flow s t a r t s at any latitude along the Somali Coast a few days after the l o c a l wind -" possesses a southerly component. Numerical modelling by Cox (1976) suggests that after the i n i t i a l , l o c a l response the 31 a r r i v a l of oceanic Rossby waves, generated by the large-scale (monsoon) wind s h i f t throughout the Indian Ocean, enhances the current buildup (the L i g h t h i l l (1969) mechanism). 32 II.5 Properties and Origin of the 40- to 50-day Oceanic O s c i l l a t i o n There has been, to date, considerably less observational work done on the oceanic 40- to 50-day o s c i l l a t i o n compared to that on i t s atmospheric counterpart. Only recently has the link between the oceanic and atmospheric o s c i l l a t i o n s been elucidated. Erikson, Blumenthal, Hayes, and Ripa (1983) have shown that the oceanic 40- to 50-day o s c i l l a t i o n in the P a c i f i c appears to be generated by zonal wind fluctuations with this period range. In this section I w i l l b r i e f l y review the results of previous observational programs concerned with the oceanic 40- to 50-day o s c i l l a t i o n . There is l i t t l e evidence for the existence of the 40- to 50-day o s c i l l a t i o n in the A t l a n t i c Ocean. Weisberg, Horigan, and Colin (1979) found a s i g n i f i c a n t but rather broad-banded peak centered on 31 days in ve l o c i t y and temperature spectra from data acquired in the Gulf of Guinea [00°01.1'N, 04°16.0'W]. These fluctuations are, however, probably generated by surface current i n s t a b i l i t i e s (e.g., see Philander, 1976,1978). There are, perhaps, two reasons for th i s lack of evidence: (i) The atmospheri'c 40- to 50-day o s c i l l a t i o n i s apparently weak over the A t l a n t i c Ocean; ( i i ) Most of the observational programs concentrating on the equatorial A t l a n t i c (such as the GATE3 experiments) 3 GATE i s the GARP At l a n t i c Tropical Experiment, where GARP i s the Global Atmospheric Research Program. 33 were of r e l a t i v e l y short duration and generally concerned with higher frequency motions. The existence of the 40- to 50-day o s c i l l a t i o n in the P a c i f i c Ocean has been established by Luther (1980). In his Ph.D. thesis, Luther discusses the oceanic 40- to 50-day o s c i l l a t i o n in some d e t a i l and finds: (i) The atmospheric o s c i l l a t i o n exists as an eastward propagating wave of planetary wavenumber 1 to 2. ( i i ) The oceanic o s c i l l a t i o n in the t r o p i c a l P a c i f i c Ocean i s eastward propagating with a wavelength of about 11,000 km. ( i i i ) The 40- to 50-day sea l e v e l fluctuations from stations within 3° of the equator are not coherent with 40- to 50-day sea le v e l fluctuations outside the equatorial waveguide. (iv) Points ( i i ) and ( i i i ) suggest that the oceanic o s c i l l a t i o n i s an equatorial Kelvin wave since the only other eastward propagating modes (i n e r t i a - g r a v i t y waves) would have sea le v e l displacements of about 2xl0-"m and would thus be undetectable. (v) Luther finds the equivalent depth (hft) for this wave to correspond to the equivalent depth for the f i r s t b a r o c l i n i c mode (n=l) in the equatorial P a c i f i c (h l = .71m implying a phase speed of ^h~A = 2.68 m/s). (vi) The rms amplitude of the sea le v e l deflection (^cm) gives a corresponding rms surface zonal current 34 of ~8cm/s. There are observations of the 40- to 50-day o s c i l l a t i o n in the Indian Ocean which may be consistent with Kelvin wave propagation. McPhaden (1981) discusses wind stress, ocean temperature and velocity time series from the island of Gan (00°41'S, 73°10'E) in the Equatorial Indian Ocean for the period January 1973 - May 1975. McPhaden concludes that "zonal winds at periods 30-60 days are highly coherent with zonal currents down to 100 m" and that "these coherence and phase structures may indicate the presence of an atmospherically forced oceanic Kelvin wave similar to that described by Luther (1980)". The power spectra of the oceanic v e l o c i t y measurements are shown in Fig.II.11. Only the eastward (zonal) v e l o c i t y component at 0-20m depth shows evidence of a spectral peak in the 6-12 cpy. (cycles per year) band (30 to 60 days period). This feature i s consistent with the notion of an oceanic Kelvin wave being responsible for the velocity fluctuations ( i . e . , the meridional component i s very small). The power in t h i s band (at 0-20m depth) i s 10 2cm 2s _ 2cpy - 1xbandwidth =10 2cm 2s' 2cpy" 1x6cpy =600cm2s"2. The corresponding velocity amplitude in t h i s 30- to 60-day band i s J600cm 2s _ 2 = 25cms _ 1. The implications of t h i s large v e l o c i t y w i l l be discussed l a t e r . Luyten and Roemmich (1982) made current measurements in the western equatorial Indian Ocean from A p r i l 1979 to June 1980. 35 FREQUENCY (cpy) Fig.11.11. Power spectra of (a) v e l o c i t y , and (b) meridional vel o c i t y , measured on the Island of Gan. (From McPhaden, 1982). Results of these current measurements are summarized in the spectra of Fig.II.12. Referring to t h i s figure, one can calculate the power in the 50-day band at the 200 meter depth. This power for both east (Fig.II.12a) and north (Fig.II.12c) spectra i s 200cm2s" 2x Ain/u, (AOL> i s the bandwidth) = 200cm2s~ 2x3x10" 7s" 1/1.2x10' *s~ 1 = 60cm 2s" 2. The v e l o c i t y amplitude corresponding to this power is J60cm 2s~ 2 Bcm/s. Note that in t h i s case the north and east components of the v e l o c i t y are approximately equal , implying that the fluctuations cannot be of a Kelvin wave nature. Reference to 36 EAST 200 M 5 PIECES i " " 1 ' 1 '—r 1 0 0 0 0 . 1000. 100. 10. 1. PERIOD (DAYS) Fig.II.12. Power spectra from the Indian Ocean array for: (a) zonal v e l o c i t y at 200m, (b) zonal v e l o c i t y at 750 m, (c) meridional velocity at 200m. (From Luyten and Roemmich, 1982). Fig.II.4 shows that the fluctuation observed by Luyten and Roemmich (1982) could s t i l l be eastward propagating i f i t i s a Yanai wave ("0" mode in Fig.II.4. Near the equator, a Yanai wave 37 can have a zonal v e l o c i t y component only i f k=ou , that i s , the dispersion r e l a t i o n i s approximately that for a Kelvin wave. Perhaps the most remarkable observation of the oceanic 40-to 50-day o s c i l l a t i o n i s that due to Quadfasel and Swallow (1984). Their investigation, consisting of current meter measurements during 1975 (see Fig.II.13) and s a t e l l i t e tracking of buoys during 1976 and 1977 (see Fig.II.14). Their 1975 measurements reveal fluctuations with a 50-day period having an amplitude of 20cm/s at depths down to 500m (si t e "188" in Fig.II.13) and surface values of 45cm/s ! By way of comparison, the strongest current fluctuations seen in the 1976 Somali Current program were about 10cm/s at a depth of 125m (see Chapter I I I ) , at least a factor of two weaker than the currents observed by Quadfasel and Swallow. In addition to these 1975 observations, the authors have found 50-day v a r i a b i l i t y of the South Equatorial Current in the tracks of s a t e l l i t e - t r a c k e d d r i f t e r s in the i n t e r i o r of the Indian Ocean (see Fig.II.14) during the 1976-77 program. Evidence i s presented to show that the fluctuations exhibit westward wavelike propagation (with X =400km in the 1975 experiment, X =1100km in the 1976-77 observations). The detection of these strong o s c i l l a t i o n s i s p o t e n t i a l l y very s i g n i f i c a n t in that: (i) The westward propagating waves, upon running into the coast, could leak energy into the coastal waveguide, and "these (shear and topographic) modes could be responsible for the fluctuations in the Somali Current observations discussed in Chapter I I I . 38 Fig.II.13. The area of Quadfasel's and Swallow's 1975 observations. Sites 188 and 189 are positions of moored current meters. (Redrawn from Quadfasel and Swallow, 1984). ( i i ) Such strong current o s c i l l a t i o n s at 50-days have only been observed in the Southern Hemisphere. This could be indir e c t evidence for the Southern Hemisphere ori g i n of the atmospheric 40- to 50-day o s c i l l a t i o n s , discussed in section II (the idea of "cold surges" in the Southern Hemisphere modulating the t r o p i c a l c i r c u l a t i o n ) . 39 F i g . I I . 14. Meandering of s a t e l l i t e - t r a c k e d buoys in the South Equatorial Current during 1976 and 1977. The wavelength of the meanders (1000 to 1300km) corresponds to about a ~ 50-day o s c i l l a t i o n at the average d r i f t speed of the buoys. (From Quadfasel and Swallow ,1984). A possible explanation for the strength of the noted current fluctuations l i e s in the material presented in the meteorological observations discussed in Chapter II. It was noted that the t r o p i c a l component of the 40- to 50-day o s c i l l a t i o n can be modelled as an equatorial Kelvin wave (Madden and J u l i a n , 1971; Chang, 1977). That i s , the zonal v e l o c i t y of the wind (u) s a t i s f i e s (II.1 ) where y e i s the width of the gaussian envelope estimated by Chang to be about 1000km ( S e c t i o n d l . 3 ) ) . Thus the zonal wind stress i s given by 40 = 7r 0 e x P [ : 2 ( f o n (C^ i s a drag c o e f f i c i e n t , T 0 - j3*/*? c O u o ) Thus the wind stress c u r l i s given by ''a 3c = " r ^ " {no meridional component of C ) .ay ~ (II.2) (II.3) where (curl C ) = 4 t o / y a Note that r e l a t i v e l y weak (I0cm/s) curent fluctuations are observed off the Kenyan Coast at y=ys=250km (see Chapter I I I ) . Denoting the strength of the wind stress c u r l at t h i s point by C s , I find , >toM ^ (II .4) Since « * p C - a ( % n = ^ p f r ^ H = 1' o n e h a s = 2J?C S (II.5) A r b i t r a r i l y setting Cs=1, I now plot the t h e o r e t i c a l wind stress c u r l (Equation (II.3)) as a function of l a t i t u d e in Fig.II.15a. 41 CURLJT (Ci) y 0 = 1 0 0 0 k m Fig.II.15. The theoretical wind stress c u r l specified by e g u a t i o n d l .3) for (a) yo=l000km, and (b) yo=2000km. Note that the peak wind stress c u r l value occurs at y/y o=0«5 (about 5°S) and at th i s peak the c u r l i s more than twice as 42 strong as i t i s at the point of observation ( y s ) . Further, i f I take y 0 =2000km (possibly not inconsistent with the rough estimate of y e provided by Chang), then and F i g . I I . 15b re s u l t s . Now, the peak occurs closer to y/y0=1 (about 10°S) with a strength of over 5 times the value off Kenya. Wind stress c u r l strengths of this magnitude might explain the s t r i k i n g l y large oceanic current fluctuations (up to 45cm/s) observed by Quadfasel and Swallow. Further, with yo=2000km, the belt of peak wind stress c u r l l i e s at about the same lati t u d e as the strong current o s c i l l a t i o n s observed by Quadfasel and Swallow. In the next chapter I w i l l examine some equatorial measurements which support the idea that an atmospherically forced equatorial Kelvin wave gives r i s e to observed current and sea-level fluctuations. The extremely energetic disturbances seen by Quadfasel and Swallow could be the ultimate source of the equatorial o s c i l l a t i o n via a leakage from the westward propagating South Equatorial Current o s c i l l a t i o n s into the coastal waveguide, from where the coastal o s c i l l a t i o n s leak into the equatorial waveguide. I w i l l discuss this notion in more d e t a i l l a t e r . Another set of observations which, however, cannot be explained by an equatorial Kelvin wave model are those due to (II.6) 43 Hayes (1979). Hayes observed benthic currents in the t r o p i c a l north P a c i f i c Ocean (see Fig..II. 16). He found that current meters 30m above the bottom (at depths of about 2500 fathoms) I B O ' 120* 7<t II I \ a_ •c \ \ ^ A* B I I I 1 2 0 - 1 0 0 ' BO* 8°27'N, 1 5 0 ' 4 9 ' W 11°42'N, 138°24'W 1 4 « 3 8 ' N , 1 2 5 ' 2 9 ' W Fig.II.16. Locations of moored current meters in the t r o p i c a l P a c i f i c Ocean with nearby bathymetry. Contours are in fathoms. (From Hayes, 1 979) . detected a near 50-day o s c i l l a t i o n strong enough that "...low frequency fluctuations dominated a l l the records. These fluctuations had dominant periods for the meridional component of two months at a l l s i t e s . . . " . S p e c i f i c a l l y , r e l a t i v e l y l i t t l e 50-day energy was found at s i t e A, while 50-day fluctuations of 2 to 3cm/s amplitude were found in the meridional component of 44 the v e l o c i t y at sit e s B, C, with the zonal ve l o c i t y component at 50-days very weak. Since the 50-day fluctuations are found in mid-ocean well outside the equatorial waveguide (which Luther estimates to be about 3° wide) dynamics other than Kelvin wave must be invoked to explain these observations. I w i l l return to this point in Chapter IV. 45 I I I . The 40- to 50-day O s c i l l a t i o n in the Source Region  of the Somali Current during 1976 and 1979 III.1 Introduction In this chapter the emphasis changes from global-scale to an examination of the oceanic and atmospheric 40- to 50-day o s c i l l a t i o n s near the source region of the Somali Current (2°S, 42°E). This region has been of longstanding interest to oceanographers due to the annual reversal of the Somali Current (and thus considerable amounts of data have been gathered there). From the point of view of a study of the 40- to 50-day o s c i l l a t i o n , t h i s region i s also of great i n t e r e s t . For example, the Western Indian Ocean i s a region of strong land-sea contrast, which i s important in dr i v i n g monsoons and other convective phenomena. The Indian Ocean and/or the Western P a c i f i c Ocean are thought to be the o r i g i n of the 40- to 50-day o s c i l l a t i o n , so i t i s important to determine what happens to the o s c i l l a t i o n on the western boundary of the Indian Ocean. A large mountain range which l i e s near the east coast of A f r i c a (the East African Highlands) is known to profoundly a f f e c t the general atmospheric c i r c u l a t i o n of the region and l i k e l y also a f f e c t s the atmospheric 40- to 50-day o s c i l l a t i o n in this region. The coast i t s e l f must modify the propagation of any oceanic 40- to 50-day fluctuations here, making an analysis of oceanic data from th i s region d i s t i n c t from the e a r l i e r work. In previous studies, the 40- to 50-day o s c i l l a t i o n has been examined in "unbounded", mid-ocean equatorial regions. 47 III.2 The 40- to 50-day O s c i l l a t i o n  over the Western Indian Ocean  during 1976 and 1979 In this section I w i l l present new evidence of the 40- to 50-day atmospheric o s c i l l a t i o n . This evidence comes from an analysis of data contained in Fernandez-Partagas and Diiing (1977), and Fernandez-Partagas, Samuels and Schott (1980). The 1977 report contains one hundred three-day surface wind maps, as shown in F i g . I I I . 1, extending from January to October 1 976. These wind maps were compiled primarily from wind reports from ships (on record at the U.S. National Climatic Center, Asheville, North Carolina), and were supplemented with coastal and island station reports. The Western Indian Ocean i s best covered, due to the presence of tanker lanes off the African Coast. The maps were compiled to aid in the understanding of the oceanic Monsoon processes, as represented by data accumulated in various INDEX studies. The analysis presented in t h i s thesis i s motivated by the observation of the 40- to 50-day o s c i l l a t i o n s in the Somali Current in i t s source region (3°S,41°E) during 1976. Accordingly, Southern Hemisphere wind data were analyzed on the grid (points A,B,C,D,E) shown in Fig.III.1. The 1980 report contains ninety three-day wind stress maps (covering January to September 1979) which were analyzed at d i f f e r e n t s p a t i a l locations (see points A',B',....H' in Fig.III.2). In both cases I w i l l analyze the zonal and meridional components of the wind stress. The spectra are calculated via the maximum 48 Fig.III.1. Sample surface wind map from Fernandez-Partagas and Diiing ( 1 977). Isotachs are in knots. l i k e l i h o o d technique, as i s discussed in Appendix I. The grid (A,B,C,D,E) in Fig.III.1 was used to calculate the zonal and meridional components of the wind stress at each of the points A, B, C, D, E and the c u r l of the wind stress (and 49 6. L a + + u d 2. e O O 1979 8^53 278 e +;* •a o> •3 <n _ . * (Dynes»c«t* -2) / « - «? V V 05 yC/*> «,(> «? E •G' "FT m ' « NO ^  O. * •> a D i i . 46. *B. 52. 54. 54. 