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The response of the upper ocean to meteorological forcing Denman, Kenneth Leslie 1972

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THE RESPONSE OF THE UPPER OCEAN TO METEOROLOGICAL FORCING by KENNETH LESLIE DENMAN B . S c , U n i v e r s i t y of Calgary, 1968 A Thesis Submitted i n P a r t i a l F u l f i l m e n t o the Requirements f o r the Degree of Doctor of Philosophy i n the Department of Physics and Oceanography We accept t h i s t h e s i s as conforming to th re q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA May, 1972 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 U n i v e r s i t y of B r i t i s h Columbia, I agree that the L ibrary s h a l l make i t f r e e l y a v a i l a b l e for reference and study. I fur ther agree that permission f o r extensive copying of t h i s thesis f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s representat ives . It i s understood that copying or p u b l i c a t i o n of t h i s thes is f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Department of P h y s i c s anrf Organography The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8 , Canada Date 26 May 197? ABSTRACT A model d e s c r i b i n g the time dependent response of the upper mixed l a y e r of the ocean to m e t e o r o l o g i c a l f o r c i n g i s developed. The t u r b u l e n t mixing and the r a d i a t i v e heating are expressed so that only simple input parameters a v a i l a b l e from r o u t i n e m e t e o r o l o g i c a l measure-ments are r e q u i r e d . The model i s s e n s i t i v e to the r a t e of production by the wind s t r e s s of energy a v a i l a b l e f o r mixing, and to the r a t e of absorption w i t h depth of the s o l a r r a d i a t i o n . Observations obtained at Ocean S t a t i o n 'Papa' i n d i c a t e the r a t e and extent of deepening of the wind mixed l a y e r of the ocean. The model a c c u r a t e l y simulates the behavior of the upper ocean during a 12 day p e r i o d f o r which observed values of wind speed, s o l a r r a d i a t i o n , and back r a d i a t i o n are used as in p u t . To o b t a i n r e a l i s t i c r e s u l t s , a value of 0.0012 f o r the r a t i o of the p o t e n t i a l energy increase of the water column to the downward t r a n s f e r r a t e of t u r b u l e n t energy by the wind s t r e s s i s used. This value l i e s w i t h i n the range, 0.0007 to 0.003, determined from observed data obtained w i t h an STD during 3 storms. Wind s t r e s s estimates are c a l c u l a t e d from p r o p e l l e r anemometer data gathered on the Weathership during the storms. The drag c o e f f i c i e n t , C^Q, remains constant at (1.63 ± 0.28) x 1 0 - 3 f o r wind speeds up to 17 meters per second. The time behaviors of the mean wind speed, the s t r e s s , and the wave energy are examined r e l a t i v e to the e x i s t i n g l a r g e s c a l e weather p a t t e r n s . TABLE OF CONTENTS Page ABSTRACT i i LIST OF TABLES v i LIST OF FIGURES v i i ACKNOWLEDGEMENTS x Chapter 1. INTRODUCTION 1 2. THEORETICAL MODEL OF THE SEASONAL THERMOCLINE 4 2.1 Description of the System 4 The Synoptic Meteorological Scale 4 Air-Sea Transfers 6 Partition of the Wind Stress Energy . 8 2.2 Derivation of the Integrated Equations 9 Assumptions 9 Conservation of Mass and Heat 10 Conservation of mechanical energy 11 Formulation of the Mixing Entrainment 13 The Turbulent Fluxes 14 The Bottom Boundary Condition 15 Integration Over the Layer 16 2.3 Relative Importance of the Surface Boundary Conditions 19 The Wind Dominated Regime. 19 The Heat Dominated Regime. 21 iv Chapter Page Transition Between Regimes 22 2.4 Solutions for the Wind Dominated Regime 23 Concept of 'Mixing Energy' 23 Analytic Solution 24 Numerical Solutions. . . 27 2.5 Solutions for the Heat Dominated Regime 31 2.6 Solutions for Diurnal Heating 34 Typical Daily Cycle 34 Effects of Wind Speed 36 2.7 Conclusions of the Theoretical Model 41 3. OBSERVATIONS AND ANALYSIS 44 3.1 Introduction 44 3.2 Routine Observations at the Weathership 45 3.3 Station 'Papa' Background 48 Influence of Advection 48 Vertical Structure 49 Time Dependent Behavior 31 Ten Year Means 31 3.4 Oceanographic Time Series Observations 53 Isopleth Contours 54 3.5 Analysis of Micrometeorological Data 61 Introduction 61 Experimental Setup °2 Digital Analysis 6 ^ Determination of the Stress. 65 Estimates of Stress and Drag Coefficient . . . . 66 V Chapter Page Time Variation of Mean Wind, Stress, and Wave Energy 72 Conclusion 81 4. MODEL WITH OBSERVED DATA INPUTS 82 4.1 Introduction 82 4.2 Heat and Mass Balance in the Upper Layer . . . . . 82 Net Radiative Heat Input 83 Storage of Heat and Mass 85 4.3 Experimental Determination of the Mixing Energy 87 4.4 Results 95 The Input Data 95 The Twelve Day Period 96 A Two Day Storm Period 99 4.5 Possible Sources of Discrepancy 101 Ekman Divergences 101 Turbulent and Advective Heat Fluxes 105 Internal Waves 107 Turbulent Dissipation. . 108 5. CONCLUSIONS HO REFERENCES 113 LIST OF TABLES Table Page I Model results for the constant input mixing regime . . . 28 II Heating regime with constant inputs. 33 III Model results for the diurnal heating at various wind speeds 38 IV Wind stress energy used in increasing the potential energy of the upper layer of the ocean from 2100 21/6/70 . 93 L I S T OF FIGURES Figure Page t 1. Schematic diagram of the relevant physical processes that make up the ocean-atmosphere system 5 2. Specification of the thermocline that i s assumed in the model along with the various boundary inputs 7 3. Results of the model for a constant wind 29 4. Results of the model with diurnal heating for 30 hours 35 5. Results of the diurnal heating for 4 days with different wind speeds ' 37 6. The location of Ocean Station 'Papa' (50° North, 145° West) 46 7. The Canadian weathership CCGS Vancouver 47 8. An STD trace taken at Station 'Papa' at 2100 GMT 23 June, 1970 50 9. Ten year daily means of routinely measured meteorological quantities at Ocean Station 'Papa' for 1958-1967 inclusive 52 ; 10. Contours of constant temperature at Station 'Papa' for the period 19 May to 28 June, 1970 . . . . . . . . 55 11. Contours of constant salinity 56 12. Contours of constant sigma-t 57 13. Standard marine meteorological parameters for the period IS May to 28 June, 1970 58 14. Contours of constant temperature at Station 'Papa' for the period 0000 GMT 22 June to 0900 GMT 23 June, 1970 60 15. The instrument array mounted on the forward mast . . . . 63 v i i i F i g u r e Page 16. Spectra of h o r i z o n t a l wind ( $ u u ) , waves (^nn)> ship r o l l and p i t c h f o r the approximate pe r i o d 2100-2200 GMT, 22 June 1970 67 17. Spectra of h o r i z o n t a l wind ( $ u u ) , v e r t i c a l wind (^ ww)» a n c* w a v e s ($nn)» a n <* t' i e cospectrum ($uw) °f t n e h o r i z o n t a l and v e r t i c a l wind components f o r the p e r i o d 2345 - 0110 GMT, 21-22 June 1970 69 18. The drag c o e f f i c i e n t , C-2l> a s a f u n c t i o n of wind speed, 71 19. Time s e r i e s of estimates of drag c o e f f i c i e n t ( C 2 1 ) , mean energy of the wave f i e l d 0 l 2 ) > s t r e s s ( u * 2 ) , and mean wind speed (U21) f o r Storm A 73 20. Time s e r i e s of estimates of drag c o e f f i c i e n t ( C 2 1 ) , mean energy of the wave f i e l d ( n 2 ) , s t r e s s ( u * 2 ) , and mean wind speed (U21) f o r Storm B 74 21. Twice d a i l y s urface pressure charts of the N.E. P a c i f i c f o r Storm A 76 22. Twice d a i l y s urface pressure charts of the N.E. P a c i f i c f o r Storm B 77 23. Comparison of wave sp e c t r a a t s i m i l a r stages of development of the two Storms A and B 79 24. The e v o l u t i o n of the wave sp e c t r a i n the two Storms A and B 80 25. The heat and mass balance of the upper 60 meters of the ocean at S t a t i o n 'Papa' f o r the pe r i o d 20 May to 29 June, 1970 84 26. A s e r i e s of temperature p r o f i l e s taken w i t h an STD during Storm B 92 27. Ex p e r i m e n t a l l y determined values f o r m 94 28. Input and r e s u l t s of the model (Run #3) f o r the p e r i o d 13-24 June, 1970 . 97 29. Comparison of the model output w i t h the observed data f o r the storm p e r i o d 1200 GMT June 21 to 1200 GMT June 23, 1970 100 i x F igure Page 30. Comparison of the s a l t d e f i c i t of the upper 60 meters w i t h the cumulative v e r t i c a l a dvection r e s u l t i n g from Ekman divergence at S t a t i o n 'Papa' f o r the p e r i o d 20 May to 28 June, 1970 103 31. G r i d used f o r the v e r t i c a l v e l o c i t y c a l c u l a t i o n s 104 ACKNOWLEDGEMENTS . Many people c o n t r i b u t e to a t h e s i s . F i r s t , I would l i k e to thank Dr. Miyake, my s u p e r v i s o r , f o r suggesting the problem and f o r p r o v i d i n g guidance and support throughout the study. Dr. B u r l i n g , Dr. Stewart and Dr. Pond have a l s o given h e l p f u l comments and c r i t i c i s m s . Without the cooperation of Captain Thomas, the o f f i c e r s , and the crew of the Canadian weathership, C.G.S. Vancouver, the data could not have been obtained. S p e c i a l thanks go to Mr. Bernard Minkley (Marine Sciences Branch) and Mr. Ron Johnson (U.B.C.) f o r enduring w i t h me and a s s i s t i n g on the seven week c r u i s e to S t a t i o n Papa. Mr. P. Wickett (F.R.B., Nanaimo) was extremely h e l p f u l by c a l c u l a t i n g the Ekman divergences f o r the c r u i s e p e r i o d from h i s model. I would l i k e to acknowledge the va r i o u s government agencies who supported t h i s research f i n a n c i a l l y . Of s p e c i a l mention are the Marine Sciences Branch and the M e t e o r o l o g i c a l Branch (now A.E.S.) who have made equipment, data, and personnel a v a i l a b l e f o r t h i s study. F i n a l l y , the N a t i o n a l Research C o u n c i l of Canada has supported me p e r s o n a l l y throughout my four years at U.B.C. Chapter 1 INTRODUCTION Recent works concerned w i t h long range atmospheric p r e d i c t i o n , such as those of Namias (1969, 1970), and Manabe and Bryan (1969), make one strong c o n c l u s i o n : the c l i m a t e and i t s f l u c t u a t i o n s are profoundly i n f l u e n c e d by the i n t e r a c t i o n of the ocean w i t h the atmosphere. As the p h y s i c a l processes i n v o l v e d are much more int e n s e at high wind speeds, the energy exchanges at the a i r - s e a i n t e r f a c e are d i s p r o p o r t i o n a t e l y l a r g e during the passage of storms. These l a r g e energy exchanges, which u s u a l l y occur over periods of 1 to 5 days, are the c o n t r o l l i n g f a c t o r s which determine the time-dependent behavior of the upper ocean. Although the study of the upper l a y e r of the ocean was pioneered by Ekman (1905), the f i r s t r e a l i s t i c study was that of Munk and Anderson (1948). In t h e i r model, they sought a homogeneous mixed l a y e r as a s o l u t i o n by using eddy c o e f f i c i e n t s of v i s c o s i t y and c o n d u c t i v i t y which v a r i e d w i t h depth according to the v e r t i c a l gradients of cur r e n t and temperature (or d e n s i t y ) . The steady s t a t e s o l u t i o n s they obtained ; pr e d i c t e d the depth of the thermocline to w i t h i n a f a c t o r of two or three. The eddy c o e f f i c i e n t approach, used i n Munk and Anderson, has the f u r t h e r disadvantage of r e q u i r i n g knowledge of the curr e n t shear which, even today, i s impossible to o b t a i n . K i t a y g o r o d s k i y (1961) a l s o formulated a one dimensional model of the upper l a y e r . By assuming that the current shear was j u s t that of the o r b i t a l wave v e l o c i t i e s , he was able to c a l c u l a t e c u r r e n t shear 2 from the measured wave f i e l d r a t h e r than from a c t u a l c u r r e n t measurements. Although he too obtained a steady s t a t e s o l u t i o n , he d i d recognize the n e c e s s i t y of i n c l u d i n g turbulence i n h i s f o r m u l a t i o n . Kraus and Turner (1967) developed a mixed l a y e r model which e l i m i n a t e d the need to use eddy c o e f f i c i e n t s . They accomplished t h i s by c o n s i d e r i n g the l a y e r to be homogeneous: heat inputs to the l a y e r and mass ent r a i n e d at the bottom of the l a y e r were ins t a n t a n e o u s l y mixed uniforml y through the l a y e r . By d r i v i n g the system w i t h a sawtooth s o l a r heating f u n c t i o n , they were able to o b t a i n a time dependent s o l u t i o n which e x p l a i n s q u a l i t a t i v e l y the annual c y c l e of the buildup and d e s t r u c t i o n of the seasonal thermocline. In a companion paper (Turner and Kraus, 1967), they obtained s i m i l a r r e s u l t s w i t h a l a b o r a -t o r y model. In both papers, the wind mixing ( s t i r r i n g ) was constant; i n the t h e o r e t i c a l model i t was equivalent to the mixing of l i g h t winds of about f i v e meters per second. Kato and P h i l l i p s (1969) examined the formation of a homogeneous l a y e r , and the downward mixing of i t s bottom i n t e r f a c e i n an annular tank w i t h a r o t a t i n g s t i r r e r . They found the mixing to be e s s e n t i a l l y a conservative process: the r a t e of increase of p o t e n t i a l energy of the s t r a t i f i e d f l u i d was equal to the r a t e of d i s s i p a t i o n of k i n e t i c energy per u n i t area i n the t u r b u l e n t l a y e r . They a l s o found that f o r constant s t i r r i n g and f o r an i n i t i a l l i n e a r d e n s i t y p r o f i l e , the t h i c k -ness of the mixed l a y e r increased as the elapsed time to the one t h i r d power. The f i n d i n g s of Kato and P h i l l i p s should be a p p l i c a b l e to the open ocean, provided that the r a t e of work being done against the buoyancy forces by mixing w i t h i n the homogeneous l a y e r can be a c c u r a t e l y estimated. 3 The large scale thermocline structure has been extensively studied i n a series of theoretical models examined in Needier (1971) and Veronis (1969). These steady state solutions, based on advective flow usually with some vertical diffusion, require as upper boundary conditions that the vertical velocity and either the temperature or the heat flux be matched with those of some upper boundary layer solution. Therefore, any model of the upper mixed layer necessarily accommodates the theories of the main thermocline in two ways. F i r s t , the tempera-ture gradient immediately below the mixed layer i s just that appro-priate to the oceanic thermohaline circulation. Second, the vertical velocity, which i s determined independently from the large scale wind stress curl applied at the ocean surface, must be continuous across the boundary between the top of the main thermocline and the bottom of the mixed layer. In this thesis, the time dependent behavior of the upper mixed layer of the ocean in response to varying meteorological inputs i s studied. Specifically, the processes which determine the behavior of the seasonal thermocline during the passage of a storm are identified and expressed in terms of observable parameters. A theoretical model is developed which includes the relevant physical processes. With input boundary parameters obtained by measurement at Ocean Station 'Papa', the model is solved numerically to simulate the time dependent behavior of the seasonal thermocline. The results are compared with actual observations. Chapter 2 THEORETICAL MODEL OF THE SEASONAL THERMOCLINE 2.1 D e s c r i p t i o n of the System The Synoptic Meteorological Scale The main processes o c c u r r i n g i n the ocean-atmosphere boundary l a y e r are shown, along w i t h t h e i r time s c a l e s , i n Figure 1. G e n e r a l l y , the l a r g e r the time s c a l e , the greater the s p a t i a l extent of a process. The Synoptic M e t e o r o l o g i c a l Scale i s seen to be intermediate between the t u r b u l e n t m i c r o s c a l e (times l e s s than an hour) and the g l o b a l or seasonal s c a l e (times greater than s e v e r a l weeks). Mesoscale phenomena, such as thundershowers which are of an hour or two d u r a t i o n , o f t e n bridge the gap between the micro- and synoptic s c a l e s . High turbulence l e v e l s are u s u a l l y a s s o c i a t e d w i t h mesoscale phenomena. In the upper l a y e r of the ocean, the s i t u a t i o n i s complicated by t i d a l and i n e r t i a l motions at d e f i n i t e f requencies. Nevertheless, three f a i r l y w e l l - d e f i n e d time s c a l e s , analogous to those i n the atmos-phere, can s t i l l be described. At times smaller than the B r u n t - V a i s a l a p e r i o d , which i s one to ten minutes i n the upper l a y e r , surface waves and s m a l l s c a l e turbulence are the dominant motions. At times g r e a t e r than the B r u n t - V a i s a l a p e r i o d , the many motions which, are f o r c e d , d i r e c t l y or i n d i r e c t l y , by sy n o p t i c s c a l e weather patterns are found. They i n c l u d e d r i f t c u r r e n t s , Rossby waves, i n t e r n a l waves, i n e r t i a l c urrent o s c i l l a t i o n s , and Ekman divergences (or convergences). These motions i n t e r a c t w i t h the mean oceanic c i r c u l a t i o n s and the General TURBULENT MICROSCALE MESOSCALE AND SYNOPTIC SCALE METEOROLOGICAL • DISTURBANCES SEASONAL SCALE O g CO CO FLUXES OF - MOMENTUM ... T - LATENT HEAT Hc - SENSIBLE HEAT H s SURFACE WAVES -if SOLAR RADIATION R BACK RADIATION ~B MWD STRESS CURL SMALL SCALE] TURBULENCE DRIFT CURRENT Ud MIXING INERTIAL OSCILLATIONS (15-7 HOURS) INTERNAL WAVES TIDAL FORCING ANNUAL CYCLE APPARENT IN MOST METEOROLOGICAL VARIABLES ROSSBY WAVES GEOSTROPHIC CURRENTS ~l second Brunt-Vdistila period I-10 minutes -I week Figure 1: Schematic diagram of the relevant physical processes that make up the ocean-atmosphere system. The processes are categorized into approximate time scales and some of the more important interactions are indicated by arrows. 