OCEAN THERMAL STRUCTURE RESPONSE TO A MOVING HURRICANE MODEL Robert Norris Trapnell K^O* \ftf ueB c sc^o°u ESS*"1** NAVAL POSTGRADUATE SCHOOL Monterey, California THES! S OCEAN THERMAL STRUCTURE RESPONSE TO A MOVING HURRICANE MODEL by Robert Norris Trapnell, Jr. September 1974 Thesis Advisors : R. L. El« 5 berry D. F. Le: Lpper Approved for public release; distribution unlimited. 1164091 UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When D««» En«er»d) REPORT DOCUMENTATION PAGE READ INSTRUCTIONS BEFORE COMPLETING FORM 1. REPORT NUMBER 2. GOVT ACCESSION NO 3. RECIPIENT'S CATALOG NUMBER 4. TITLE ("and Subtitle) Ocean Thermal Structure Response To A Moving Hurricane Model 5. TYPE OF REPORT a PERIOD COVERED Master's Thesis; September 1974 6. PERFORMING ORG. REPORT NUMBER 7. AUTHORC»> Robert Norris Trapnell, Jr. «. CONTRACT OR GRANT NUM8ER('ij 9. PERFORMING ORGANIZATION NAME AND ADDRESS Naval Postgraduate School Monterey, California 93940 10. PROGRAM ELEMENT. PROJECT, TASK AREA ft WORK UNIT NUMBERS II. CONTROLLING OFFICE NAME AND ADDRESS Naval Postgraduate School Monterey, California 93940 12. REPORT DATE September 1974 13 NUMBER OF PAGES 53 14. MONITORING AGENCY NAME ft ADDRESS*-/' different from Controlling Office) Naval Pos'tgraduate School Monterey, California 93940 15. SECURITY CLASS, (ot thle report) Unclassified 15a. OECLASSIFI CATION/ DOWN GRADING SCHEDULE 16. DISTRIBUTION STATEMENT (ot thle Report) Approved for public release; distribution unlimited. 17. DISTRIBUTION STATEMENT (ot the abetract entered In Block 20, II different from Report) 18. SUPPLEMENTARY NOTES 19. KEY WORDS (Continue on reveree elde it neceeemry end Identity by block number) Air-Sea Interaction Hurricane Model Ocean Model Convective Mixing Wind Mixing Advection 20. ABSTRACT (Continue on reveree elde It neceeemry end Identity by block number) The time-dependent mixed-layer depth and temperature response of a two-layer ocean model to a moving hurricane model was investigated. The hurricane model was that of Elsberry, Pearson, and Corgnati (1974). The maximum hur- ricane wind was bounded at 50 m/sec in order to isolate ocean reactions from hurricane variance. The ocean model was based on that of Fraim (1973) which was derived from the earlier mixed-layer models of Kraus and Turner (19671 DD t'™n 1473 (Page 1) EDITION OF I NOV 65 IS OBSOLETE S/N 0 102-014-660 1 | -i IJNCT.ASSTFTFn SECURITY CLASSIFICATION OF THIS PAGE (When Date Knterod) UNCLASSIFIED CuCU«*1TY CLASSIFICATION OF THIS PAOECWim D»(« En«»r»tf. Block #20 Continued. and Denman (1973). The ocean model was an idealized, quasi-three-dimensional model designed to examine the effects along the path of movement due to the passage of a moving hurricane. Results indicated that slower- moving storms engendered the greatest change in mixed- layer depth and temperature. Away from the center a combination of entrainment mixing and convection was responsible for the majority of the depth and tempera- ture changes. Near the eye, in the upwelling zone, a combination of entrainment mixing, convection, and advection caused temperature change while advection alone dominated layer depth change. DDr 525^3 1473 (BACK) UNCLASSIFIED S/N 0102-014-G601 2 SECURITY CLASSIFICATION OF THIS PAGEfWh»n Dmtm Enfnd) 5> Ocean Thermal Structure Response To A Moving Hurricane Model by Robert Norris Trapnell, Jr. Captain, United States Air Force B.S., University of Texas, 1969 Submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN OCEANOGRAPHY from the NAVAL POSTGRADUATE SCHOOL September 1974 #0& *^ 7"? ABSTRACT The time-dependent mixed-layer depth and temperature response of a two-layer ocean model to a moving hurricane model was investigated. The hurricane model was that of Elsberry, Pearson, and Corgnati (1974). The maximum hurri- cane wind was bounded at 50 m/sec in order to isolate ocean reactions from hurricane variance. The ocean model was based on that of Fraim (1973) which was derived from the earlier mixed-layer models of Kraus and Turner (1967) and Denman (1973). The ocean model was an idealized, quasi- three-dimensional model designed to examine the effects along the path of movement due to the passage of a moving hurricane. Results indicated that slower-moving storms engendered the greatest change in mixed-layer depth and temperature. Away from the center a combination of entrain- ment mixing and convection was responsible for the majority of the depth and temperature changes. Near the eye, in the upwelling zone, a combination of entrainment mixing, convec- tion, and advection caused temperature change while advec- tion alone dominated layer depth change. TABLE OF CONTENTS I . INTRODUCTION 9 II. DESCRIPTION OF THE MODEL 12 A. THE HURRICANE MODEL , 12 B . THE OCEAN MODEL 14 1. Description of the Model 14 2. Derivation of the Model Equations 19 III . DISCUSSION OF RESULTS 26 A. COMPARISON WITH FRAIM'S RESULTS 27 B. DISCUSSION OF COOLING MECHANISMS BENEATH A MOVING STORM 33 C. EFFECT OF STORM MOVEMENT ON OCEAN RESPONSE 41 IV. SUMMARY AND CONCLUSIONS 48 BIBLIOGRAPHY 50 INITIAL DISTRIBUTION LIST 52 LIST OF TABLES Table I. Comparison Between Some 12 -Hour Values of the Fraim Radial Stationary Hurricane Model and the Diametrical Stationary Hurricane Model 29 II. Actual Values of Mixed-Layer Temperature for Key Points of Passage at the 750 km Point of the 4.75 Knot Storm 36 III. Actual Values of Mixed-Layer Depth for Key Points of Passage at the 750 km Point of the 4.75 Knot Storm 36 IV. Comparison of Parameters for Several Storm Translational Speeds at t = 18 Hours 46 LIST OF FIGURES Figure 1. Schematic of Hurricane Moving Along the 200 X 3 Grid in the Open Ocean 16 2. 