5e. Fig.III.2. Sample surface wind stress map from Fernandez-Partagas, Samuels and Schott (1980). Arrows indicate the wind stress at each point on the 2° g r i d ; the wind stress magnitudes are in dynes/cm 2, the directions (forward) are given in degrees from true north. i t s spectrum) at the point E. The wind stress c u r l c a l c u l a t i o n i s based on a f i n i t e difference approximation on the grid A,B,C,D,E s p e c i f i e d by the formula c u r f 3 - z S D AC 50 where ^ i s the x-component of the wind stress at point A, etc., £ * B D i s the separation between points B and D, and i s the separation between the points A and G. Time series of the zonal and meridional stress components for the point E are shown in Fig.III.3a,b. The long period trends (seasonal or Monsoonal effects) are removed via a s p l i n e - f i t routine, and the resulting de-trended time series are analyzed. The corresponding spectra are shown in Fig.III.4a,b. The only major peak in the zonal wind stress i s at 60 days, which roughly corresponds to the 40- to 50-day o s c i l l a t i o n found by Madden and Julian (1971, 1972a). As expected there i s no s i g n i f i c a n t 60-day energy in the spectrum of the meridional component of the wind stress. I find the same results at each of the points A, B, C, D; that i s , a strong peak near 60 days for the zonal component with l i t t l e or no energy at t h i s time scale in the meridional component of the wind stress. The wind stress c u r l also shows a strong peak in t h i s same frequency regime (Fig.III.5). Madden and Julian (1971) present evidence of a modulation in the amplitude of the 40- to 50-day o s c i l l a t i o n with a noticeable increase in amplitude (in Fig.III.6) in January to A p r i l of the years 1961, 1962, and 1964. I have investigated this p o s s i b i l i t y by low passing the zonal stress and stress c u r l records (with a cutoff period of 25 days). The r e s u l t s , shown in Fig.III.7, do indicate the presence of a modulation; t h i s i s p a r t i c u l a r l y evident in the low-passed wind stress c u r l record during days 100 to 200. I have also analyzed the 1979 data, i l l u s t r a t e d in Fig.III.2. I n i t i a l l y , I analysed data on the gr i d A', B', C , D' (Oi) Zonal Component 51 100 150 200 TIME (DAYS) (£>) Meridional Component 300 100 150 200 TIME (DAYS) 300 Fig.III.3. Components of the wind stress f i e l d at point E. Dashed lines represent a s p l i n e - f i t to the mean stress f i e l d . (of Fig.III.2), in a similar s p a t i a l region as the grid in Fig.III.1. However, i t proves d i f f i c u l t to obtain good zonal 52 o <M o'n i r i i i i i i 0.0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 16 FREQUENCY (CPD) Fig.III.4 Autospectra of the components of the wind stress f i e l d at point E. The spectral density is in units of (dynes/cm 2) 2/cpd. For a discussion of confidence l i m i t s , see Appendix I. spectra in t h i s region, because the wind blow i s almost meridional during large parts of the record, and when the wind 53 CD CD _ 0.0 0.02 0 .04 0.06 0 .08 0.1 0. 12 0 .14 0.16 FREQUENCY (CPD) Fig.III.5. Autospectrum of the c u r l of the wind stress calculated about point E. The spectral density i s in units of 10" 1 4(N/m 3) 2/cpd. is near north, there seems to be a tendency to round out the di r e c t i o n to zero degrees (due north). That i s , angles of zero degrees are over-represented in the record, re s u l t i n g in large " f l a t spots" in the zonal wind stress record. Good zonal records can, however, be obtained in more northern regions, where the wind almost always has a strong zonal component. Accordingly, I analyzed the stress components at the points E', F', G', H' of Fig.III.2 (which are s t i l l r e l a t i v e l y near the equator). The res u l t i n g spectra are shown in Fig.Ill.8a,b,c,d for points E' and H'. The 40- to 50-day o s c i l l a t i o n now appears somewhat more broad-banded, but i t s t i l l dominates the zonal spectra. 54 •I960 Fig.III.6. Zonal wind and station pressure from Canton Island treated with a 47-day band pass f i l t e r . (From Madden and Julian, 1971). Surprisingly, I f i n d a strong meridional component of the stress at point E'; there i s l i t t l e or no meridional energy evident at point H'. In fact, I find that the meridional component generally weakens in comparison to the zonal component as one moves eastward from station to station along the l i n e E'-H'. This may indicate a deflection or d i s t o r t i o n of the c e l l s near the East African Highlands, which are known to af f e c t profoundly the general atmospheric c i r c u l a t i o n in t h i s region. In F i g . I l l . 8 ( d ) , one sees a strong peak centered on 11 days period. This may be the o s c i l l a t i o n described by Krishnamurti et a l . (1985) as "a 10 to 20 day westward propagating wave.... noted to influence monsoon a c t i v i t y " . F i n a l l y , I examine the low-passed record of the zonal wind stress component from the point H', in Fig.III.9. I note that there i s again an amplitude modulation during days 100 to 200. In conclusion, the 40- to 50-day o s c i l l a t i o n s discovered by 55 Fig.III.7. (a) Low passed wind stress record from point E. (b) Low passed wind stress c u r l record (in units of 10'8 N/m3). 56 0.0 0 .02 0 . 0 4 0 .06 0 .08 0.1 0 .12 0 . 1 4 0 16 FREQUENCY (CPD) 0.0 0 .02 0 . 0 4 0 .06 0 .08 0.1 0 .12 0 . 1 4 0 .16 FRFQIIFNCY f CPD 1 Fig.III.8. (a) Autospectrum of the zonal component of the wind stress at point E'. The spectral density i s in units of (dynes/cm 2) 2/cpd. (b) As in (a), except for the meridional component. 57 Fig.III.8 (cont.) (c) Autospectrum of the zonal component of the wind stress at point H'. Units as in (a). (d) As in (c), except for the meridional component. CD 0 50 100 150 200 250 300 TIME (DAYS) Fig.III.9. Low passed zonal wind stress record from point H'. Madden and Julian (primarily in atmospheric data from the equatorial P a c i f i c Ocean region for years from 1957 to 1967) were also present in the zonal wind stresses over the Western Indian Ocean in the years 1976 and 1979. 59 III.3 Fluctuations of the Western Indian Ocean The world's oceans exhibit fluctuations of their characterizing f i e l d s on a wide variety of time scales. One of the most s i g n i f i c a n t signals i s that on the interannual time scale. The variations so described tend to be quasiperiodic in nature. For example, the Southern O s c i l l a t i o n - E l Nino phenomenon i s marked by unusually high sea-surface temperatures and r a i n f a l l and unusually weak tradewinds over the t r o p i c a l P a c i f i c Ocean, recurring at 2 to 10 year i n t e r v a l s . Annual variations of oceanic f i e l d s are prominent in Monsoonal regions, p a r t i c u l a r l y the Indian Ocean. Fluctuations at shorter time scales are often described by r e l a t i o n to the l o c a l i n e r t i a l frequency. At periods longer than i n e r t i a l , one has s u b i n e r t i a l fluctuations, which often propagate as waves of the second class (depending on rotation for their existence). At periods shorter than i n e r t i a l , one has s u p e r i n e r t i a l fluctuations, which often propagate as waves of the f i r s t class (not dependent upon rotation for their existence). In t h i s chapter I w i l l analyze variations of western Indian Ocean f i e l d s on annual, s u b i n e r t i a l and s u p e r i n e r t i a l timescales, with emphasis on the 40- to 50-day (subinertial) fluctuations. The discussion of the annual variations w i l l be general, largely based on the oceanographic l i t e r a t u r e . The analysis of the s u b i n e r t i a l and s u p e r i n e r t i a l fluctuations w i l l be based on the observations of Diiing and Schott ( 1978) in the source region of the Somali Current. Diiing and Schott ( 1978) made extensive current and 60 temperature measurements in a region centered at 3°S and 41°E (see Fig.III.10) and extending over a six-month period (January-July, 1976) which spans the springtime monsoon reversal. During t h i s period the large-scale (monsoon) winds change from northeast to southwest and the intense northward flowing Somali Current is formed (see Fig.III.12 below). The data used in this study were obtained from the southern moorings K1 and K2 (Fig.111.10); the mooring d e t a i l s are given in Table 1. A summary of the current measurements from these two moorings i s given in Fig.III.11 in the form of stick diagrams, which were obtained from d a i l y averages. The stations K1 and K2 represent quite d i f f e r e n t regimes for three reasons: 1) The bathymetry changes considerably in going northward from K1 to K2. For example, Fig.(III.10) reveals a much greater bottom slope at K2. 2) The C o r i o l i s parameter at K1 i s about 50% larger than at K2. 3) The mean flow conditions at the two stations d i f f e r considerably (see Fig.III.11). Station K1 i s located at the northern end of the northward-flowing East African Coast Current, which is r e l a t i v e l y steady throughout the six-month observation period. At K2 on the other hand, the current i s weak and variable (that i s , the fluctuations are comparable to the mean) up to the middle of A p r i l . After t h i s , in response to the Monsoon reversal, the near-surface, northward-flowing Table 1 De ta i l s about moorings and instrument performance (from Dliing and Schott , 1978) Mooring i d e n t i f i c a t i o n Pos i t i on and water depth Mooring Deployed Recovered (1976) Instrument depths (m) Pressure Mean Min/Max. Yes/No Malfunct ions Kl 0 = 3 ° 5 8 . 1 ' S X = 4 0 ° 2 0 . 0 ' E z = 885m Jan 11 June 21 180 235 445 159/264 yes no no ro tor l o s t on March 13 none vane broken on May 1 K2 0 = 2 ° 4 6 . 5 * S X = 4 1 ° 1 . 3 * E z = 286m Jan 11 Ju ly 8 125 176 261 124/128 yes yes no none none none 62 Fig.III.10. Mooring array off Kenya from mid-January to mid-July, 1 976 (from Diiing and Schott, 1978). Somali Current is formed (see Fig.III.12). Apparent in the stick diagrams of Fig.III.11 are fluctuations with time scales shorter than the seasonal variations. Autospectra of the current fluctuations show that most of the—^var-i-a-nce- i s at s u b i n e r t i a l frequencies, with d i s t i n c t i v e peaks at around 50 and 17 days. The analysis of the wind stress data contained in Fernandez-Partagas and Diiing 63 Fig.III.11. Vector time ser i e s of currents at locations K2 and K1. Vectors point in d i r e c t i o n of flow with north upward. Average depth and standard deviation of i n d i v i d a l sensors are indicated. For cases where the speed i s missing an asterisk indicates that the values are obtained from regression with speed record at l e v e l below (from Diiing and Schott, 1 978). (1977) indicates that the 50-day oceanic signal i s probably forced by wind flu c t u a t i o n s of t h i s time scale (see F i g . ( I l l . 1 ) ) . 64 20° E 40° 60° 20° E 40° 60° FEBRUAt WIND \ ^ N E Or • — » • SE^ J ' / It ^^^^ AUGUST V/lNO i MS Ae/<<' —<* . mi — • — * 20° N r s 20° 40° 20° 40° 60° 20° 40° 60° Ag • Agulhas Current wt • Westerlies - - - — So " Somali Current NM • Northeast Monsoon Current SE • South Equatorial Current Mo • Mozambique Current SM • Southwest Monsoon Current EC • Equatorial Counter Current EA - East African Coast Current CONSTANCY > very constant *• fairly constant — -*• variable SPEED 1/4 -3/4 knots 1-11/2 -> 2 Fig.III.12. Schematic d i s t r i b u t i o n of surface currents in the western Indian Ocean during both monsoons and related nomenclature (from Diiing and Schott, 1978). Fluctuations in the Somali Current During 1976 In t h i s section I w i l l describe the fluctuations referred to in the introduction to this chapter (see Fig.III.11). I am primarily concerned with the 40- to 50-day fluctuations, but w i l l b r i e f l y discuss the s u b i n e r t i a l 17-day o s c i l l a t i o n as well as the 3- to 5-day (subinertial) fluctuations. The discussion to follow w i l l be based on an analysis that employs f i l t e r i n g and spectral techniques. The f i l t e r i n g is accomplished in a conventional manner, v i z . , using a Butterworth f i l t e r (see e.g. Kanasewich, 1977). Because the records are short, maximum 65 l i k e l i h o o d has been used in the spectral analysis; the merits of t h i s technique are discussed in Appendix I. Description of the Data The current and temperature data were acquired j o i n t l y by the U n i v e r s i t i e s of K i e l and Miami, as part of the INDEX p i l o t studies. A description of the (1976) experiment is found in Duing and Schott (1978). The current and temperature data from moorings K1 and K2 supplied to the author were h a l f - d a i l y means which were interpolated from low-passed data (in which t i d a l energy was removed). The K1 records analyzed started on January 15 (1976) and extended to June 18 (1976), y i e l d i n g approximately a 5-month time span. The records from K2 started January 16 and extended to July 3. Note, however, that some current records from K1 are truncated due to rotor malfunction (see Table 1). The rotor loss at the top sensor of K1 was due to a "high speed burst" of current that induced s u f f i c i e n t mooring t i l t to l i f t the rotor out of i t s bearings. Thus, at K1, the p o s s i b i l i t i e s for comprehensive data analysis are limited, not only due to data loss, but also due to contamination of the records by the burst and i t s transient e f f e c t s . In fact, the burst appears in the longshore current at a l l sensor depths, causing a sudden 66 temperature drop at each sensor; only the offshore component of the current was unaffected by the burst. In the discussion to follow, I w i l l concentrate on the data from station K2. Physical Situation at K1 and K2 Stations K1 and K2 span an area known as the source region of the Somali Current. The climatic mean surface current f i e l d near K1 and K2 is shown in Fig.III.12. During the Northeast Monsoon (February in Fig.III.12), K1 i s in the zone of the East African Coast Current, which has a strong northward flow throughout the observation period. During t h i s period, station K2 i s near the convergence zone of the East African Coast Current and the southward flowing Somali Current. However, after the t r a n s i t i o n to the Southwest Monsoon (August in Fig.III.12) the flow at both stations becomes part of the strong, northward flowing Somali Current system. The analysis of the 1976 records shows that the deep flow at K1 i s weaker than the surface flow, but i s always northward in d i r e c t i o n . The deep flow at K2 i s , however, somewhat more complicated. In Fi g . III. 13 one sees that the longshore mean flow at K2 is always northeast at the top sensor, occasionally southwest at the middle sensor, and almost steadily southwest at the bottom sensor. Fig.III.14 shows thi s southwest flow 67 a 20 40 60 RO j 00 \20 \0Q 160 180 TIME (DflYSl Fig.III.13. (a) The longshore v e l o c i t y record from the top sensor of station K2. The dashed l i n e represents a s p l i n e - f i t to the mean current. (b) As in (a), except for the middle sensor at K2. (c) As in (a), except for the bottom sensor at K2. Note that the v e r t i c a l scale is one-quarter that in (a). condition to be t y p i c a l of the deep flows near K2 (we see a reversal in the p r o f i l e at about 250m, which is similar in depth 68 Fig.III.14. Current p r o f i l e s , ( l e f t ) and l a t e r a l d i s t r i b u t i o n (right) of the deep flows at s i t e B, a station not far from K2. Each consecutive p r o f i l e has been sh i f t e d by 60 cm/s to the right of the previous p r o f i l e . The data were c o l l e c t e d during INDEX (1979). (From Leetmaa, Rossby, Saunders, and Wilson, 1980) . to the bottom sensor of K2). It is suggested here that the deep southwest flow i s topographically deflected seaward before reaching K1. One sees in Fig.III.15 that station K2 i s near the shelf break; the flow at the upper sensor i s c h a r a c t e r i s t i c of the continental shelf regime (northeast flow), while the flow at the bottom sensor seems to represent the shelf/slope flow regime. An idealized impression of the s i t u a t i o n i s given in F i g . I l l . 16. I w i l l return to these considerations in Chapter IV when model parameter f i t t i n g i s discussed. 69 Fig.III.15. The bottom topography at stations K1 and K2. CONTI v V \ ABYSS Fig.III.16. A'conceptual model of the mean flow near station K2 during the period of strongest flow (late May). 70 o o _, 0.0 0.2 0 . 4 0.6 0.8 1,0 FREQUENCY (CPD) Fig.III.17. Autospectrum of the temperature fluctuations at the top sensor of station K1. The spectral density i s in units of (°C) 2/cpd. Fluctuations at K1 At the beginning of Section 2, I noted that the "high speed burst" at K1 in early March caused data loss and complicates the data interpretation. Thus, although I find spectral evidence for low-frequency fluctuations (in a 30- to 75-day band; see Fig.III.17), i t i s not clear whether or not these low-frequency o s c i l l a t i o n s are just transients associated with the burst. To c l a r i f y matters, i t i s useful to examine the low-passed (again using a Butterworth f i l t e r ) temperature and current records. Fig.Ill. I 8 a,b,c,d,e,f shows the low-passed (cutoff period of 25 71 IT) Fig.III.18. (a) Low-passed longshore v e l o c i t y record from the top sensor at station K1. The time of the high speed burst i s indicated by an arrow. (b) As in (a), except for the middle sensor at K1 (c) As in (a), except for the bottom sensor at K1 days) longshore ve l o c i t y and temperature records. The longshore v e l o c i t y records (Fig.Ill.18a,b,c) show evidence of a long 72 TIME (DAYS) 60 80 100 TIME (QRYS) Fig.III.18 (cont.). (d) Low-passed temperature record from the top sensor at station K1. The time of high speed burst is indicated by an arrow. (e) As in (d), except for the middle sensor at K1. (f) As in (d), except for the bottom sensor at K1. period fluctuation before the high-speed burst (which occurs after about 50 days have elapsed from the start of the record). The o s c i l l a t i o n s are best seen in the records from the middle 73 and bottom sensors which both show about a 4 cm/s amplitude with no apparent phase s h i f t between these sensors. The low-passed temperature records c l e a r l y show a constancy of phase with depth in the pre-burst fluctuations. These fluctuations may be wind-induced; the strong longshore v e l o c i t y fluctuations seen at the bottom sensor (at a depth of 445m) indicate a barotropic response. The r e l a t i v e l y weak current shear at station K1 may allow this apparent barotropic response, in contrast to the situation at station K2 where the current shear i s stronger and the response is d e f i n i t e l y not barotropic. 74 Fluctuations at K2 Evidence for a 40- to 50-day o s c i l l a t i o n i s most c l e a r l y found in the K2 temperature spectra (Fig.Ill.19a,b,c). These spectra are calculated over the f i r s t 150 days of the 170 day records (150 days y i e l d s a spectral band centered on 50 days). This figure shows that the temperature records have their variance concentrated at low frequencies: the dominant spectral peaks occur at periods of 38 days, 50 days and 50 days in the records of the top, middle and bottom sensors, respectively. [I w i l l refer to the low-frequency peak, defined in frequency only approximately, as the 40- to 50-day peak]. To a i d in interpretation of these low-frequency fluctuations, i t i s helpful to examine the low-passed versions of the records under consideration. The low-passed temperature time series are shown in Fig.Ill.20a,b,c for each sensor depth at station K2. The low-frequency temperature fluctuations are apparently quite transient at the top sensor; the lower two sensors show a comparatively steady low-frequency record. The bottom sensor shows four peaks over an approximate 150-day span, implying an average 50-day period for the fluctuations. One expects the low-frequency temperature fluctuations to be related to any low-frequency v e l o c i t y fluctuations present; to t h i s end I now examine the low-passed longshore ve l o c i t y records ( F i g . I l l . 