6 Atmospheric Circulation to produce seasonal and climatic effects. Apart from seasonal fluctuations, most of the variance i n the air-sea energy exchanges i s related to the synoptic scale storms, which characteristically take two or three days to pass. Although many large scale responses to storms occur in the ocean, the previous works suggest that the qualitative behavior of the upper layer i s determined predominantly by local effects. These include radiation, turbulent heat transfers with the atmosphere, and wind mixing. Air>-Sea Transfers The input boundary conditions for the upper layer of the ocean are shown in Figure 2. They include the radiative heat fluxes, the turbulent fluxes of sensible and latent heat, and the wind stress (turbulent flux of momentum). Associated with the latent heat flux i s a flux of moisture, which is usually unimportant in the balances to be discussed for the ocean's upper layer. Solar radiation incident on the sea surface i s confined to the vi s i b l e and infrared regions: the spectrum has wavelengths ranging from 0.3 to 1.0 microns (1 micron = 10 ^  meters) with a peak at about 0.5 microns (Jerlov, 1968). For wavelengths greater than 1.0 microns, most of the radiation i s absorbed within the upper few centimeters of the ocean. Much of this absorbed heat i s reradiated back to the atmosphere. For wavelengths between 0.3 and 1.0 microns, the radiation penetrates to much greater depths. Although the absorption within this band varies with wavelength, an average extinction coefficient, y Ccm *) can be defined for wavelengths from 0.3 to 1.0 microns such that the rate of solar heating at some depth z = -z can be expressed as 7 Z = 0 a z=-h • z=-d w Figure 2 : S p e c i f i c a t i o n of the thermocline that i s assumed i n the model along w i t h the v a r i o u s boundary i n p u t s . The mixed l a y e r parameters are h, the t h i c k n e s s ; p s , the d e n s i t y ; and y, the e x t i n c t i o n c o e f f i c i e n t f o r the i n c i d e n t s o l a r r a d i a t i o n , R. Other boundary c o n d i t i o n s besides R. are T, the wind s t r e s s ; -B, the back r a d i a t i o n ; - ( H e + H g ) , the t u r b u l e n t heat f l u x e s at the upper s u r f a c e ; p ( - h ) , the d e n s i t y immediately below the l a y e r , and w, the v e r t i c a l v e l o c i t y below the l a y e r . 8 R£z ) = R(0)e ^ Zo. The ocean also acts as an approximate black body o radiator. The resulting back radiation to the atmosphere, -B, is assumed to occur at the surface within a layer of water of negligible thickness. The turbulent fluxes from the ocean to the atmosphere of sensible heat-H , latent heat-H , and horizontal momentum T(the wind s' e stress in the lower atmospheric boundary layer) also occur at the surface. Partition of the Wind Stress Energy According to Dobson (1971), most of the momentum transported into the ocean by the wind stress i s used to generate surface waves. Some of the wave energy is advected away, some is dissipated or trans-formed into turbulence through wave breaking i n the upper few meters, and some is transferred into a d r i f t current. The breaking waves and vertical shear of this d r i f t current provide sources for turbulent energy production on small scales. The turbulent energy may be dissipated; i t may be used to increase the potential energy of the water column by doing work against the buoyancy forces; or i t may be used to increase the kinetic energy of the mean flow. If the wind stress has a large scale curl, then Ekman divergence or convergence may occur in the surface layer, resulting in an associated vertical advective flow at the bottom of the mixed layer. The physical processes just discussed can be specified i n terms of differential equations which describe the conservation of mass, heat, and mechanical energy. 9 2 . 2 Derivation of the Integrated Equations The time dependent behavior of the upper mixed layer at a single point is formulated in this section. I shall derive two coupled equa-tions from the conservation equations of heat and mechanical energy. In the f i n a l equations, time derivatives of h, the mixed layer thickness; T g, the mixed layer temperature; and T ^, the temperature immediately below the mixed layer are expressed in terms of the time dependent boundary conditions. These include wind stress, solar and back radia-tion, turbulent heat fluxes at the upper boundary, and the temperature gradient below the layer. Assumptions -The ocean is assumed to be an incompressible, stably s t r a t i f i e d f l u i d obeying the Boussinesq approximation. Wave-like dynamical effects such as internal, i n e r t i a l , and Rossby waves are ignored. Further, the ocean is assumed to be horizontally homogeneous; horizontal inhomo-geneity enters only through Ekman divergence (and convergence) resulting from non-zero curl in the large scale wind stress f i e l d . The upper mixed layer is an idealized vertically homogeneous layer bounded at the bottom by a density discontinuity. Heat and mechanical energy inputs at the upper and lower boundaries, or at any point within the mixed layer, are assumed to be redistributed uniformly throughout the layer by turbulent diffusion. The times required for this redistribution are small compared to the times over which the processes of interest in this model are assumed to occur. Below the lower interface, a stable density profile Is specified. Also below the mixed layer, an advective vertical velocity, w, may be 10 specified at some arbitrary depth, z = -d (< - h). In Figure 2, the relevant boundary conditions for the model are illustrated. Conservation of Mass and Heat The equation of state for sea water (with the salini t y held essentially constant) Is (according to P h i l l i p s , 1966): d T = 9 T d p 3 T d ? dt 9p dt 9C dt where T is the temperature, p is the density, and £ is the entropy per unit mass defined such that ^ (QT - vg) QT P d t = - ^ T " f 7 . <2-2> A A Here Q^, i s a heat source term, H_ is the molecular heat flux term (assumed to be negligible), and T^ is the absolute temperature. By definition, the specific heat (at constant volume) is c = T 8T In the upper layer c^ - c^, the specific heat of sea water at constant pressure (taken here to be 0.96 cal g - 1 K° - 1). The conservation of mass for an incompressible f l u i d i s ^£ = V » u = 0 so that equation (2-1) can be written as — = — (2-3) dt p c p ' <2 3) If the absorption of solar radiation can be represented by a single extinction coefficient, y» then the heat source term i s just Q T YR*eYz — - = — — — in units of (K sec ) where R* is the solar radiation P P incident on the sea surface (in cal cm - 2 s e c - 1 ) . Substituting the expression for Q T into the turbulent form of (2-3) gives the following heat conservation equation: dt dz dz P c O p yz 11 (2-4) where T(z) and T'(z) are the mean and fluctuating components of tem-perature, w is the mean vertical velocity, and p is the mean density. The term (w'T1) represents the local divergence of the turbulent flux of heat. When this term is.integrated between upper and lower boundaries, i t becomes equal to the net turbulent flux of heat across those boundaries into the layer. Vertical advection by the mean 9T flow, w -g^ -, is assumed to be zero within the completely mixed layer because observations show the vertical temperature gradient to be very small there. Conservation of mechanical energy The turbulent mechanical energy equation has been used by many authors. For the case of horizontal homogeneity with a mean horizontal current u, Phillips (1966) has simplified the equation to: where c 9_ f c 2 3t [2 2 = W ' 2 — j — r 8u 3 3z 3z g - £ (2-5) + y' 2 + W,z, £ is the rate of dissipation of turbulent ,|2 energy, g i s the acceleration due to gravity, p' is the pressure fluctuation, and p ' is the density fluctuation, which according to the Boussinesq assumption, enters only through buoyancy effects. The term TTT dt represents the local rate of change of turbulent kinetic energy per unit mass. In this model, dt w i l l be assumed to be zero according to the following argument. Suppose that dt is as large as the dissipation, e. Then 1, a ( c 2 ) , 2e c 2 dt w But c 2 3w, = 3x/p where w. is the f r i c t i o n velocity in the water 12 corresponding to the wind stress, x. For winds near 10 meters per second, w i s about 1 cm sec Grant, Moill i e t , and Vogel (1968) found the average value for the dissipation, e, within the mixed layer -2 2 -3 but below the wave breaking zone to be about 10 cm sec . With 1 9 ( 2) 2 —1 the above values, =^-zr£° ~ 10 sec . If the doubling time for the turbulent kinetic energy, c 2, were even as large as 1 hour, this would represent in 1 day a completely unrealistic increase of a factor of must, therefore, be much smaller than the dis-24 9 2 . The term dt sipation, e. 3u The term -u'w' 75— of (2-5) represents the rate of production of dz turbulent energy by the turbulent Reynolds stresses acting on the mean current shear. The rate of energy loss (in the case of a stably s t r a t i f i e d fluid) from the turbulence by work done against the density gradient to increase the potential energy of the mean density f i e l d i s —w 0 given by the term — — T h e flux divergence term, o^ -9_ 9z P 2 represents the local divergence of the vertical transports of the turbu-lent energy produced mainly by breaking waves at the upper surface. Similarly to the divergence term for the turbulent heat flux in (2-4), the divergence term for mechanical energy is assumed to redistribute the turbulent energy uniformly throughout the layer within a time > which i s short compared to the times of interest in the model. The resulting steady state equation for turbulent mechanical energy i s —1— r 9u 9 -U W 7C TT" dz dz p 2 = agw'T' + e (2-6) 13 where I have replaced p' with T' by use of dp = apQdT and a = 7~ |§ • o The terms on the l e f t are the source terms f o r turbulent mechanical energy; the terms on the r i g h t are the sink terms. Formulation of the Mixing Entrainment I f the eddy c o e f f i c i e n t approach i s to be avoided, some other method of mathematically expressing the turbulent mixing must be found. According to Kraus and Turner (1967), w'T'(-h), the mixing term at the bottom of the la y e r , may be expressed as w'T'(-h) = H (-w ) (T . - T ) (2-7) m -n s where T g i s the temperature of the la y e r , T ^ i s the temperature immediately below the la y e r , and -w i s the entrainment v e l o c i t y of the m mixed layer into the water beneath i t . H i s the Heaviside step function having the properties H = 0 i f -w £ 0: no entrainment mixing m 1 i f -w > 0; entrainment mixing at z = -h. dh The entrainment v e l o c i t y can be written as -w = w + - r — , where w, m dt the mean v e r t i c a l advective v e l o c i t y immediately below the mixed la y e r , i s determined predominately by the l o c a l c u r l of the wind s t r e s s . Equation (2-7) then becomes W'T» (-h) = -H w 4. d h  V + dT T - T , s -h (2-8) This formulation, -w = w + ^ 7 , was used as the upper boundary condition f o r a model of the well-mixed atmospheric boundary layer by G e i s l e r and Kraus (1969). 14 The Turbulent Fluxes The mixing entrainment term w'T'(-h) represents an upward f l u x of c o l d water across the bottom boundary of the l a y e r . I t can be thought of as a heat l o s s due to the t u r b u l e n t heat f l u x across the bottom boundary. S i m i l a r l y , at the upper boundary, tt'T'CO) = (2-9) p c_ o p where F. i s the downward f l u x of heat across the a i r - s e a i n t e r f a c e •k -2 -1 ( i n u n i t s of c a l cm sec ). I f H . and H . are the t u r b u l e n t f l u x e s s* e* of s e n s i b l e and l a t e n t heat at the sea s u r f a c e , and -B^ i s the net heat l o s s by long wave r a d i a t i o n from the sea s u r f a c e , then F * = H e * + H s * + B * ' ( 2 - 1 0 ) From t h i s p o i n t on, the v a r i a b l e t r a n s f o r m a t i o n (R, B, H , H ) = 6 S 1 (R., B., H ., H .) w i l l be assumed. p c *' *' e*' s* o p W i t h i n the upper l a y e r , the heat inputs are instantaneously d i s t r i b u t e d u n i f o r m l y throughout the l a y e r by the t u r b u l e n t d i f f u s i o n w'T'. The appropriate form of w'T' which matches the boundary con-d i t i o n s (2-8) and (2-9) and which e l i m i n a t e s the z-dependence of the p o i n t heat equation (2-4) w i t h i n the l a y e r i s : H(w + dh/dt)(T - T_ h) - F - R ( l - e Y h ) h w'T'(z) = + - R ( l - e Y Z ) - F f o r (-h < z < 0) . (2-11) 15 The exact form of this function does not enter into the integrated equations, but i t is essential that the redistribution be there. The divergence term for the mechanical energy flux, f t 21 k 2 J -3z may also be expressed in a similar form: i t redistributes the turbulent kinetic energy produced by breaking waves and by the Reynolds stress on the mean flow. The Bottom Boundary Condition To specify the system completely, one must know the temperature 3T gradient below the mixed layer. If the assumption is made that no turbulent energy penetrates below z = -h, the time rate of change of temperature at some depth z, where -d < z < -h, is given by the heat conservation equation, o-T(z) _ yz 3t = y Re - w 3T(z) 3z (2-12) At the interface, z = -h, the total derivative must be used: g T(-h) - | j T(-h) - £ | f yRe -yh - h w + dh dt 3T 3z - h (2-13) Here, I have assumed that there may be heating in the lower layer as a result of the penetration of solar radiation. The bottom boundary condition, together with the expression for the turbulent entrainment mixing, effectively decouples the lower thermocline region from the mixed layer above. Because a l l the available turbulent energy i s assumed to be used up in mixing at or above the interface, the turbulence level below the layer i s zero. The only surface influence acting on the ocean at depth z < -h is the small fraction of solar radiation which penetrates to that depth. 16 Integration Over the Layer The conservation equations of heat and mechanical energy can f r° now be integrated verti c a l l y over the homogeneous mixed layer J dz I -h 3T The heat equation C2-4) becomes (with, w 7 -— neglected) o z h ' £ E i + H dt w + dh dt (T - T , ) = F + R(l - e ~Y h ) . s -h (2-14) Upon division by h, this equation i s identical to the point equation (2-4) with (w'T') evaluated from (2-11). Integration of the mechanical energy equation (2-6) yields -h LP_ 2 J 8z fp' c 2 , where w' ]*— + y^ — =ag z=0 f-O W'T' dz + -h edz (2-15) J-h ro is assumed to be zero. The term ag z=-h w'T1 dz -h can be eliminated by the use of the double integral of (2-4): ro h i d T s _ ^ _ R ( h _ - i j _ Y - i R e - y h -h w'T'(z)dz = 2 dt If (G, D) = --p^r- (G^, D^) in units of cm2 K° sec" 1 where Poag ro — i — r 3u , . u'w' 7c dz — p w' dz o IP. 2 > z=0 and ro e dz (2-16) then (2-15), into which (2-16) has been substituted, becomes dT s 2 dt •(G - D) + Fh + R(h -y ) + y Re - Yh (2-17) 17 I t i s through t h i s s u b s t i t u t i o n of (2-16) i n t o (2-15) that the mechanism f o r convective mixing enters i n t o the mechanical energy equation. Only f o r very low wind speeds or f o r l a r g e evaporation does the convective mixing become Important. Equation (2-17) can be used to e l i m i n a t e dT /dt from the heat s equation (2-14): H( dh I \ G - D + Ry~J (1 - e " Y h ) ] -h [F + R ( l + e Y h ) ] W d V " h(T - T J s -h (2-18) The complete set of equations d e f i n i n g the upper mixed l a y e r i s now expressed e x p l i c i t l y i n terms of parameters which can be measured or can be estimated from measurements. Using equations (2-13), (2-17) and (2-18), w i t h (2-10) s u b s t i t u t e d i n t o (2-17) and (2-18), one obtains f o r the model the f o l l o w i n g set of f i r s t order o r d i n a r y d i f f e r e n t i a l equations i n h, T g, and T ^. dT dt -M- (G - D) +h (B + H + H ) +R (h - Y" 1 + Y - 1 e ~ Y h ) 6 S (2-19) H w + dh dt :[G - D + R Y _ 1 ( 1 - e -yh B + HE + HR + R ( l + e' h ( T s - T_ h) (2-20) ^ T_ h = Y R e _ Y h - (w + dh/dt) |f (2-21) -h As the temperature gradient below the l a y e r , 9T(z)/3z, v a r i e s w i t h time BT(z) because of s o l a r heating and v e r t i c a l a dvection, ^ must be known f o r any time t i n order to evaluate 9T 9z 9z i n (2-21). Equation -h (2-12), then, i s the necessary boundary c o n d i t i o n : 18 ^ = Y R j Z - w f i (2-12) f o r z < -h. Let us examine equations (2-19), (2-20), and (2-21). Except f o r H, the Heaviside step function defined e a r l i e r , and y, the e x t i n c t i o n c o e f f i c i e n t , a l l the c o e f f i c i e n t s e x p l i c i t l y depend on the time t , as they are e i t h e r dependent va r i a b l e s (h, T g, T_^) or time-varying inputs. These f o r c i n g functions, which are d i r e c t l y r e l a t e d to processes e i t h e r acting at the boundaries or o r i g i n a t i n g beyond the boundaries, a l l occur on the r i g h t hand side of the equations. They are G, the rate of turbulent energy production by the wind stress acting at the upper boundary; D, the t o t a l d i s s i p a t i o n within the layer; R, the incident s o l a r r a d i a t i o n ; w, the imposed v e r t i c a l v e l o c i t y at the bottom of the layer; and B, H g, H g, the heat transfers at the a i r - s e a i n t e r f a c e . Such a set of coupled equations would not be expected to have a n a l y t i c solutions except i n s p e c i a l instances. Further, the properties of t h i s system of equations change d r a s t i c a l l y according to the value of dh dh the Heaviside step function (0 f o r w + — € Oj 1 f o r w + > 0). The model represented by equations (2-12), (2-19) - (2-21), while s i m i l a r to that of Kraus and Turner (1967), d i f f e r s i n several basic aspects. F i r s t , an a r b i t r a r y density p r o f i l e below the mixed layer i s allowed. Second, the penetration of s o l a r r a d i a t i o n below the la y e r has not been neglected. F i n a l l y , the e f f e c t of v a r i a t i o n s i n the rate of working by the wind stress r e s u l t i n g from synoptic s c a l e v a r i a t i o n s i n the wind can be investigated. In the following sections, I w i l l i l l u s t r a t e that these three properties are e s s e n t i a l f o r a quantitative model simulation of the time-dependent response of the upper mixed 19 l a y e r to Synoptic M e t e o r o l o g i c a l Scale i n p u t s . 