12-Hour Predicted Values of Tangential Wind, Ocean Radial Velocity, and Ocean Vertical Velocity for the Stationary Diametrical Hurricane Model on a 3 km Grid 28 3. 12-Hour Predicted Values of Mixed-Layer Depth, Mixed-Layer Temperature, and Thermocline Temperature for the Stationary Diametrical Hurricane Model on a 3 km Grid 32 4. Time Cross -Section of Mixed-Layer Temperature at the 750 km Point for the 4.75 Knot Storm 34 5. Time Cross -Section of Mixed-Layer Depth at the 750 km Point for the 4.75 Knot Storm 35 6. Change in Thermal Structure at the 750 km Point After Passage of the 4.75 Knot Storm 39 7. Time Cross -Section of Below-Layer Gradient at the 750 km Point for the 4.75 Knot Storm 40 8. 18-Hour Predicted Values of Tangential Wind, Ocean Radial Velocity, and Ocean Vertical Velocity for a Hurricane Moving at 4.75 Knots... 42 9. 18-Hour Predicted Values of Mixed-Layer Depth, Mixed-Layer Temperature, and Thermocline Tem- perature for a Hurricane Moving at 4.75 Knots... 43 10. Same as Figure 8 Except for a Stationary Hurricane 44 11. Same as Figure 9 Except for a Stationary Hurricane 45 ACKNOWLEDGEMENTS The author wishes to express his thanks to Dr. R. L. Elsberry for his time, patience, and guidance; Dr. D. F. Leipper for his time and comments; and Dr. R. L. Haney for his comments. The author also wishes to thank the personnel of the William R. Church Computer Center, especially: Mr. W. D. Ehrman, consultant, for his time and help in writing the model, and Mr. E. Donnellan, swing-shift supervisor, for his expedition of many jobs and unfailing humor. I. INTRODUCTION There exists a large body of evidence which demonstrates that there is a complex interaction between a hurricane and the ocean. For example it appears that sea-surface tempera- tures in excess of 26 C are required to produce hurricane- force winds. Through these winds, the hurricane imparts stress to the ocean, which acts as a sink for the hurricane's momentum. Interactions have been modeled in several ways. It is apparent that for an ocean-interacting hurricane model to be valid, the heat-momentum energy-exchange balance must be satisfied. Elsberry, Pearson, and Corgnati (1974) developed an axi- symmetric, ocean-interacting hurricane model which met the energy balance conditions. A description of the air-sea interaction within the boundary-layer of a hurricane was the output of and the primary purpose for the design of the model For this purpose, a stationary, mature hurricane over a con- stant, isothermal ocean was assumed. The capability for in- teraction with a time-dependent ocean model was included in the design. One of the uses envisioned for the model described above was to investigate the cause of low sea-surface temperatures found in the wake of a hurricane. Based on limited observa- tions following tropical cyclones, wind mixing, convective mixing, and upwelling appear to be the main cooling mechan- isms. Jordan (1964) concluded that mixing was dominant. Fischer (1958) mentioned upwelling as being important. Leipper (1967) found upwelling to be dominant near the eye with a transition to convective and/or mechanical mixing dominance at greater radii. The classical model by O'Brien and Reid (1967) and O'Brien (1967) simulated a stationary hurricane with constant forcing over a one-layer ocean. Re- sults of the model emphasized the importance of upwelling and downwelling. Gilbert (1974) modeled a stationary, time- dependent hurricane over a multi-layer ocean. Results of the Gilbert model indicate transitory inertial oscillations exert as much influence as the steady-state wind. Fraim (1973) developed a time-dependent, two-layer model for the ocean only based on the earlier mixed-layer models of Kraus and Turner (1967) and Denman (1973). The model was driven by the time -dependent , axi -symmetric , stationary hurricane model of Elsberry, et. al. (1974). The axi-sym- metric assumption permitted representation of the hurricane and the ocean by a radial cross -section assumed to be the integrated average of all radial cross-sections. Fraim (1973) concluded that a combination of entrainment mixing and upwelling was the major factor in changing the mixed- layer temperature and depth. One of several logical extensions of the model was to use it to examine the effects on the ocean induced by a moving hurricane. One purpose for moving the storm was to consider Geisler's (1970) internal wake theory. The theory involves a comparison of the hurricane translational speed 10 (U) with the baroclinic wave speed (C) . For a slow moving storm (U/C < /l) , Geisler postulated upwelling and resultant surface cooling. For a fast moving storm (U/C > 2) , Geisler postulated an internal wave wake at the interface between the mixed layer and thermocline. It should be emphasized that while Geisler's model and a similar model by O'Brien (1968) were primarily hydrodynamic , the model under study here was thermodynamic in nature. The present model was originally designed to be fully a three-dimensional model. However, due to computer time and storage constraints, the model was made to represent a ver- tical cross-section of the ocean along which the storm was moved. The storm was translated at several typical speeds (including U = 0) in order to determine the effects of move- ment on changes in the mixed- layer depth and temperature. 