2 l a , b , c ) . At the top sensor, the longshore ve l o c i t y record, unlike the corresponding temperature record, i n i t i a l l y shows l i t t l e low-frequency fluctuation, with a rapid 75 r— , ,co CO Q I T J C E U . r— (_> L U ° 0 _ C N J ' CO • o. >-a o CO " CO L U . h -c_) CO 0.0 0 2 0.4 0.6 0.8 FREQUENCY (CPD) l .o : W W . I I 0.0 0.2 0.* I 0.4 0.6 0.8 FREQUENCY (CPD) 1 .0 LD CO a < = > r— C_) Q-o' CO T T 0 .0 T 0.2 0,4 0.6 0.8 FREQUENCY (CPD) 1 .0 Fig.III.19. (a) Autospectrum of the temperature fluctuations at the top sensor of station K2. The spectral density i s in units of (°C) 2/cpd. (b) As in (a), except for the middle sensor at K2. (c) As in (a), except for the bottom sensor at K2. Note that the v e r t i c a l scale i s about—one—six-t-h that in (a). build up of low-frequency energy about 50 days into the record. 76 CO CC LD Ujin L L J Q c e o . Q_ n 1 1 1 r 40 60 80 100 120 TIME (DRYS) 180' "i r 20 40 60 80 100 120 TIME (DAYS) 140 160 180 ~~1 1 1 1— 60 80 100 120 TIME (DAYS) Fig.III.20. (a) Low-passed temperature record from the top sensor at station K2. (b) As in (a), except for the middle sensor at K2. (c) As in (a), except for the bottom sensor at K2. The middle and bottom sensors show a rapid increase in low-frequency variance at considerably later times. The power spectrum of the longshore ve l o c i t y fluctuations at the top sensor (Fig.III.22a, calculated over the same 150-day segment as the temperature fluctuations, shows a strong 40- to 50-day peak. The 40- to 50-day peak also appears in the spectra of the l a s t 77 0""> -—I 1 1 1 1 1 1 I I 20 40 6 0 80 100 120 1 40 160 180 TIME (DRYS) Fig.III.21. (a) Low-passed longshore velocity record from the top sensor at station K2. (b) As in (a) , except for the middle sensor at K2. (c) as in (a), except for the bottom sensor at K2. Note that the scale i s o n e - f i f t h that in (a). 100 days of the lower two longshore ve l o c i t y records (Fig . I l l . 2 2 b , c ) . Apparent in Fig.III.22a i s evidence for fluctuations at a 78 0 . 2 0 . 4 0 FREQUENCY 6 0 . 8 I CPD) 1 ] .0 o o _ , —* ro ' — ' C M > - ° . h - o , ,00 _ C O " Z ° . U J o Q C M -CE c r o _ | o b T 0 , 0 0 . 2 0 . 4 0 . 6 0 . 8 FREQUENCY.(CPD) 1.0 o _ IT) 0 .0 0 . 2 0 . 4 0 . 6 0 . 8 1.0 FREQUENCY (CPD) Fig.III.22. (a) Autospectrum of longshore v e l o c i t y fluctuations at the top sensor of station K2. The spectral density i s in units of (cm/s) 2/cpd. (b) As in (a), except for the middle sensor at K2. Note that the v e r t i c a l scale i s twice that in (a). The spectrum was calculated over the l a s t one hundred days of the record. (c) As in (a), except for the bottom sensor at K2. Note that the v e r t i c a l scale i s on e - t h i r t i e t h that in (a). The spectrum was calculated over the last one hundred days of the record. period of 4 days. These o s c i l l a t i o n s would be s u p e r i n e r t i a l at the latitude of K2, where the i n e r t i a l period i s 10.3 days. Schott and Quadfasel (1982) have found evidence for the 79 existence of 3- to 5-day o s c i l l a t i o n s at stations several hundred kilometers north of K2 (about 5°N). The authors suggest that these o s c i l l a t i o n s are due to barotropic i n s t a b i l i t y of the mean current. One also sees in Fig.III.22a an additional prominent low-frequency peak at about 17 days. This peak also seems to be present in the spectrum of the top sensor temperature fluctuations ( F i g . I l l . 1 9 a ) , and i t appears c l e a r l y in the band-passed (12.5 to 25 days) top sensor longshore ve l o c i t y records ( F i g . I l l . 2 3 ) . These fluctuations have been T 60 80 100 120 TIME (DRYS) 180 Fig.III.23. Band-passed (12.5 to 25 days) longshore velocity record from the top sensor at station K2. observed by Schumann (1981) at various s i t e s off the Natal coast (see Fig.III.24). As shown in Fig.III.25, the 17-day signal i s strongly coherent, for the temperature and offshore velocity f i e l d s , between stations along the coast. The signal also shows strong coherence between the longshore wind v e l o c i t y and the offshore current v e l o c i t y . Schumann also shows that the 17-day 80 Fig.III.24. Location of measurement s i t e s for Schumann's Natal experiment.(From Schumann, 1981). winds exhibit .seasonality, with the spectral energy of 17-day fluctuation decreasing from the A p r i l to September segment analysed to the October to February segment analysed. This seasonality of the wind energy at 17-days may account for the transient nature of the o s c i l l a t i o n seen in Fig.III.23 (note that o s c i l l a t i o n i s strongest in the months from A p r i l to June). The 17-day o s c i l l a t i o n has been found in other midlatitude wind analyses; see Speth and Madden (1973). Apparently,, l i t t l e i s known about th i s o s c i l l a t i o n in the tropics; Fig.III.4 of th i s 81 0 95 -i -90 J Fig.III.25. (a) Coherency and phase spectra between measurements made at moorings off Port Edward and Southbroom (see Fig.III.24). S o l i d l i n e indicates longshore current components, dashed l i n e indicates offshore current, components, and dotted l i n e indicates temperature. (b) Coherency and phase spectra between the longshore wind measured at Louis Botha airport and the longshore current ( s o l i d line) and the offshore current (dotted line) measured at Richard's Bay. (From Schumann, 1981). thesis suggests that the o s c i l l a t i o n i s not present in the t r o p i c a l atmosphere" (over the Western Indian Ocean) and thus that the oceanic signal must have propagated northward off the coast of Natal. Of the offshore v e l o c i t y spectra, only that of the top * Madden (1978) has shown that the l a t i t u d i n a l structure of the atmospheric 16-day o s c i l l a t i o n shows l i t t l e energy at low latitudes (see Fig.III.26). 82 H;(T<»l2.5Doys) Hj(T»8.3Days) 0 -20 0 20 40 60 .80 100 o (m) Fig.III.26. Amplitude of Hough functions ( H 3 ,HJ, ) versus lati t u d e and the measured amplitude of a 16-day wave at 100 mb. (From Madden , 1978). sensor's record shows any energy near 50 days (the lowest frequency peak in Fig.III.27 at 30 days). If one examines the low-passed record from t h i s sensor (Fig.111.28), one sees a s t r i k i n g s i m i l a r i t y of the record to that of the longshore fluctuations (Fig.111.21 a). S p e c i f i c a l l y , after about 50 days the records show high (visual) c o r r e l a t i o n with l i t t l e phase difference. These facts seem to indicate a highly r e c t i l i n e a r o s c i l l a t i o n , directed somewhat eastward of the chosen axis (the longshore o s c i l l a t i o n s have about twice the amplitude of the offshore o s c i l l a t i o n s ) . Comparison of Fluctuations at K2 and K1. If one compares Fig.111.19d,e,f with Fig.Ill.21a,b,c one sees a strong resemblance in low frequency temperature fluctuations in the time in t e r v a l p r i o r to the high speed burst 83 o o ID _, CM O 0.0 0 .2 0.4 0.6 0.8 1.0 FREQUENCY (CPD) Fig.III.27. Autospectrum of the offshore v e l o c i t y fluctuations at the top sensor of station K2. The spectral density 'is in units of (cm/s) 2/cpd. CO i in 0 20 40 60 80 100 120 140 160 180 ' T I M E (DAYS)-Fig.III.28. Low passed offshore velocity"record from the top sensor of station K2. (around day 50) at K1. To obtain a crude estimate of the longshore wavelength one can compare the phase of the fluctuations at the top sensor of K1 with the phase of the fluctuations at the mid-sensor of K2 since they are at similar 84 depths (180m vs. 176m). The large peak at K1 (prior to the burst) occurs about five days e a r l i e r than the corresponding peak at K2. If t h i s represents a delay due to wave propagation (from K1 to K2), a phase speed of (154km/5days) ^ 30km/day (K1 is about 154 km from K2) i s implied. This phase speed estimate corresponds to a wavelength of (50 daysx30km/day)=1500km, which has a large uncertainty, being based on one cycle of data. 85 III.4 Summary In Chapters II and III I have presented the body of information regarding the 40- to 50-day o s c i l l a t i o n . The review of the l i t e r a t u r e (Chapter II) showed the atmospheric o s c i l l a t i o n to be global in nature. The t r o p i c a l component, propagating eastward with planetary wavenumber 1, and modulating the Walker c e l l , i s connected to a midlatitude component by poleward propagating fluctuations. While there i s r e l a t i v e l y l i t t l e information in the l i t e r a t u r e on the oceanic 40- to 50-day o s c i l l a t i o n , i t has been suggested that the o s c i l l a t i o n propagates as an equatorial Kelvin wave (confined to within 3° of the equator) in the P a c i f i c Ocean and probably also in the Indian Ocean. The contribution of th i s thesis, as described in t h i s chapter, has been to show the existence of the 40- to 50-day fluc t u a t i o n s in an inter e s t i n g and important region, namely the source of the Somali Current. It is important that the existence of the o s c i l l a t i o n be shown there, since that region i s on the western boundary of the Indian Ocean, which i s thought to be the source- of the (eastward propagating) fluctuations. It may be that the land-sea contrast in this area i s necessary for generating the fluctuations, and indeed the strength of the fl u c t u a t i o n s , as established in this chapter, may lend credence to t h i s notion. The nearby East African Highlands are known to have an important influence on the atmospheric c i r c u l a t i o n over the Western Indian Ocean, and appropriately, I noted in Chapter 86 II and showed in this chapter that the 40- to 50-day fluctuations propagate p a r a l l e l to the coast (and t h i s mountain range). The most important contribution of t h i s thesis regarding the atmospheric 40- to 50-day o s c i l l a t i o n , i s the demonstration, for the f i r s t time, of i t s existence in wind stress c u r l data. The wind stress c u r l i s very important since i t i s the forcing term for oceanic low-frequency motions. I w i l l make extensive use of the wind stress c u r l analysis in Chapter IV. The analysis of the oceanic data has established, for the f i r s t time, the existence of the 40- to 50-day flu c t u a t i o n as a coastal equatorial phenomenon. Further, some evidence has been presented which indicates that t h i s o s c i l l a t i o n t r a v e l s along the coast toward the equator. I have shown that large fluctuations exist in the oceanic temperature f i e l d (of the order of 2°C). This i s an important observation in that air-sea interaction i s thought to play an important role in the generation of the 40- to 50-day o s c i l l a t i o n s , via heat exchange (see Chapter IV). 87 IV. The Oceanic Response to Atmospheric Forcing IV.1. Introduction The preceding chapters have presented considerable evidence which shows that: (i) An o s c i l l a t i o n of 40- to 50-days period e x i s t s in the atmosphere over a wide meridional belt of the tropics, being weak or undetectable only over the A t l a n t i c . ( i i . ) An o s c i l l a t i o n of 40- to 50-days period exists in both the t r o p i c a l Indian and P a c i f i c Oceans. These points suggest that there is a r e l a t i o n between the atmospheric and oceanic 40- to 50-day o s c i l l a t i o n s . It is possible that: ( i . ) the atmosphere drives the ocean, ( i i . ) the ocean drives the atmosphere, or ( i i i . ) the o s c i l l a t i o n i s due to an ocean-atmosphere feedback. I w i l l examine the f i r s t p o s s i b i l i t y in some d e t a i l in t h i s chapter by presenting some sp e c i f i c wind-driven models of equatorial and coastal c i r c u l a t i o n and applying these models to the observations discussed e a r l i e r . In Appendix III mechanisms ( i i . ) and ( i i i . ) are b r i e f l y discussed. The time-dependent modelling of an atmospherically forced ocean has a long history. For example, Veronis and Stommel (1956) looked at the mid-latitude oceanic response to time-harmonic atmospheric forcing. Large scales were also examined in L i g h t h i l l ' s (1969) treatment of the response of the Indian Ocean to seasonal variations of the wind stess. At intermediate 88 scales, atmospherically generated continental shelf waves have been predicted and detected. To properly model the response of the ocean to the 40- to 50- day atmospheric o s c i l l a t i o n s one should employ the f u l l nonlinear Navier-Stokes equations for a rotating s t r a t i f i e d , f l u i d . However, such a treatment i s beyond the scope of t h i s thesis. To make the general problem tractable the most important features of the dynamics w i l l be modelled separately. Thus, two models w i l l be used to calculate the response of the ocean on the largest scale, and in these models variations of the c o r i o l i s parameter w i l l be included. S p e c i f i c a l l y , the basin and equatorial Kelvin wave responses to the global-scale atmospheric c i r c u l a t i o n c e l l s w i l l be calculated. Next, the effects of a strong horizontal mean current shear w i l l determined via a reduced gravity model of the upper-layer , where current shears are strongest. The v e r t i c a l structure of the response w i l l then be determined with a quasi-geostrophic three-layer model (now ignoring horizontal v a r i a t i o n s ) . In a l l cases the problems w i l l be treated as li n e a r . Preliminary Considerations Before I proceed to introduce models of the atmospherically driven ocean, i t i s useful to determine the relevant length, time, and amplitude scales assoc-i-a-ted—with- the fluctuations. The most important problem i s to explain the magnitude of the observed oceanic fluctuations. That i s , one must f i r s t establish 89 that wind forcing can account for the strength of the oceanic o s c i l l a t i o n s before attemting to explain d e t a i l s of the observations. Accordingly, I w i l l now examine the fundamental scales for v e l o c i t y and temperature: There are two natural scales for the wind-driven current v e l o c i t y ,V (see, e.g., Pond and Pickard, 1978): OK) ( i . ) an Ekman scale \ ~ ^ (IV.1a) ( i i . ) a Sverdrup scale \j~ - C c " ^ z £ (IV.1b) The maximum values of T =.3 dyne/cm2, cur l - t : =6x10"8 N/m3 are obtained from Fig . ( I l l .7a, b) and with Ifj'XlO - 5 s _ 1 (at about 2.5°S), fix. 2x1 0" 1 1nr 1 s " 1 , HptlOOm, one finds from (IV.1) the maximum response of response averaged over the wind mixed layer (upper 100 meters) i s ^ ^ 3 cm/s, V s^3 cm/s (IV.2) In section (III.3a) i t was shown that much larger v e l o c i t i e s than t h i s (up to 10 cm/s) were found at depths considerably greater than 100m. This fact suggests that di r e c t wind forcing ( i . e . , an Ekman or Sverdrup response can account for only a small part of the observed 40- to 50-day current v e l o c i t i e s . If the oceanic signal i s wind-driven, then a mechanism must be invoked to explain the enhancement of the response over d i r e c t forcing scales; e.g., a wave-resonance mechanism. 90 Simple scaling arguments allow one to account for the temperature fluctuations with d i r e c t wind forcing; i . e . , without resort to wave models. The i n i t i a l temperature fluctuation amplitudes at station K2 are shown in the second column of Table 2. From the second and t h i r d columns, the fl u c t a t i o n s are seen to vary roughly as the mean temperature gradients do. This suggests that the f l u c t a t i o n s are induced by a r e l a t i v e l y depth-independent (over 125-251 m) v e r t i c a l v e l o c i t y . This i s based on the assumption that the temperature fluctuations (T') are related to the mean temperature gradient (^ ) and the isotherm displacement ( °l ) by T ' ^ ^ ^ l . The Information in Table 2 allows one to calculate an estimate for ?l : the geostrophic equations for a reduced gravity model ( i . e . , as in Pedlosky, 1979, equation(3.12.4), the time variations are much slower than the i n e r t i a l frequency so that an approximate geostrophic balance holds throughout the f l u i d ) 1? (IV.3) A low-frequency wave off the equator w i l l approximately s a t i s f y ~f V (IV.4a) (IV.4b) (the equations are v e r t i c a l l y averaged over the upper mixed layer; g' i s the reduced g r a v i t y ) . Table 2. I n i t i a l temperature fluctuations at K2. Sensor Fluctuation Amplitude Mean temperature gradient (see Figs.IV.13, IV.16) 1 z=125 m o.9°c 0.1°C/m 2 z=176 m 0.75°C 0.07°C/m 3 z=261 m 0.35°C 0.03°C/m 92 Thus a scale for the i n t e r f a c i a l displacement at the base of the mixed layer due to a guasi-geostrophic wave would be *7 ~ ~ £U_Ly (IV.5) 3 3 where U,V are scales for the x,y components of the velocity and L x and Ly are length scales for the x,y d i r e c t i o n s . Diiing and Schott (1976) give 100 km as the scale width of the Somali Current, thus I take L x =100 km. In Chapter II I deduced a longshore wavelength for the 40- to 50- day o s c i l l a t i o n s of 3200 km (see Fig.II.9). Thus, I take 3200/27T km as L y . From section III.3, one finds that approximate depth averaged values for U, V are V « 5 cm/s U « 1 cm/s (IV.6) Thus one finds, using g'« 2.5x10"2ms~2, |f|=7x10~6s~ 1 17 „ £i 1-5m • (IV.7) Since t h i s i s much less than the 10m displacement required (see IV.3), one must now look at an Ekman pumping model. From the Ekman and continuity equations integrated over the wind forced layer, v i z . , £ v - — (IV.8a) 93 - f w = '^—> (IV.8b) V 07^ (!V.8c) ( evaluated at the layer bottom) one can e a s i l y show -77 _ cLurl^ ^ (assuming f=constant) (IV.9) The Ekman balance specified by (IV.8a,b) i s v a l i d , as Pedlosy (1979, p.176) notes, when the layer under consideration i s homogeneous (allowing separarion of Ekman and geostrophic v e l o c i t i e s ) as i s approximately the case for the wind-mixed layer. Using F i g . (III.7b) one finds maximum wind stress c u r l amplitudes of about 6x10~8 Nt/m3. Thus equation (IV.9) gives (with f=-7x 10'6 s _ 1 and j>=l0 3kg/m 3) l n l ~ ^^hl^Sm for a 50-day o s c i l l a t i o n (IV.10) Thus the Ekman pumping model accounts for the order of magnitude of the temperature fluctuations. It i s now helpful to compare the time h i s t o r i e s of the temperature f i e l d (Fig. III.20a,b,c) with that of the wind stress c u r l (Fig.111.7b). F i r s t note that the temperature time series at the top two sensors are very much al i k e , except that the peak near 140 days i s suppressed at the 94 upper sensor. The response at the upper sensor may be complicated by dynamical effects other than the Ekman pumping discussed above (I w i l l return to this b r i e f l y in section IV.6). Thus, the second sensor represents the "purest" response to the wind stress c u r l . Accordingly, the two time series are plotted together in F i g . IV.1. The s t r i k i n g s i m i l a r i t y of the two series lends credence to the simple Ekman pumping model and establishes the role of the atmosphere in forcing the ocean at 40- to 50-days. In the remainder of this chapter I w i l l look at three simple models of the oceanic response to a 40- to 50-day o s c i l l a t i o n of the wind f i e l d . As noted in the introduction to th i s chapter, one expects the atmosphere to drive a basin-wide response as well as creating l o c a l i z e d responses where restoring forces are concentrated (in t h i s case forces due to mean current shears and bottom topography). F i r s t , the response of the ocean to the global-scale c i r c u l a t i o n c e l l s w i l l be examined. Next, two models w i l l be employed to analyze the response of the coastal ocean s t r i c t l y in the region of interest. 