2.3 R e l a t i v e Importance of the Surface Boundary Condit ions For any given set of boundary c o n d i t i o n s imposed at some time t , the system i s e i t h e r i n a Wind Dominated Regime or a Heat Dominated Regime, depending on whether the Heavis ide step f u n c t i o n H , i n equation (2-20), i s equal to 1 or 0. When H = 1, the entrainment mixing term i s i n c l u d e d i n the equations so they r e t a i n t h e i r o r i g i n a l form. Under these c o n d i t i o n s , water from below i s being entrained i n t o the mixed l a y e r as a r e s u l t of work being done by the turbulence against the buoyancy f o r c e s . When H = 0, however, the term represent ing the entrainment m i x i n g , (w + d h / d t ) ( T - T , ) , which contains the time s - h d e r i v a t i v e of the l a y e r t h i c k n e s s , does not appear i n the equations . Under such c o n d i t i o n s , s o l a r heat ing i s greater near the top of the l a y e r . Therefore , some of the a v a i l a b l e turbulent energy i s used to mix the l i g h t e r water downwards w i t h i n the surface l a y e r . Since the turbulence l e v e l w i t h i n the l a y e r i s assumed to remain constant , the net r e s u l t i s that i n s u f f i c i e n t turbulent energy may remain to mix the bottom i n t e r f a c e downwards. The l a y e r thickness thus remains r e l a t i v e l y constant during warming p e r i o d s . I f the combination of low winds and intense s o l a r heat ing should a r i s e , the o l d l a y e r i s l e f t unchanged and a new, shallower l a y e r of warm water i s superimposed on the o l d one. The Wind Dominated Regime The equations f o r the Wind Dominated Regime (when w + dh/dt > 0) are j u s t equations (2-19) and (2-20), where H has been set equal to 1 i n (2-20). Some i n t e r p r e t a t i o n of these equations i s necessary. F i r s t , the equations are coupled : h appears i n the equation f o r dT / d t , 20 and T appears i n the equation f o r dh/dt. Both equations, i n f a c t , s have h i n the denominator; As the la y e r deepens, the capacity f o r heat increases so that dT /dt decreases. With a thic k e r l a y e r , more t u r -s bulent energy i s l o s t i n d i s s i p a t i o n or used i n r e d i s t r i b u t i n g the heat inputs uniformly throughout the l a y e r . Thus, as the layer thickens, the energy a v a i l a b l e f o r mixing decreases, the rate of entrainment decreases, and the rate of thickening, dh/dt, also decreases correspond-i n g l y . The influence of the boundary conditions on the wind dominated layer i s more straightforward. The greater the temperature d i f f e r e n c e at the bottom of the lay e r , the greater the amount of work done i n entraining a given amount of water into the mixed l a y e r . Thus, the rate of thickening, dh/dt, i s inv e r s e l y proportional to the temperature di f f e r e n c e , T - T , , i n equation (2 -20 ). The wind energy a v a i l a b l e s ~n fo r mixing, G - D, acts to thicken the lay e r , and subsequently to cool i t upon entrainment of colder water from below. The heat f l u x losses at the upper boundary, - (B + H + H ), have a s i m i l a r , but much smaller, 6 S e f f e c t . In the event of very l i t t l e wind, one would expect the con-vec t i v e mixing energy derived from the surface heat losses to be almost balanced by the d i s s i p a t i o n , D. The e f f e c t of the s o l a r r a d i a t i o n i s s l i g h t l y more complicated since I t i s absorbed exponentially throughout the lay e r . Because i t represents a heat source, the e f f e c t of solar heating i s almost always to increase the temperature of the l a y e r , T . Provided that h i s large s enough (h > 2 y i s s u f f i c i e n t ) , the net e f f e c t of the s o l a r heating i s to s l i g h t l y decrease the rate of thickening of the l a y e r . As h, the thickness of the l a y e r , i s usually greater than 20 meters and Y - 1 » 21 the reciprocal of the extinction coefficient, i s usually less than 10 meters, the condition, h > 2 y _ 1> i s usually satisfied. Equations (2-19) and (2-20) can be integrated numerically by a Runge-Kutta technique for systems of ordinary differential equations; T . is evaluated from (2-12) and (2-21) at each time step and sub-—n stituted into equations (2-19) and (2-20). The number of equations describing the model could be reduced by one i f (2-21) were eliminated by definition of a new variable T = T g - T ^ . However, i t is preferable to leave the equations in a form where the measurable thermocline parameters are readily discernible. The Beat Dominated Regime dh The equations for the Heat Dominated Regime (when w + < 0) dh are simplified by the absence of the term involving -— in (2-20). By setting H = 0 in (2-20), one obtains a single nonlinear equation in h: (G - D) + Ry"1 ( 1 - e Y h ) = ^ h|R(l + e y n ) + B + H g + H g . (2-22) This equation can be solved numerically for h by a Newton's iterative technique (Henrici, 1966). Using the value of h so obtained, one can determine T by direct integration of (2-19) over the time step used s in the calculations: -yh, AT = =-s h R(l - e -Yh ) + B + H + H e s At (2-23) where (G - D) has been eliminated from (2-19) by use of (2-22). In equation (2-23), the time step At is small enough so that the boundary conditions can be put in as constant values over the time step. Under these conditions (H = 0), a new shallower mixed layer 22 c o n s i s t i n g of warmer water i s superimposed on the o l d l a y e r . The temperature immediately below the bottom i n t e r f a c e , T_^ ( t + A t ) , i s equal to the temperature of the mixed l a y e r a t the previous time, T g ( t ) , plus y R e Y ^ t + ^ f c ^ A t , the change due to s o l a r heating at that depth during the time At. Thus, a past h i s t o r y of the upward m i g r a t i o n of the mixed l a y e r i s imprinted on the temperature s t r u c t u r e below i t during any extended p e r i o d of intense s o l a r h e a t i n g . Transition Between Regimes The t r a n s i t i o n between a wind dominated and a heat dominated regime occurs smoothly ( i n e i t h e r d i r e c t i o n ) and i s a f u n c t i o n of the time dependent behavior of the boundary c o n d i t i o n s . In the numerical c a l c u l a t i o n s , the system i s assumed to be i n i t i a l l y i n the wind dh dominated regime. I f w + - j j - < 0 a f t e r the i t e r a t i o n s f o r a s i n g l e time step have been c a r r i e d out, then the c a l c u l a t i o n s f o r that step are redone w i t h the system i n the heat dominated regime. The time step used depended on the boundary c o n d i t i o n s ; u s u a l l y i t was one hour f o r the heat dominated regime. For the wind dominated regime, one minute i t e r a t i o n steps were necessary. According to equation ( 2 - 2 0 ) , + w > 0 f o r zero or negative net heat input i n t o the l a y e r . Thus, the system i s always i n the wind dominated regime under such c o n d i t i o n s . However, given p o s i t i v e heat input and any i n i t i a l s t a b l e d e n s i t y p r o f i l e below the l a y e r , i t i s not c l e a r whether there i s a given wind mixing energy production r a t e above which the system i s always i n the wind dominated regime. In dh equation ( 2 - 2 0 ) , consider l i m h-*>° W + dt f o r R + B + H + H > 0 . The e s dh entrainment mixing v e l o c i t y , w + — , i s greater than zero i f and only i f dt 23 2CG-D) , R + B + H e + H s hCT-T J ' T - T . a S h + " ' s -h s -h implying that G -*• 0 0 as h -> 0 0. Thus, for a positive heat input, there is no f i n i t e wind speed above which the system i s always i n the wind dominated regime. The given i n i t i a l value of h w i l l determine from which regime the system proceeds. If the system should be in the wind dominated regime i n i t i a l l y , the mixed layer w i l l thicken at a decreasing rate (for constant inputs), as w + dh/dt asymptotically approaches zero. 2.4 Solutions for the Wind Dominated Regime In this section, various solutions of the model in the wind dominated regime are presented for boundary inputs constant with time, and the sensitivity of the system to different values of the input parameters is investigated. The Concept of 'Mixing Energy ' The rate of turbulent energy input by the wind stress, M'W* dz - p W* , dz o -h Po 2 in equation (2-15) , z=0 J may be specified exp l i c i t l y i n terms of the wind stress, T. The rate of working by the wind stress at ten meters height i s given by — —3 — E = TIL- = p C._ U n where C i n i s the drag coefficient, IL n i s the a 10 a xu 1U 1U 1U mean wind speed, and p i s the density of a i r . Si On the assumption that the wind and wave fields are st a t i s -t i c a l l y stationary, the same wind stress T acts on the water below. A velocity scale appropriate to the underlying water i s then: 24 V* - C c / p / - (P a/P 0) , S c J Q U 1 Q • C2-24) This can be used to estimate the rate of turbulent energy transfer downwards some depth below the surface: E w * - ( p a / p o ) * TJ 1 0 T - (p a/p o)^ C* E a . (2-25) Putting P a/P Q = 0.0012 and C1Q = 0.0013, we find that E w - 0.0012 E f l . This constant ratio corresponds to the assumption of Turner (1969) that the turbulent energy available for mixing within the layer is produced at a rate that i s some constant fraction, m, of the rate of downwards transfer of turbulent energy from the wind f i e l d at ten meters height: - D A = m U 1 0 T = mEa . (2-26) From observed temperature profiles of the upper ocean taken before and after a storm, i t i s possible to obtain an estimate for m by calculating the rate of increase of potential energy in the water column. The sensitivity of the system to the value chosen for m is illustrated in the following sections. Analytic Solution For the simple case of no heat exchanges, no dissipation, and no imposed vertical velocity (R = B = H = H = D = w = 0), I have 6 S obtained an analytic solution for the wind dominated regime with a 3T constant wind and a linear temperature profile, L = TJ^ -, below the mixed layer. With the above assumptions, equations (2-19), (2-20) and (2-21) can be combined into a single equation: 25 h" + bh(h') 3 = 0 (2-27) where b = L/2G and ( )' = d( )/dt. An exact analytic solution can be obtained i f the substitution p = h 1 i s made in (2-27): & + bhp 2 = 0 • dh (2-28a) A single integration yields the equation: 2 = h'(bh + c x) (2-28b) and a second integration gives the following cubic equation in h as a function of t: o 3c.h h + — 7 - — — + 6t 3c, b - + — = 0 . (2-29) The constants of integration, c^ and C2» can be evaluated from the i n i t i a l conditions: 1-1 dh dt t=0 bh bh 2 > c l + (2-30) The cubic equation given by (2-29) i s already in the reduced form,x + px + q = 0, so that the roots can be evaluated directly by Cardan's Formulae (see Beaumont and Pierce, 1963). 3 2 When (p/3) + (-q/2) > 0, there exists one real root and a pair of complex conjugate roots. The appropriate root must necessarily be the real one in order that the real boundary conditions be satisfied: 26 The same equation a l s o gives the root of i n t e r e s t when = 0. For the case ( p / 3 ) 3 + C-q/2) 2 < 0, there are three s o l u t i o n s : h = 2 cos | + 120n (2-32) one f o r each of n equal to 0, 1, and 2, where <(> = cos" 1 -q / [-P3] h 2 / [27 J The ap p r o p r i a t e s o l u t i o n , the one w i t h n = 0, i s found by matching the s o l u t i o n w i t h that from (2-31) as ( p / 3 ) 3 + (-q/2) 2 0. For a t y p i c a l case, (2-32) i s the s o l u t i o n f o r times l e s s than s e v e r a l hours, but (2-31) becomes the s o l u t i o n f o r times greater than s e v e r a l hours. For l a r g e times, the expression (2-31) becomes (with the model parameters s u b s t i t u t e d f o r p and q ) : 3 12G b L V. J 3 3 t (2-33) This l i m i t i n g form provides i n s i g h t i n t o the dependence of the mixing model on c e r t a i n thermocline parameters. For l a r g e t , the depth depends on the o n e - t h i r d power of the mixing energy; thus, i t depends l i n e a r l y on the wind speed. This simple r e l a t i o n s h i p gives some t h e o r e t i c a l j u s t i f i c a t i o n to the e m p i r i c a l r e s u l t of Tabata, Boston, and Boyce (1965) that the depth of the mixed l a y e r i s l i n e a r l y r e l a t e d to the wind speed averaged over the previous twelve hour p e r i o d . That the depth depends on the r a t i o G/L i s not s u r p r i s i n g s i n c e G represents the mixing energy a v a i l a b l e , and the temperature gradient L i s 27 proportional to the buoyancy forces against which work is done. The expression (2-33) i s , in fact, identical in form to the following expression derived by Kato and Ph i l l i p s (1969) directly by a potential energy argument for a constant stress and a linear density profile: In the tank experiment of Kato and P h i l l i p s , there was no mixed layer i n i t i a l l y , only a linear density profile. In the present model, the layer thickness and the density difference at the bottom of the layer are independently specified as i n i t i a l conditions. At large times however, the i n i t i a l conditions become unimportant; thus, the two systems become equal in the asymptote. Numerical Solutions An analytic solution to (2-19) - (2-21) (with H = 1) could not be found for solar heating, or for time-varying inputs. It was there-fore necessary to integrate equations (2-19) and (2-20) numerically by a Runge-Kutta technique. The solution obtained numerically was checked against the analytic solution for the input parameters just discussed: both gave the same answers for time steps less than an hour. The results of the numerical integrations for the wind dominated regime with several values of input parameters given in Table I are presented i n Figure 3. The value used for the mixing energy i s appro-priate to a wind speed U 1 Q of about 12.5 m s e c _ i (except for Run #2). According to equation (2-33), one might expect the layer depth to be equally dependent on G, the rate at which energy from the wind (2-34) where N 2 = g-*r-. o °9z TABLE I Model r e s u l t s f o r the constant input mixing regime. - i s the wind energy a v a i l a b l e f o r mixing; and -B^ are the s o l a r and back r a d i a t i o n a t the sea s u r f a c e ; -3T/3z i s the temperature gra d i e n t immediately below the l a y e r ; T s t i s the sea surface temperature at time t ( h r s ) ; and h i s the thickness of the mixed l a y e r . Ts and h were 1 0 meters and 8 . 5 ° C . The e x t i n c t i o n c o e f f i c i e n t , y, was 0 . 0 0 2 cm" , and the drag c o e f f i c i e n t , C I N , was 0 . 0 0 1 3 . Run R* B* - 3 T / 3 z T S 2 4 T S 4 8 H 2 4 H 4 8 (ergs cm sec ) ( c a l cm " 2 A " I N day ) ( C ° m"1) ( ° C ) (meters) 1 3 . 0 5 0 0 0 . 0 3 8 5 7 . 7 5 7 . 5 3 3 7 . 1 4 7 . 8 2 6 . 1 0 0 0 0 . 0 3 8 5 7 . 5 3 7 . 2 6 4 7 . 8 6 1 . 2 3 . 3 . 0 5 4 0 0 - 8 0 0 . 0 3 8 5 7 . 9 0 7 . 7 8 3 4 . 5 4 3 . 1 4 3 . 0 5 0 0 0 . 0 1 9 2 7 . 8 6 7 . 7 1 4 3 . 9 5 7 . 8 29 Figure 3: R e s u l t s of the model f o r a constant wind. Run #1: G* = 3.05 ergs cm - 2 s e c - 1 , R* = B* = 0, 9 T / 3 z = 0.0385 C° m - 1. Run #2: G* = 6.1 ergs cm - 2 s e c - 1 . Run #3: R* = 400 c a l cm - 2 d a y - 1 , B* = -80 c a l cm - 2 d a y - 1 . Run #4: 8 T / 9 z = 0.0192 C° m - 1. In Runs 2-4, parameters not mentioned above are unchanged from Run #1. G* = 3.05 ergs cm 2 s e c - 1 corresponds to a mean wind speed of 12.5 m sec 1. 30 becomes available for mixing, and on L \ the reciprocal of the temperature gradient below the layer. However, doubling the wind energy (Run #2) and halving the slope of the temperature profile (Run #4) do not both result in the same behavior of the solution at small times. The behavior of the layer at small times can be examined by substituting the i n i t i a l values of the parameters into the inter-mediate analytic solution (2-28a). We obtain £ ^ . (2-35) a t L(h 2 - h 2) + 2h f"T (0) - T , (0)~| o o[_ s -h J Thus, on the assumption used to obtain (2-27), the rate of deepening i s directly dependent on the mixing energy but not on the slope of the temperature gradient below; thus, doubling the rate of production of turbulent mixing energy increases the i n i t i a l rate of deepening while halving the slope has no effect. After only a few hours, the rates of deepening in the two cases (Runs #2 and #4) are about equal. In Run #1, the rate at which turbulent energy from the wind becomes available for mixing within the layer, G A - = -m U^Q T , has been scaled with m set equal to 0.0012. This corresponds to a scale velocity, HIU^Q, equal to WA, the f r i c t i o n velocity in the water (defined in equation (2-26)). According to (2-33), doubling m, i.e. doubling the rate of availability of mixing energy, increases the limiting depth of the mixed layer by 26%. As the thickness of the mixed layer during a storm can be determined from observations to within 20%, this result indicates that physically r e a l i s t i c solutions are obtained only for values of m differing by less than a factor of 2. Kato and Phillips (1969) obtained an empirical value for m of 0.0015 31 with an uncertainty of about 30%. The empirical value m = 0.01, estimated by Turner (1969) from two cases of salinity-temperature-depth profiles in the open ocean, i s too large to allow this model to yield r e a l i s t i c deepening. The effect of heating on the wind dominated regime is i l l u s t r a -ted by Run #3. During a typical summer day in mid-latitudes, the solar -2 -1 radiation i s about 400 cal cm day and the surface back radiation i s -2 -1 about -80 cal cm day . With these input values, the results of Run #3, which are given in Table I, indicate that the deepening i s only slightly less (4.7 meters less after 48 hours) than for no heat exchanges. This difference i s well within the uncertainty of most contemporary measure-ments of the depth of the mixed layer. The corresponding increase in sea surface temperature T g, for the case with heating, i s 0.25 C°; such a trend may be observable from ordinary bucket measurements. During the winter, large evaporative heat losses at the sea surface often accompany high winds. The resulting convective mixing might have a more appreciable effect on the wind dominated regime than the case just investigated in Run #3. 2.5 Solutions for the Heat Dominated Regime The heat dominated regime i s represented by equation (2-22) in which there i s no explicit time dependence. Thus, for inputs constant in time, equation (2-22) shows that the mixed layer thickness, h, i s also constant in time. The mixed layer temperature, T g, obtained from (2-23), increases linearly with time. With the assumptions H = H = w = 0, calculations were carried 6 S out to determine the sensitivity of the heat dominated regime to the relative rates of solar heating, R, and long wave radiative cooling, -B. The effects of including or neglecting the solar heating below the layer, and of varying y» the extinction coefficient of solar radiation in sea water, were also examined. The results of these calculations, where the wind speed was 4 m sec are summarized in Table II. In Runs #1 and #3, the mixed layer depth was calculated f i r s t where the solar heating had been confined to the upper layer, and second where the extinction term had been retained in the layer below. Where the radiative transfer below the mixed layer was neglected, the mixed layer thickness was over-estimated by about 15%. Comparison of the depths obtained in Runs #1 and #3 shows that the back radiation must be included exp l i c i t l y in the model. In Run #1, a l l the heat exchange was by solar heating; In Run #3, the same total heat exchange was composed of solar heating penetrating the whole layer and of long wave radiational cooling from the sea surface. The net result of specifying the back radiation separately was to increase the thickness of the mixed layer by about 25%. In Run #2, the extinction coefficient, y, was 0.001 cm half the value in Run #1. Decreasing y by a factor of two (allowing the solar radiation to penetrate to greater depths) increased the mixed layer thickness by 70%. Kraus and Turner (1967) concluded that their calculated mixed layer was too deep, a result which may be attributed, at least partially, to their neglecting the radiative heat input below the mixed layer. Their extinction coefficient, y = 0.0005 cm was smaller than the values used here, which would also result in a larger mixed layer thickness. However, they were interested in TABLE II Heating regime with constant inputs. The wind speed was 4m sec corresponding -2 -1 to a mixing energy GA - = 0.123 ergs cm sec . R^  and -B^ are the solar and back radiation at the sea surface; y is the extinction coefficient of the solar radiation; h^ is the mixed layer depth when the solar radiation below the layer i s neglected; and h_ is the mixed layer depth when i t i s included. Run R* B* Y h l h2 (cal cm 2 day ^ ) (cm 1) (meters) 1 350 0 0.002 14.1 12.2 2 350 0 0.001 24.1 3 440 -90 0.002 16.7 15.2 34 a subtropical ocean; here, we are interested in the subarctic water found at Ocean Station 'Papa'. According to Figure 51 of Jerlov (1968), a subtropical ocean tends to be optically more transparent than a mid-latitude ocean, and thus has a smaller extinction coefficient. The results of this section indicate that, at low wind speeds, the upper ocean is sensitive not only to the total amount of heat energy received, but also to the relative amounts of heat transferred by solar and back radiation. The depth of the layer is sensitive to the value used for y, the extinction coefficient of solar radiation, so this parameter must be well known. Further, with the shallow mixed layer characteristic of a heat dominated regime, the small amount of solar radiation which does penetrate below the layer cannot be neglected. 