11 II. DESCRIPTION OF THE MODEL A. THE HURRICANE MODEL The hurricane model was a two-layer, time -dependent model by Elsberry, Pearson, and Corgnati (1974). By assuming the hurricane to be mature, an assumption of axi-symmetry could be made. A surface inflow layer and an upper outflow layer constituted the model. The tangential wind (VQ) profile at the top of the inflow layer was taken to be u VQr2 = constant (1) at radii outside the radius of maximum wind r. . Inside r. the winds were assumed to obey VQr = constant. (2) Equations (1) and (2) were empirically derived from observa- tions by Riehl (1963) and supported by observations by Gray and Shea (1973) . The maximum extent of the storm was de- fined to be the radius (r ) at which flow in the outflow layer turned from cyclonic to anti-cyclonic, i.e. VQ = 0. The radius of maximum wind was assumed to coincide with the eyewall radius, which was taken to be the radius at which the average of five adjacent values of equivalent potential temperature (0 ) was a maximum. Since the eye-wall has been observed to be a region of massive latent heat flux due to evaporation, the maximum 0 was strongly tied to the sea- surface temperature of the underlying water. An expression 12 for the maximum wind speed was determined by assuming a gradient balance between the wind field and the pressure field V2 p 8r r v6 LDJ (where f is the coriolis parameter) and using the relation vl.ri V| - -§- . (4) . Integrating equation (3) from r to r. gives 0.5r ve. " T~^ t-2f^T^- ri 1 ° 1 (S) / r -r . -rV4f2(/rTr— - r-)2 + 106 -2 — - (6 - 9 )]• v ^ l o iJ pr v e- e ' J o 1 o From r to 2r the wind profile in (1) was modified to de- oo r v J crease exponentially to 5 meters/second as a transition zone to realistic boundary conditions in order to allow downwelling From observations by Riehl (1963) , the surface pressure at the eyewall was defined by the relation 6p = 2.5669 (6) where the 6 symbols indicate a deviation from p = 1005 milli- bars and 9 =350 K. The radial pressure field was computed by integrating the gradient wind over the radius, using the wind speed from (1) and (2). The atmospheric boundary-layer model consisted of a sur- face layer and a spiral layer above. Specification of the surface layer was by Monin-Obukhov similarity theory. Within 13 the surface layer, the surface roughness was a function of the stress at the surface. The spiral layer, a transition layer from radial to tangential flow, was specified by a modification of Ekman-layer theory. An iterative solution was used to match the two layers. Further details of the iterative solution can be found in Cardone (1969) . A simple model for the hurricane was required to keep computer time and storage requirements to a minimum. Compar- atively simple specification of the boundary layer and lack of specification of upper layers enabled compliance with those requirements. The axi-symmetric assumption allowed representation of the storm by a radial cross-section. Again, computational considerations required such an assumption. A friction velocity for the computation of water velocities and kinetic energy gain, and the heat flux were provided to the ocean model by the hurricane model. A new radial distri- bution of mixed-layer temperature was predicted by the ocean model, and then provided to the hurricane model. This type of interaction can lead to storm intensification. However, for the purposes of this study, the hurricane maximum wind was bounded at 50 m/sec. This enabled isolation of changes in the ocean. B. THE OCEAN MODEL 1. Description of the Model The ocean model was a modification of that by Fraim (1973), which was based on earlier mixed-layer models by Kraus and Turner (1967) and Denman (1972). The ocean was 14 assumed to have an initial mixed-layer depth of 30 m with a mixed-layer temperature of 30 C. The below-layer gradient was initially 0.1 C/meter to the top of an undisturbed layer defined by T = 20 C at Z = 130 meters. These values are typical of those found in the Gulf of Mexico and the Caribbean Sea in June, July, and August. The model was idealistic in that truly isothermal mixed layers and linear below-layer gradients are seldom found. Also, it would not be expected that the same thermal structure would exist over the whole ocean, or even any sizeable fraction of it. There were several important modifications of the Fraim (1973) model. The Fraim model examined a radial cross- section of the ocean beneath a stationary hurricane. Equa- tions for the ocean model were written in R-6 co-ordinates. The present model examined a vertical cross -section of the ocean along which the diametrical cross -section of the hur- ricane was allowed to move. The diametrical cross -section consisted of the radial cross -section used by Fraim and the mirror image of that radial cross -section. This was made possible by the assumption of axi-symmetry in the hurricane model. The ocean equations were written in cartesian co- ordinates on a 200 X 3 grid (see Figure 1) with a 6 kilo- meter grid whereas Fraim used a 100 X 1 grid with a 3 kilometer grid spacing. The time increment was 8 minutes compared to the 10 minute increment used by Fraim. A time increment of 8 minutes was used because it gave finer con- trol over available translational speeds. The movement of 15 O C Rj O •H 5h Jh m o o •H ■P O •H H «" 16 the storm was achieved by moving the center of the storm to successive grid-points at time increments required to ap- proximate the desired speed. Another modification was the addition of a horizontal return flow within the thermocline layer extending down to a specified layer of no motion. This closed the ocean system except for the radial' advection, and thus improved the heat balance calculation for the stationary storm. Lastly, there was added to the model a convective adjustment for local convective instabilities. Several important assumptions were made about the manner in which the ocean and hurricane would interact. First, since the storm was