9 5 F i g . I V . 1 . Temperature fluctuations ( s o l i d l ine) at the mid-sensor of station K 2 compared to the l o c a l wind stress c u r l (dashed l i n e ; in units of 1 0 ~ 8 N/m3). 96 IV.3 Response of the Equatorial Ocean to the Global-Scale  C i r c u l a t i o n C e l l s : Generation of Equatorial Kelvin Waves. In section II.3 , i t was shown that the 40- to 50-day o s c i l l a t i o n of the atmosphere has a t r o p i c a l component which propagates throughout the Indian and P a c i f i c Oceans as a (planetary wavenumber 1) equatorial Kelvin wave . In section II.5 i t was shown that the oceanic 40- to 50-day o s c i l l a t i o n propagates throughout the P a c i f i c (and p o s s i b i l y the Indian Ocean) as a f i r s t b aroclinic mode equatorial Kelvin wave with equivalent depth = .71 m. It i s interesting, then, to see whether the atmospheric (global-scale) c i r c u l a t i o n c e l l s can deliver s u f f i c i e n t forcing to account for the oceanic observations. The analysis w i l l u t i l i z e the estimates of the wind stress c u r l from Chapter III to est a b l i s h the stength of the forcing. From the momentum equations for an Kelvin wave on an equatorial plane (invoking an equatorial plane results in an error of only 14% even at 30° north or south l a t i t u d e , according to G i l l , 1982, p.434) a . '-c (IV.11a) / 3 y u (IV.11b) one forms the v o r t i c i t y equation 97 - / 3 y M x = i 6 . ( c « r / ^ ) (IV. 12) which involves the known forcing function curlg'cT . One must project t h i s forcing function onto the v e r t i c a l and horizontal eigenmodes. Following Luther- (1980), the equivalent depth i s based on a constant Brunt-Vaisala frequency ocean (of N«0.0023s" 1). Luther finds the 40-to 50-day Kelvin wave to be f i r s t b a r o c l i n i c mode in nature. This f i r s t b a r o c l i n i c mode (of a constant N ocean ) has depth dependence given by the function cos (if z/H)exp ( -N 2z/2g) . For N 2 « 5 X 1 0 " 6 s" 2 the factor exp(-N 2z/2g) i s approximately equal to 1 over the entire depth of about 2500m. Thus the projected forcing F Ef£ e c+. i s given by (IV.13) Since c u r l 2 t r exists only over the shallow mixed layer of depth d_ , where d <<H, one has J ~ « ^ ~ ^ J ,** (IV.14) This l a t t e r function c u r 1 ^ ) was given in e q u a t i o n d l .5) as 98 where periodic forcing ( e ) has been asssumed. Thus, j>H r° Now , for a Kelvin wave of the f i r s t b a r o c l i n i c mode (of equivalent depth h t ) ,1. U k x - U J $ -JTNI y'~ U ^ u Q ^ e (IV.15) so that equation(IV.12) becomes (IV.16) I now project the function e x p [ - 2 ( y / y Q ) 2 ] onto the l a t i t u d i n a l structure function for the f i r s t b a r o c l i n i c mode by ca l c u l a t i n g the projection c o e f f i c i e n t a A so that the e f f e c t i v e forcing i s The projection c o e f f i c i e n t a t is given by -oo I \ v„»- ' 99 If one now Fourier transforms (IV.16) zonally, and with Tjik)- \ Uo*L Ax O-no (IV.17) and -OO one has (j(/< ) = i . r - o (iv.19) The Fourier inversion of (IV.19) yie l d s u „ = I — : — d < (IV.20) o It was noted e a r l i e r in this section that Luther finds the 40-to 50- day o s c i l l a t i o n to propagate eastward at planetary wavenumber s=1 or 2 , with s=1 dominating (Madden and Julian (1971) find S=1), where s= (40 , 000km/2 7T ) k (40,000 km i s roughly the circumference of the earth). Thus, one might assume F(s) has a gaussian structure (a standard model for spectral l i n e shapes) with halfwidth at s= 2 (see F i g . I V . 2 ) . Thus, I assume - l n i ( s - l ) 1 F ( s ) = F0 e ( i v . 2 1 ) To estimate the magnitude of u Q f r i c t i o n i s introduced since the 100 Fig.IV.2. The gaussian d i s t r i b u t i o n of amplitude s p e c i f i e d by equation(IV.21). 101 denominator in the integral of equation (IV.20) i s zero when k= u j y j ^ . T h i s is done in an approximate manner so as not to unduly complicate the model. With f r i c t i o n , becomes complex , i . e . , where UJI represents the f r i c t i o n a l terms. V e r t i c a l f r i c t i o n i s represented by a v e r t i c a l eddy d i f f u s i o n c o e f f i c i e n t , A z , and horizontal f r i c t i o n by a horizontal eddy d i f f u s i o n c o e f f i c i e n t , A^ . Dimensionally, one requires an inverse time for the f r i c t i o n term O J ^ . For the horizontal f r i c t i o n write = ' ( ^ A h ( I V * 2 2 ) and for the v e r t i c a l f r i c t i o n write Ou.) rr ( S= verf-.cc/ Utxfr (IV.23) If one desires v e r t i c a l f r i c t i o n to be represented by a bottom drag one chooses (see section (IV. 6)) S~ SEKMAV= J^?« In LeBlond and Mysak (1978), estimates are given for bounds on A^ given by and bounds on A z given by 3 x / o " S < Az < 2.^/0°" m*>^ Thus the ranges for uoi are: c J n 102 One may now (somewhat a r b i t r a r l y ) 5 choose an intermediate value of cvi = 5xl0- 8m 2/s. Equation(IV.20) i s then • oo - Z i a. An estimate of the magnitude Ju 0J i s given by I Vol 2.0, Tr ,00 -cu F0<) d k (IV.24) The integral has a value of about 17, N/m2, so that with a =1.4, A. /2> ^ 2x10- 1 1 m-1s-1 , one finds / u e | cc 1 6cm/s This amplitude is s u f f i c i e n t to explain Luther's observed value of 8 cm/s. If the f r i c t i o n c o e f f i c i e n t i s made ten times larger the response drops to about 5 cm/s, s t i l l close to the observed 5Eddy c o e f f i c i e n t s tend to increase with the scale of phenomenon. Thus , I expect the horizontal c o e f f i c i e n t to be large because the horizontal scale i s large (<—' 1000 km) but not too large since the strong gradient of planetary v o r t i c i t y near the equator w i l l tend to i n h i b i t horizonta-1--mi-x-i-og-. The v e r t i c a l c o e f f i c i e n t should tend toward the large end of the range since I am concerned with bottom f r i c i t i o n where s t r a t i f i c a t i o n i s weak. 103 strength. If the f r i c t i o n c o e f f i c i e n t i s decreased by a factor of ten the response increases to about 27cm/s, which i s comparable to the strength observed by McPhaden in the Indian Ocean (about 25 cm/s). Thus, a link i s established between the equatorial oceanic 40- to 50- day fluctuations and the global-scale atmospheric c i r c u l a t i o n c e l l s f i r s t described by Madden and J u l i a n (1972a) . 1 04 IV.4 The Basin Response Two sets of observations have established s i g n i f i c a n t non-coastal and o f f - e q u a t o r i a l (outside the equatorial waveguide) fluctuations at 40- to 50-days: Hayes' (1979) observations and Duing and Schott's (1978) Somali Current observations. I now wish to try and explain these observations in terms of a basin-wide forced model. That i s , I w i l l look at the barotropic response of the whole Indian Ocean to the global-scale atmospheric c i r c u l a t i o n c e l l s in the hope that a western-i n t e n s i f i e d response w i l l be s u f f i c i e n t l y strong to account for these oceanic o s c i l l a t i o n s . I use Duing's (1970) model of the Indian Ocean basin with periodic forcing (he examined the annual response ; I w i l l specify a 50-day period ). The model i s i l l u s t r a t e d in Fig.IV.3. This model i s limited in that only the depth averaged (barotropic) responses are calculated. However, i t i s interesting to f i n d the strength of the oceanic responses on the largest scales; that i s , throughout the extent of a large oceanic basin and throughout the depth of the ocean (Hayes' observations were made near the bottom of the P a c i f i c ) . Following Duing I proceed from the v e r t i c a l l y averaged equations of motion: 105 X = -L Fig.IV.3. The basin model. Parameters are fixed at L=1000 B=6000 km, H (depth)= 4000m, for purposes of c a l c u l a t i o n . km, 106 where U= — u d s is the v e r t i c a l l y averaged (barotropic) x-component of the v e l o c i t y , V, P are the corresponding averaged quantities, k i s the bottom f r i c t i o n c o e f f i c i e n t , and , are the x and y components of the wind stress , respectively. Pedlosky (1979, p.238) notes that homogeneous models generally adequately reproduce the horizontal motions of large-scale oceanic c i r c u l a t i o n . Using the assumption of nondivergent flow (perhaps the most serious r e s t r i c t i o n for an equatorial model), a stream function °^ is introduced v - = ^ * 1/ = - ^ * ( I V - 2 6 ) 6 y and cross d i f f e r e n t i a t i o n of (IV.25) leads to (IV.27) I assume the forcing to be periodic with frequency Oo= ^/goorv/s r and accordingly I write V*Cx,y,tO = V-Cxjy) CCUj£. (IV.28) From equation ( I I .5) I note that e (the x-dependence of the forcing i s small since the wavelength of the global-scale o s c i l l a t i o n i s about 40,000 km). Thus the 1 07 governing dynamics are described by with corresponding boundary conditions (dictated by the no flow condition through the basin walls) <fi(o,y) = ^ ( 8 , x)= 4 6 ^ - ^ ) ^ 0 ^ - 0 = 0 (IV.30) I now decompose the forcing into a Fourier series: ( « ^ E U e *T= ( c - U E ) . f W 2 ? ^ ] <IV-3,) n-z. 1 where ^ \L i e s , 4 - r f — J d * ( I V - 3 2 ) The solution i s now the sum of the responses to each Fourier component of the forcing (since t h i s i s a linear problem). Thus I find (using equation 29 , page 39 of Diiing, 1970) a n r ^ ' " ( e ^ - l l ^ ^ d - ^ 8 ) ] (IV.33) 7T-L e ^ 8 _ e A , 8 J 108 where / l 1 | X = " f ± \ £ + c , ^ = ^ 7 ^ > ^ = Numerical computation shows that a n decreases approximately as 1/n , so that the factor a n/n 2 causes the sum to converge rapidly (5 terms are used to obtain the results that follow). The sums (IV.31) are calculated and the results displayed in Fig.IV.4 . I have used a f r i c t i o n c o e f f i c i e n t of k=10"7 s" 1 in these computations , which i s in the mid-range of reasonable values suggested by Dliing (parameter s e n s i t i v i t y studies indicate that the maximum response amplitudes are not too sensitive to the value of k , changing by a factor of 2 with a ten-fold variation of k). The results shown are both the stream function and the v e l o c i t y magnitude (U 2+V 2) 2 for three di f f e r e n t times: at t = 0, t= (^/4) (1 /«.) , and t= (TT/2) (1 /o») . The units for the streamfunction plots (lV.4a,c,e) are m2/s and 10~1 cm/s. for the v e l o c i t y magnitudes (IV.4b,d,f). The c e l l u l a r patterns characterizing the stream function (IV.4a,c,e) s h i f t from completely anti-symmetric at t=0 to f u l l y symmetric at t = ( f / 2 ) ( 1 /O J ). Many c e l l s are also evident in the v e l o c i t y magnitude plots (IV.4b,d,f) , with maximum v e l o c i t i e s of 2 cm/s evident in the westernmost c e l l s . This maximum ve l o c i t y i s too small to account for the Somali Current observations at stations K1 and K2. These results may , however , be useful in explaining the observations of Hayes (1979) i f the model can be applied to the P a c i f i c . Examination of Fig.IV.5 shows that the region of Hayes' observations i s roughly bounded by a basin formed by the Line Islands and East P a c i f i c Rise on one side and the Americas on the other (the basin i s about 6000 km long). If the model was Fig.IV.4 (a ) . The basin model streamfunctions f o r t = 0 O 110 112 1 1 5 applied here the results might explain the observation of 40- to 50-day energy at great depths. Hayes finds (see Fig.10, Table 1 of Hayes, 1979) meridional current o s c i l l a t i o n s near the bottom of the P a c i f i c at s i t e s B and C (see Fig.IV.5) with an approximate 2-month period and a strength of about 2 cm/s. There i s much less 2-month energy at s i t e A. The meridional orientation of the o s c i l l a t i o n s i s unusual but f i t s with the basin model results where the strongest v e l o c i t i e s are meridional. The basin model also predicts that 2 to 3 cm/s current o s c i l l a t i o n s can exist in the very deep ocean. The low amplitudes found at s i t e A might indicate that t h i s point i s in the f r i c t i o n a l boundary layer which would associated with the westward i n t e n s i f i e d currents. 1 16 Fig.IV.5. Ocean ridges near s i t e s A,B,C of Hayes' program. (Adapted from King, 1.962). 1 17 IV.5 Upper Layer Dynamics In t h i s section I w i l l examine the dynamics of a surface layer ( i . e . , a reduced-gravity model) forced by a 50-day wind o s c i l l a t i o n in the coastal region. S p e c i f i c a l l y , in t h i s section I w i l l ignore coupling of the wind-driven layer to the deeper ocean ( i t w i l l be shown that the r o t a t i o n a l Froude number characterizing interlayer coupling i s small). I w i l l show, using Lee's (1975) forced solution of the Niiler-Mysak (1971) model, that the waveguide constituted by the Somali Current with l a t e r a l current shear responds to a 50-day atmospheric o s c i l l a t i o n with more than s u f f i c i e n t strength to account for the observed current o s c i l l a t i o n s . I start with the momentum and conservation of mass equations integrated over the surface layer which rests on f l u i d of s l i g h t l y greater density below. Under these conditions a reduced gravity model applies (see G i l l , 1982, pp.121,122) and the relevant equations are: (IV.34a) (IV.34b) (IV.34c) where x,y are the coordinates aligned with the current (and are thus not north-south), so that f= x+/S^^ y 118 ( =5>f/ax, i f Ay) H i s the mean layer thickness (about 100m) h i s the t o t a l layer thickness (=H+layer perturbation) g' = 4f g, with A P being the density difference between the top and bottom layers. I w i l l now show that the dynamics are e s s e n t i a l l y non-divergent; that i s , the right hand side of (IV.34c) i s n e g l i g i b l e . The motions may be regarded as hor i z o n t a l l y nondivergent i f (Lee,1975) the time scale of the motion i s large compared to L/j/g'H or equivalently, G O L/Jq'H<< 1 (this parameter is e f f e c t i v e l y a rotational Froude number). If one chooses L as the current width, which Duing and Schott (1978) give as 100 km, and with u-> corresponding to a 50-day o s c i l l a t i o n ( o_)~l 0~ 6 s ~ 1) one finds ooL/J g' H ^ 0.1 « 1 . Thus, the motions may be regarded as horizontally nondivergent and from t h i s i t follows that interlayer coupling i s weak. Next, l i n e a r i z a t i o n i s performed according to where V(x) i s the mean flow , possessing l a t e r a l shear, and then form the v o r t i c i t y equation S u -* u, v -»V( x) +v (IV.35) 119 From the data presented in Chapter III I now wish to estimate the size of various restoring force terms, that is a l l the terms balancing the term, (|^ + "^|^) 1 , in equation(lV.35). The relevant scales are u - O ( O.ol m / s ) V = 0 ( 0-1 m /s) (IV.36) - i i -i /Sfco > fib) 2 . * /O 5 (for a 45° i n c l i n a t i o n of the coastal waveguide /3c*) and ^>ty) are equal) To estimate the size of V x and V x x , I assume the mean flow to a have a hal f - s i n e p r o f i l e of width 100 km, i . e . , V(x)=1(m/s)sin(frx/l00km). Thus, and V \tOOkml M 0 0 k » v J I thus choose as a scale for V x x The estimates of the strength ^f the restoring force terms can now be completed: U A*>" = 0Cz*i°''™{s~'o.ol^/r)~Q(2.