2.6 Solutions for Diurnal Heating In this section, diurnally varying solar heating is applied to the model: the resulting response of the mixed layer is illustrated for different wind speeds. Typical Daily Cycle The solar radiation at Station 'Papa' is routinely measured with a pyranometer and presented as hourly integrated values. From the Canadian Monthly Radiation Summary (1970), the average daily solar _2 radiation at 'Papa' in May and June, 1970 was 374 and 359 cal cm day For input into the model, the daily solar radiation was 400 -2 -1 ,cal cm day composed of hourly values distributed as shown in Figure 4. Using the tables of Laevastu (1960, 1963), I have calculated twelve hour averages for back radiation at 'Papa', which have an overall -2 -1 average of -82 cal cm day for May and June, 1970. A constant rate Figure 4: Results of the model with diurnal heating for 30 hours. The solar radiation and back radiation per day are 400 and -80 cal cm - 2 day - 1. -The wind speed i s 4 m sec" . £ 36 -2 -1 of back radiation, B A = -80 cal cm day , has been applied to the model. With the large relative humidities Cover 85%) and the small air-sea temperature differences usually observed at 'Papa' during the summer months, the turbulent heat flux, + Hg, is usually less than half the back radiation, so has been ignored. The resulting behavior of the upper mixed layer for a 30 hour period i s presented in Figure 4: the wind speed was constant at 4 m sec ^ . The model requires that an i n i t i a l mixed layer thickness, h, be specified. In this case i t was 70 meters, below the range of depths shown in Figure 4. The profiles plotted every two hours show that a shallow layer of warm water b u i l t up during the day un t i l about 1600 local time. At that time, the solar heating had decreased sufficiently for the layer to start mixing downwards. The layer descended from about 8 meters, reaching a maximum depth of about 16 meters at 0600 local apparent time. During the second day, a new warm layer again started to build up superimposed on the previous day's layer. The structure below was affected only slightly by the small fraction of solar radiation penetrating below the active mixed layer. The step structure appearing below the mixed layer resulted from the hourly steps in solar heating. Effects of Wind Speed Results of the diurnal heating model for different wind speeds are shown in Figure 5 and summarized in Table III. Each run was for a 4 day period. The heat input was the same as in Figure 4, but the wind speed was 0, 4, and 8 meters per second. The temperature profiles, plotted every 6 hours at the bottom of the figure, are for the 37 Q LU X — I 5 2 u. «> O J= I S O -I I t -8-5 801-7-5 0 10 20 .30 40 50 60 RUN Uio (m sec"') 1 0 2 4 3 8 C O 7 9. 36 48 60 TIME (HOURS) CL j " loJ O J Q E 0 20-40-60 F i g u r e 5: Res u l t s of the d i u r n a l heating f o r 4 days w i t h d i f f e r e n t wind speeds. Run #2, f o r which the temperature p r o f i l e s are p l o t t e d , i s the same case i l l u s t r a t e d i n Figure 4. The heat input i s i d e n t i c a l w i t h t h a t i n Figure 4. TABLE III Model results for the diurnal heating at various wind speeds. U^Q i s the wind speed: h . , h are the minimum and maximum thicknesses of the min max diurnally varying mixed layer; h c is the mixed layer thickness for constant heating; ATgg^r x s the increase in sea surface temperature after 96 hours; A^jiurnal i s the amplitude of the diurnal variation (with the trend removed) of the sea surface temperature. The extinction coefficient, y, was 0.002 cm \ and the available mixing energy coefficient, m, was 0.0012. U10 h . min h max h c AT 96hr AT diurnal Run (m sec ) (meters) (C degrees) 1 0.0 3.86 7.92 7.90 1.27 ± 0.27 2 4.0 8.64 16.28 16.24 0.80 ± 0.19 3 8.0 21.86 52.49 52.46 0.26 ± 0.09 LO CO intermediate wind speed of Run #2 (which corresponds to the case shown in Figure 4). The traces in the middle plot of Figure 5 represent the bottom interface depth, z = -h, as a function of time. While the interface oscillated over a range of about 30 meters for a wind speed, of 8 m sec * (Run #3), the actual temperature difference, T g - T_^, at the active interface was only of the order of 0.1 C°. The profiles for Run #2, plotted at the bottom of Figure 4, i l l u s t r a t e the behavior of the whole layer more clearly. Notice that while the mixed layer thickness oscillated markedly during each day, the formation of the main temperature step (at about 16 meters) occurred at the maximum depth to which the diurnal effects penetrated. As one would expect, at higher winds the layer penetrated to greater depths and i t s thickness had a marked diurnal oscillation, but the mean temperature increase and the diurnal variation in sea surface temperature were correspondingly smaller. The layer depths for the three runs have also been calculated for the same daily solar heat input but put into the layer at a constant rate. The mixed layer thicknesses for constant solar heating are given in Table III under h^. The maximum layer thickness for diurnal heating was about the same as the thickness for constant heating. In the diurnal case, when the maximum depth was reached at 0600, the total heat input up to that time had been less than for the constant heating case. The results for the zero wind case (Run #1) must be inter-preted with great caution. During the night, the solar radiation R^  is zero so that for zero winds, only the back radiation at the sea surface, -B^ ., drives the mixing. Under these conditions, equation (2-20) for the rate of thickening of the mixed layer becomes: 40 dh -2D - hB dt ~ h(T - T , ) s -h C2-20a) This equation indicates that the layer thickens at a rate dependent upon the amount of convective energy (generated by -B) that i s not dissipated within the layer. However, through the parameterization of the mixing energy generated by the wind stress (equation (2-26)), the dissipation D was also set equal to zero when the wind speed was assumed to be zero. While this parameterization is effective for larger wind speeds (when the convective energy is negligible compared to the wind generated mixing energy), the dissipation i s almost certainly important in balancing any convective mixing at zero wind speed. The magnitude of the convective mixing energy generated by the back radiation can be estimated from the mechanical energy equation (2-15) into which (2-16) has been substituted. The mechanical energy generated by the back radiation, -B^ i s just: E p ag c o JTYT d z = _ OghB* -h B c p For h = 103cm, g - 10 3 cm sec 2, c p = 1 cal g" 1 °K _ 1, and a - -0.13 x 10~3 V 1 (for T ^ 8°C, S - 33°/ 0 0) ; a back radiation = -80 cal cm 2 day 1 (- -1.0 x 10 3 cal cm 2 sec *) generates a mixing energy of the order of 0.1 ergs cm 2 sec 1. According to the formulation of the mixing energy generated by the wind stress used in this model (equation (2-26)), this estimate of the convective mixing energy i s roughly equal to the mixing energy generated by a wind of 4 m sec 1. The total dissipation i n the upper 10 meters can be estimated from previous observations. Stewart and Grant (1962), and Grant, Moilliet and Vogel (1968) obtained values for the dissipation 41 which yield an estimate for the total dissipation within the upper 10 meters of at least 1 erg cm 2 sec 1. While their measurements were at wind speeds considerably greater than 4 m sec 1 , the dissipation obviously i s comparable with the convective energy estimate. In the case discussed here, the effect of neglecting dissipation at low wind speeds was to give a much larger than expected diurnal oscillation in both sea surface temperature and mixed layer thickness. The diurnal oscillation (with the trend removed) obtained in Run #1 for zero wind speed was ± 0.27 C°, a difference which would be d i f f i c u l t to measure with any confidence in the open ocean. The diurnal o s c i l l a t i o n for 8 m sec 1 winds was ± 0.09 C°, a difference which definitely would not be observable in the open ocean. It is not surprising, then, that series of sea surface temperature measurements, taken every three hours at Station 'Papa', rarely have a discernible diurnal variation. The upper mixed layer i s not well-defined at very low wind speeds because the turbulent energy that i s produced is mostly dis-sipated within the layer. Without a well-defined mixed layer, horizontal "patchiness" in the sea surface temperature and large gradients near the surface probably develop. Such effects cannot be predicted by a model such as the one developed here. 2.7 Conclusions of the Theoretical Model During the wind dominated regime discussed in Section 2.4, the fraction of wind stress energy available to do work against the buoyancy forces i s instrumental in determining both the rate of deepening of the mixed layer and i t s eventual thickness. The density stratification which exists i n i t i a l l y below the mixed layer has just as 42 great an influence as the wind energy on the eventual depth of pene-tration of the layer. However, the rate at which mixing energy becomes available varies as the third power of the wind speed: changing the wind speed by a factor of 2 has 8 times the effect on the eventual depth as changing the slope below the layer by a factor of 2. For stable summer conditions, heat inputs are relatively unimportant at high wind speeds. For unstable winter conditions when the heat loss at the sea surface may be as large as -400 cal cm sec , the convective mixing energy may become comparable with the wind mixing energy. Some parameterization, based on observations, of the pene-trative convection similar to that of Turner (1969) for the wind mixing would then be required by this model. Under the conditions of low wind speeds and substantial heating discussed in Section 2.5, the layer thickness becomes extremely sensitive to the value of the extinction coefficient. Surface heat exchanges and radiative heating below the mixed layer are of lesser importance in determining the mixed layer thickness for the conditions which usually exist at Station 'Papa' during May and June. The limitations of the model for very low wind speeds^ discussed in Section 2.6, do not severely r e s t r i c t i t s usefulness in describing the seasonal thermocline in the open ocean. At Ocean Station 'Papa', the average wind speed during the summer months is s t i l l 7 or 8 m sec 1 (Figure 9) which, in this model, represents a flux of energy available for mixing of about 1 erg cm sec , roughly ten times the flux of convective mixing energy generated by a surface back radiation of -80 cal cm 2 day 1. The model predicts that the expected diurnal variations in 4 3 sea s u r f a c e temperature i n the open ocean are u s u a l l y too small to be observable. Only during very low winds, w i t h instruments that have a r e l a t i v e accuracy of b e t t e r than ± 0.05 C°, would one expect to observe d i u r n a l v a r i a t i o n s i n the sea surface temperature. Even then, such v a r i a t i o n s are l i k e l y to be p a r t i a l l y masked out by the h o r i z o n t a l " p a t c h i n e s s " which occurs i n low wind s i t u a t i o n s where there i s no w e l l - d e f i n e d homogeneous mixed l a y e r . D i u r n a l e f f e c t s i n the open ocean are not as pronounced as the e f f e c t s caused by the h i g h wind speeds a s s o c i a t e d w i t h synoptic s c a l e m e t e o r o l o g i c a l disturbances. Chapter 3 OBSERVATIONS AND ANALYSIS 3.1 Introduction Routine oceanographic and meteorological observations have been taken every 3 hours at Ocean Station 'Papa' for over ten years. Although these observations have been used in previous studies of the upper ocean, more refined measurements are required to delineate the physical processes involved. F i r s t , continuous micrometeorological measurements are necessary to assess the mechanical energy input to the ocean. Second, more frequent sampling of the upper ocean is essential for investigation of the time dependent behavior of the upper mixed layer. The time of year best suited for a study of this type was determined from the routine data and from previous studies. The Canadian Meteorological Service (now Atmospheric Environment Service) made the routine observations immediately available during the cruise and arranged for the output from a wave gauge to be recorded continuously for the duration of the cruise. The Canadian Marine Sciences Branch assisted i n the program by providing expendable bathythermographs and a Bissett Berman 9040 STD so that hourly soundings of the upper ocean could be obtained during storms. In this chapter the previous studies are summarized, and the data obtained for this study at Ocean Station 'Papa' during May and June 1970 are presented. 45 3.2 Routine Observations at the Weathership A weathership has been located at Ocean Station 'Papa', 50°N, 145°W (as i n Figure 6), since 1950. The Canadian Meteorological Service has conducted routine marine meteorological observations at the surface as well as upper a i r soundings by radiosonde. Presently, they have two sister ships, the CCGS Quadra and the CCGSVancouver (shown in Figure 7). The routine surface observations taken every 3 hours include atmospheric pressure; wind speed and direction; a i r , sea, wet bulb and dew point temperatures; visual wave height and direction; and cloudiness. The routine surface observations relevant to this study, for the period 18 May to 28 June 1970, are presented later i n Figure 13. In addition to the visual estimate, the waves are recorded intermittently in chart form from an NIO wave measuring device (Tucker, 1956). The incoming solar radiation is measured continuously by a Kipp CM3 pyranometer located on the forward mast of the ship (see Figure 7). Since 1956, the Canadian Marine Sciences Branch has conducted an oceanographic program from the weathership. Mechanical bathythermo-graphs every 3 hours, bottle casts to 4200 meters every week, and salinity-temperature-depth (STD)profiles twice a week are the basic physical oceanographic measurements of this program. The standard oceanographic measurements for the cruise period 15 May - 1 July, 1970, along with the STD traces obtained especially for this study, are presented in a Marine Sciences Branch Data Report (Minkley, 1971). 4 6 4 7 Figure 7: The Canadian weathership CCGS Vancouver. The l o c a t i o n of the instrument mast on which the pro p e l l e r anemometers were mounted i s indicated. 48 3.3 Station 'Papa' Background Ocean Station 'Papa' i s located in the eastern part of the Subarctic Pacific Region on the southern edge of the Arctic Gyre. According to Thomson (1971), typical geostrophic currents are in an ENE direction, roughly parallel to surface isotherms and isopycnals, with a speed of the order of one cm sec 1.. The largest currents to be expected are those associated with storm-induced i n e r t i a l o s c i l l a -tions. Hasselmann (1970) estimates such currents to have speeds of the order of 10 cm sec 1. Influence of Advection Typical surface temperature gradients in the NE Pacific are about 1°C/100 km. For currents of 1 to 10 km per day, the maximum temperature change to be expected from advective effects would be about 1 C° in ten days. Because the mean flow tends to be along rather than across isotherms, the actual advective effects are probably at least an order of magnitude smaller. From geostrophic transport calculations, Tabata (1965) attempted to estimate the temperature and salinity changes at 'Papa' attributable to horizontal advection. He concluded that the average monthly temperature change at 'Papa' due to advective effects for a five year period was 0.26 C° month 1, with a maximum of 0.78 C° month 1. The average monthly salinity change was 0.02 °/oo month with a maximum of 0.05 °/oo month 1 . During the heating season, changes in sea surface temperature associated with synoptic scale weather patterns are of the order of 1 C° over several days. This i s at least an order of magnitude greater than changes expected from advective effects over the same time period. 49 Therefore, one does not expect horizontal advection to affect s i g n i f i -cantly the heat budget of the upper layer of the ocean for periods less than several weeks. Vertical Structure Tabata (1961, 1965) and Thomson (1971) examined the vertical structure of the ocean at 'Papa' i n some detail. In any dynamic process in the ocean, i t i s the density structure, as determined by the temperature and salinity through the equation of state, which significantly influences the sequence of physical events. At 'Papa' the main pycnocline occurs between 100 and 200 meters: there, the salinity increases from 32.8°/oo to 33.8°/oo, the temperature decreases from 4.5 to 4.0 °C, and cr increases from 26.0 to 26.8 (see Figure 8). The density in this region is determined primarily by the salinity. In summer, the seasonal thermocline forms in the upper 75 meters: there, the temperature variations control the density variations. Typical summer variations in the upper layer as a result of synoptic scale meteorological influences are AT = 2 C° and AS = 0.06 °/oo. For mean values T = 7 °C and S = 32.7 °/oo, the resulting variations in sigma-t are Aafc = -0.286 and +0.047. The temperature effect i s obviously larger. In the winter months, the salinity variations associated with the large evaporation at the sea surface may become significant. With increasing depth below the main thermocline, the gradients and curvatures of the temperature and salin i t y traces in Figure 8 decrease. Below about 125 meters, exponential curves could be f i t t e d to the traces. The vertical structure in and below the 50 Figure 8: An STD trace taken at Station 'Papa' at 2100 GMT 23 June, 1970. The hatched areas represent the change i n T, S, and at that occurred since 19 May 1970. 