assumed to be mature at initia- tion, the wind stress was imparted to the ocean exponential- ly so that after 9 hours the ocean felt 95% of the stress. Second, some assumption had to be made as to what fraction of the stress would be used in inducing turbulence for en- trainment mixing. The balance of the stress would be used in the creation of horizontal currents. With no pertinent data available it was decided to let entrainment dominate initially. An exponential transition dependent on the mixed- layer depth shifted the dominance to currents as the mixed layer deepened. Third, it was assumed that ocean currents began in Ekman balance and adjusted immediately to changes in the stress pattern. This assumption eliminates inertial waves which must be resolved using smaller time steps. Fourth, in examining only a vertical cross -section it was assumed that there was no variation in temperature or ocean 17 velocity across the path of the hurricane. This simplifica- tion was made because the net Ekman current should be normal to wind flowing inward at a specified angle, rather than being directed radially. Because of this, there was a re- sultant lack of symmetry across the path. This lack of sym- metry introduced instabilities into the model.. Fifth, it was assumed that the storm was wind dominated, implying that downward heat flux due to solar radiation was negligible. The ocean model was forced by kinetic energy input due to wind stress, upward sensible heat fluxes, and upward latent heat flux due to evaporation. Horizontal currents due to wind stress were computed. Resultant upwelling and downwelling due to divergent and convergent current regimes were then determined. A new mixed-layer depth was calculated through kinetic energy, heat flux, and advection considera- tions. Based on the new mixed-layer depth, a new mixed-layer temperature was calculated. The below-layer gradient and temperature at the top of the thermocline were then deter- mined. In this model the mixed-layer temperature and the temperature at the top of the thermocline can be different. If the temperature at the top of the thermocline was greater than the mixed-layer temperature a convective overturning process was initiated in order to stabilize the thermal structure. Following these calculations a radially averaged mixed-layer temperature was passed to the radial cross -section of the hurricane. 18 2 . Derivation of the Model Equations Following Denman (1973) , the first law of thermo dynamics can be written dT Q-p *fR*e (7) dfit p C p C Kw p w p where QT = RAe ' , the heat source term, p = ocean water density, Kw } * Y = average extinction coefficient, C = specific heat at constant pressure, R* = solar radiation incident on the sea-surface, and z = depth in the ocean, positive downward. Expanding the total derivative in (7) yields 3T 3T 3T 3T ro. + u-~— + v-~— + w-^— (8) PwCp 3t "3x v3y 3z dz where w = -rr is positive for downward motion. Time-averaging the turbulent form of (8) gives or XT 3T 3uT 3vT x 3wT 3w'T' P c w p 3t 3x + 3y ' 3z 3z dT QT 3w'T' ar P c Kw p 3z £ + j^f w (10) where the terms u'T' and v'T' were neglected because it was assumed that there was no systematic correlation between u' and v' and T» . Turning now to an examination of the boundary condi tions , firstly at the ocean surface, z = 0, a heat loss at 19 the sea surface will induce convective overturning which cools the mixed layer. Therefore, the downward turbulent heat flux (w'T') must equal the non-radiative heat loss, i.e. t",T'>o - - (PC- (") Kw p where QA = QE + Qs » and Qp = upward latent heat flux due to evaporation, and Q<, = upward sensible heat flux. At the base of the mixed layer downward turbulent heat flux is caused by wind-induced mixing and convective overturning. The turbulent heat flux must equal the heat transfer due to entrainment of water into the mixed layer. This balance is expressed by dh ^'T'h = C IK ' w HTS - Th) (12) o where h = mixed-layer depth, T = mixed-layer temperature, T, = temperature at the top of the thermocline, and dh — -rr- = the time variation of the mixed-layer due to wind mixing and convective overturning. For entrainment to occur (dh /dt) > 0, that is, there can be no shallowing of the mixed layer unless w is upward. Inte- grating (10) over the mixed layer and substituting the bound- ary conditions gives L dTs ^T dho QA h~ -XT = — k- h - ( —r£ - w )(T - T, ) £~.(13) o ^t PwCp o l Tt s hJ pwCp 20 Expanding the total derivatives, neglecting solar radiation, and solving for 3h /at gives 8h -h 8T 9uT 3vT 3wT O = 0 r S + S_ S S -, ^T (T -Th) L at ax ay az J QA 3hQ ahQ - p c (t -f. ) + w - u ~af - v"ay (14) w p w s h- which is a predictive equation for deepening shallowing of the mixed layer in terms of the local change in mixed-layer temperature . To derive a similar predictive equation for the mixed-layer temperature Denman (1973) integrated (10) to an arbitrary depth z, and then over the mixed- layer depth yielding K at fi° QAh hc rS--f c^)z^ -¥£ (is) i w p J»0 where -J (w'T1) dz is the conversion of potential energy to kinetic energy by convection. If the turbulent kinetic energy is not varying in time, we may write W + G - D = 0 (16) where G = kinetic energy input from the wind, D = dissipation within the mixed layer, and h W = S° (w'T') dz then substituting (16) into (15), expanding the total deriva tives, and solving for aT /3t yields 21 8T 20. nfr n^ 8uT 8vT 8wT s = XA 2 (G-D) s s s ri7>. at ' h p C " hz 8x " 9y 8z l J o w p o 7 which is the predictive equation for change in the mixed- layer temperature due to convection, entrainment mixing, or advection. Setting Q . = 0 , and/or (G-D) = 0, and/or u, v, and