*iQ3s~:L) ( I V.37) 120 One might note that the above argument i s p a r t i c u l a r l y sensitive to the value of the current width chosen since t h i s quantity i s squared in the estimate of the term V y ) < . In fact the width of the Somali Current i s variable. Leetmaa (1973) finds a current width of 200 km in late A p r i l of 1971 based on dir e c t current measurements across the Somali Current at 2°S. However, at this time the current in the upper layer has a strenth of about 2m/s (the width tends to increase with the current strength) so that even in th i s case the term uV x x i s s t i l l about two and a half times stronger than the next largest term /3(y)V. Thus, the dominant restoring force term i s uV x x so the e f f e c t i v e balance becomes / <L -r V~&_\ I" + u V -^u^l^/fH (IV.38) This equation obviously ignores variations of f despite the fact that one i s near the equator. The scaling that eliminated ^3 (variation of f) i s in no way a "gimmick". The restoring force provided by the horizontal shear allows the waves to traverse a waveguide extending from 10°S to 10°N (at the peak strength of the Somali Current ). One should emphasize that t h i s waveguide crosses the equator with minimal leakage (see section (IV.7)). In contrast, waves depending on f for th e i r restoring forces w i l l leak into the equatorial waveguide and l i t t l e energy w i l l cross the equator (Anderson, 1981). I now wish to introduce a streamfunction ( ^  ) defined by 121 Hu=~¥ > Hst = ^ > (IV.39) (H i s assumed constant) „ This procedure i s j u s t i f i e d since the motions have been shown to be e f f e c t i v e l y horizontally nondivergent. Equation(IV.38) now becomes ( i + v " U -Vx*D~ = r < ^ / ^ (iv.40) v<o~t 6y J X : > c a S e - ^ . I w i l l approach this problem via a model due to N i i l e r and Mysak (1971), whose forced solution was given by Lee (1975) 6. The model i s i l l u s t r a t e d in Fig.IV.6. The term V equation(IV.40) now vanishes except at the discontinuites in V'(x) (at x=L,2L). Thus equation(IV.40) yi e l d s Ch + v & ) v ^ = i - " ^ £ ( I V - 4 , ) in the regions 0<x<L, L<x<2L, x>2L. In studying forced models i t i s often necessary to include the e f f e c t s of f r i c t i o n , since, without f r i c t i o n , response amplitudes become i n f i n i t e when the natural frequency of the driven system corresponds to the driving frequency (resonance condition) . Here, I w i l l include a simple interlayer drag that 6 Lee included the role of i n s t a b i l i t y in the growth of forced waves. In the situation considered here, the waves are stable. 1 22 Fig.IV.6. The upper layer dynamics model. 1 23 requires no major modification of the governing dynamics (equation(IV.41)). With th i s drag the primitive momentum equations are (hi +V~h-)u +cfv+Ru--f=>x4' (IV. 42a) <}> ' j>H (*- +VL)v~£o + >?V - ~ K + - ^ } (IV.42b) Vctti dy / J° W where R i s the drag c o e f f i c i e n t which is related to a nondimensional drag c o e f f i c i e n t (C x ) by the equation (see p.145 in section IV.6) fZ = where V 5 is a vel o c i t y scale.. With the inclusion of f r i c t i o n equation(IV.42) becomes For the purposes of the ca l c u l a t i o n here I take (see p.145 below) Cx =3x10"". Equation (IV.43) i s v a l i d in each of the regions marked 1,2,3. The required boundary condition at the coastal edge of region 1 i s (IV.44) since the coast must be a streamline. Matching conditions at x=L and x=2L are required . There the mass transport perpendicular to the coast and the surface displacements (pressure) must be 124 continuous (the matching condition for pressure can be found by integrating equation(IV.40) across a discontinuity in mean current v e l o c i t y ) . These conditions may be expressed, respectively, as (IV.45) and [ t £ - * * * v ! . ) £ + ( f + £ > j _ + = o (IV.46) In addition one requires that ^ be bounded as x—> oo . Before proceeding, the equations are nondimensionalized as follows: V =V. V * , (IV.47) Then equation(IV.43) (after dropping the asterisks) becomes 125 where ^ = ^ © / V / - is the Rossby number and the nondimensional f r i c t i o n and forcing terms are given by R) - R /V A/on J'-nr>eK>s 'a»a I 717& The boundary and matching conditions are ^ ( c v A i c ) = O For the model in F i g . IV.6 one has (IV.49) (IV.50) (IV.51) I w i l l now look for plane wave solutions of the problem. For a forcing function that represents a plane wave t r a v e l l i n g p a r a l l e l to the coast (see Chapter I I ) , i t follows that cur-4, tr = c 0 e (IV.52) 126 (assuming the wind forcing i s constant in the x-direction). Thus, I look for corresponding solutions of the form y t x t f j - L ) ^ ^0(*) e y (iv.53) With these s p e c i f i c a t i o n s equations (IV.48) and (IV.50) give 37* = " C C ° (IV.54a) Coo - IR -Asyk) ( ^ - f c * * ) = -L Cc [ C a , - ^ - ^ V O s J ^ k C t ^ A ^ W j _ + = O (IV.54b) at x=1,2 with now a function of x,k, and OJ , i . e . , = . In Lee (1975) , these equations are solved for the model spec i f i e d by equation(IV.51). Lee proceeds as follows: Denote the solutions of equation(IV.54) in region i by ^/ c . It i s possible to f i n d p a r t i c u l a r solutions of equation(IV.54) that s a t i s f y cf to s <4± ( i , M = 4- C l ^ t o ) (IV.55) and which are bounded as x-» oo . They are given by l tl . X L C o [ k k f s . n k [ k g - i ) 3 <JX 127 ^ P < ^ = ^ (iv.5 6 b) ^ 3 p ^ M = - ^ ^ e x p f - l k K x - a ) ] (IV.56c) • [<W, CkCx-z)] - s\^{ktx-xj\^ + j. Co e*p C"«<l6c-2.)] Now the solutions of equation(IV.54) that are continuous at x=1 and x--2 are ^ ^ A ^ c ^ ^ ^ s r o k ^ +t/-lt> CxjkjJ) (IV.57a) , , . (IV.57b) L+ C A , kjO)) s e A p £ - M Cx-a.) J (IV.57c) The functions A(k,oj ) and B(k,<^ ) can be found by applying the matching conditions at x=1 and x=2. The equations thus derived may be written I (IV.58) where 128 (X ^ = - d u j - c £ -Ak) 21 Equations(IV.58) have solutions where (IV.59) (IV.60) A (kjs) = o ^ 4 A o < 2 X - o ( z l c<i2_ . ( i v . 6 1 ) The solutions are i l l u s t r a t e d in F i g . IV.7a,b,c,d,e,f,g,h. Shown here are the nondimensional streamfunction ( </ ) and the nondimensional longshore v e l o c i t y (= ) calculated on the basis of L=l0 5m , c u r l c ^ ( ( l l B u J w / u_ =5x10"8 Nt/m3 (the peak value in Fig.III.7b) for the cases k=0.2,0.4,0.6,0.8. To obtain the dimensional longshore v e l o c i t y one m u l t i p l i e s the nondimensional velocity by the scale f a c t o r — L > £ U r | r £;0.5cm/s. The response curves show f a i r l y complicated behavior a r i s i n g from the superposition of the p a r t i c u l a r and homogeneous solutions. The resonant wavenumbers for 50-day forcing (A =0 in IV.60) are k=0.2,0.6, and at these wavenumbers the longshore ve l o c i t y response is largest, although strong responses are observed in a l l cases. Indeed, the response i s generally of the order of Fig.IV.7 (a). Nondimensional streamfunction , amplitude ( s o l i d l i n e ) and phase (dashed line) for the forced shear modes for the case k=0.2. The current p r o f i l e i s given in equation (IV.51); the peaks of the amplitude at x=1,2 correspond to the di s c o n t i n u i t i e s in V 80. 60. . o rH X '—• 40." 20. o.- — i — .5 180 -. 90 0 1.0 1.5 2.0 2.5 3.0 X 3.5 4.0 4.5 - 9 0 -4-180 5.0 s M W CO Fig.IV.7 (b). Nondimensional longshore v e l o c i t y , amplitude ( s o l i d line)and phase (dashed l i n e ) for the forced shear modes for the case k=0.2. 130 Fig.IV.7 (d). As in (b), except for the case k=0.4. 131 Fig.IV.7 ( f ) . As in (b), except for the case k=0.6. 132 Fig.IV.7 (h). As in (b), except for the case k=0.8. 1 33 I00cm/s or more at the peaks. This i s undoubtedly an overestimate of the response , due to the inclusion of an excessively large potential v o r t i c i t y gradient in the model. That i s , the wind induced transports across the potential v o r t i c i t y gradient give r i s e to e f f e c t i v e forces ( l i k e the topographic forcing caused by Ekman transports) and I have assumed a delta function spike in the potential v o r t i c i t y gradient (at x=L,2L where the derivative of the mean ve l o c i t y changes sharply). One sees that t h i s effect gives r i s e to very sharp jumps in the longshore v e l o c i t y response, so that horizontal f r i c t i o n would be very e f f e c t i v e in l i m i t i n g the response in t h i s model. A rough estimate of the effect can be made by noting that with horizontal f r i c t i o n the governing dynamics i s described by c i t - •+ A h L*7Y = j ~* j (IV.62a) V t - f f u + / l ^ V x y = -t ^_ (IV.62b) s f where A^ i s the horizontal eddy c o e f f i c i e n t . The v o r t i c i t y equation- would then become LUJ ~tA^ = c w r " ' ^ £ . (IV.63) I now approximate the sharp change in the wavefunction across the mean flow region with a half sine p r o f i l e ,sin(TTx/2L). The f r i c t i o n term can now be written as £'<*" "t^)*"~\ V** H , so 134 that (IV.63) becomes Thus the required inverse time scale representing the f r i c t i o n i s (TT/2L) 2A^ , with the range of A^ sp e c i f i e d by (see section IV.3) IO^r^/5 < / \ < IOL -Vs It i s appropriate to look at the large end of t h i s range since I have minimized the estimate of large shear. The inverse time scale for f r i c t i o n becomes ( — 0 * 1 1 0 = / 0 S This i s much larger than the scale provided by interlayer drag, which has a value of about 3 x l 0 ~ 7 s ~ 1 . Thus the inclusion of horizontal f r i c t i o n would s i g n i f i c a n t l y decrease the (excessive) model responses giving current amplitudes more l i k e those observed by Duing and Schott (1978) in the coastal region. 135 (IV.6) Multilayer Dynamics In t h i s section I w i l l concentrate on the v e r t i c a l structure of the response of the Somali waveguide to l o c a l atmospheric forcing . Accordingly, I w i l l develop a multi-layer model in order that the v e r t i c a l variations of the response be modelled. I w i l l now derive the multi-layer equations necessary to describe the quasi-geostrophic response of the ocean to atmospheric forcing. The equations for an incompressible, non-d i f f u s i v e , Boussinesq f l u i d in hydrostatic equilibrium w i l l be derived in considerable generality: I w i l l allow an a r b i t r a r y number of layers and w i l l include v e r t i c a l f r i c t i o n , although I w i l l ignore horizontal f r i c t i o n and the /2> -effect (the reasons for ignoring the /<3 -effect w i l l be discussed a f t e r the model formulation). The physical situation i s i l l u s t r a t e d in Fig.IV.8. The v e r t i c a l l y integrated equations governing the motion of each layer are, following McNider and O'Brien (1973): Conservation of horizontal momentum: £ U J - ^ J = " + - r j x ) ^ j S . ( I v . 6 4 a ) a-v: ^ u ; = ~ ^ + t t ? - ~ * r V (IV 64b) Conservation of v e r t i c a l momentum (hydrostatic assumption): n-l bn = J X J \ j h j + 3 J ^ n - * + H » l • (IV.64C) 136 Fig.IV.8. The N-layer model. 137 Continuity of an incompressible nondiffusive f l u i d : where, uj ,vj are the x,y components of v e l o c i t y ( v e r t i c a l l y averaged in the layer j , )j i s the pressure in the layer j , H • i s the thickness of the layer j in the absence of motion, j i s the displacement of the interface j , h • i s the thickness of the layer j (=H- + '7l\ t-j > Lj are the x,y components of the stress applied to the top of the layer j , ; S x, t a r e the x,y components of the stress applied to the bottom of the layer j , JDJ i s the density of the layer j , 138 For convenience , I w i l l subtract the s t a t i c part of the pressure from the t o t a l pressure , i . e . the contribution 27 °ifjHj-t^fnC^^-^) i s subtracted from p n . I w i l l now specify the Boussinesq approximation ; that i s , density differences between layers are assumed to be small so that the j?nV can be replaced by a reference density j>0 except when density differences occur. The hydrostatic assumption (IV.64c) can then be written as where If one also assumes ^n/H^^ 1 , then one can eliminate the TL^s- except where they occur in derivatives. That i s , I w i l l replace h- by H; in (IV.64a,b,c). Thus by eliminating the J J oZnS in the continuity equation 3 / = l . J L . . . A/-X 139 one can close the equations of motion. That i s , there are 3N equations for the two v e l o c i t i e s and the pressure in each of the N layers. I w i l l now introduce the assumption of quasi-geostrophy by following the scaling of Pedlosky (1979) . Velocit y , length and stress scales (U,L, 'CTQ ) are introduced as follows: ( u ' . v ' ) = ( u/-c > v A r ) where the primed variables are non-dimensional. The horizontal momentum equations become (after dropping primes) (IV.65b) 1 40 If I assume that the magnitude of the time variation and stress terms , characterized by R Q and ^ ° / £ » 0 / i f j £f ;f e respectively, are small, then I have geostrophic balance to a f i r s t approximation: - V j == - K i * , These equations are non-divergent horizontally (u. +v:v =0), and this has consequences when considering the non-dimensional continuity equation: (IV.65c) It i s required that the terms on the right hand side are small, that i s , of order R Q . Thus , L^h = OCR.) , H i &-*iHsi 141 which implies that With the notations I am demanding that the rotational Froude numbers (F-k ) and the J topographic parameter (T) are of order unity. I w i l l now formalize these considerations by introducing the following asymptotic expansions in the Rossby number R Q : Co) _ a) (?) 1 4 2 Thus , to lowest order I obtain ct - s 5 o ( ^ ) fry V Co) do) (°) (I have included the factor sgn(f Q) to allow for the p o s s i b i l i t y of working in the Southern Hemisphere), and to f i r s t order Co) (1) GO t r 0 D i L (IV.66a) D Coj fc). lil ^-o (IV.66b) , (d) tt> J * j / (p5^-i-r) - F j j tPj>- > 1 A . . . ^ (o) Co) to) to') (IV.66c) _ / t o ) CP) -j Note that the f i r s t order v e l o c i t i e s and pressures in equation(IV.66a,b) can be eliminated by forming a v o r t i c i t y equation [ 5L (IV.66b)- ^_(lV.66a)]. Also noting that the lowest order pressures (p ) are streamf unctions, I write J to) 143 and thus find the v o r t i c i t y equation to be given by ( I V ' 6 7 a ) (IV.67b) . , *p . fi \ where A, sr ^ x - t . i t i s interesting to note that the l e f t hand sides of IV.67a,b are not invariant under a change of sign of f G : I now wish to l i n e a r i z e equations(IV.67) . That i s , I w i l l assume small perturbations about a basic state and retain only terms of f i r s t order in the perturbations in the equations of motion. The basic state w i l l be assumed to be one of steady motion in the positive y d i r e c t i o n ; the v e l o c i t i e s may vary from layer to layer. These motions are assumed to be in geostrophic balance so that (remembering that i s a geostrophic pressure) where i s the mean pressure and ff^j i s the perturbation pressure (Vj i s the mean ve l o c i t y in the layer j ) . Thus, 1 44 equation(IV.67) becomes (after dropping primes) > Ovj- + -*,•) - - V j . t ) i - Fw<r^-v4) - F ] j (v. - v ^ ) ] ^ ( i v 6 8 a ) (IV.68b) where I have now assumed h_ =h Q (x). The l e f t hand side of IV.68a is invariant under a change of sign of f Q while the right hand sides of (IV.68a,b) change sign when f Q does. The l e f t hand side of IV.68b i s invariant under a change of sign of f D only i f ^ 3<yx also changes sign. The stresses appearing in equation(IV.68) are of three types: ( i . ) The stress 1Tt i s the wind stress acting on the ocean. It may be calculated from the formula C ^ w - o ^ M C f t L^wo ( C o i s a d r a 9 c o e f f i c i e n t ) . ( i i . ) The i n t e r f a c i a l stresses )'L\ ... L" i which represent the turbulent momentum exchange between layers moving at d i f f e r e n t v e l o c i t i e s . ( i i i . ) The bottom stress <~ Q which represents the drag induced by the f l u i d moving over a s o l i d boundary. The usual formulation for bottom f r i c t i o n i s in terms of Ekman layer dynamics (Pond and Pickard, 1978). One supposes that the 1 4 5 (dimensional) f r i c t i o n a l stress given by ^ F R I C . = A * ( A g i s the c o e f f i c i e n t of eddy v i s c o s i t y ) is largely confined to an Ekman layer. If one uses the e-folding distance <^ =" J^—1 of the Ekman layer , then one can approximate the bottom stress by A similar approach can be used to estimate the i n t e r f a c i a l stresses, but the stresses are not confined to an Ekman layer , at least not in the continuous case I am trying to represent Presumably, the scale depth for the stress should be no greater than the t o t a l layer thickness . Thus I write the i n t e r f a c i a l stress in the form of an inequality J , ^ T (IV.69) where I have invoked continuity of stress at an interface to write cT: = TJT. A . An alternative formulation for the i n t e r f a c i a l stresses is given by O'Brien and Hurlburt (1972). They invoke the familiar quadratic stress law: C J ~" L J + J J J ' 0 y (IV.70) 146 % where Cj. i s a dimensionless i n t e r f a c i a l drag c o e f f i c i e n t and u is the modulus of the mean speeds represented by u- and u ^ . To l i n e a r i z e t h i s stress law, I w i l l assume u to be constant Quoted values for Cx range from 3x10"" (O'Brien and Hurlburt, 1972) to 10"5 (McNider and O'Brien, 1973). I have found values of A 4 (for equatorial thermoclines) ranging from 1 cm2/s (Robinson , 1966 ; Jones, 1973) to 10 cm2/s (Philander , 1973). Using these values in (IV.69), one can f i n d the extremes possible for the stress using an eddy v i s c o s i t y c o e f f i c i e n t : J 0 = 1 ,2, . . . . A/-1 where I have assumed f£il0" 5 s" 1, H £ 2 100m. This range is approximately matched i f one chooses "TT in (IV.70) to be about 1Ocm/s which i s a reasonable scale for the fluctuations I wish to model. I w i l l thus explore the range where C^ . =C*u and = = S C l ^ ^  ~ +1) • For the bottom f r i c t i o n ( TT^f ) I w i l l choose the largest value in the range: With these considerations , I may now write (IV.68) as 147 (k + ' % ) L****^ - F u ^ o ) J . £ F a i C^-)-Fu\}% (IV. 71 a) 7 , (IV.71b) & + V v i - y ) t ^ - ^ t ^ - - < ^ - a ) ] + r r w ^ - v ; j d v . 7 i c ) I w i l l now discuss the suppression of the ^3 -e f f e c t in the preceeding considerations. In the modelling to follow I am interested in two things: ( i . ) the amplitudes of the responses in the various layers and ( i i . ) the longshore wavelength of the oceanic response . It w i l l be seen that the upper-layer response is determined p r i n c i p a l l y by the value of the f r i c t i o n parameter; the response in the lower layers depends on the coupling to the upper layer (via the ro t a t i o n a l Froude numbers and f r i c t i o n parameters ) . The following simple argument assesses the role of /2> in interlayer coupling: For geostrophic balance with variable f, / 3 y " i = - f l y /f -The divergence no longer vanishes to lowest order since 148 implying that thus From the continuity equation I find I now have an interlayer coupling term (analogous to the ro t a t i o n a l Froude number) given by ^jf . Thus the "/s -coupling" w i l l be important i f the term is comparable to the dimensional coupling paramter , that i s In fact I am interested in latitudes of about 2°S, so that with L=100km (the Somali Current width), I have //3y1^0.