51 main thermocline is determined by the large scale thermohaline circula-tion. An examination of the influence of the upper boundary conditions on the thermohaline circulation was carried out by Needier 0.971). Time Dependent Behavior The annual growth and decay of the seasonal thermocline in the open ocean is well understood: Tully and Giovando (1963) described the annual cycle qualitatively, and Kraus and Turner (1967) (also Turner and Kraus, 1967) modelled i t both theoretically and in the laboratory. Effects correlated with the passage of synoptic scale weather systems (periods from 1 to 5 days) have not, however, been adequately investi-gated. Tabata (1961, 1965) studied the heat budget of the upper layer of the ocean at 'Papa' on time scales greater than a month. The average solar radiation input varies from about 50 cal cm 2 day 1 in December to over 300 cal cm 2 day 1 in June. Surface back radiation is relatively constant over the whole year at-80 to-90 cal cm 2 day 1. The average turbulent heat losses due to evaporation and conduction, which Tabata estimated using the standard bulk aerodynamic formulae of Jacobs (1951), vary from over -150 cal cm 2 day 1 during the winter months down to about-30 cal cm 2 day 1 during May and June. Ten Year Means In Figure 9 are plotted ten year daily means of the wind, the air-sea temperature difference, the temperature difference between wet and dry bulbs, and the sea surface temperature. The cruise period of interest l i e s between the heavy vertical lines. The ten year means support the findings of Tabata: May and June are months of relatively TEN YEAR DAILY MEANS FOR 1958-67 20-1 Wind 10-(m sec-') o Tair- Tsea (C°) ^/•^-.^.••.jt.,'.. ivH''v^..'4,''1 Tair "^ wet (C°) Time (months) Figure 9: Ten year daily means of routinely measured meteorological quantities at Ocean Station 'Papa' for 1958-1967 inclusive. The sampling interval was three hours. 53 low winds, s m a l l a i r - s e a and wet-dry temperature d i f f e r e n c e s , and a l a r g e p o s i t i v e time r a t e of change i n sea s u r f a c e temperature. Thus, the evaporative heat l o s s i s a minimum then, and the conductive or s e n s i b l e heat t r a n s f e r i s a c t u a l l y of the opposite sign, f u r t h e r reducing the net t u r b u l e n t heat l o s s at the s u r f a c e . As the sea surface temperature increases almost continuously at that time of year, the l a y e r must be e s s e n t i a l l y s t a b l y s t r a t i f i e d . From these previous s t u d i e s , the f o l l o w i n g t e n t a t i v e conclusions can be drawn about the a i r - s e a i n t e r a c t i o n at Ocean S t a t i o n 'Papa' during the summer months. F i r s t , the upper mixed l a y e r i s most s t a b l e from May to August, i n d i c a t i n g that the mixing i s e s s e n t i a l l y a wind generated r a t h e r than a convective phenomenon. Second, the t u r b u l e n t f l u x e s of heat are a minimum then, u s u a l l y s e v e r a l times s m a l l e r than the back r a d i a t i o n , and I have ignored them. T h i r d , because the r a t e of h e a t i n g i s g r e a t e s t i n the summer months, the temperature f l u c t u a t i o n s i n the upper l a y e r a s s o c i a t e d w i t h passing weather disturbances have the l a r g e s t ' s i g n a l to n o i s e r a t i o ' of any time during the year. F i n a l l y , the work of Tabata i n d i c a t e s t h a t the advection of heat by ocean currents i s not important f o r time s c a l e s much l e s s than one month. 3.4 Oceanographic Time S e r i e s Observations During the p a t r o l at Ocean S t a t i o n 'Papa' i n May-June, 1970, a 6 week long s e r i e s of STD p r o f i l e s was obtained. The upper 150 meters of the ocean was sampled 8 times each day during storms and twice a day otherwise. S i p p i c a n XBTs (expendable bathythermographs) were a l s o r e l e a s e d at one hour i n t e r v a l s during storms. These s e r i e s i l l u s t r a t e the time dependent behavior of the seasonal thermocline 54 as i t was affected by intense summer heating and by the large variations in mean wind speed that accompanied synoptic scale weather disturbances. Isopleth Contours The outputs from the STD observations to 150 meters were digitized at one meter intervals. Contours of isopleths of temperature, sali n i t y , and sigma-t [- 103 ( p - l ) ] were drawn i n a depth versus time graph with a standard contouring program; these are shown in Figures 10, 11, and 12. The sigma-t values were calculated from temperature and salini t y by use of the empirical formulae given in the US Naval Hydrographic Tables (1952). The time of each observation is indicated by a vertical line immediately above the time axis. No smoothing has been done. The relevant surface meteorological observa-tions for the same 6 week period (including the wind speed shown in Figure 12) are plotted in Figure 13. Notice, f i r s t , that the isopleths between 40 and 80 meters, particularly of temperature and sigma-t^ converge with time during the period of high winds up to May 25. At the end of this period, the upper mixed layer was essentially homogeneous i n both temperature and sigma-t down to sixty meters, not unlike the upper layer during winter conditions. After May 25, the seasonal thermocline started to form: shallow "tongues" of warm water built up during relatively calm periods only to be mixed down to about 45 meters during each storm. The seasonal trend, evident in both temperature and sigma-t, i s re l a -tively absent i n the salini t y contours. Therefore, slow fluctuations in sigma-t above the permanent halocline (100-150 meters) are essentially of thermal origin. The data plotted in Figure 8 support this TEMPERATURE in II II II i II II II in II II II II II inn III i i i i II i i i i i I  i II i i i i II iiiiiiinii III i i i i i i mi i mini II i i i i i i i i i i — i — i — i — I — i — I — i — i — i — i — i — i — i — I i ! i i i i ' • i • i i i i . . I . I . i 18 22 26 30 11 5 9 13 17 21 25 29 MAY . JUNE 1970 TIME (GMT) Figure 10: Contours of constant temperature at Station 'Papa' for the period 19 May to 28 June 1970. The traces were obtained with a Bissett-Berman 9040 STD at the times marked by vertical bars just above the time axis. Ul C L L U Q 20 40 60 80-100-120-140-SALINITY 32.70 /32.90 —33.00 l! U j v \ ) ^ ^ V ^ - - 3 3 . 6 0 III i i ii ii i i i ii ii III ii ii i i ii ii mn mi i i i II i i i i i n i i i i i II IIIHIIIIINI in i i i i i i minium n i i i i i i i i i i - i — i — i — i — i — i — i — i — i — i i i i I i i • i • i • i • i i i i i • i • i • i •- i • i _i L 18 22 26 MAY 30 '1 9 13 17 21 25 29 TIME (GMT) JUNE 1970 Figure 11: Contours of constant s a l i n i t y . WIND S P E E D SIGMA-T — 2 6 . 6 i n 11 i i i i i 11 i i i i III i i i i 11 i i i i m n in I i i M I i i i i i n i i i i i i i -L—J 1—i 1 1—I 1—I 1 1 1—1 i I I I ' I . I . I III i i i I i i i l l i n i u m n i i i i i i i i J—i I i I i I i I i I i I i l i I i i 18 22 26 MAY 30 '1 13 17 JUNE 21 5 9 TIME (GMT) Figure 12: Contours of constant sigma-t. Routine three h o u r l y wind speed. 25 29 1970 Figure 13: Standard marine m e t e o r o l o g i c a l parameters measured every three hours at S t a t i o n 'Papa' by the Canadian M e t e o r o l o g i c a l Branch f o r the p e r i o d 18 May to 28 June, 1970. 59 statement: the hatched areas represent the changes near the surface from 19 May to 23 June, 1970. The s a l i n i t y change a f f e c t s sigma-t about one e i g h t h as much as the change i n temperature a f f e c t s i t . The frequency of sampling w i t h the STD was greater during the three h i g h wind periods 5-6 June, 15-17 June, and 21-23 June. That the contours show much more s t r u c t u r e during these periods i s obvious; during the periods of twice d a i l y sampling, there must have been some a l i a s i n g of higher frequency o s c i l l a t i o n s . Figure 14 i l l u s t r a t e s the extent of t h i s a l i a s i n g f o r the storm p e r i o d 22-23 June, 1970. Isotherm contours were p l o t t e d from the h o u r l y XBT p r o f i l e s i n e x a c t l y the same manner as from the STD p r o f i l e s . High frequency a l i a s i n g i n t h i s p l o t i s evident only around 200 meters depth; there, i t r e s u l t s from the temperature change i n 20 meters being l e s s than the temperature accuracy of the measurements. Near the seasonal thermocline, the major f l u c t u a t i o n s occurred over periods of 8 to 9 hours. From 0000 to 1200 GMT on June 22, the mixed l a y e r thickened markedly and i t s temperature decreased as c o o l e r water from below was entrained upwards i n t o the l a y e r . The s t r a t i f i c a t i o n at the bottom of the l a y e r became more in t e n s e , as i s i n d i c a t e d by the convergence of the isotherms between 20 and 60 meters during the time up to 1200. As the B r u n t - V a i s a l a p e r i o d at the bottom of the mixed l a y e r was l e s s than 10 minutes, high frequency a l i a s i n g of i n t e r n a l waves w i t h periods l e s s than an hour must be present i n F i g u r e 14. However, the smoothness of the curves i n d i c a t e s that these were s m a l l compared to the f l u c t u a t i o n s of 8 to 9 hours d u r a t i o n . The s i x week long s e r i e s of Figure 10 i n d i c a t e s that the seasonal and s y n o p t i c s c a l e e f f e c t s f o r periods longer than a day are much l a r g e r and thus e a s i l y d i s t i n g u i s h a b l e X B T I S O T H E R M S 8.3 8.2 8.1 8.0 °C I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I J 1 1 1 1 1 I , I I ' ' I ' ' I ' i 0 0 0 6 12 18 0 0 0 6 12 22 JUNE 23 JUNE 1970 T I M E ( G M T ) Figure 14: Contours of constant temperature at Station 'Papa' for the period 0000 GMT 22 June to 0900 GMT 23 June 1970. The traces were obtained hourly (as indicated by the vertical bars) with Sippican XBTs. 61 from the s h o r t e r p e r i o d f l u c t u a t i o n s . 3.5 A n a l y s i s of Mi c r d m e t e o r o l o g i c a l Data Introduction In order to s p e c i f y the l a r g e s c a l e dynamics of the ocean, f l u x e s of momentum, heat and moisture across the a i r - s e a i n t e r f a c e must be estimated from bulk aerodynamic formulae (Jacobs, 1951) which s p e c i f y these f l u x e s i n terms of r o u t i n e l y a v a i l a b l e mean wind speed, and a i r - s e a temperature and humidity d i f f e r e n c e s . Such formulae r e q u i r e well-founded constants or a d j u s t a b l e c o e f f i c i e n t s that depend on s t a b i l i t y , which i n tu r n may be defined i n terms of mean wind speed and the a i r - s e a temperature d i f f e r e n c e . Recently, a great amount of e f f o r t has gone i n t o attempts to measure these f l u x e s d i r e c t l y . For the most p a r t , these s t u d i e s took place e i t h e r over s h e l t e r e d c o a s t a l waters, or from q u a s i s t a t i o n a r y platforms such as instrumented a i r p l a n e s (Miyake, Donelan, and M i t s u t a , 1970) and FLIP (Pond et a l , 1971). No b e l i e v a b l e measurements have been made i n open ocean sea c o n d i t i o n s f o r wind speeds much over 10 meters per second. Attempts at such measurements continue to be made by groups such as those at Hamburg (Brocks and Krugermeyer, 1970); at C.S.I.R.O., A u s t r a l i a (Hicks and Dyer, 1970); at Moscow (Volkov, 1970); and at H a l i f a x (Smith, 1970). The importance of high wind c o n d i t i o n s i n determining the wind s t r e s s i n the open ocean must be emphasized. In m i d - l a t i t u d e s , most of the momentum t r a n s f e r from the atmosphere to the open oceans occurs at wind speeds greater than ten meters per second. An average wind speed of 10.1 m sec * was determined from the synoptic observations 62 at Ocean Station 'Papa' for the period 1958-1967 inclusive. The stress, however, i s proportional to the square of the mean wind speed. The averaging time to be used must be of the order of hours i f a l l the effects associated with the four day synoptic peak found in the horizontal wind spectra (Figure 1 in Millard, 1971) are to be included. The long term (say monthly or yearly) average i s strongly weighted by contributions from higher wind speeds and is considerably larger than a stress based on monthly or annual averages of the wind. For example, during the six week period at Station 'Papa' from 18 May to 28 June 1970, the average wind speed from the routine three hourly -1 . 2 - 2 observations was 9.7 m sec with a square of 94.2 m sec . The mean 2 -2 of the squares of the three hourly observations was 150 m sec . If the drag coefficient i s assumed to be constant, the latter mean square estimate would represent a stress over 50% greater than one calculated from the average wind. In this section, micrometeorological data obtained during the cruise at Station 'Papa' w i l l be analyzed to determine the variation of the stress, drag coefficient, and wave energy i n relation to the synoptic weather system. Two periods associated with the passage of high wind speed disturbances are examined. Experimental Setup Horizontal and vertical propeller anemometers were mounted on the forward mast of the ship at a height of about 21 meters above the water surface (Figure 15). While this was not an ideal location to measure wind free of influence from the ship's superstructure, i t offered the best alternative on the ship and had exposure for 180 degrees 6 3 Gill propeller anemometers Figure 15: The instrument array which was mounted on the forward mast as indicated in Figure 7. The horizontal and ver t ica l G i l l propeller anemometers were used in this study. The instrument on the right , a one dimensional sonic anemometer, was not used in this study. of direction relative to the ship. At this location, the mean winds should not suffer too much from blockage effects as the instruments were placed at a higher level than any part of the ship's superstructure except for masts and a spherical radar dome which was well aft of the instrument mast. A pitch and r o l l gyroscope and a vertical accelero-meter of the geophysical type (Miyake, Donelan, and Mitsuta, 1970), as well as a shipborne NIO wave recorder (Tucker, 1956) were located inside the lab. Their output signals together with those of the anemometers were recorded almost continuously for a six week period on a 3090b Hewlett Packard tape recorder at a tape speed of 15/16 inches per second. Digital Analysis The data were s p l i t up into runs of length one to one and a half hours and digitized at either 31.25 or 33.33 Hertz. Before digitizing, the horizontal and vertical wind signals were low-pass fi l t e r e d (the half power cut off was at 20 hertz which i s well past the upper frequency limit of the response of the G i l l anemometers) in order to remove high frequency noise from electronic apparatus associated with the ship's role as a navigational aid to air and sea transport. From the digitized data, Fourier coefficients and spectra and cross spectra were estimated using the IOUBC time series software package described by Dobson (±971) and, i n more detail, by Garrett (1970). The mean wind and direction relative to the ship was obtained for each run from the sum of an offset voltage, which was used during each recording to maintain a high signal to noise ratio, plus the zeroeth harmonic (the mean) of the recorded horizontal wind spectrum. 65 As the ship was usually drifting or steaming at speeds between 0.5 and 1.5 meters per second, i t was necessary to vectorially correct for this to obtain a true mean wind velocity. Although the ship's heading was recorded continuously in graphical form, ship speeds were recorded in the ship's log only every 15 minutes or so. The estimated error in the true mean wind i s considered to be less than 0.5 m sec determination of the Stress Estimates of wind stress were obtained using the dissipation technique suggested by E.L. Deacon (1959). In this technique, i t is assumed that within the atmospheric boundary layer the production of turbulent energy by the mean wind shear 9 U / 9 z is balanced by the local small scale dissipation e. Buoyancy production resulting from sensible heat and moisture fluxes, and flux divergences are thus considered to be negligible. The equation which states this assumption is (see also equation (2.6) which applies to a case which includes buoyancy): - u ^ |H = e . (3-1) 9z If Kolmogoroff's hypotheses are valid (i.e., that there is a range of scales in which the turbulence is isotropic and in which there is no energy production or dissipation such that the energy is transferred from larger to smaller scales through this range); then the downwind spectrum, ® n u> can be shown to be related to the dissipation e by 2- - J L $ (k) = A'e 3 k 3 (3-2) uu 2frf where k = is the downwind component of wave number and f is the U 66 frequency in Hertz. The Kolmogoroff constant, A', is taken equal to 0.55 after Nasmyth (1970) and McBean, Stewart, and Miyake (1971). Assuming a logarithmic wind profile, we get e = U ^ 3 / K Z where the von Karman constant K i s 0.40, and the f r i c t i o n velocity u^ is equal to (-u'w')? It follows that the wind stress i s z u / = A1 (2TTK) 3 or (for A 1 = 0.55, K = 0.40) L JL 3 $ . ( f ) fs (3-3) uu u # 2 = 3.36 3 f $ (f) • (3-4) uu Provided that a well-defined frequency range exists in which the representation (3-2) is valid, then the stress can be calculated from (3-2) using an estimate of $ u u in this frequency range. From the stress so obtained, a drag coefficient can be calculated: C D = u//U 2 (3-5) where and U are specified for the same height. Estimates of Stress and Brag Coefficient The stress and the drag coefficient were calculated from the observed data using equations (3-4) and (3-5). A typical set of spectra for wind, waves, and ship motion ( r o l l , pitch, and vertical acceleration) are shown in Figure 16. It should be noted the peak due to ship motion has a period of about 10 seconds (f - 0.1 Hz) which appears in the horizontal wind spectrum at the low frequency end of the linear region on this plot of log^Q (f $ u y) against log-^ Q f« This peak i s very narrow-banded in a l l spectra. At higher 67 10-i 0.0-o L U Q_ C0 X -ion HORIZONTAL WIND (m s e c - 1 ) • i ~ o - 2 . 0 --3.0-i 2-1 0 --1 -- 2 - i 1 1 1-ROLL (degrees) -i r WAVES (m) oo-o LU X O - -2D-O -30-1-0--1-PITCH (degrees) <| . i i i i . 4 • - 4 0 - 3 0 - 2 . 0 -1.0 OO 1.0 - 3 . 0 - 2 . 0 - 1 0 0X3 1.0 2 0 Figure 16: Spectra of h o r i z o n t a l wind ( $ u u ) , waves ($nn)» s n i p r o l l and p i t c h f o r the approximate p e r i o d 2100 - 2200 GMT 22 June 1970. The v e r t i c a l l i n e designated ' f w ' represents the frequency of the peak i n the wave spectrum. The l i n e designated '-2/3' represents the f~2/3 slope i n the $ u u spectrum expected from Kolmogoroff theory. The mean wind (U~2i) was 15.1 m s e c - l . 