w = 0 allows isolation of the cooling mechanisms for use individually and in combination. Substitution of (17) into (14) yields 8ho Qa 2(G-D) /ho 9ho rijn ~Tt " p c (T — ^TTT h~TT — =~T7T ' uTx" " v~ay + w (18} Kw p v s hJ oK s hJ J which is the predictive equation for changes in the mixed- layer depth as used in the present model. A partitioning procedure was used to divide the total available stress, t , from the boundary layer into one frac- cl tion for the production of turbulence, x , and the remainder (x - t ), for the generation of wind-driven currents. h x = x EXP ( - pA- ) (19) o a v Cz J K J o where x = total stress available from boundary-layer model, h = mixed-layer depth, z = roughness length, and C = a constant (101*) (after Fraim 1973). Kraus and Turner (1967) assuming a value for x , specified the mechanical energy input (G) in equations (16) , (17), and (18) as 1 p* 3/2 G = -~- ( — ) u*x (20) gotpa pw ° 22 where g = gravitational acceleration, a » (1/p ) (dp /dt) , coefficient of expansion p = air density, a p = sea water density, Kw J * U* = atmospheric friction velocity, = /x / p , and a a x = that part of atmospheric surface stress used to produce turbulent mixing. The dissipation (D) was set equal to zero. The calculation for the ocean velocities was by means of Ekman-spiral theory. The theory specifies a depth (Z ) at which the current direction has reversed from its surface direction, the current is EXP(-tt) of its surface value, and below which the wind has no direct influence. The velocity components of the ocean within the spiral layer are (t -t )cosa u= — - — - EXP(-BZ) [cos(-BZ)-sin(-BZ)] (21) P C2f£-)Js w pw (t -t )sina v= — - — % EXP(-BZ) [cos(-BZ)-sin(-BZ)] (22) p (2fM^ wv p * *w where Z = tt/B, specifies the Ekman depth, B = [£/2v/pv]h, f = coriolis parameter, p = sea water density, w y = vertical eddy coefficient of viscosity, z = depth, and a = atmospheric inflow angle. 23 Integration of (21) and (22) over the Ekman depth (Z ) gives the net mass transport in the Ekman layer M = (x - t )cosa/£ (23) M = (t - t )sina/f. (24) y v a oJ v ' From these mass transport equations it is possible to calculate the mean ocean velocities in the mixed layer uZ = M /p (25) e x w K J vZ = M /p . (26) e y w *■ * The horizontal velocities in the layer are related to the ver- tical velocity w by the continuity equation |w= . ( |u + |v j ■ ( dz v dx 9y J K } Equations (17) and (18) give two equations for three unknowns (T ,h ,T\). A return flow is calculated for the v s ' o ' rr layer between 130 meters and the new mixed-layer depth. It is assumed to have the same mass transport as the opposing current in the mixed layer, and to be centered on a point 2/3 of the distance from 130 meters to h . The below-layer o ] gradient is adjusted by the return flow, upwelling, and down- welling. The new T, is the temperature at the intersection of the new below-layer gradient and the new mixed-layer depth. This relation is given by 24 where 3T/3z = is the below-layer gradient, TR = temperature at top of the undisturbed layer = 20 C, and ZR = depth of top of undistirbued layer = 130 m. 25 III. DISCUSSION OF RESULTS It was the intent of this study to use a sea-air inter- acting numerical model to simulate the effects upon the ocean of a hurricane moving across it. The reasons for con- ducting the study were: 1) to investigate the cause of low sea-surface temperatures in the wake of a hurricane, and 2) to determine the effects on the ocean for different speeds of storm movement. Changes to the mixed-layer depth are controlled by Equa- tions (17) and (18) . Convective mixing is represented by the first term in both equations. Entrainment mixing in- volves the second term in both equations. Advection is described by the remaining terms in the equations. Accord- ing to the two equations entrainment mixing and convection can only cool and deepen the mixed layer. It has been as- sumed that Q. is positive by assuming a wind-dominated regime. On the other hand, since ocean velocity values can be both negative and positive, the advection terms can both warm and cool, shallow and deepen the mixed layer. This discussion is divided into three parts: 1) a com- parison with the results of Fraim (1973) , 2) a discussion of the relative importance in this model of the mechanisms dis- cussed above, and 3) a discussion of the effect of the hurri- cane translational speed on changes in the ocean. 26 A. COMPARISON WITH FRAIM'S RESULTS Before attempting to move the hurricane, it was necessary to establish that the stationary version of the present model yielded reasonable results. For the initial experiment, the 200 x 3 grid (Figure 1) was defined with a 3 km grid spacing as was the 100 x 1 grid in Fraim's model. Profiles of values at 12 hours for the present model are shown in Figure 2. A comparison of maximum and minimum values for Fraim's model and the present model are listed in Table I. The first point of comparison between the two studies is the hurricane portion of the model. The initial conditions for both hurricanes were the same: air temperature T = 29C, radius of maximum wind r. = 27km, and maximum tangential wind V. = 30m/sec. There were some differences. Fraim did not l constrain the maximum tangential wind or the radius of maxi- mum wind. In the present model, for this experiment, the maximum tangential wind was forced to increase sinsuoidally from 30m/sec to 50m/sec at 12 hours. Also, a limit of r. < 60 km was set on the radius of maximum wind. l - The tangential wind profile (bottom, Figure 2) is char- acterized by a rapid increase with radius to its maximum value at r =60 km (Equation (2)), a slow decline to its value at r (Equation (1)), and a more rapid decline to the outer boundary value. From r to 2r the wind is allowed to } oo drop off to 5m/sec. At r the change in the wind profile is abrupt and therefore easily detected. The radius r tends to be the limit of applicability of the assumption of axi- symmetry at upper levels, although the assumption is good to 27 CD C\J _J CC r-y LJ cc i— t ~~> t. rv. til \ * z> n. O CE VI i — i \ h- z: 21 w U_i O o 71 2: cr i— i h- :z 240 i 3B0 DI5TRNCE (KM.) 600 TIME 12.0 HOURS Figure 2. 