&m/s whereas £UZ< i m/s Thus the -coupling e f f e c t i s comparable to the dimensional coupling parameter. However, parameter s e n s i t i v i t y studies of the model to be presented indicate that model properties only vary weakly with the coupling parameter. The f r i c t i o n a l terms turn out to be much more important in interlayer coupling. 149 I w i l l now address point ( i i . ) above: The wavelength is determined by the dispersion r e l a t i o n which, including the /S effect (but ignoring the rot a t i o n a l Froude number terms which I w i l l show to be small) i s given by For the situation of interest one finds k z<<7T so that k~ c^Tk with /2= fio««=»f-o««- _ X o Thus k ^  r j ^ ^ r where U i s of order unity. It seems that the effect w i l l not make a s i g n i f i c a n t difference to the longshore wavenumber c a l c u l a t i o n . An important consequence of the variations in the c o r i o l i s parameter i s the leakage of shear wave energy into the equatorial waveguide (as the decrease of f reduces the i n t e r f a c i a l slopes and thus the restoring force for shear waves). This leakage i s s i g n i f i c a n t as the shear waves pass through the equatorial waveguide (which extends 2° to 3° on either side of the equator). Since the region of interest i s at about 2.5°S the waves are s t i l l on the border of the equatorial waveguide and should not have leaked s i g n i f i c a n t energy at this l a t i t u d e . To represent the si t u a t i o n of interest , that i s , the Somali Current near station K2, I now choose a 3-layer model, in order that the three current meters at K2 be represented. To simplify the calculations I w i l l assume the flow to be confined to a channel , as i l l u s t r a t e d in F i g . IV.9. The use of channel 1 50 Fig.IV.9. Model of the channel flow considered in the response analysis. V 1,V i,V 3, are assumed constant. The i n t e r f a c i a l slopes are drawn appropriate to a Southern Hemisphere situation with the mean v e l o c i t y decreasing monotonically with depth. 151 models in i n s t a b i l i t y calculations has been j u s t i f i e d by Mysak, Johnson , and Hsieh (1981). They showed that b a r o c l i n i c shear modes (with which I s h a l l be primarily concerned) decay away from the coast in an f-plane model. Physically, t h i s i s reasonable in that the restoring forces for the wave modes (current shears) exist only in the narrow shear zones near the coast; to a f i r s t approximation the waves are confined to the region where restoring forces exist, and wave energy found in the ocean's i n t e r i o r i s due to leakage from the current zone. Thus, I consider the presence of an outer r i g i d wall to be reasonable, c e r t a i n l y at 3~4°S. The equations of motion are: (SE + V 4 ) £ A i V F * i ^ ~ ' 0 > (&i-vO % (IV.72a) (IV.72c) 1 52 where Vj i s the mean velocity in layer j (j=1,2,3) = ° (internal rotational Froude numbers for J ~ ' /-/• u each layer) 3 ^ = 3Ov-j>,) /f* , 33 = G>, J /j>s , T"1 = 01 1 (topographic parameter) D- — _ ( i n t e r f a c i a l f r i c t i o n parameters) R> = < ~ B - (bottom f r i c t i o n parameter) In the above , R 0 (the Rossby number) =(U//f0|L), oc i s the bottom slope , and C j . and C B are i n t e r f a c i a l and bottom f r i c t i o n c o e f f i c i e n t s respectively. For the channel model , the boundary conditions are ^n=0 <*+ * = 0 , 1 < n = - ^ 3 ) . < I V ' 7 3> I w i l l now solve the equations for quite general forcing functions; I w i l l require only that the solution be periodic in time. The solution w i l l be expressed as a sum of along-channel propagating waves. To t h i s end , i t i s convenient to Fourier transform equations (IV.72) in the along- channel coordinate (y): 153 (- c e +k vJ^Oxx-fe1) <Ft(*,k)+ ( <tlC*,k)-t?tU,k))3 (IV. 74a) + '<Cr3 i(v--v;)- F ^Cv>vjn S^x,*) ( I V ' 7 4 b ) ^ ^ (IV.74c) where I have assumed o s c i l l a t o r y forcing functions and responses, i . e . and ^ Cx,y) e CO oo -Lky The solutions to the homogeneous version of equations (IV.74) 54 are: J J ~ lj'2.,'2> « Since the cross channel eigenfunctions sin(m7T"x) are orthogonal, the forcing function can be expanded in terms of them, v i z . , oo \ I Thus , the inhomogeneous equations may be written as follows: = fm0<) 5 (IV.75a) - V ^ ~ V ^ ) J •* ' < ( f 3 ^ - V ; ) - F ^ ^ ; ) ) ^ J (iv.75b) 155 where K 2=k 2+m 2fr 2. If these equations are written in matrix form , i . e . , A ^ GO 1 £fe)\ o { o 1 where (the c o e f f i c i e n t s a Q 4 1 0 ^ o are given in Appendix II ), then one finds the solutions ) J > ) (IV.76a) A e + L A - - ) ~ ' ^ ^ " ^ (IV.76b) ^ < ^ = a ^ ^ ' . 5 " T^vJOi'jJc) (iv.76c) 156 where det(A •• ) i s the determinant of the matrix A.-: and T j ^ (k) are the transfer functions for the the layer j . one can now write the solution as „ e * " * ' Y_fr\ Ti*f*) l i n m l r x (IV.77) Application to the Somali Current To apply the model developed , the available data must be used to calculate the model parameters . For example, one must specify: ( i . ) The bottom slope (to calculate the topographic parameter T). ( i i . ) The density s t r a t i f i c a t i o n (to calculate the appropiate layer depths and then the rotational Froude numbers). ( i i i . ) The mean current in each layer. I am primarily concerned with applying the model near K2 ,since I have more detailed information from t h i s station. As seen in Chapter III , the mean current f i e l d and bathymetry near K2 are quite complicated , making i t rather challenging to f i t a three-157 layer model to this regime. S p e c i f i c a l l y , the concept of a bottom layer i s somewhat contrived in that topography and bottom flow change greatly in going from the continental shelf to the contintal slope. However , in most of the considerations to follow, i t w i l l be seen that the upper-layer shear mode dominates the amplitude response of the system. This makes accurate representation of the bottom regime less important, and stresses the importance of f i t t i n g the channel to the near-surface flow. Accordingly, I w i l l adopt the "scale" width of 100km for the Somali Current , given by Duing and Schott (1978) , as the channel width "L". The question i s then where to position the center of the channel r e l a t i v e to station K2. Because of the importance of the current f i e l d for the model one should now examine Figure IV.10. It i s seen that the surface currents south of Chisimaio att a i n t h e i r maximum value (with respect to transverse extent) near K2. It then seems reasonable to position the center of the channel near R2 . With these general considerations in mind, I can now proceed to f i t the parameters of the channel model. Parameter F i t t i n g The choice of layer depths w i l l be based on the s t r a t i f i c a t i o n . The only dir e c t information I have on the s t r a t i f i c a t i o n i s that obtained from the mean temperature curves (dashed l i n e s in Fig.IV.11a,b,c ) constructed from the temperature data supplied by the K2 sensors. The three sensors 1 58 Fig.IV.10. Surface current data c o l l e c t e d during INDEX (1979). Current arrows are centered on the observation point. (From Duing, Molinari, and Swallow ,1980). F i g . I V . 1 1 . (a) The temperature record from the top sensor of station K 2 . The dashed l i n e represents a s p l i n e - f i t to the mean temperature. (b) As in (a), except for the middle sensor. Note that the ordinate has been shifted by 5°C. (c) As in (a), except for the bottom sensor. Note that the ordinate has been shifted by 8°C. 160 provide l i t t l e v e r t i c a l resolution . A better temperature p r o f i l e was obtained at station M1 (see Fig.III.10) which i s about 50 km north of K2. This p r o f i l e i s shown in Fig.IV.12. From i t the depth of the mixed layer i s estimated to be about 75m , and thi s allows one to draw a rough mean temperature p r o f i l e for K2 as i l l u s t r a t e d in Fig.IV.13. These plotted values of temperature are intermediate between the maximum and minimum extremes seen in the K2 temperature records (this p r o f i l e represents the situation in late A p r i l ) . Also shown in this figure i s a temperature p r o f i l e taken from Cox (1976) which i s intended to represent the mean Indian Ocean conditions. The correspondence i s reasonable except that Cox apparently has a somewhat shallower mixed layer. This correspondence encourages one to assume that Cox's density and temperature p r o f i l e s (Fig. IV.14) are reasonably applicable to the situation of interest. If Cox's density data i s plotted versus temperature (Fig. IV.15) , I find a reasonably linear relationship, suggesting that I can use the re l a t i o n C^ , = O~o ( i -o< >rP ) (where CT0='30.3 CK '=r f _x x /o"3 °c ^ ) to infer densities from the temperature p r o f i l e s . In a layer model, the interfaces should represent rapid changes in the properties of the actual (continuous) f l u i d . This suggests putting the f i r s t interface at the middle of the observed pycnocline. The preceeding considerations suggest that the pycnocline and the thermocline coincide, so that Fig.IV.13 indicates placement of the f i r s t layer deep enough to include the top current meter of the f i r s t interface at about 130m, (I 161 Fig.IV.12. Reconstruction of v e r t i c a l temperature d i s t r i b u t i o n by using simultaneous temperature and pressure records from the top sensor at mooring M1. (From Duing and Schott ,1978). 162 TEMPERRTURE CO 10. 12. 14. 16, 18, 20. 22. 24. 26. 28. Fig.IV.13. Temperature p r o f i l e s : the dashed curve i s from Cox (1976) and represents mean Indian Ocean conditions; the s o l i d curve i s from K2 data and represents l o c a l conditions in late A p r i l . The surface point on the K2 curve i s estimated from data in Brown, Bruce, and Evans (1980). 163 Fig.IV.14. Potential temperature (Q) and density ( O Q ) p r o f i l e s from Cox (1976). 164 Fig.IV.15. Sigma e vs. 0 for the upper 300 meters of the s t r a t i f i c a t i o n data in Cox (1976) (see Fig.IV.14). 1 65 also want the f i r s t layer deep enough to include the top current meter of K2 at 125m). The depth of the second layer i s somewhat more arb i t r a r y ; I w i l l include the remainder of the thermocline in the second layer, giving i t a depth range of 130 to 200 meters. I w i l l make the t o t a l depth representative of K2, that i s , a t o t a l depth of 300 meters (making the bottom layer 100 meters t h i c k ) . With these s p e c i f i c a t i o n s , i t i s now possible to obtain the representative densities of each layer by averaging the p r o f i l e d temperatures (Fig.IV.13) over each layer. I f i n d , using CJt = CTc£±-c*'V) with C T o - ^ O . S and o<'= o.oo'rx'C"'1: (layer 1) T"^ = 2 5 . £ a C C7^  ) ± = 23. (layer 2) T r = '7 a C ( T j ^ ^ . S (layer 3) 7~'3 - l^-H °CL o03~2."7. The s t r a t i f i c a t i o n i s , of course, subject to seasonal change. Fig.IV.16a,b shows the temperature s t r a t i f i c a t i o n at the high and low extremes in early March and late May , respectively. Representing the s t r a t i f i c a t i o n by (dT/dz) (at z=l76m, since I know the p r o f i l e shape best here) I find t h i s value only varies from 0.06°C/m to 0.07°C/m in going from the low to the high temperature case. Thus, seasonal variation of s t r a t i f i c a t i o n i s unlikely to be important in the model parameterization. Having decided on layer depths, I can now average the mean longshore v e l o c i t i e s (based on Fig.111.13a,b,c) over the v e r t i c a l extent of each layer. In averaging over the top layer , 166 TEMPERATURE C O O o _J m TEMPERATURE C O 10. 12. 14. 16. 18. 20. 22. 24. 26. 28. Fig.IV.16. (a) Temperature s t r a t i f i c a t i o n at station K2 during period of lowest temperatures (late May). The point at the surface i s estimated from data in Brown, Bruce and Evans (1980). (b) Temperature s t r a t i f i c a t i o n at station K2 during the period of maximum temperatures (early March). The point at the surface i s estimated from data in Brown , Bruce, and Evans (1980). an assumption must be made as to the magnitude of the near-surface v e l o c i t y . Here, I have l i t t l e information as to the 1 67 seasonal v a r i a t i o n of th i s quantity. Based on Fig.III.12 I assume a minimum speed of about 25 cm/s, during the Northeast Monsoon. Data from Duing, Molinari, and Swallow (1980) suggests a maximum speed of about 120 cm/s , during the Southwest Monsoon. Between these extremes, I assume a v a r i a t i o n similar in shape to the mean curve in Fig.III.13a. This allows me to draw approximate p r o f i l e s and perform the depth averages. The time variations of the average v e l o c i t i e s are i l l u s t r a t e d in Fig.IV.17 . A constant bottom slope i s , as indicated e a r l i e r (Fig.III.15) i n a p p r o p r i a t e . The actual bottom slope (&C ) varies from about -4x10"3 (shelf) to -50x10"3 (slope) over the channel width; thus I w i l l assume an average slope of about 26x10"3. I now summarize the preceeding discussion regarding parameter values: L^= IOSr^ , /- ) i = 1 3 0 m , H x — TO rr\ , W a = I O O " i , f = - "7.1 -x ICf 6 s"1 f (J ~ O- 5 m/s } 0< - -2.6 * )0^J 0;)! = **. 7 - g 3 = /. S * /O i ^ S (IV.78) - 36-«7 168 Fig.IV.17. Depth averaged longshore v e l o c i t i e s as a function of t ime. 169 The chosen layer thicknesses give one current meter per layer (see Table 1). The Rossby number (R D) appears to be too large (£r0.7) to j u s t i f y a quasi-geostrophic treatment. However, the expansion can be j u s t i f i e d a p o s t e r i o r i . The Rossby number multiplies both the advective term kV and the time variation oO ( i . e . , the Rossby number multi p l i e s the Lagrangian time variations co-kV). In the calculations to follow i t w i l l be found that for Somali Current conditions the near resonant waves of interest approximately s a t i s f y the advective condition oo=kV, so that the Lagrangian time variations are in fact small and a quasi-geostrophic treatment i s j u s t i f i e d . Model Properties With the parameters fixed as l i s t e d in (IV.78) the dispersive properties of the three-layer system may be investigated. The relevant dispersion relation (that i s , the rela t i o n between oo and k which s a t i s f i e s det(A;-. ) = 0, see J (IV.76), when = Cg =0), i s shown in Fig.IV.18 for the case of maximum mean v e l o c i t i e s V x =l00cm/s, =25cm/s, V 3 =8cm/s. The upper curved l i n e represents the topographic mode . The two almost straight l i n e s are shear modes (waves depending on the i n t e r f a c i a l slope for their restoring force , much as topographic waves depend on bottom slope for a restoring force). These li n e s are straight because the waves are moving almost e n t i r e l y due to advection by the mean flow. That i s , because restoring forces are small (the rotational Froude numbers F • 1 70 Fig.IV.18. The dispersion r e l a t i o n for the maximum shear case Vj =(100,25,8)cm/s, with the other parameters having values given in equation(IV.78). The three modes evident here are described in the text. 171 are small) the waves s a t i s f y ou^kVj for the two upper layer v e l o c i t i e s Vj . It is also seen that no i n s t a b i l i t i e s occur in the wavenumber region of inte r e s t . (I am interested in wavenumbers near those that s a t i s f y the dispersion r e l a t i o n for <-uo (<-&o = .29 , corresponding to a period of 50 days). Since these curves represent the maximum shear case, one need not be concerned with the complicating e f f e c t s of i n s t a b i l i t y (which occurs when the top two curves coalesce). I w i l l now specify the sp a t i a l structure of the forcing which appears in (IV.77): I assume that the atmospheric 40- to 50- day o s c i l l a t i o n s propagate as waves p a r a l l e l to the coast. Information presented in Chapter II showed that the 40- to 50-day o s c i l l a t i o n s do in fact propagate almost p a r a l l e l to the coast (at low altitudes) and a wavelength of about 3200km was deduced (based on the observations of Murakami, Nakazawa and He, 1984). The amplitude of the o s c i l l a t i o n i s assumed constant over the narrow (100km) channel. That i s , c ( k<>y -<-*•> so that (IV.79) 1 72 With this forcing the v e l o c i t y response in each mode (m) decreases approximately with the square of the mode number. Since the even modes are not excited, the response of the f i r s t mode is almost ten times that of the next excited mode (the t h i r d ) . Accordingly, I only consider the f i r s t mode here. I can now write ^ • ( V y ^ ) - cons-/, e T": M < o ) s'.nrnlTX , (IV. 80) J 1 / 1 ' JJ- 1 -Thus the phase and amplitude of the response are determined by the transfer functions T ; ( k j . The transfer functions (phase and amplitude) are i l l u s t r a t e d in F i g . IV.l9a,b,c for the same maximum shear case discussed above with regard to the dispersion r e l a t i o n . I have chosen a r e l a t i v e l y small f r i c t i o n c o e f f i c i e n t , C x=l0~ 5m/s, in order that a l l three modes be resolved for purposes of i l l u s t r a t i o n . In the surface layer, only the upper-layer shear mode (k~.145) i s apparent in the amplitude reponse . In the second layer , the upper-layer shear mode dominates but the second-layer shear mode (k~0.55) is apparent. The third-layer response i s also dominated by the upper-layer shear mode, but the topographic mode (k~0.09) i s well resolved, too. In t h i s low f r i c t i o n case, the resonant peaks are situated close to the resonant wavenumbers found in Fig.IV.18 (corresponding to OJ o=0.29). The