68 frequencies, the linear region i s limited by the frequency response cutoff of the instrument; i t may be expected to extend for another half decade into higher frequencies i f the spectra were corrected for instrument response (McDonald, 1972). Subtracting the contamination due to ship motion at the low frequency end and that due to propeller frequency response (near 12 Hz) at the high end, one can recognize that the linear region of this plot represents a well-defined -2/3 power law region where the dissipation technique may be applied. (Note that in this region $ « f 3 ) . & uu Miyake et a l (1970b) found that the dissipation technique (using (3 - 4 ) ) gave stress estimates about 25% greater than those given by the more direct eddy correlation method. Use of the more recent value for the Kolmogoroff constant, 0.55, (they used A' = 0 . 4 8 ) reduces their stress estimates by about one half this difference (13%). This corresponds to the technique used by McBean, Stewart, and Miyake (1971), who obtained estimates of stress by the dissipation technique which were much closer to those calculated by the eddy correlation technique. Businger et a l (1971) contend that von Karman's constant should be 0.35 rather than 0 . 4 0 , but the evidence for such a change is inconclusive. In one case, the estimate by the dissipation technique was compared with an estimate by the eddy correlation technique. For this comparison, signals from the horizontal and vertical propeller anemometers were combined to form the cospectrum shown in Figure 17. The integral under the cospectrum provides a direct estimate of the wind stress. Corrections to the anemometer data for ship motion and for instrument t i l t (see, for example, Pond et a l , 1971) brought the eddy correlation 69 f-2/3 log(f"$ u u ) -1 - - H -log(f"$ww) -1--2--r -2-AAA * • . A . t A - Corrected for tilt *-Corrected for tilt and ship motion 0--1-l O ^ ( f x $ n ) + + -2 -1 l o g f 0 Figure 17: Spectra of horizontal wind ($Uu)» vertical wind (S^) , a n < * waves ($nr|) > a n ^ t^ie cospectrum ($ u w) of the horizontal and vertical wind components for the period 2345 - 0110 GMT, 21-22 June 1970. The integral of the cospectrum, corrected for t i l t and ship motion,yielded an estimate of the stress about 25% less than that obtained by the dissipation technique. With the mean wind speed ( U 2 1 ) of 14.9 m s e c - 1 , the respective values for the drag coefficient were C 2 1 (u'w') = 1.41 x IO"3 and C 2i(e) = 1.85 x 10 - 3. 7 0 estimate to within 2 5 % of the dissipation estimate. The dissipation technique was also used to calculate the drag coefficient C ^ j for the duration of two storms. The hourly estimates of which were for mean winds of 4 . 3 to 1 7 . 2 m sec \ are displayed in Figure 1 8 . For each 2 m sec * interval of wind speed, a mean and a standard deviation were calculated for C^.. Up to 1 4 m sec there was a tendency for to increase with wind speed. However, these data with their scatter show that the drag coefficient i s essentially constant for the given range of observed wind speeds. The mean value with i t s standard deviation i s C 2 1 = ( 1 . 4 1 ± 0 . 2 3 ) x I O - 3 . ( 3 - 6 ) For comparison with the estimates of other workers, the drag coefficient at 2 1 meters was corrected to that appropriate to a height of 1 0 meters. Assuming neutral s t a b i l i t y and a logarithmic wind profile, * 1 0 = ? 2 1 - f l n <§> ' ( 3' 7> so that the drag coefficient, C ^ Q, becomes C 1 Q « ( 1 . 6 3 ± 0 . 2 8 ) x I O - 3 . ( 3 - 8 ) The s t a b i l i t y i s shown to influence the structure of the tur-bulence in the atmospheric boundary layer; i t affects both the wind stress and the wind profile. To test the effect of st a b i l i t y on the drag coefficient, a Richardson number was estimated from the stress and the routine meteorological measurements: r g R i = 0(z) - T s 9 ( 3 - 9 ) z 6 u * 2 / ( K 2 z 2 ) where TQ i s a p r o f i l e c o e f f i c i e n t (approximately equal to 71 Neutral Unstable Storm 'A' A A Storm B O C2 1 = (1.41 ±0.23)x10~3 ro + O CM u _ 8 10 U21 (m sec ' 12 •1) 14 16 18 20 JjLgure 18: The drag c o e f f i c i e n t , C 2 1 , as a f u n c t i o n of wind speed, U2i> f o r the two storm p e r i o d s , 16-17 June and 22-24 June, 1970. The bars represent the mean and standard d e v i a t i o n f o r each 2 m s e c - 1 i n t e r v a l of mean wind speed. The reference height i s 21 meters. 72 0.1), 9(z) is the potential air temperature at a height z, and T g i s the sea surface temperature. In Figure 18, the points designated as being for unstable conditions are those from occasions for which Ri < -0.01 (in almost a l l cases, |R±| < 0.1 indicating neutral conditions). No significant effects of s t a b i l i t y were observed. Time Variation of Mean Wind, Stress, and Wave Energy — 2 Estimates of the mean wind I 1 ] / t ^ e s t r e s s u * » a n ^ t n e wave energy r\2, which were a l l calculated for data records roughly 1 hour in length, are plotted as functions of time in Figures 19 and 20. Figure 19 covers the period 16-17 June, designated as Storm A; Figure 20 covers the period 22-24 June, 1970, designated at Storm B. Inspection shows that the time behavior of the stress estimates, which were derived from the small scale fluctuations in the wind, corresponds closely with the time behavior of the mean wind speed estimates. A noticeable feature i s the apparent change with time of the drag coefficient. The drag coefficient appears to increase on the leading side of both storms and then remains constant or possibly decreases slightly. This time dependency of indicates that the observed scatters in estimates of drag coefficient in most experiments may not be completely a result of s t a t i s t i c a l errors of measurement but may depend partially on physical conditions. The f i r s t explanation that comes to mind for the time dependency of the drag coefficient i s that i t is a s t a b i l i t y effect. The only tendency which could be observed from the Richardson numbers calculated from equation (3-9) was one indicating a transition from unstable to neutral conditions at the start of each storm. The apparent increase 73 — i 1 1 i 1 i — 0000 0600 1200 1800 0000 0600 ieu, is*. T I M E ( G M T ) Figure 19: Time series of estimates of drag coefficient (C21), mean energy of the wave f i e l d ( n 2 ) , stress (u* 2), and mean wind speed (U21) for Storm A (21z 15 June to 18z 17 June, 1970). The arrows at the bottom are unit vectors of the true mean wind at 3 hour intervals. The wave energy calibration i s (2.1 meters v o l t - 1 ) . Figure 20: Time series of estimates of drag coefficient (C21) , mean energy of the wave (n 2), stress (u* 2), and mean wind speed (&21) for Storm B (18z 21 June to 18z 24 June, 1970). 75 i n the drag c o e f f i c i e n t , , which occurred a t the s t a r t of each storm i s the opposite e f f e c t to that expected from the non-dimensional formulae f o r the mean wind speed examined i n Businger et a l (1971). For unstable c o n d i t i o n s , they gave the f o l l o w i n g expression r e l a t i n g the mean wind speed at a given height (taken to be 21 meters here) to the Richardson number: U 0 1 = 2 4 In 21 K 21 (1 - u R i ) " ^ (3-10) where z Q i s the roughness l e n g t h of the boundary surface and p i s a 2 p o s i t i v e constant. For a constant s t r e s s u A , equation (3-10) i m p l i e s that an in c r e a s e i n the Richardson number R i ( f o r i n s t a n c e going from unstable to n e u t r a l c o n d i t i o n s ) would r e s u l t i n an in c r e a s e i n the mean 2 —2 wind speed. As C„, = u. /U „,, the drag c o e f f i c i e n t C„, would decrease, z l * 21 21 Thus, the observed i n c r e a s e i n drag c o e f f i c i e n t on the f r o n t s i d e of the storms cannot be explained i n terms of s t a b i l i t y e f f e c t s . Another f e a t u r e of Figures 19 and 20 i s that the maximum wave energy observed i n Storm A i s only a quarter of that observed i n Storm B. While the l o c a l winds were m a r g i n a l l y l e s s i n Storm A, t h i s d i f f e r e n c e does not seem to account f o r a f o u r - f o l d d i f f e r e n c e i n maximum wave energy. I t can be expl a i n e d , q u a l i t a t i v e l y at l e a s t , by reference to the synoptic s i t u a t i o n s , which are shown i n Figures 21 and 22. Storm A passed w e l l to the south of S t a t i o n 'Papa'. There, the winds were from the n o r t h f o r the d u r a t i o n of the storm. The l o c a l l y produced waves at 'Papa' were v i s u a l l y observed to be t r a v e l l i n g towards the south during t h i s time. As S t a t i o n 'Papa' was on the northern edge of the storm, no s i g n i f i c a n t lower frequency wave energy was advected i n t o the re g i o n (near 'Papa') from t h i s storm. Storm A Figure 21: Twice d a i l y s urface pressure charts of the N.E. P a c i f i c f o r Storm A. ON Storm B F i g u r e 2 2 : T w i c e d a i l y s u r f a c e p r e s s u r e c h a r t s o f t h e N . E . P a c i f i c f o r S t o r m B . ^ 78 During Storm B the p i c t u r e was d i f f e r e n t . This storm approached S t a t i o n 'Papa' from the WSW and during t h i s time the wind and waves were a l s o from t h a t general d i r e c t i o n . Therefore, the low frequency waves produced e a r l i e r would be propagating i n t o the re g i o n w h i l e the storm was reaching i t s maximum at 'Papa'. Thus, the l o c a l wave f i e l d at the peak of the storm should c o n s i s t o f . l o c a l l y generated wind waves plus the lower frequency advected system. Wave sp e c t r a observed at d i f f e r e n t stages of these storms (which are i d e n t i f i e d by the n o t a t i o n s 1, 2, 3, 4 i n Figures 19 and 20) are compared i n Fi g u r e 23. The e v o l u t i o n of the wave sp e c t r a during the two storms i s shown i n Figure 24; i n each storm, the wave energy increased as the energy peak migrated from higher to lower frequencies. The expected f form at h i g h frequencies i s evident i n both F i g u r e s . During Storm A, the e v o l u t i o n of the wave f i e l d c o r -responded to that which one would expect f o r the growth of a l o c a l wave system. On the other hand, at an e a r l y stage of Storm B a s i g n i f i c a n t amount of low frequency energy was already present. This low f r e -quency energy was maintained w h i l e the l o c a l wave system b u i l t up; consequently, the t o t a l wave energy was greater than i n Storm A. I t seems p o s s i b l e that the e v o l u t i o n of the drag c o e f f i c i e n t shown i n Figures 19 and 20 i s a f u n c t i o n of the stage of the development of the wave f i e l d . A r e l e v a n t parameter could be the r a t i o of the phase speed of the dominant waves e i t h e r to the f r i c t i o n v e l o c i t y u^, or to the mean wind speed 1^0" Volkov (1970), and Volkov and Mordukhovich 2 (1971) d i d p l o t the s t r e s s u^ versus C^/u^, but the dependence observed 2 was e s s e n t i a l l y j u s t a u^ versus l / u A dependence. R e l a t i v e changes i n the wave phase speed C, are sm a l l compared to those i n the f r i c t i o n S t o r m A A - - A — S t o r m B O O -Log 1 0 f Figure 23: Comparison of wave spec t r a at s i m i l a r stages of development of the two Storms A and B. Each p a i r of spec t r a are f o r the corresponding estimates i d e n t i f i e d by the notations 1, 2, 3, or 4 i n Figures 19 and 20. 80 Figure 24: The evolution of the wave spectra i n the two storms, A and B. The spectra i d e n t i f i e d by the numbers 1, 2 and 3 are the same spectra as those s i m i l a r l y i d e n t i f i e d i n Figure 23. 81 v e l o c i t y u^; thus, the r a t i o C^/u^ appears to be inadequate as a measure of the i n f l u e n c e of the wave f i e l d on the s t r e s s . Q u a l i t a t i v e l y , i t can be seen, from Figure 20 e s p e c i a l l y , t h a t the apparent maxima i n drag c o e f f i c i e n t estimates do tend to occur when the r a t e of increase of energy i n the wave f i e l d i s a l s o a maximum. Perhaps a more s e n s i t i v e r a t i o to i n d i c a t e the e f f e c t of the wave 1 8 — % — f i e l d on the s t r e s s would be — ir— ( n 2 ) where n 2 i s the estimate of u^ dt X L. the mean energy i n the l o c a l l y generated wave f i e l d . Conclusion The main c o n c l u s i o n from the micrometeorological observations t h a t i s r e l e v a n t to modelling the upper ocean i s the constancy of the drag c o e f f i c i e n t over the observed range of mean wind speeds (4 to 17 m sec ^ ) . This r e s u l t , although not unexpected, lends confidence to the par a m e t e r i z a t i o n of the wind s t r e s s which was used i n the model described i n Chapter 2. Chapter 4 MODEL WITH OBSERVED DATA INPUTS 4.1 I n t r o d u c t i o n In Chapter 2, a model of the upper ocean was developed from the three equations f o r conservation of heat, mass and mechanical energy. The r e s u l t i n g equations d e s c r i b i n g the upper mixed l a y e r are (2-12), (2-19), (2-20), and (2-21). In these equations, the time r a t e change of the upper l a y e r depth i s s p e c i f i e d i n terms of the t u r b u l e n t energy de r i v e d from the wind s t r e s s , the net r a d i a t i v e heat input through the ocean s u r f a c e , and other heat exchanges at the upper boundary. The major a d j u s t a b l e parameters are y, the e x t i n c t i o n c o e f f i c i e n t of the s o l a r r a d i a t i o n , and m, the f r a c t i o n of the wind s t r e s s energy a t ten meters height which i s e v e n t u a l l y used to increase the p o t e n t i a l energy of the water column. In t h i s chapter, r e a l data f o r the s i x week c r u i s e p e r i o d are examined to determine appropriate values of the two a d j u s t a b l e para-meters. Using these parameter v a l u e s , the numerical model Is run w i t h r e a l input data c o n s i s t i n g of h o u r l y values of wind speed, s o l a r r a d i a t i o n , and back r a d i a t i o n from the sea s u r f a c e . One p e r i o d of 12 days and one two day storm p e r i o d are examined. 4.2 Heat and Mass Balance i n the Upper Layer In t h i s s e c t i o n , the observed d i s t r i b u t i o n s of heat and mass i n the upper l a y e r are examined i n order to assess the r e l a t i v e importance of the heat input terms. Net Radiative Heat Input During the summer, the heat content of the upper l a y e r i s determined l a r g e l y by r a d i a t i v e heat exchanges w i t h the atmospheric boundary l a y e r . In Figure 25a, the d a i l y incoming s o l a r r a d i a t i o n and long wave back r a d i a t i o n from the sea surface are p l o t t e d . The data on s o l a r r a d i a t i o n have been published i n the Canadian Monthly R a d i a t i o n Summary (1970); i t i s assumed that 7% of the i n c i d e n t s o l a r r a d i a t i o n i s r e f l e c t e d from the sea sur f a c e . Estimates of the e f f e c t i v e back r a d i a t i o n were c a l c u l a t e d f o r 12 hour periods from the e m p i r i c a l formulae summarized i n Laevastu (1960). The back r a d i a t i o n f o r a c l e a r sky i s expressed i n terms of the sea surface temperature, T g (°C), and the r e l a t i v e humidity, r ( i n % ) : 14.28 - 0.09T - 0.046r -B - § . (4-1) 69.72 The e f f e c t i v e back r a d i a t i o n i s obtained by c o r r e c t i n g f o r the cl o u d i n e s s , C l (expressed on a s c a l e from 1 to 10): B^ = B (1 - 0.076 Cl) ( U n i t s : c a l cm" 2 min" 1) . (4-2) Nomographs f o r o b t a i n i n g B^ based on the above formulae appear i n Laevastu (1965). The r e l a t i v e constancy of the back r a d i a t i o n i n Figure 25a, compared w i t h the v a r i a t i o n s i n s o l a r r a d i a t i o n , r e s u l t s from the assumption i m p l i c i t i n (4-1) that the sea surface a c t s roughly as a bla c k body r a d i a t o r : the r a d i a t i o n from the surface v a r i e s as the f o u r t h power of the sea surface temperature expressed i n °Kelvin. -2 -1 The average d a i l y i n c i d e n t s o l a r r a d i a t i o n ( i n c a l cm day ) during 84 SOLAR RADIATION R ~2L N LU < I— E -Z. o 8 g LU — O I LU " Q "E o < -125 150 125 100 75 50 25 0 -25 2 5 2 0 1-5 h 10 0-5 f-0 0 b ^ ^ - 1 — - -v~ -BACK RADIATION B M A S S 20 24 MAY 28 13 17 21 25 29 JUNE IS70 Figure 25: The heat and mass balance of the upper 60 meters of the ocean at Station 'Papa' for the period 20 May to 29 June, 1970. The daily totals of incoming solar radiation, R (with a reflection of 7%), and net back radiation, -B, are displayed in Part (a). In Part (b), the cumulative sums of these values are plotted against the heat content of the upper 60 meters calculated from STDs. In Part (c), the mass defi c i t of the upper 60 meters i s plotted. 85 the c r u i s e p e r i o d was 337 dur ing May and 359 during June. The c o r r e s -ponding average of the e f f e c t i v e back r a d i a t i o n was -81 (May) and -84 (June). These values agree w i t h those c a l c u l a t e d by Tabata (1961). During May-June, 1970, the u s u a l seasonal increase i n i n c i d e n t s o l a r r a d i a t i o n was absent because of above normal c loudiness i n June; c loudiness at 'Papa' during the c r u i s e was t y p i c a l l y 8/10. Storage of Heat and Mass The heat storage term i n the heat budget of the upper mixed l a y e r can be c a l c u l a t e d from temperature p r o f i l e s . The heat content above a given depth z = - d , r e l a t i v e to some reference temperature T , i s j u s t the i n t e g r a l of the observed temperature over depth: ro H = p Q C p j [T(z) - T j dz (4-3) - 2 where H i s the heat content i n c a l cm and c^ i s the heat capac i ty at constant pressure i n c a l g * °K The change i n heat content between two traces taken at times t^ and i s : A H C t j , t 2 ) = P Q c p | ^[T.