12-Hour Predicted Values of Tangential Wind, Ocean Radial Velocity, and Ocean Vertical Velocity for the Stationary Diametrical Hurricane Model on a 3 km Grid. 28 Table I. Comparison Between Some 12-Hour Values of the Fraim Radial Stationary Hurricane Model and the Diametrical Stationary Hurricane Model. Fraim Model Diametrical Model Max ORV .48 .28 Max DW .03 .21 Max UW 3.35 1.88 Min MLD 4.31 21.23 Max MLD 51.23 58.20 Min MLT 25.26 28.84 Max MLT 30.43 30.00 Max TW 55.58 50 RMW 42 60 Area average ocean heat loss 930 371 Min TT 19.80 26.90 Max TT 29.99 30 ORV - ocean radial velocity (m/sec) DW, UW - downwelling, upwelling (m/hr) MLD - mixed-layer depth (m) MLT - mixed-layer temperature (C) TW - tangential wind (m/sec) RMW - radius of maximum wind (km) TT - thermocline temperature (C) Heat loss (cal/cm2) 29 a greater radius at lower levels. Because r expands with intensification (r = 225 km at 12 hours vs. 150 km at 0 o hours) , the wind velocity value at the outer boundary of the hurricane model has increased from 5m/sec to 12.5m/sec. The maximum tangential wind (Table I) was about 10% greater in Fraim's model than that of the present model due to lack of constraint on the maximum wind value. The ocean radial velocity (actually the x-component of the horizontal velocity) (middle , Figure 2) responds directly to the stress imparted by the tangential wind (Equations (21) and (22)). Ocean velocity values are calculated by integrating the x-component of the Ekman current over the mixed-layer depth. Fraim integrated the mass transport over a shallower depth thus giving a larger velocity (Fraim's results , Table I) . As the ocean radial velocity directly follows the tan- gential wind, the ocean vertical velocity (top, Figure 2) follows the gradient of the radial velocity. The result is that the maximum upwelling occurs at the point of maximum increase of ocean radial velocity, i.e., between the center of the storm and the radius of maximum wind. Maximum down- welling occurs at r where the ocean radial velocity drops sharply. Fraim's model upwelling (Table I) is stronger be- cause the radius of maximum wind was smaller. The Fraim model had no major downwelling because tangential wind pro- file was not altered in the r to 2r region. o o ° The radius of the maximum mixed layer depth, minimum mixed-layer temperature, and minimum thermocline temperature, 30 (Figure 3) is the same, and coincides with the radius of maximum wind. Coinciding with the radius of maximum upwell- ing are the minimum mixed layer depth, and pips of warming in the mixed-layer (T ) and thermocline (Tv) temperatures. The pips appear to be due to a lack of cooling due to upwell- ing. The minimum mixed-layer temperature (Table I) is much smaller in Fraim's model because it did not have the thermo- cline layer return flow discussed in Chapter II B. 1. Lack of a return flow of water allowed strong cooling of the thermocline layer in the Fraim model which created a large T -T, difference (Table I) and retarded deepening of the mixed layer. Consequently, the area-averaged heat loss in Fraim's model was more than twice the loss in the present model (see Table I). Most of the heat loss is concentrated within 100 km of the center. Advection of water both hori- zontally and vertically acts to cool or warm the thermo- cline layer. Warming in the upper portion of this layer increases the below-layer gradient while cooling decreases the gradient. A change in the below-layer gradient is thus associated with the temperature at the top of the thermo- cline, that is, the thermocline temperature, since the tem- perature at the level of no-motion is held constant. Stronger upwelling in Fraim's model (Table I) accounts for the shallower mixed-layer depth minimum and seems to indicate a higher maximum mixed-layer temperature. The comparison be- tween the models shows the results are similar, however, due to the differences mentioned above the present model yielded more. reasonable results. 31 UJ 1 — 1 00 _J 0 <.\j u ^^ o -5— CC Q_ UJ UJ h- 1— CC l-U >- CE LJ CM Q UJ X Q. 1 — 1 21 UJ 1— [V. 0 CC Uj > /■— ■> CC z: _i w CD 31 UJ h- X Cl. 1 — 1 UJ z: tn mo DI3TRNCE (KM, ) TIME = 12-0 HOURS Figure 3. 12-Hour Predicted Values of Mixed-Layer Depth, Mixed-Layer Temperature, and Thermocline Tempera ture for the Stationary Diametrical Hurricane Model on a 3 km Grid. 32 B. DISCUSSION OF COOLING MECHANISMS BENEATH A MOVING STORM As the comparison of the present model with Fraim's was considered satisfactory, it was decided that the results warranted progressing to a moving storm. The storm was al- lowed to interact with the ocean until a maximum wind of 50 m/sec was achieved so as to allow examination of the ocean's response to the same storm. A new limit of r. < 45 r 1 - km was placed on the radius of maximum wind. The 200 x 3 grid spacing was expanded to 6 km giving a 12 km by 1200 km rectangle in the open ocean. Starting point for the center of the storm was the 900 km point (Figure 8) . Movement of the storm was from right to left at 1 grid increment per 2, 3, 5, and 7 time increments, or 6 km per 16, 24, 40, and 56 minutes which corresponds