amplitude response decreases rapidly with increasing depth, in this low f r i c t i o n case. One sees that the upper-layer dynamics dominate the response in a l l layers. The 1 73 CM _ . r -a o _ a -RESPONSE .0 6.0 8.0 I I I j -o r\i — A o J V . o i i i i 7<i i i i r i 0.0 0.2 0.4 0.6 0.8 1.0 WAVENUMBER k Fig.IV.19. (a) The nondimensional amplitude ( s o l i d l ine) and phase (dashed line) of the upper layer transfer function under conditions of maximum shear , with other parameters as l i s t e d in equation(lV.78) and CI=10- 5 (m/s), and C q=3X10 - 5 (m/s). The phase shown is that of the longshore v e l o c i t y fluctuations r e l a t i v e to the forcing. (b) As in (a), except for the second layer. Note that the amplitude scale i s one tenth that in (a). (c) As in (a), except for the bottom layer. Note that the amplitude scale i s one hundredth that in (a). phase curve in the top layer exhibits the c l a s s i c behavior in going through resonance; that i s , one sees a 180° phase s h i f t in going from one side of resonance to the other. The phase 174 ch a r a c t e r i s t i c of the middle layer i s more complicated , with an additional phase s h i f t due to the second-layer shear mode. It i s interesting to note that the second layer actually leads the top layer at resonance in the upper-layer shear mode (k~0.145). The bottom layer shows phase s h i f t s associated with a l l three modes; the t o t a l phase change in going from k=0 to k large and positive is much greater than the 180° phase s h i f t ' seen in the top layer. As expected, increasing f r i c t i o n has the effect of broadening the response peaks and making the phase changes less sharp as resonance i s approached . The second-layer shear mode and the topographic mode tend to be suppressed as f r i c t i o n increases. The amplitude of the upper-layer shear mode (at resonance) in each layer has an interesting behavior, as i l l u s t r a t e d in Fig.IV.20 (note that separate scales are used for the upper and two lower layer amplitudes). The response in the top layer decreases very rapidly as i n t e r f a c i a l f r i c t i o n increases. The second layer shows an i n i t i a l rapid decrease , then levels o f f . In the bottom layer a slow decrease i s followed by a clear increase in amplitude. Thus, there are two c o n f l i c t i n g tendencies in the lower layers: ( i . ) increasing f r i c t i o n "detunes" the system, tending to decrease response, while ( i i . ) the increased f r i c t i o n a l drag from the layer above tends to force the response to higher values. How do other parameter changes aff e c t the transfer functions? It was shown that the s t r a t i f i c a t i o n changes seasonally , and this w i l l affect the rotational Froude numbers. 1 75 10- s 3.16xlO- 6 10"3 3J6x10" 3 10~+ FRICTION COEFFICIENT (M/S) Fig.IV.20. Layer amplitudes for the upper layer shear mode as a function of the i n t e r f a c i a l f r i c t i o n parameter C^. This figure applies to the maximum shear case, with the other parameters as l i s t e d in equation(IV.78). Note that the horizontal axis i s logarithmic and that separate v e r t i c a l scales are used for the upper layer and the two lower layer amplitudes. However, the transfer functions rotational Froude number changes, are and quite insensitive to i t was previously small found 176 that the temperature gradients change l i t t l e from one seasonal extreme to the other. Changes in mean current v e l o c i t y are much more important , given their large magnitude (see Fig.III.11). Generally speaking, as the mean current v e l o c i t i e s decrease the response curve of the dominant upper layer broadens, and i t s amplitude at resonance decreases. More importantly , however, i s the change of the resonant wavenumber of the upper-layer shear mode with (time) change in current speed (Fig.IV.21) . Since the waves are propagating primarily by advection, the resonant wavenumber i s simply given by ^eeS~ t A J°Ar i (where V a i s the ve l o c i t y of the surface l a y e r ) . With these considerations in mind one can now apply the model to the Somali Current s i t u a t i o n . I w i l l now compare the low-passed c u r l record with the top sensor longshore v e l o c i t y record (Fig.IV.22). After about 40 days (from the record's start) the two time series show a tendency to be anti-correlated (that i s , they show an apparent 180° phase s h i f t ) u n t i l about 120 days (after which they s l i p into phase). The model presented e a r l i e r predicts a 180° phase s h i f t between the forcing and the v e l o c i t y response (see, eg., Fig.IV.19a) when resonance i s achieved. I believe that t h i s period of large v e l o c i t y amplitude and 180° phase s h i f t represents a period of near resonance between the c u r l of the wind stress forcing and the ve l o c i t y response . Examination of Fig.IV.21 shows that t h i s near resonance can l a s t only for a limited period of time, as the upper-layer v e l o c i t y slowly changes in time'. That i s , assuming narrow band wavenumber 177 60 80 TIME 100 120 (DAYS) 140 160 180 Fig.IV.21. layer shear The resonant wavenumber and mode as a function of time. wavelength for the upper 178 y> —, Fig.IV.22. Comparison of longshore v e l o c i t y fluctuations from the top sensor at station K2 ( s o l i d line) with wind stress c u r l (dashed-line; the c u r l i s in units of 10"8 N/m3). 1 7 9 forcing, the resonant wavenumber of the system is f i r s t greater than the input wavenumber, then as time elapses i t becomes 7 smaller than the input wavenumber. If one takes the input wavenumber as corresponding to the middle of the resonant i n t e r v a l , one finds a value of k i A / P t r r~0. 25, corresponding to wind forcing with a wavelength of about 2500km8. This value of 2500 km is in reasonable agreement with the value of 1500 km obtained from the longshore phase s h i f t of the temperature time series, and i s in very good agreement with the wavelength of atmospheric o s c i l l a t i o n s ( A =3200 km) established in Chapter II . The system transfer functions for thi s k R ex 0NWAn- =0.25 situa t i o n are shown in Fig.IV.23a,b,c ( f r i c t i o n c o e f f i c i e n t s of C^ =C Q =3x10"5m/s have been used). Note that the surface-layer v e l o c i t y amplitude i s about 8cm/s, in quite good agreement with the values observed in the low-passed v e l o c i t y record (Fig.111.21 a). V e l o c i t i e s of about 2cm/s are predicted in the middle layer. The low-passed record from the second sensor (Fig.Ill.21b) shows fluctuations on t h i s scale (before 120 days); however, i t i s not clear that these r e a l l y are 50-day 7For a wave period of 50 days, the relevant t h e o r e t i c a l time scale i s U J D / „ 1 ~ 8 days. Since t h i s i s much shorter than the time scale of mean v e l o c i t y changes shown in Fig.IV.17 (40-80 days), t h i s type of analysis is meaningful in a WKB sense. 8The corresponding relevant theoretical length scale i s ko.yyT1 ^ 400km, which i s less than 4° of l a t i t u d e . Thus i t appears reasonable to apply our channel model to the region of consideration (2°-4°S). 180 Fig.IV.23. (a) The amplitude ( s o l i d line) and phase (dashed line) of the wind induced longshore v e l o c i t y as a function of the resonant wavenumber for the upper layer. The physical situation i s described in the text. (b) As in (a), except for the middle layer. Note that the v e r t i c a l scale i s one-quarter that in (a). (c) As in (a) , except for the bottom layer. Note that the v e r t i c a l scale i s one-fourtieth that in (a). o s c i l l a t i o n s . The same statement applies to the bottom sensor fluctuations; the amplitude i s in good agreement (before 120 days) with that predicted, but observed fluctuations seem 181 noiselike in character. What then happens at 120 days and after? The system i s no longer resonant, and, thus, the wind sress c u r l can no longer excite substantial o s c i l l a t i o n s at the sensor depths. Yet , the comparison in Fig.IV.24 c l e a r l y shows the peak at about 130 days to be in phase with a peak in the wind stress . Also the pulses observed at the second sensor and at the bottom sensor (characterized by negative velocity) occur at about the same time. Comparison of Fig.IV.22 with Fig.IV.24 i s helpful here. One sees that just after 130 days the wind stess i t s e l f has a strong peak , coinciding in time with the above mentioned events. The wind stress can only generate s i g n i f i c a n t forcing i f i t interacts with the bottom topography; that i s the longshore component of the wind stress sets up a near-bottom return flow for the Ekman transport at the surface. This transport then takes place over rapidly varying topography and can thus induce vortex stretching. If one imagines the return flow to be confined to a bottom layer, then one obtains a bottom forcing term J _ 0K B S h8 bx where h i s the thickness of the bottom layer. If one applies this forcing to the bottom layer of the 3-layer model (while neglecting the surface forcing) one can calculate the expected responses in each layer. While the response w i l l now be o f f -resonant, one may expect substantial amplitudes because the bottom forcing term i s 20 times larger than than the c u r l of the 182 Fig.IV.24. Comparison of longshore ve l o c i t y fluctuations ( s o l i d l i ne) with wind stress (dashed l i n e ) . wind stress term. The results of such a calculation are shown in Fig.IV.25a,b,c . Amplitudes and phases are calculated on the curves at k=0.25. One sees that a reasonable amplitude and phase 183 ro -o o _ CO — o CP o - o RESPQNSE (CM, 10.0 15.0 20.0 \ o CO c o i n - 7 Q _ o r-_ CM o ID _ O o _ ro o in T o 1 1 1 1 i i i i i r ' 0 .0 0.1 0 .2 0 .3 0.4 0 .5 WRVENUMBER k Fig.IV.25. (a) The amplitude ( s o l i d line) and' phase (dashed li n e ) of the wind induced longshore v e l o c i t y as a function of the resonant wavenumber for the upper layer in the case of bottom forced motion. The physical s i t u a t i o n i s described in the text. (b) As in (a), except for the middle layer. (c) As in (a) , except for the bottom layer. i s obtained for the top layer: a speed of about 3 cm/s i s low for the ve l o c i t y peak observed in Fig.IV.24 at 130 days, but the calculated phase of about -310° i s close to the phase re l a t i o n s h i p observed. It seems plausible then , that the upper 184 sensor peak at 130 days i s due to the longshore wind stress, and that divergence e f f e c t s associated with t h i s increase of wind stress give r i s e to the aforementioned suppression of the temperature peak in Fig.III.20. The amplitude in the second layer i s larger than that in that in the f i r s t , in accordance with observation (the pulse in Fig.III.21b at 140 days has an amplitude of 16cm/s , as compared to the 8cm/s peak in Fig.III.21a. The phase of t h i s motion, however, i s in disagreement with observation . The model predicts the second layer to lead the top layer by almost 90°, whereas they are observed to be v i r t u a l l y in phase. The bottom layer has a large predicted amplitude (^lOcm/s) in contrast to observation ( F i g . I l l . 2 1 c ) . Further the model predicts a 270° phase s h i f t between wind stress and response in contrast to the clear 180° phase difference observed (the stress in Fig.IV.24 vs. Fig.III.21c). The bottom-forced model makes reasonable q u a l i t a t i v e predictions for t h i s part of the record. In contrast to the surface-forced model i t y i e l d s strong current responses in the lower two layers which i s as required by observation. The quantitative predictions are poor as may be anticipated from the fact that the topographic forcing varies greatly along the channel. Thus, the response at station K2 w i l l be the sum of many wave responses, each a r r i v i n g from a region of d i f f e r e n t topography. For example, a p a r t i c u l a r l y strong contribution may come from the region where the topographic mode i s resonant with the wind forcing (near station K1). This would give r i s e to a wave exhibiting a 180° phase s h i f t r e l a t i v e to the forcing, and 185 this i s just what i s observed at the bottom sensor, as noted above. 186 IV.7 Leakage of Shear Waves into the Equatorial Waveguide It was noted in section IV.5 that waves propagating on the restoring force provided by the horizontal mean current shear can cross the equator , since the restoring force does not depend on f. However, the disturbance these waves create in the equatorial waveguide w i l l give r i s e to some equatorial wave propagation. The question i s , how much energy leaks from shear modes to equatorial modes? Evidence discussed in Chapter II leads one to believe that the 40- to 50-day o s c i l l a t i o n propagates as an equatorial Kelvin wave in the Indian and P a c i f i c Oceans. Thus , I examine the leakage of shear waves crossing the equator into an equatorial Kelvin mode. To examine th i s s i t u a t i o n I need the momentum equations for the free equatorial Kelvin wave (IV.81a) (IV.81b) with boundary condition (IV.82) the function U representing the zonal component of the velocity of the shear wave crossing the equator . This problem i s analogous to the treatment of current-generated trench waves 187 presented by Mysak and Willmott (1981). This approach w i l l be consistent provided the leakage i s r e l a t i v e l y small so that i t is not necessary to consider the loss rate in determining U(y,z,t). The forcing function, U, i s here considered to be confined to a thin upper layer (~l00m, as in section IV.5) lying over deep ocean (of about 3500m depth). I w i l l also assume constant N 2 type s t r a t i f i c a t i o n , so that one has simple v e r t i c a l modes cos(n T^z/H). With these assumptions I write (I consider here only the f i r s t b a r o c l i n i c mode, of equivalent depth h t, since Luther's observations indicate that the 40- to 50-day o s c i l l a t i o n propagates as a f i r s t b a r o c l i n i c mode Kelvin wave). The e f f e c t i v e forcing function U(t) must be projected onto the v e r t i c a l and l a t i t u d i n a l eigenmodes. Since U(t) represents a wave one has U (IV.83) (IV.84) where the Heavyside step function, is given by The projection c o f f i c i e n t for the l a t i t u d i n a l dependence i s 188 given by •oo •4 ~~ „oo e ^ i * y Jy oo ' (IV.85) The projection coeffient for the v e r t i c a l dependence i s given by J%o^JV(W,) <^ - w (IV.86) and since h<<H Thus a 2 TT From equations (IV.81a,b) , one finds with uCx-o Jd)= s'.nco-tOo^ ^ ! * ) * k^ J From Mysak and Willmott (1981) , one knows the solution to be (IV.87) 189 I take U 0 to be about 1cm/s (much less than the meridional component of about I0cm/s) , and k= (2 77-/3000km). L i g h t i l l (1969) gives h ± to be .75m for Indian Ocean conditions. Thus, I have <Jo£x>t)= i^ls e*p(-0.3) s'.nC*tr<± ~(^T)Ol ~ O O S ^ / s ^ o / C ^ C * - (^7)0] . Hence, I fi n d that the leakage of shear waves into the equatorial Kelvin mode is n e g l i g i b l e . 190 V. Summary and conclusions In chapter II I reviewed the l i t e r a t u r e establishing the existence and properties of the 40- to 50-day o s c i l l a t i o n in the atmosphere and to a lesser extent , in the ocean. The atmospheric o s c i l l a t i o n i s now known to be global in nature. The t r o p i c a l component , propagating eastward and modulating the Walker c i r c u l a t i o n , is connected to a midlatitude component by poleward propagating fluctuations. Although less is known about i t , the oceanic o s c i l l a t i o n probably propagates as an equatorial Kelvin wave (confined to within 3° of the equator) in the P a c i f i c Ocean and in the Indian Ocean. In Chapter III i s described an o r i g i n a l analysis of data from the source region of the Somali Current , which showed the 40- to 50-day o s c i l l a t i o n to exist in the atmosphere and ocean and to have unique properties there. Some of the data were consistent with the results from Chapter II showing the atmospheric fluctuations to propagate p a r a l l e l to the coast , probably due to the effect of the East African Highlands. The existence of 40- to 50-day fluctuations in the wind stress c u r l has been shown for the f i r s t time in any region. This i s p a r t i c u l a r l y important since the wind stress c u r l i s a major forcing term for low-frequency oceanic motions. Previous work has detected oceanic fluctuations at 40- to 50-days period only as mid-ocean equatorial motions. I have shown here that the o s c i l l a t i o n exists as a coastal signal in the Somali region, 191 propagating p a r a l l e l to the coast and toward the equator. The strength of the current fluctuations i s impressive, over 1Ocm/s at depths in excess of 100m. These fluctuations are seen to vary in amplitude with time , posing the challenge of r e l a t i n g these variations to changes in the mean flow conditions and possible atmospheric forcing. I have demonstrated the existence of large temperature fluctuations in the 40- to 50-day period range. These fluctuat ions (of the order of 2°C at the upper sensor of station K2) may play an important role in the air-sea interactions thought to be ultimately responsible for the 40- to 50-day o s c i l l a t i o n . In Chapter IV I showed the atmospheric 40- to 50-day fluctuations are energetic enough to drive the observed oceanic f l u c t a t i o n s . In particular , I demonstrated that the wind stress c u r l fluctuations d i r e c t l y pump the oceanic temperature fluctuations . The strength of the wind stress c u r l fluctuations i s s u f f i c i e n t to drive the observed equatorial Kelvin waves of 40- to 50-days period in the P a c i f i c . I modelled waves propagating p a r a l l e l to the Kenyan coast, restored by forces associated with v e r t i c a l and horizontal current shears , and showed that atmospheric forcing i s s u f f i c i e n t to account for the oceanic 40- to 50-day fluctuations observed in the source region of the Somali Current. Waves propagating on the horizontal shear may cross the equator with minimal leakage of energy into the equatorial waveguide . I found that v e r t i c a l shear modes experience resonance with the atmospheric forcing at a wavelength that corresponds to the known wavelength of 1 92 atmospheric 40- to 50-day fluctuations ( ~3000km) . This resonance condition gives the best f i t between model and observed properties. The energy I believe to exist in these v e r t i c a l shear modes w i l l , in contrast to the horizontal shear modes, be strongly leaked into the equatorial waveguide and may be an important source of the mid-ocean equatorial 40- to 50-day waves seen in the Indian Ocean. Clearly, the real problem of interest is the leakage of energy into the equatorial waveguide when both horizontal and v e r t i c a l current shear are present in the modelled current which l i e s in an unbounded ocean. 