(Z, t 2 ) - T ( z , t x ) ] dz . (4-4) - 2 The mass d e f i c i t i n the upper l a y e r , M^ ( i n g cm ) , i s given by M d = | ^ [ p d - p(z)] dz (4-5) where p^ i s a constant reference d e n s i t y at z = - d . S i m i l a r l y , the - 2 s a l t d e f i c i t , ( i n mg cm ) , of the upper l a y e r r e l a t i v e to a reference value i s : d o 86 [sCz) - S.] dz . C4-6) -d d As we are i n t e r e s t e d only i n temporal changes, the a c t u a l reference values are unimportant provided they are constant. The heat content, and the mass and s a l t d e f i c i t s of the upper 60 meters were c a l c u l a t e d from the STD data contoured i n Figures 11, 1?, and 13. The heat and mass i n t e g r a l s are p l o t t e d as f u n c t i o n s of time i n Figures 25b and 25c. The s a l t d e f i c i t s are p l o t t e d i n Figure 30 and are discussed i n a l a t e r s e c t i o n . A l s o p l o t t e d i n Figure 25b are the cumulative sums of s o l a r r a d i a t i o n , back r a d i a t i o n , and the net heat input from these two types of r a d i a t i v e exchange. On time s c a l e s longer than about two weeks, the net r a d i a t i v e heat input q u i t e c l o s e l y balances the measured i n c r e a s e i n heat content of the l a y e r . On s h o r t e r time s c a l e s , two f a c t o r s cause the l a c k of agreement. P r i m a r i l y , the disagreement r e s u l t s from short p e r i o d i n t e r -n a l waves and Ekman divergences (or convergences) being a l i a s e d i n t o the heat storage i n t e g r a l s . A l s o , the t u r b u l e n t heat f l u x e s , which have not been i n c l u d e d i n the heat balance, may be important i n some in s t a n c e s . Changes i n the mass d e f i c i t (shown i n Figure 25c) c o r r e l a t e c l o s e l y w i t h those i n the heat content (shown i n Figure 25b). The equation of s t a t e , can be used to estimate the r e l a t i v e importance of the heat and s a l t i n t e g r a l s on the mass i n t e g r a l . I f 9p/9T and 9p/9S are considered to be r e l a t i v e l y constant w i t h i n the upper s i x t y meters, the equation of state (4-7) becomes AM = — — | ^ AH + ' i - ' | f i - AN, . (4-8) d p c 9T p oS d Ko p o For T - 7°C, S - 33 0 / q o , equation (4-8) i s AMj = (0.14 x 10~ 3 g c a l " 1 ) AH + (0.8) ANj . (4-8a) 4 -2 The trends over the s i x week period f o r heat, AH - 10 c a l cm -2 (Figure 25b), and f o r s a l t , AN, - 0.2 g cm (Figure 30), contributed d -2 roughly 1.4 and 0.16 g cm r e s p e c t i v e l y to the trend i n the density _ 2 anomaly, AM, - 1.6 g cm . Thus, the seasonal trend i n the density d anomaly did depend mainly on the trend i n the heat content. The short term o s c i l l a t i o n s , which were approximately the same magnitude as the trend i n the s a l t d e f i c i t (Figure 30), were about ten times smaller i n magnitude than the trend i n the heat content (Figure 25b). The r e s u l t i n g contributions of s a l t and heat to the o s c i l l a t i o n s i n the density anomaly (Figure 25c) were therefore roughly equal. These short period o s c i l l a t i o n s from observation to observation evident i n a l l three traces are c e r t a i n l y the r e s u l t of i n t e r n a l waves i n the region below the mixed l a y e r , and of short term Ekman divergences and convergences. 4.3 Experimental Determination of the Mixing Energy The rate of increase of p o t e n t i a l energy i n the upper layer of the ocean as the mixed layer thickens i s calcu l a t e d , i n t h i s s e c t i o n , from the s e r i e s of STD traces obtained during a storm. The changes of p o t e n t i a l energy are determined f o r three storms, and are correlated with the energy inputs derived from the wind s t r e s s . Turner (1969) 8 8 assumed that the r a t e of i n c r e a s e i n p o t e n t i a l energy of the upper ocean as a r e s u l t of work done by mixing i s a constant f r a c t i o n , m, of the r a t e of t u r b u l e n t energy t r a n s f e r r e d downwards by the wind s t r e s s at ten meters above the s u r f a c e . I f - D A i s the r a t e at which wind s t r e s s energy i s made a v a i l a b l e f o r mixing w i t h i n the upper l a y e r , then m = - ^3- . ( 4 - 9 ) U 1 0 T p a C 1 0 U 1 0 I m p l i c i t i n any assumptions about m i s a l s o some assumption concerning the drag c o e f f i c i e n t , C ^ Q- The measurements presented i n the previous chapter, along w i t h those of other workers, i n d i c a t e that C ^ Q i s a constant f o r a l l mean wind speeds up to at l e a s t 1 8 m sec In t h i s study, i s taken to be equal to 0 . 0 0 1 3 . Equation ( 4 - 9 ) shows that I f C ^ Q i s constant, then the p a r a m e t e r i z a t i o n of the energy a v a i l a b l e f o r mixing i n v o l v e s only one a d j u s t a b l e constant, m. The shallow l a y e r of warm water formed during a p e r i o d of s o l a r heating and low winds soon a l t e r s i t s appearance upon the onset of the h i g h winds a s s o c i a t e d w i t h the passage of a weather disturbance. I t becomes an isothermal l a y e r of i n c r e a s i n g thickness and decreasing temperature. As c o l d e r , heavier water below the l a y e r i s entrained and mixed upwards, the center of mass of the system i s r a i s e d w i t h an accompanying increase i n p o t e n t i a l energy. This sequence of events takes p l a c e over a few days; however, s i m i l a r changes might a l s o be produced by some other process such as h o r i z o n t a l convection. Therefore* the question a r i s e s , 'how can one determine from a set of d e n s i t y p r o f i l e s the change i n p o t e n t i a l energy due to wind mixing alone?' 89 -2, The potential energy anomaly (in ergs cm ) of the upper layer of the ocean relative to the surface i s defined as: ft ro ro • g[p(z) - p ] zdz = he [p(z) - p ] d(z ) -d J-d ° C4-10) Turner (1969) calculated Aft using traces chosen from before and after a period of high winds such that the measured change in total heat content was small (less than 15%). However, i f the potential energy changes are to be calculated throughout a storm, then some method of correcting for apparent changes of heat content resulting from internal waves and short term convergence or divergence must be found. For a homogeneous upper layer of thickness h, equation (4-10) becomes r-h ft = g ro -d [p(z) - p ] zdz + g (p - p ) zdz . (4-11) -h The time derivative of this equation (assuming that p(z) does not change with time for z < -h) can be written as 2 dp dft , , . dh eh s — = gh (p_h - po) 7 7 - " V — . s' dt 2 dt Using the simplified equation of state, dp = P Qa dT, where a one can express (4-12) in terms of temperature: 1_ P, (4-12) dft dt -gPoa ,2 dT n s dh ^ d f " + h C T s - T-h> dT (4-13) The mechanical energy equation appropriate to the mixed layer thickening can be written from equation (2-17) (in which equation (2-16) has been used to eliminate the net heat input terms}: ^ ^ L _ ) dh, = -(G* - D*) + R*£i 2 dt s -h/ dt gp a p c e K o Ko p (4-14) 90 where the assumptions are that there i s no ver t i c a l advection (w = 0) and no solar heating beneath the bottom of the mixed layer. Substitution of (4-14) into (4-13) gives the rate of potential energy change between two STD traces taken at times At apart: in potential energy due to work being done against the buoyancy forces; the energy for this process i s derived from the wind. The second term represents the rate of increase in potential energy (since a is negative) due to solar heating. The depth, y i s a weighted average of depth in which the weighting function i s the rate of absorption at each depth; i t may be considered as the 'moment' of that density sink about the surface. The surface sensible and latent heat transfers and the surface back radiation do not enter expl i c i t l y into (4-15) because they have a 'moment' of zero about the ocean surface (z = 0), a property which allowed them to be eliminated from (2-17) by use of heat storage differences due to both internal waves at or near the bottom interface and short term convergences (or divergences) can be corrected for. If the assumption Is made that these heat changes emanate from a source, A^, located at the bottom of the upper mixed layer, this effect can be included in (4-15) as a term completely analogous to the radiation term: Aft At (4-15) The f i r s t term on the right represents the rate of increase (2-16). Apparent potential energy changes that result from measured c (y 1 + hAA) (4-16) P 91 The r a t e of change i n the heat content, which can be estimated from (4-4), i s j u s t . H - R* + B* + A* ' «-17) S u b s t i t u t i o n of (4-17) i n t o (4-16) y i e l d s the f o l l o w i n g formula f o r the r a t e of working by the a v a i l a b l e t u r b u l e n t energy d e r i v e d from the wind i n terms of experimentally measurable parameters: G D . AO * a * * At c Y \ + h At * * (4-18) P Estimates of (G A - D A) were c a l c u l a t e d f o r three separate storm periods from the STD t r a c e s . A s e r i e s of four STD temperature t r a c e s taken during one of the storms i s shown i n Figure 26. The p o t e n t i a l energy changes were c a l c u l a t e d r e l a t i v e to an i n i t i a l p r o f i l e f o r each storm. To get m from equation (4-9), cumulative means of U^Q T were c a l c u l a t e d using the r o u t i n e v i s u a l estimates of wind speed taken every three hours from a cup anemometer. A summary of the c a l c u l a t i o n s f o r the storm d e p i c t e d i n Figure 26 i s given i n Table IV, and values of m are p l o t t e d i n Figure 27 as a f u n c t i o n of the time elapsed from the s t a r t of each storm p e r i o d . The range of m i n Figure 27 i s roughly 0.0007 to 0.003, w i t h a mean of 0.0018 (the one anomalously hi g h value at OOOOz on 22/6/70 has been neg l e c t e d ) . The value obtained by Turner (1969) f o r two i n d i v i d u a l c a l c u l a t i o n s was 0.01. The values of m presented i n Figure 27 represent the f r a c t i o n of wind s t r e s s energy e v e n t u a l l y used to do work against the buoyancy f o r c e s . While m e x h i b i t s some s c a t t e r , these r e s u l t s are the only q u a n t i t a t i v e estimates of t h i s parameter derived from a s e r i e s of \ 92 TEMPERATURE (°C) Figure 26: A s e r i e s of temperature p r o f i l e s taken w i t h an STD during Storm B. The Greenwich Mean Time f o r each p r o f i l e i s : #1, 0430 , June 21; #2, 0045 June 22; #3, 1500 June 22; and #4, 0900 June 23, 1970. TABLE IV Wind stress energy used in increasing the potential energy (fi) of the upper layer of the ocean from 2100 21/6/70. Stn Time h 't R* 0 ft B* 0 AH* ^STD ^heat Afi . . mixing _Afim G * D * At <^o> m (GMT) (m) (10 1 0 ergs cm 2 -1, sec ) (10 5 -2 ergs cm ) (10 ergs -2 -1, cm sec ) 132 -136 0000 22/6/70 28.7 0.231 -0.046 7.9 12.3 -7.06 5.20 4.8 270 0.0178 137 0300 22/6/70 25 0.339 -0.092 -0.102 1.10 0.223 1.32 0.61 319 0.0019 139 0900 22/6/70 39 0.377 -0.184 0.82 4.03 -0.84 3.19 0.98 326 0.0030 140 1500 22/6/70 43 0.381 -0.276 4.31 7.99 -5.80 2.19 0.41 358 0.0014 141 1800 22/6/70 38 0.516 -0.322 0 3.29 0.152 3.44 0.54 360 0.0015 142 2100 22/6/70 47 1.11 -0.37 5.53 11.8 -7.42 4.38 0.58 356 0.0016 143 0000 23/6/70 40 1.67 -0.41 1.13 4.52 -0.10 4.42 0.51 361 0.0014 144 0300 23/6/70 44 2.09 -0.46 3.28 7.93 -2.78 5.15 0.53 355 0.0015 145 0600 23/6/70 50 2.16 -0.51 5.64 14.6 -6.67 7.9 0.73 341 0.0021 146 0900 23/6/70 51 2.16 -0.55 7.48 15.7 -9.85 5.9 0.53 323 0.0016 10 -1 94 10" 10 -4 — 1 — i 1 1 T " 1 : o from 2100(GMT) 4 June 7 0 -• from 1800 '" 15 " -• from 2100 21 " -• - -- • -o 0 A O A 6 A • • • A a A a o A O — -O A -i « I 1 1 0 12 18 24 30 T I M E ( h o u r s ) 36 42 48 Figure 27: Experimentally determined values for m, the fraction of the downward transfer of energy by the wind stress at 10 meters height that becomes available for mixing within the upper layer. Each point represents the change from the start of that particular storm: they are actually running means of m. 95 actual STD profiles other than Turner's attempt from the Bermuda data presented in Stommel et a l (1969). 4.4 Results In this section, the results of the model calculations for the period OOOOz 13 June to OOOOz 25 June, 1970 are discussed. During this period there were two well-defined storms superimposed on a background of relatively low winds and intense solar heating. The Input Data The oceanographic observations described in the previous sections provide two results necessary to run the model with real input data. Fi r s t , the heat budget study indicates that none of the turbulent heat fluxes across the air-sea interface, the horizontal advection of heat by ocean currents, and the vertical advection from below the layer were important to the heat balance of the upper mixed during May and June, 1970. Second, a range of 0.0007 to 0.003 for m, the ratio of potential energy change to the rate of production of wind stress energy, has been determined from the STD observations. The value of m = 0.0012 that gave r e a l i s t i c results in the idealized studies of wind mixing and diurnal heating discussed earlier l i e s within this range. The other adjustable parameter in the model is y, the extinction coefficient of solar radiation in sea water. From Jerlov (1968), a reasonable range for the extinction coefficient at Ocean Station 'Papa' is 0.001 to 0.003 cm \ Dr. T.R. Parsons (personal communication) has used photometer measurements obtained at 'Papa' (see McAllister, Parsons, and Strickland, 1959; and Parsons and Anderson, 1970) to calibrate Secchi disk readings. He estimates the extinction coefficient, y, 96 to be in the range 0.001 to 0.002 cm * at 'Papa' during May and June. The value of y = 0.002 cm ^ used in the diurnal model discussed earlier is within the ranges just presented. If the turbulent and advective heat fluxes are Ignored, then the boundary conditions are derived from hourly values of solar radiation ( R A ) , back radiation ( - B ^ ) , and wind speed at ten meters height ( U J Q ) • The hourly values of solar radiation are just those obtained from the pyranometer output. The routine estimates of wind speed taken every three hours were corrected by a factor obtained from the G i l l propeller measurements. They were then smoothed and interpolated to one hour intervals. The estimates of back radiation for 12 hour periods, which were calculated using the formulae of Laevastu, were also interpolated and smoothed to give hourly values. The boundary conditions, U J Q > and B A , are shown in the upper two traces of Figure 28. The i n i t i a l temperature profile i s also shown at the bottom l e f t of the figure; i t was estimated from the STD profiles taken near OOOOz on June 13. The Twelve Day Period The results of the model calculations for the 12 day period, 13 through 24 June, 1970 are presented here. The values of the parameters m and y were progressively adjusted for Runs #1, #2, and #3 as described below. Only the results for Run #3 are plotted. Run Hi (m = 0.0025, y = 0.002 cm - 1). The mixed layer thickness calculated by the model slowly departed from the observed mixed layer thickness u n t i l , on the twelfth day, i t exceeded 75 meters. At that time, the observed thickness of the mixed layer was only about 50 meters. 9 7 8 in o 13" 12 8 4 0 M 30 0 . O r - < M 2 1 cc < CD UJ 8 5 s l 8 0 WH 7 5 . BACK RADIATION SOLAR RADIATION A A BUCKET MODEL 15 16 17 18 19 20 21 TIME IN DAYS (GMT) JUNE 1970 23 24 OBSERVED STD MODEL PROFILE 'ftfrrrfrrrrm 7 9°c Figure 28: Input and results of the model (Run #3) for the period 13-24 June,_1970. Parts (a) and (b) are the observed data inputs to the model: UJQ ^ s t n e mean wind estimated from the 3 hourly observa-tions from cup anemometers, and R and -B are the measured incident solar radiation and calculated back radiation at the sea surface. Part (c) is the comparison of the predicted sea surface temperature with that observed from bucket measurements. Part (d) is the comparison of the temperature profiles of the model (each 12 hours) with those obtained with an STD (where available). The parameter values were m = 0.0012 and \ = 0.003 cm - 1. 98 Run #2 (m = 0 . 0 0 1 2 , y = 0 . 0 0 2 cm" 1). S e t t i n g m = 0 . 0 0 1 2 was e q u i v a l e n t to u s i n g the f r i c t i o n v e l o c i t y as the s c a l i n g v e l o c i t y i n G^, the energy a v a i l a b l e f o r mixing ( r e f e r to Chapter 2 , S e c t i o n 2 . 4 ) . The computed behavior of the mixed l a y e r during the stormy periods reproduced q u i t e c l o s e l y the observed behavior. During the h e a t i n g p e r i o d s , however, the l a y e r depth was about 60% too l a r g e , and the sea surface temperature d i d not reach the pronounced peaks present i n the r e a l data. From the t h e o r e t i c a l r e s u l t s of Chapter 2 , we know t h a t , i n the model, the thickness of the mixed l a y e r during low winds had been decreased s i g n i f i c a n t l y by i n c r e a s i n g the e x t i n c t i o n c o e f f i c i e n t . The e x t i n c t i o n c o e f f i c i e n t was t h e r e f o r e increased from 0 . 0 0 2 to 0 . 0 0 3 cm * i n the next run. Run #3 (m = 0 . 0 0 1 2 , y = 0 . 0 0 3 cm - 1). The r e s u l t i n g sea surface temperature and twice d a i l y temperature p r o f i l e s c a l c u l a t e d from the model are p l o t t e d i n Figure 28 f o r Run #3. The observed temperature p r o f i l e s superimposed on the model p r o f i l e s were obtained, i n a l l cases, w i t h i n one hour of the time f o r which they are p l o t t e d . The agreement between the p r e d i c t e d and observed sea surface temperature i s very good on time s c a l e s greater than one day. The model p r o f i l e s , w h i l e they are o b v i o u s l y i d e a l i z e d , do reproduce the time dependent behavior of the mixed l a y e r . Deepening of the mixed l a y e r r e s u l t i n g from the h i g h winds during 1 5 - 1 7 and 2 1 - 2 3 June i s simulated s a t i s f a c t o r i l y . The shallow warm l a y e r which b u i l t up during 1 4 - 1 5 June and again during 1 8 - 2 1 June i s a l s o evident i n the model p r o f i l e s . The general agreement i n d i c a t e s t h a t the model r e p r e s e n t a t i o n of the time-dependent behavior of the upper mixed l a y e r i s q u i t e good. 99 The observed sea surface temperature, represented by the dashed line in Figure 28, was obtained from standard bucket tamperatures taken every three hours; the error i n these measurements i s at least ± 0.05 C°. Time variations observed i n sea surface temperatures that were faster than about 12 hours are not predicted by the model. These probably result from the frequency and accuracy of the sampling technique and so can be neglected. On the other hand, the larger variations in sea surface temperature on scales of two or three days are associated with the synoptic meteorological inputs, and are accurately reproduced by the model to within 0.1 C°. One can observe a diurnal variation in the model estimate of the sea surface temperature. However, any such variation in the observa-tions i s obscured by measurement errors; thus, i t might be recovered by a Fourier analysis of observed summer sea surface temperatures. A Two Day Storm Period The model was also run, with the same parameter values used in Run #3, for the 48 hour storm period 21 to 23 June marked by the vertical bars in Figure 28. During this storm, salinity-temperature-depth profiles were taken every three hours, and temperature-depth profiles were taken every hour so that the rate of deepening of the mixed layer as well as i t s eventual depth of penetration might be determined. The results of the model calculations are plotted in Figure 29. Surface temperature measurements from the STD traces, from the XBT traces, and from bucket values, and the surface temperature calculated using the model are plotted in Figure 29a. Figure 29b shows observed 100 C O U J < s LU LU CO r-UJ a >- a) S UJ 5 Q o BUCKET ± 0 0 5 C° A STD ±001 C° a XBT ±005 C° OOOOZ GMT 18 JUNE 2! * STD a XBT • a a 1 • J • I "1 I t _. 06 12 18 JUNE 22 00 06 12 JUNE 23 o 20 1 40-.£ 60 • H 80|-CL. hi a 7 9°C J • i ' I k | I L I t_.Jy.-l I .