to 11.84, 7.89, 4.75, and 3.39 knots respectively. A time cross -section of the 4.75 knot storm was drawn for the mixed-layer temperature response (Figure 4, Table II) and mixed-layer depth response (Figure 5, Table III) of the ocean to the mechanisms indicated in the figures. The 750 km point was chosen for this section because during the 30 hour forecast period it experienced passage of the most in- teresting parts of the storm i.e. the radii of maximum wind, the radii of maximum upwelling, and the no-stress zone of the eye. Figure 8 can be used to see the relative locations of these zones. As shown in Figure 4, advection alone has virtually no impact on the mixed-layer temperature, though in the absence of deepening (Figure 5) shows a slight warming as mentioned 33 o II < o o o o II o ii CO s CO CO p o m P a •H o PL, o o P P Rj H ?H D* •H m o o •H ■P O CJ CO 6 o •H £ in in CA in O pq (K H W in in in in GO •H 35 CO to 6 e VI to £ •H NUIO rH r- 1 P •H t t>>\0 \D X .— 1 CT3 O^ r^i t— 1 CT) \D •H rH aj vO\OH vO o <^ Q OK7> CO CO O <^ LO \0 rH r^ 0 LO u (ni rg cni r-i CNJ Oh O tO LO LO "d- \£> f- a) 2: £ 0) P • H > • • • Oh aj C cn cn cn en CO a> aj P! \0 vO vO VO LO ^-1 0 r~j (ni {ni OJ rg bo fH 0 tO LO LO LO \0 m P CJ c3 P U o C to 0! W WD" to W uD" to aj P Ph Pi •H <+H O O Oh to X P P P CD C C c « CD •H (D s CO LO LO LD -3- O 6 CO cn o 4-H •H aj 1— 1 cr> "^t ^t- ^3" i— 1 Oh •H aj rH N vO >0 \o l>- cn cn cn CTi en X tO t3- -^t ^t C^] u o (nj rsj oj rj CnJ E-i . o Q 0 $H S •H K)Nrf N rH rH •H to CO to rsi 0 > rH r~- ^t 0-1 r- 0j 4-> CT> Cn CT> CT, cn -*cno 0 to J CO > o CN) cni rq og CN] J > O to to "=t ■^ *tf- 1 C 1 • C •d -p o -d £ O a> o u cu r-l u * c X O •H X W •H P to S c S 0 C LO 0 O m r^ •H <4H P •H o ■«^ P aj O O P aj tf) c •H to ^ c •H a» 0 \0 LO ^t cn a^ > sx •H X 0 0 O 0) 2 LO •H X a> co cr> og r-~ CD rH aj p p u 5h X rH a$ l>- p u rH c rH 000 0 O cn 0 r- LO LO > 4-1 CD O to to to to to X > ^ CD O cni to r-a tNi rsi X) o > 03 > aj 1— 1 13 rH aj aj cd rt OS +-> C CD • H cti U i o i CO m in +-> o o cj ^; • H • O O O r- lf\ o s • Eh • i 0 oj M * — ■. o PQ ^ K a 0 O en o o • o •H 40 thermocline layer indicating a more intense upwelling than for advection alone. C. EFFECT OF STORM MOVEMENT ON OCEAN RESPONSE The second purpose listed for the study was to determine the effect of movement of the storm on the ocean's response to the storm. In Table IV is listed some maximum and minimum values of key parameters for several storm translational speeds . Changes in the mixed layer due to storm movement are comparatively easy to explain. Slower-moving storms have more time at a given location for entrainment mixing and con- vection to cool and deepen the mixed layer. For the sta- tionary storm the downwelling maximum nearest the storm center (see Figure 10) has added to the deepening but per- haps retarded the cooling. The anomalous spikes of down- welling (for instance at the 900 km point in Figure 10) are due to the sharp drop in the tangential wind at the storm boundary and the mixed- layer depth spikes generated due to the stationarity of the storm. Increasing heat flux to the storm with increasing storm speed (Table IV) is due to fast moving storms extracting more heat because of continuously encountering uncooled 30 C water. Radial heat flux (Table IV) is due to heat flow in currents out through the mixed layer and in through the thermocline layer. For a stationary storm advection supplies more heat than it removes. This seems to be the source of warm water for the mixed-layer temperature pips (Figure 11). 41 o m _J cc . — . cj CC i — i Z) en LO cc x rv 1:1 s 4 Z> ZI L) 21 CL \ UJ _J U LU LO o Z> 1 | _J CL CJ i — i LiJ n D CO i cr. \ ro (X z: z: CL * LU .j O LU cr O Z> o i o LO O j u : cr to LO i — i \ ''•nJ i— z: z: ^_y LU CD Ll z: z: CE i — i h- 1Z « o 0 240 480 720 DISTRNCE. (KM> 960 1200 TIME -- 18,0 HOURS Figure 8. 18-Hour Predicted Values of Tangential Wind, Ocean Radial Velocity, and Ocean Vertical Velocity for a Hurricane Moving at 4.75 Knots. 42 en LU » — 1 . CD _J o ^M u xw CD 27. .. CC D. LlJ z: 11 LU \~ h- ID C\J en cc CE U) ^ CD UJ Cl x z: •— • UJ ZI I— CV O CC ^J ~ LOJ, CE E m Q HI LU I— X Q_ i— i UJ I G D 0 STORM CENTER 240 480 ?2t) DISTRNCE I KM. ) 960 1200 TIME = 18*0 HOURS Figure 9. 18-Hour Predicted Values of Mixed-Layer Depth, Mixed-Layer Temperature, and Thermocline Tempera ture for a Hurricane Moving at 4.75 Knots. 43 t>oo *0 o m _J CE . — 1 LJ cc i — i ZD h- CD Lj3 (X m r\_ UJ \ % (D UJ CD Z> 12G0 DJSTRNCE (KM TIM r 18 »n U HOUR e J Figure 10. Same as Figure 8 Except for a Stationary Hurricane 44 m UJ t — ( CO ._) vJ (VI u ^_y o 27. % cc D.. UJ z: n Li_i \— 1— CM p~> cc CD CC LJ (vi _) fc-- CD , UJ CL X z: 1 — 1 U_l JZ h— (VI CD CC ci r ^ CD X LU r— x: cl LjU CD O CD 0 240 480 i ?20 DISTRNCE (KM, 1200 TIME = 18*0 HOURS Figure 11. Same as Figure 9 Except for a Stationary Hurricane 45 Table IV. Comparison of Parameters for Several Storm Translational Speeds at t = 18 Hours. 