193 BIBLIOGRAPHY Anderson, D. L. T., 1981: Cross-equatorial waves with application to the low-level East-African j e t . Geophys. F l u i d Dynamics, J_6 , 267,284. Anderson, J. R., 1984: Ph.D. Thesis, Colorado State University, Fort C o l l i n s . Anderson, J., and R. D. Rosen, 1983: The latitude-height structure of 40-50 day variations in atmospheric angular momentum. J. Atmos. S c i . , 40 , 1584-1591 . Bjerknes, J., 1969: Atmospheric teleconnections from the equatorial P a c i f i c . Mon. Wea. Rev., 9_7 , 163,172. Block, R. G., 1984: E l Nino and world climate: Piecing together the puzzle. Environment, 26 , 14-39. Boyd, J. P., 1977: A review of equatorial waves in the atmosphere. Review  Papers of Equatorial Oceanography, FINE Workshop  Proceedings . Nova/N.Y.I.T. University Press. Brown, 0. B., J . G. Bruce, and R. H. Evans, 1980: Evolution of sea surface temperature in the Somali Basin during the Southwest Monsoon of 1981. Science, 209 , 595-597. Chang, C. P., 1977 : Viscous internal gravity waves and low- frequency o s c i l l a t i o n s in the t r o p i c s . J. Atmos. S c i . , 34 , 901-910. Charney, J. G., 1964: On the growth of the hurricane depression. J. Atmos. S c i . , 21 , 68-75. Cox, M. D., 1976: Equatorially trapped waves and the generation of the Somali Current. Deep-Sea Res., 2^3 , 1139-1152. Duing, W., 1970: The Monsoon Regime of the Currents of the Indian Ocean. International Indian Ocean Expedition oceanographic monographs, no. 1 , Hawaii Institute of Geophysics, Honolulu, 68 pp. — Duing, W., R. L. Molinari, and J. C. Swallow, 1980: Somali Current: Evolution of surface flow. Science, 209 , 588- 590. 1 94 Diiing, W. , and F. Schott, 1978: Measurements in the source region of the Somali Current during the Monsoon reversal. J. Phys. Oceanogr., 8 , 278-289. Erikson, C. C , M. B. Blumenthal, and P. Ripa, 1983: Wind- generated equatorial Kelvin waves observed across the P a c i f i c Ocean. J. Phys. Oceanogr., J_3 , 1622-1639. Fernandez-Partagas, J., and W. D'uing, 1977: Surface Wind Maps for the Western Indian from August 1975  to October 1976. Technical Report, Rosentiel School of Marine and Atmospheric Science, Univ. of Miami, Miami, F l o r i d a , 33149. Fernandez-Partagas, J., G. Samuels, and F. Schott, 1980: Surface Wind Maps for the Western Indian Ocean from January  to September 1979. Technical Report, TR80-4, Rosentiel School of Marine and Atmospheric Science, University of Miami, Miami, F l o r i d a , 33149. G i l l , A. E., 1982: Atmosphere-Ocean Dynamics. Academic Press, New York, 661 pp. Goswami, B. N., and J. Shukla, 1984: Quasi-periodic o s c i l l a t i o n s in a symmetric general c i r c u l a t i o n model. J. Atmos. S c i . , 4J_ ,20-38. Hayashi, Y., 1974: Spectral analysis of t r o p i c a l disturbances appearing in a GFDL general c i r c u l a t i o n model. J. Atmos. S c i . , 31 , 180-218. Hayes, S. P., 1979: Benthic current observations at DOMES sit e s A, B, and C in the t r o p i c a l North P a c i f i c Ocean. Marine Geology and  Oceanography of the Central P a c i f i c Manganese Nodule  Province , Plenum Press. Holton, J., 1979: An Introduction to Dynamic Meteorology. Academic Press, New York, 391 pp. Jones, J. H., 1973: V e r t i c a l mixing in the Equatorial Undercurrent. J. Phys. Oceanogr., 3 , 286-296. Jones, R., 1972: The use and abuse of spectral analysis in t r o p i c a l -meteorology. Dynamics of the Tropical Atmosphere: Notes from a Colloquiam, 252,263. National Center for Atmospheric Research, Boulder, Colorado. 195 Ju l i a n , P. R., and R. A. Madden, 1981: Comments on a paper by T. Yasunari, a quasi-stationary appearance of 30 to 40-day period in the cloudiness fluctuations during the summer monsoon over India. J. Met. Soc. Japan, 59, 435-437. Kanasewich, E. R., 1981: Time Sequence Analysis in Geophysics . The University of Alberta Press, Edmonton, 480 pp. Kerr, R. A., 1984: Slow Atmospheric O s c i l l a t i o n s Confirmed. Science, 225 , 1010-1011. King, C. A. M., 1962: An Introduction to Oceanography. McGraw H i l l , New York, 337 pp. Krishnamurti, T. N., P. K. Jayakumar, J. Sheng, N. Surgi, and A. Kumar, 1985: Divergent c i r c u l a t i o n s on the 30 to 50 day time scale. J . Atmos. S c i . , 42 ,364-375. Lau, K., and H. Lim, 1982: Thermally driven motions in an equatorial beta-plane: Hadley and Walker c i r c u l a t i o n s during the winter monsoon. Mon. Wea. Rev., 110 , 336-353. LeBlond, P. H., and L. A. Mysak, 1978: Waves in the Ocean. E l s e v i e r , New York, 602 pp. Lee, C. A., 1975: The generation of unstable waves and the transverse upwelling two problems in geophysical f l u i d dynamics. Ph.D. Thesis, Institute of Oceanography and Institute of Applied Mathematics, University of B r i t i s h Columbia, 157 pp. Leetmaa, A., 1972: The response of the Somali Current to the Southwest Monsoon of 1970. Deep-Sea Res., J_9 , 31 9-325. Leetmaa, A., 1973: The response of the Somali Current at 2 S to the Southwest Monsoon of 1971. Deep-Sea Res., 20 , 397-400. Leetma, A., H. T. Rossby, P. M. Saunders, and P. Wilson, 1980: Subsurface c i r c u l a t i o n of the Somali Current. Science, 209 , 590-592. L i g h t h i l l , M. J . , 1969: Dynamic response of the Indian Ocean to onset of the Southwest Monsoon. Philos. Trans. R. Soc. Lond., A, 265 , 45-92. 196 Lindzen, R. S., 1974: Wave-CISK in the trop i c s . J . Atmos. S c i . , 3 J _ , 156-179. Luther, D. S., 1980: Observations of long period waves in the t r o p i c a l oceans and atmosphere. Ph.D. Thesis, Massachusetts Institute of Technology and Woods Hole Oceanographic I n s i t u t i o n , 210 pp. Luyten, J. R., and D. H. Roemmich, 1982: Equatorial currents at semi-annual period in the Indian Ocean. J . Phys. Oceanogr., j_2 , 406-41 3. Madden, R. A., 1978: Further evidence of t r a v e l l i n g planetary waves. J . Atmos. S c i . , 35 , 1605-1618. Madden, R. A., and P. R. Ju l i a n , 1971: Detection of a 40- to 50 day o s c i l l a t i o n in the zonal wind of the t r o p i c a l P a c i f i c . J . Atmos. S c i . , 28 , 702-708. Madden, R. A., and P. R. Ju l i a n , 1972a: Description of global-scale c i r c u l a t i o n c e l l s in the tropics with a 40- 50 day period. J . Atmos. S c i . , 29 , 1109-1123. Madden, R. A., and P. R. Ju l i a n , 1972b: Further evidence of global-scale 5-day pressure waves. J . Atmos. S c i . , 29 , 1464-1469. McNider, R. T., and J . J. O'Brien, 1973: A multi-layer transient model of coastal upwelling. J . Phys. Oceanogr., 3 , 258-273. McPhaden, M. J. ,1982: V a r i a b i l i t y in the equatorial Indian Ocean, Part I: Ocean Dynamics. J. Mar. Res., 40 , 157-176. Murakami, T., T. Nakazawa, and J. He, 1984: On the 40- 50 day o s c i l l a t i o n s during the 1979 Northern Hemisphere summer. Part I: Phase propagation. J. Met. Soc. Japan, 62 , 440-469. Mysak, L. A., E. R. Johnson, and W. W. Hsieh, 1981: Baroclinic and barotropic i n s t a b i l i t i e s of coastal currents. J. Phys. Oceanogr., jj_ , 209-230. Mysak, L. A., and A. J . Willmott, 1981: Forced trench waves. J. Phys. Oceanogr., JJ_ , 1482-1502. N i i l e r , P. P., and L. A. Mysak, 1971: Barotropic waves along an eastern contintental shelf. Geophysical F l u i d Dynamics, 2 , 273-288. O'Brien, J. J . , and H. E. Hurlburt, 1972: 1 97 A numerical model of coastal upwelling. J. Phys. Oceanogr. , 2 , 14-26. Quadfasel, D. R. , and J. C. Swallow, 1984: Personal Communication. Pedlosky, J., 1979: Geophysical F l u i d Dynamics. Springer- Verlag, New York. 624 pp. Philander, S. G. H., 1973: The equatorial thermocline. Deep-Sea Res., 20 , 69-86. Philander, S. G. H., 1976: I n s t a b i l i t i e s of zonal equatorial currents: I. J. Geophys. Res., 8_[ , 3725-3735. Philander, S. G. H., 1978: I n s t a b i l i t i e s of zonal equatorial currents: I I . J . Geophys. Res. , 8J3 , 3679-3682. Philander, S. G. H., T. Yamagata, and R. C. Pacanowski, 1984: Unstable air-sea interactions in the trop i c s . J . Atmos. S c i . , 4J_ , 604-613. Pond, S., and G. L. Pickard, 1978: Introductory Dynamic Oceanography. Pergammon Press, Toronto, 241 pp. Robinson, A. R., 1966: An investigation of the wind as the cause of the Equatorial Undercurrent. J. Mar. Res., 2_4 , 179-203. Schott, F., and D. R. Quadfasel, 1982: V a r i a b i l i t y of the Somali Current system during the onset of the Southwest Monsoon, 1979. J. Phys. Oceanogr., _1_2 , 1343-1357. Schumann, E. H., 1981: Low frequency fluctuations off the Natal Coast. J . Geophys. Res., 86 6499-6508. Speth , P., and R. A. Madden, 1983: Space-time spectral analyses of Northern Hemisphere geopotential heights. J. Atmos. S c i . , 40 , 1086-1100. Thompson, J. D., and J. J . O'Brien, 1973: Time dependent coastal upwelling. J. Phys. Oceangr., 3 , 33-46. Veronis, G., and H. Stommel, 1956: The action of variable wind stress on a s t r a t i f i e d ocean. J. Mar. Res. J_5 , 43-75. 198 Weickmann, K. M., 1983: Intraseasonal c i r c u l a t i o n and outgoing longwave radiation modes during Northern Hemisphere winter. Mon. Wea. Rev., 111 , 1838-1858. Weisberg, R., A. Horigan, and C. Colin, 1979: Equatorially trapped Rossby-Gravity wave propagation in the .Gulf of Guinea. J . Mar. Res., 3_7 , 67-86. Yasunari, T., 1981: Structure of an Indian summer monsoon system with a period of around 40 days. J. Met. Soc. Japan, 59 , 336-354. 199 Appendix I; The Maximum Likelihood Method. The data analyzed in t h i s thesis presents two problems: ( i . ) the shortness of the time series compared to the period under consideration. ( i i . ) the e s s e n t i a l l y quasi-periodic nature of the 40- to 50-day o s c i l l a t i o n ( potentially leading to rather broad-banded instead of sharp spectral peaks). These d i f f i c u l t i e s lead to problems in choosing a spectral window . The window must do two things: ( i . ) i t must suppress the side-lobes of any major peak (especially important for short time s e r i e s ) , ( i i . ) i t must smooth the data , i . e . , enhance the signal to noise r a t i o . Given a short time series the method of side-lobe suppression i s very important. T r a d i t i o n a l l y , one simply tapers the data at both ends with a cosine-bell to reduce the ringing (side-lobes). This can , however, unduly reduce the variance of long-period signals. For example, suppose one has only one cycle in the series, as i l l u s t r a t e d in Fig.AI.1. Then , tapering with a cosine b e l l (one multiplies the series by the taper function shown in Fig.AI.2 strongly reduces the o s c i l l a t i o n at i t s peak Fig.AII»3. Time series with peak in middle. 201 values. Note that a much di f f e r e n t result might be obtained i f the cycle has i t s peak in mid-series (see Fig.AI.3- the taper has l i t t l e e f f e c t ) . Thus , one wishes to use an optimally chosen window. In the maximum l i k l i h o o d case , the window optimally passes sinusoids. In the maximum entropy case one obtains an optimally smooth spectrum. However, there seems to be the p o s s i b i l i t y of over smoothing when dealing with quasi-periodic processes as i s i l l u s t r a t e d in Fig.AI.4 . In t h i s case the maximum entropy window smooths the 40- to 50-day fluctuations e n t i r e l y . The spectra presented in thi s thesis are calculated using maximum l i k e l i h o o d . In the oceanic spectra, a l l the 40- to 50-day peaks are s i g n i f i c a n t (with 90% confidence) with only two degrees of freedom (the minimum possible). The c u r l of the wind stress spectrum (Fig.III.5) also shows a 40- to 50-day peak s i g n i f i c a n t at the same le v e l of confidence (90%) while the wind stress spectra (Figs. III.4, III.8) show s i g n i f i c a n t 40- to 50-day energy at an 80% confidence l e v e l (again with two degrees of freedom). 202 Fig. All.4. Comparison of maximum entropy and conventional smoothing i n r e s o l v i n g the 40- to 50-day peak. (From Jones, 1972). 203 . APPENDIX I I . The c o e f f i c i e n t s i n equation (lV»76) '. a l x = • - ( - u Q + k V x ) ( F 2 1 + K 2 ) + k F 2 1 (V± - V 2 ) + l" K 2 R 1 2  a12 = ( - % + k V x ) F 2 i - I K 2 R 1 2 a 2 1 = ( - 0) o + k V 2 ) F 2 2 - i K 2 R 2 i a 2 2 = _ ( _ U o + k V 2 ) ( F 2 2 + F 3 2 + K 2 ) + k { F 3 2 ( V 2 - V 3 ) - F 2 2 (V! - V 2 )} + i K 2 ( R 2 1 + R 2 3 ) a 2 3 = ( - u G + k V 2 ) F 3 2 - i K 2 R 2 3 a 3 2 = ( - 0>o + k V 3 ) F 3 3 - i K 2 R 3 2 a33 = " ( " "o + k V (F33 + k 2 ) " k { T + F 3 3 < v 2 - v 3)> + i K 2 ( R 3 2 + R ^ 204 Appendix I I I . The Effect of the Ocean on the Atmosphere The purpose of th i s appendix i s to calculate the strength of the atmospheric fluctuations which may be produced by the known oceanic temperature fluctuations in the 40- to 50-day period range (of strength of about 2°C). If these temperature fluctuations can induce atmospheric fluctuations of s i g n i f i c a n t strength, then the importance of the role of the ocean in- the generation of the 40- to 50-day o s c i l l a t i o n s w i l l be established. That i s , such a finding w i l l support the idea that ocean-atmosphere interaction is responsible for the 40- to 50-day o s c i l l a t i o n s . Philander, Yamagata and Pacanowski (1984) model the atmospheric response to a heat source Q(x,y,t) via the equations -£ V +3 Hx = -A U ( A I I I . D :P 5"+ d l-ty = -AV (AIII.-2) 0 ( ^ + V y ) =-BH-Q ( A l l I. 3) The coordiates (x,y) measure distance in an eastward and northward d i r e c t i o n ; the corresponding (atmospheric) ve l o c i t y components are U and V. The one-layer atmos~phere has a depth D (I w i l l treat the o s c i l l a t i o n I wish to model as barotropic over a 10 km depth; the 40- to 50-day o s c i l l a t i o n has been found to 205 have appreciable strength at least up to 10 km. Perturbations to D are measured by H. The c o e f f i c i e n t for Rayleigh f r i c t i o n i s A, and the c o e f f i c i e n t for Newtonian cooling i s B. The motions modelled are quasi-steady in that the time var i a t i o n terms that would normally occur in equations(AI11.1,2) are small compared to the f r i c t i o n a l decay term A . That i s , A=(l/5)day- 1, Oo = (2 7r/50days) ~ ( l / 8 ) d a y " 1 . Also, one has B=(1/15)day~ 1. Coupling between the ocean and atmosphere is via the heat source term Q. That i s , changes in sea surface temperature (SST) change the water vapor content of the atmosphere and thus the latent heat release responsible for t r o p i c a l convection. The change in the p a r t i a l pressure of water vapor in a saturated atmosphere over the ocean may be found from the Clausius- Clapeyron equation: where P i s the p a r t i a l pressure of water vapor , T the absolute temperature,AE V is the latent heat of condensation of water, and R i s the gas constant. From th i s r e l a t i o n one finds that a 2°C fluctuation in ocean temperature (fluctuations of t h i s magnitude were found in Chapter III) giving r i s e to the same change in surface atmospheric ~temperature creates a p a r t i a l pressure change of about 0.02 atm (based on a background temperature of about 300°K). The p a r t i a l pressure of the water (AIII.2) 206 vapor i s equal to the saturation mixing r a t i o (q s) multiplied by the molecular weight of water (about 18 grams per mole) divided by the molecular weight of a i r (about 28 grams per mole) , so that one finds q i S = (mass of water vapor/mass of a i r in given volume) =0.02(18/28)~0.01. With th i s knowlege of the saturation mixing r a t i o change due to the temperature fluctuation , one can calculate the induced r i s i n g (due to latent heat release) ve l o c i t y from equation(12.36) of Holton(1979): w dln Q _ -L\ EIV d c ^ (AIII.3) where 0 i s the potential temperature, Cp i s the heat capacity of a i r at constant pressure , and w is the v e r t i c a l v e l o c i t y . (This equation is derived by assuming that the buoyant force due to latent heat release and consequent expansion i s balanced by entrainment). I now write equation(AI11.3) as (AIII.4) 207 where I have replaced ( d q 3 / d t ) with co q s , with cO corresponding to a 50-day period . The term (d8/dz) may be estimated from Figure 12.1 of Holton (1979); (d6 /dz)£i3°K/km on average over the lower 10 km of a t r o p i c a l atmosphere. One thus finds: w^5x10" 3 m/s, which i s the quantity Q needed in eqaution (AIII.1). Now note that the types of disturbances we are interested in here only have a zonal component (V=0), so that equation(Al11.1) becomes 3^ *= -A U c5K (AIII.5a) (AIII.5b) where I have used (c>/^x)=ik. Thus, c * D o = - ( 8 + a ) (AIII.6) and (AIII.7) 208 I now assume k corresponds to a planetary wavenumber 1 (about 40,000 km) o s c i l l a t i o n so as to model the global-scale c i r c u l a t i o n c e l l s . Then, substituting the calculated value of w for Q one finds U^3m/s. In Chapter III I found zonal wind stress fluctuations of a strength of about 0.3 dyne/cm2 corresponding to a zonal v e l o c i t y fluctuation of about 5m/s. Thus, one sees that 2°C fluctuations in SST at a 40- to 50-day period can give r i s e to a atmospheric v e l o c i t y fluctuations which are almost as strong as the observed o s c i l l a t i o n s . Thus, i t seems that the ocean has a s i g n i f i g a n t input into the dynamics of the 40- to 50-day o s c i l l a t i o n in the atmosphere. 

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