„A-,J I J I I 1 I I L—I OBSERVED STD MODEL PROFILE rrrrr Figure 29: Comparison of the model output w i t h the observed data f o r the storm p e r i o d 1200 GMT June 21 to 1200 GMT June 23, 1970. The inputs were j u s t those i n Figure 28 f o r the pe r i o d denoted by heavy v e r t i c a l bars. The parameter values were m = 0.0012 and y = 0.003 cm - 1. The e r r o r bars i n P a r t (b) f o r the mixed l a y e r depth from the STDs represent the observed depth range of strongest s t r a t i f i c a t i o n immediately below the l a y e r . The e r r o r i n the XBT estimates of mixed l a y e r depth i s about ±5 meters. 101 and computed estimates of the mixed layer depth. The error bars repre-sent the observed depth range of strongest st r a t i f i c a t i o n just below the layer. From the XBT profiles, a point estimate of the mixed layer depth has been plotted; the error in this estimate i s about ± 5 m. At the bottom of Figure 29, the available STD profiles have been superimposed on temperature profiles computed from the model at three hour intervals. The sea surface temperatures estimated by the model agree with the STD and XBT values within the uncertainty of the measurements; bucket temperatures, as one might expect, show less consistency . The model predicts correctly the mean rate of deepening within the accuracy to which i t can be determined from the observations; the scatter in the observed mixed layer depth is mostly attributable to internal waves. 4.5 Possible Sources of Discrepancy Using actual meteorological inputs, the model presents a r e a l i s t i c picture of the time dependent behavior of the upper mixed layer. The sea surface temperature derived from the model i s consis-tent with the actual observations within the limits of the errors in measurement. However, the depth and shape of the simulated mixed layer do show some differences from the depth and shape of the observed mixed layer. A discussion of the possible sources of the discrepancies between the model and the observations is presented in this section. Ekman Divergences Hasselmann (1970) described a possible sequence of responses of the upper mixed layer of the ocean to the passage of a synoptic weather 102 disturbance. The responses to m e t e o r o l o g i c a l f o r c i n g which l a s t s only a few hours are t r a n s i e n t and h i g h l y time dependent. For f o r c i n g which l a s t s about one-quarter to one-half an i n e r t i a l p e r i o d (15.7 hours at 50° l a t i t u d e ) , the response w i l l very l i k e l y i n c l u d e i n e r t i a l o s c i l l a t i o n s . For f o r c i n g which l a s t s c o n s i d e r a b l y longer than one day, Ekman-type responses w i t h t h e i r inherent h o r i z o n t a l divergence or convergence s t a r t to occur. In the model c a l c u l a t i o n s , the v e r t i c a l advective v e l o c i t y ^ w , below the l a y e r was assumed to be zero. However, given some non-zero wind s t r e s s c u r l p a t t e r n p e r s i s t i n g f o r s e v e r a l days, steady s t a t e theory p r e d i c t s r e s u l t i n g divergent or convergent f l o w i n the surface l a y e r w i t h a corresponding upwards or downwards v e r t i c a l v e l o c i t y j u s t below. According to Fofonoff (1962a), t h i s v e r t i c a l v e l o c i t y i s given by 3 T x 1 W = ^ f T + ^ f V H X i ( 4~ 1 9> o o where i s the wind s t r e s s at the ocean s u r f a c e , f i s the C o r i o l i s parameter, and 8 = ~ . Using a model which he o r i g i n a l l y developed w i t h Fofonoff (see Fofonoff, 1962b), Mr. P. Wickett of the F i s h e r i e s Research Board, Nanaimo, Canada has k i n d l y c a l c u l a t e d the d a i l y v e r t i c a l m i g r a t i o n of the upper mixed l a y e r f o r the p e r i o d during which the data f o r t h i s t h e s i s were c o l l e c t e d . H i s model uses equation (4-19) where the wind s t r e s s d i s t r i b u t i o n i s c a l c u l a t e d from d a i l y pressure charts of the N.E. P a c i f i c Ocean; the a l t e r n a t e 2° g r i d shown i n Figure 31 i s employed by the model. The cumulative sum of d a i l y values of the c a l c u l a t e d v e r t i c a l m i g r a t i o n of the bottom of the l a y e r i s p l o t t e d i n Figure 3 0 . These h ^ •z. c\j LJJ I %% o CO si if)•' 8 0 6 0 4 0 2 0 0 - 2 0 - 4 0 - 6 0 A 8 P O I N T A V E R A G E o 2 P O I N T A V E R A G E ft SALT o P O O i — ^ " c N d o _L 2 0 2 4 2 8 M A Y 13 17 21 J U N E 1 9 7 0 2 5 2 9 LU < _ l Li_ O Figure 30: Comparison of the salt deficit of the upper 60 meters ( ) with the cumulative vertical advection resulting from Ekman divergence at Station 'Papa' for the period 20 May to 28 June, 1970. The divergence estimates are averaged over the grid points which surround 'Papa' (shown in Figure 31). If the changes in salt content were entirely due to horizontal divergence within the upper 60 meters, then a 200 mg cm - 2 change in salt would represent a ve r t i c a l migration of the order of 20 meters. o 104 estimates were averaged over a surface area s e v e r a l hundred k i l o m e t e r s i n diameter to give a s p a t i a l r e s o l u t i o n e q u i v a l e n t to the once d a i l y temporal r e s o l u t i o n . The b l a c k t r i a n g l e s were obtained by averaging (weighted by the r e c i p r o c a l of the d i s t a n c e from 'Papa') over the 8 g r i d p o i n t s nearest to S t a t i o n 'Papa'; the c l e a r c i r c l e s were obtained by averaging over the two nearest g r i d p o i n t s . Only the f i r s t few p o i n t s f o r the e i g h t p o i n t averages were i n c l u d e d because they remained roughly constant a f t e r t h a t . F i gure 25 i l l u s t r a t e s that the storage i n t e g r a l s of heat and mass are s t r o n g l y c o r r e l a t e d . The d e f i n i t e seasonal trend that they both e x h i b i t i s l a r g e l y absent from the s a l t content presented as the s o l i d curve i n F i g u r e 3 0 . However, the trend i n s a l t content roughly o o • o X«-*Papa' o • o o I 1 1 1 — 148 146 144 142 °W Longitude Figure 3 1 : G r i d used f o r the v e r t i c a l v e l o c i t y c a l c u l a t i o n s . The estimates of the v e r t i c a l v e l o c i t y at S t a t i o n 'Papa' (X) were weighted averages of the values at the neighbouring g r i d p o i n t s . In Figure 3 0 , the 8 p o i n t average (A.) was over a l l the g r i d p o i n t s shown here; the 2 p o i n t average (o) was over the two p o i n t s nearest to 'Papa' (designated here as • ) . 53 -°N 51 -L a t i t u d e 49 -47 -105 parallels the calculated cumulative vertical advection of the layer. Whether a cause and effect relationship does exist between the two traces over the six week period cannot be determined from the present data. However, the large fluctuation i n salt content on May 24 appears to be correlated with the very large upward velocity and consequent salt transport into the surface layer predicted.from the Ekman calculations. The vertical migration necessary to increase the salt content by a given amount can be estimated. If in some time At, an upward advection of Az at a depth of 60 meters took place; then the increase in salt content of the upper 60 meters would be of the order of PQAS Az where AS i s the change in salinity over the distance Az AS - -5-9z Az 60 From Figure 8, 9S/9z - 5 x 10 °/oo cm * at 60 meters. Therefore, an -2 increase of 0.2 g cm in the salt content of the layer above 60 meters corresponds to an upward advection at 60 meters due to divergence of about 20 meters. Thus, in Figure 30, the vertical migration inferred from the changes in salt content i s roughly 5 times larger than the vertical migration estimated by Ekman theory. That the transient response of the salt content on May 24 was not predicted by the Ekman theory is not surprising: equation (4-19) was derived for a steady state. While the results presented here are not conclusive, they do suggest that large scale variations in the wind stress curl can influence the behavior of the upper layer of the ocean. Turbulent and Advective Heat Fluxes The agreement of the model with the observed data seems to justify neglecting turbulent transfer of heat at the air-sea interface and advection of heat by horizontal ocean currents. In a study such as 106 this one, the advection of heat by horizontal ocean currents for times less than one month can probably be neglected during any season. However, turbulent heat fluxes could possibly account for some of the variation between the model results and the observations. I estimated the turbulent heat fluxes from the routine meteorological data using the aerodynamic formulae of Jacobs (1951) with the exchange coefficients determined by Pond et a l (1971). Calculations every 12 hours for the period 13-25 June 1970 gave a mean turbulent heat loss of roughly-25 to-30 -2 -1 cal cm day which was about one third of the average back radiation. On two occasions the turbulent heat loss was the same size as the back radiation; neglecting the turbulent heat fluxes during the buildup of the summer thermocline probably causes appreciable errors only on isolated occasions. The turbulent heat fluxes do become the dominant terms in the heat budget during the autumn and winter months when high winds often occur together with low relative humidity and large air-sea temperature differences in excess of -5 C v. Tabata (1961) estimated the mean evaporative heat loss during October through December to be between -2 -1 -100 and-200 cal cm day ; also, the solar radiation input then is less than the loss due to back radiation. This large heat loss at the surface causes an unstable layer to form. The resulting penetrative mixing at the bottom of the homogeneous layer has been included in this model, but any proportionality constant, such as m used for the wind mixing, has been arbi t r a r i l y set equal to 1. In order to deter-mine the value of such a constant, laboratory experiments on convective mixing would be necessary. If such an empirically derived proportionality constant can be obtained, this model should be successful in simulating the behavior of the upper layer of the ocean under conditions where convective mixing dominates. Internal Waves Many of the variations in the depth of the mixed layer and in the shape of the temperature profile below the mixed layer can be attributed to internal waves which have periods ranging from the Brunt-Vaisala period, of about one to ten minutes in the upper ocean, to the local Inertial period of 15.7 hours (at latitude 50°). The short period waves, which may be excited by tida l or indirect meteorological forcing, have characteristic wavelengths of the order of hundreds of meters or of kilometers. The longer period waves associated with horizontal i n e r t i a l currents show coherences over about ten to a hundred kilometers (Webster, 1968). Pollard (1970), and Pollard and Millard (1970) were able to explain the time envelope of i n e r t i a l period currents measured from moored current meters in the open ocean in terms of sudden changes in the magnitude and the direction of the surface wind vector. These changes, occurring over periods of six to twelve hours, were associated with moving storm patterns. Figure 24 illustrates temperature contours in the depth-time plane obtained from hourly XBTs dropped during the storm period of Figure 29. From 1200 June 22 to 1200 June 23, 1970, apparent vertical oscillations in the thermocline were present with a period of about 8 to 9 hours. It is doubtful that these resulted from aliasing of short period, short wavelength oscillations. Although the ship was randomly positioned within about 15 kilometers of the geographical position designated to be Station 'Papa', no correlation between the ship's 108 position and the temperature profile was observed. These oscillations may have been long internal waves propagating in the lee of the moving storm system. Such a response has been studied by Geisler (1970) for a hurricane moving over a linear two layer ocean; O'Brien (1967), and O'Brien and Reid (1967) have studied the non-linear response during a stationary hurricane. The relative smoothness of the contours in Figure 24 from hour to hour indicate that the short period internal waves must be of a much smaller magnitude. Turbulent Dissipation Dissipation of energy within the upper mixed layer has been included implicitly in the determination of m, the fraction of the downward transfer of energy by the wind stress ten meters above the sea surface which i s used to increase the potential energy of the water column by mixing. The observed range in the rate of increase of potential energy of the upper 60 meters of the ocean due to wind mixing -2 -1 was about 1 to 20 ergs cm sec (see Table IV). Is the dissipation integrated over the mixed layer of a comparable magnitude? Estimates of the turbulent dissipation in the upper layer of the ocean were obtained by Stewart and Grant (1962); Grant, Moill i e t , and Vogel (1968); and Nasmyth (1970). They showed that the dissipation i s relatively constant within the well-mixed layer below the wave breaking layer. Although Grant, Moilliet and Vogel cautioned against integrating over a l l depths to obtain the total turbulent dissipation per unit surface area, they did make such a calculation for the depth range 15 -2 -1 to 90 meters, and obtained a dissipation of 30 ergs cm sec . If the layer thickness were to increase from about 15 to 45 m during a storm, 109 the total dissipation within the layer would increase by an estimated -2 -1 15 ergs cm sec . Such an increase i s comparable with the observed range in the rate of doing work against the buoyancy forces given in Table IV. The arguments suggest that i f the model is to be applicable over wide ranges of wind speeds and surface heat losses, the dissipation should be treated ex p l i c i t l y . However, before a more com-plicated assumption about the effect of the dissipation on the mixing energy i s incorporated into a model such as this, more experiments on the mechanisms of energy partition in the upper layer of the ocean must be carried out. Chapter 5 CONCLUSIONS A p h y s i c a l model which describes the time dependent behavior of the upper mixed l a y e r of the ocean r e s u l t i n g from atmospheric f o r c i n g has been developed. The t u r b u l e n t mixing and the r a d i a t i v e h e a t i n g have been e f f e c t i v e l y parameterized so th a t the model r e q u i r e s only simple input parameters a v a i l a b l e from r o u t i n e m e t e o r o l o g i c a l measure-ments. I have obtained an a n a l y t i c s o l u t i o n to the model which pre-d i c t s the r a t e and extent of the wind induced t h i c k e n i n g of the mixed l a y e r f o r zero heat input and a constant wind. Numerical s o l u t i o n s f o r the case of d i u r n a l l y p e r i o d i c heating i n d i c a t e that d i u r n a l v a r i a t i o n s i n the upper mixed l a y e r a t S t a t i o n 'Papa' should be measurable only during very low wind c o n d i t i o n s . The model i s s e n s i t i v e to the r a t e of production of energy a v a i l a b l e from the wind s t r e s s to do work against the buoyancy f o r c e s , and to the r a t e of ab s o r p t i o n w i t h depth of the s o l a r r a d i a t i o n . Observations have been obtained of the wind-driven mixing of the oceanic mixed l a y e r which i n d i c a t e the r a t e and extent of the ass o c i a t e d deepening of the mixed l a y e r . F u r t h e r , p o t e n t i a l energy c a l c u l a t i o n s using data from STD traces y i e l d estimates of the r a t e at which work i s done by the turbulence against the buoyancy f o r c e s w i t h i n and at the bottom of the upper l a y e r of the ocean. The r a t i o of the r a t e of p o t e n t i a l energy increase i n the l a y e r during a storm to the downward t r a n s f e r of energy by the tu r b u l e n t wind s t r e s s at a height of 10 meters, I l l designated as m, has been calculated to have a range from 0.0007 to 0.003. The numerical model accurately simulated the behavior of the upper mixed layer over a 12 day period; the data used as input to the model were the observed values of wind speed, solar radiation, and back radiation. The ratio, m, of the energy available for mixing to that transferred downwards by the wind at 10 meters which was needed for the model to yield results consistent with the real profiles i s shown to l i e within the range of values determined independently from observed salinity-temperature-depth (STD) profiles obtained during three storms. The radiation extinction coefficient used in the model is shown to be consistent with values observed by other authors. Other heat transfer mechanisms such as turbulent fluxes from the ocean surface ,and horizontal or vertical advection within the ocean i t s e l f appear not to be important to the spring development of the summer thermocline at Ocean Station 'Papa'. A technique for estimating the wind stress over the open ocean from a moving ship in moderate winds has been demonstrated to give meaningful results. This technique shows promise as a possible means of obtaining a body of estimates of the wind stress for winds over 20 m sec ^ which is necessary for the accurate s t a t i s t i c a l description of the air-sea interaction at high wind speeds. The drag coefficient appropriate to the wind speed at 10 meters has been shown to remain relatively constant at (1.63 ± 0.28) -3 -1 x 10 i n the open ocean for wind speeds up to 17 m sec . Observed variations of C^Q with time are not random, but appear to be correlated with the stage of a storm; these variations cannot be accounted for 112 by s t a b i l i t y effects but may be correlated with the evolution and advection of the wave system associated with the storms. For two storms with similar duration and wind speeds, the maximum wave energy differed by a factor of 4. However, the rate and extent of thickening of the upper mixed layer was approximately the same i n each case. Such a finding lends confidence to the parameteri-zation used i n this model, which expressed the flux of energy available for mixing within the layer as a constant fraction, m, of the rate of downward transfer of energy by the wind ten meters above the water. REFERENCES Beaumont, R.A. and R.S. P i e r c e (1963). The A l g e b r a i c Foundations of  Mathematics. Addison-Wesley: Reading, Mass. Brocks, K. and L. Kriigermeyer (1970). The Hydrodynamic Roughness of the Sea Surface. B e r i c h t e des I n s t i t u t s f u r Radiometeorologie und Maritime Meteorologie. U n i v e r s i t a t Hamburg, Nr. 14. Bryan, K i r k (1969). Climate and the Ocean C i r c u l a t i o n : I I I . The Ocean Model. Monthly Weather Review, November, 97_(11), pp. 806-827« Businger, J.A., J.C. Wyngaard, Y. Izumi, and E.F. Bradley (1971). F l u x - p r o f i l e r e l a t i o n s h i p s i n the atmospheric surface l a y e r . J o u r n a l Atmospheric Sciences 28(2), pp. 181-189. 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