3.39 4.75 7.89 11.84 Stat kts. kts. kts. kts . .27 .28 .29 .29 .29 .48 .32 .28 .27 • .21 1.18 3.10 2.98 2.78 2.60 17.18 30 30 30 30 72.60 61.62 57.91 54.78 53.94 28.85 28.86 28.88 29.06 29.25 30.09 30 30 30 30 max ORV max DW max UW min MLD max MLD min MLT max MLT area average ocean heat loss - 29 265 325 352 335 radial heat flux -146 58 89 103 92 heat lost to • storm 156 161 163 168 174 *see Table I for abbreviations 46 The apparent reversal of radial heat flux values for the two fastest storms in unexplained. A sustained upwelling for 18 hours duration would account for the shallow minimum mixed-layer depth (Table IV) . By the equation for continuity (Equation (27)) the integrated ver- tical velocity is equal to the horizontal velocity divergence times the integrated depth. Therefore the maximum upwelling value for the stationary storm is small because of the mixed- layer depth in which it occurs (Figures 10 and 11) . For a moving storm the leading upwelling zone the upwelling is more intense because it occurs in the deepest mixed layer (Figures 8 and 9). Maximum downwelling intensities are governed by mixed-layer depth also (Figures 8, 9, 10 and 11). Maximum values for the ocean radial velocity are much the same be- cause the mean mixed-layer depths at the radius of maximum ocean radial velocity are nearly the same. The ocean radial velocity profiles are altered slightly due to mixed-layer depth considerations (Figures 8 and 10) . 47 IV. SUMMARY AND CONCLUSION It was the intent of this study to use a numerical model to simulate the effects of a hurricane moving across the ocean. In such numerical models the inherent limitations in elude the finite differencing scheme as well as the initial and boundary conditions. Imposed limitations involve assump tions associated with applying the model. For this study a simple model was used for both the hur- ricane and the ocean. A bound was set on the hurricane maximum wind in order to standardize the storm input to the ocean. There were several key assumptions about the way in which the ocean and storm interact: 1) wind stress was par- titioned into making currents and turbulence based on the mixed-layer depth, 2) ocean currents were assumed to be in Ekman balance and to respond immediately to changes in the stress pattern, 3) it was assumed that there were no gradi- ents of temperature or ocean velocity across the path of the storm, 4) the storm was assumed to be wind-dominated, and 5) since the storm was mature at initiation the total stress imparted exponentially until 95% of the total stress was imparted to the ocean at 9 hours. In answer to the two main questions of interest: 1) cooling of the ocean appears to be primarily due to entrain- ment and convection except in intense upwelling within the region of maximum wind. In this region all mechanisms con- tribute to cooling of the mixed layer, 2) slow moving storms 48 cool and deepen the mixed layer more than do fast moving storms. Waves in the mixed- layer depth due to internal waves on the mixed-layer thermocline layer interface postu- lated by Geisler (1970) were not seen. This is due to the thermodynamic nature of the model under study here compared to the hydrodynamic nature of Geisler' s model. In summary, the results are satisfactory within the limits of the model. The logical course to follow is to reduce the number of assumptions used. In a model with as many simul- taneous interactions as this one, the task is one to be taken slowly, in small increments. 49 BIBLIOGRAPHY 1. Cardone, V. J., 1969: Specification of the Wind Dis- tribution in the Marine Boundary Layer for Wave Forecasting, New York University, School of Engineering and Sciences, Scientific Report GSL-TR69-1, University Heights, New York. 2. Denman , K. L., 1973: A Time -Dependent Model of the Upper Ocean, Journal of Physical Oceanography, 3_ 173-184. 3. Elsberry, R. L., Pearson, N. A. S., and Corgnati, L. B., 1974: A Quasi -Empirical Model of the Hurricane Boundary Layer, Journal of Geophysical Research, 79 , 3033-3040. 4. Fedorov, K. N., 1973: The Effect of Hurricane and Typhoons on the Upper Active Ocean Layers, Oceano- logy, 12, 329-332. 5. Fischer, E. L., 1958: Hurricanes and the Sea-Surface Temperature Field, J. Meteor . , 15 , 328-333. 6. Fraim, T. S., 1973: Oceanic Thermal Response to a Time-Dependent Hurricane iModel, MS Thesis, Naval Postgraduate School, Monterey, California. 7. Geisler, J. E., 1970: Linear Theory of the Response of a Two Layer Ocean to a Moving Hurricane, Geophysical Fluid Dynamics, 1, 249-272. 8. Gray, W. M. and Shea, D. J., 1973: The Hurricanes' Inner Core Region, 2, Thermal Stability and Dynamic Characteristics, J . Atmos . Sci . , 30, 1565-1576. 9. Jordan, C. L., 1964: On the Influence of Tropical Cyclones on the Sea-Surface Temperature Field, Proc. Symp . Trop. Meteor., New Zealand Meteor. Service, Wellington, 614-622. 10. Kraus, E. B., and Turner, J. S., 1967: A One-Dimen- sional Model of the Seasonal Thermocline, Tellus , 19, 88-106. 11. Leipper, D. F., 1967: Observed Ocean Conditions and Hurricane Hilda, 1964, J. Atmos. Sci. , 24, 182-196. 12. O'Brien, J. J., 1967: The Non-Linear Response of a Two-Layer, Baroclinic Ocean to a Stationary, Axially-Symmetric Hurricane: Part II, J. Atmos. Sci. , 2_4, 208-215. 50 00' 13. O'Brien, J. J., 1968: The Response of the Ocean to a Slowly Moving Cyclone: NCAR Manuscript, No. 68-57 , pp. 36 . 14. O'Brien, J. J. and Reid, R. 0., 1967: The Non-Linear Response of a Two-Layer, Baroclinic Ocean to a Stationary, Axially-Symmetric Hurricane: Part I, J. Atmos. Sci. , 24, 197-207. 15. Riehl , H., 1963: Some Relations Between Wind and Thermal Structure of Steady-State Hurricanes, J. Atmos . Sci. , 20, 276-287. 51 t)VJ INITIAL DISTRIBUTION LIST No. Copies 1. Defense Documentation Center 2 Cameron Station Alexandria, Virginia 22314 2. Library, Code 0212 2 Naval Postgraduate School Monterey, California 93940 3. Professor R. L. Elsberry, Code 51Es 7 Department of Meteorology Naval Postgraduate School Monterey, California 93940 4. Professor D. F. Leipper, Code 58Lr 1 Department of Oceanography Naval Postgraduate School Monterey, California 93940 5. CAPT R. N. Trapnell, Jr., USAF 3 Det 1, 1st WWg COMNAVMAR Box 17 FPO San Francisco 96630 6. Air Weather Service (AWVAS/TF) 1 Scott AFB, Illinois 62225 7. Professor R. L. 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