Reprinted from JOURNALOF ATMOSPHERICAmericanSCIENCES,M~Jocic:aI Vol. SocIety 24, No.2, March,. 1967, pp. 197-a)7 PriDteciIa u. s. A.
The Non-Linear Responseof a Two-Layer,Baroclinic Oceanto a Stationary,Axially- SymmetricHurricane: Part I. UpwellingInduced by Momentum Transferl
JAKESJ. O'BRIEN NatioMal C".,. for AlmDspllmcRes~.. ~, CMo. AND ROBERTO. REm TGGI A6'M U.;..,siIy, CMle" SIaliOtf (Received 21 June 1966)
ABSTRACT This study is concerned with the theoretical description of upwelling induced in & str&tified, rot&ting, twG-l&yer oce&nby momentum transfer from &n intense st&tion&ry, axi&1ly-symmetric &tmospheric vortex. The dyn&mic intema.l responseof the oceanis assumedto be ui&lly-symmetric which permits consideration of the solution in two independent vari&bles, radial dist&nceand time. NumeriC&1integmtion vi& the method of characteristics is utilized to obtain v&lues of radial velocity, tangenti&l velocity, &nd depth of the upper I&yer for & period of two d&ys. Transfer of momentum between the air &nd the se&&nd between the upper and lower layers &re allowed. Transfer of he&t and moisture with the atmosphere is not considered. A general model is derived which le&ds to a hier&rchy of models of increasing complexity. The detailed solution of the first of these is illustrated. Results agree qualitatively with observ&tions taken in the Gulf of Mexico following hurricane BUda, 1964.Intense upwelling is confined to within twice the radius of maximum winds. The displacedwarm central waters produce somedownwelling adjacent to the upwelled region. The degreeof upwelling is time-dependent &nd the hurricane-force winds must act on the ocean for several hours before significant upwelling occurs. The model indic&tes a strong coupling of the radially prop&g&Ungintema.l w&ve mode &nd the vortex mode of the system. This coupling confines the signiDcant intema.l disturbances to within the wind-forced region.
1. Introduction pheric turbulent shear stresses are prescribed. TranSfer of momentum between the air and sea and between the The present investigation is concerned with a par- ticular aspect of the general problem of air-sea inter- upper and lower layers is included. However, transfer of sensible heat or latent heat between the model ocean action, specifically, the theoretical description of up- and the atmosphere is not considered. An initial state welling induced in a stratified, rotating ocean by an intense hurricane. Attention is confined to the time- of rest is stipulated for the laterally-infinite, homo- geneous ocean layers. dependent response of a two-layer ocean over which there is a stationary, axially-symmetric atmospheric Within this framework, it is possible to consider a vortex. The dynamic response of the ,ocean to the hierarchy of models of increasing complexity which can cyclone is assumed to be axially symmetric, which be studied separately. The present paper includes the permits consideration of the solution in cylindrical co- development of the general model and describes the ordinates as a function of only two independent vari- detailed solution of one specific model. This model ables, radial distance and time. This implies that scales permits momentum transfer between the air and the sea of motion comparable to surface waves and smaller as well as between the t"f!o ocean layers. A sUbsequent scalesare ignored explicitly in the model. Effects of small paper will describe a similar model which considers the scale turbulent processes are included implicitly by additional effect of turbulent mixing of heat and salt consideration of mixing and the vertical transfer of between the ocean layers. This second paper will also momentum. compare various energy integrals of the two models: The dependent variables are depth, density, and the This investigation was inspired by observations taken by Leipper (1967) who reported an observed depth-averaged radial and tangential velocities in both decrease of over 5C of sea surface temperature in an upper and lower layers of the ocean. The radial dis- area of 15,(XK)mi2 following the passage of hurricane tnoutions of atmospheric surface pressure and atmos- Hilda, 1964, in the central Gulf of Mexico. He disPlayed 1 A part of this paper was p~ted at the 47th Annual M~tiDg vertical sections perpendicular to, the hurricane's path of the American,Geophysical Union, Washington, D. C., April which when compared to undisturbed sections, showed 1966. the changes in temperature and salinity which appeared 198 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUIm 24
of cold deep water can explain the extreme cooling found by Leipper after Hilda. The previous theoretical studies are reviewed and compared with the present work in a later section.
2. Formulation of a general model Consider a two-layer ocean .in a laterally-infinite region of uniform undisturbed depth. The fluid densities SCHEMATIC OF COORDINATES AND VELOCITY within the upper and lower layers are designated COM PONENTS respectively by PI and P2, where P2> Pl. (See list of symbols at end of paper.) A schematic drawing of the geometry of the modd is shown in Fig. 1. A cylin- drical coordinate system is chosen such that the radial coordinate r is zero at the center of the axially-sym- metric atmospheric cyclone and increases outward. The vertical coordinate s is zero at the horizontal sea bed and increases upward. The instantaneous thick- nesses of'the 'Upper and lower layers are denoted, respectivdy, by hi and hI. These thicknesses and the densities, PI and P2, are taken to be functions of r and t but not of s. The fundamental assumptions imposed on the com- plete hydrodynamic differential equations are:
a) The dynamic response to the axially-symmetric FIG. 1. Geometry of the hurricane-ocean system. The lower dia. gram representsany radiaI crosssection. In addition,"V>"I. atmospheric cyclone is assumed to be axially symmetric, i.e., independent of the azimuth coordinate 8. b) The horizontal velocities are assumed to be to be the result of the storm's action on the ocean.The independent of depth in each layer. (Thus, they can be warm surface waters were displaced to either side of regarded as corresponding to the vertically-averaged the cyclonepath and a coreof cold water appearednear values of velocity of an actual system.) the center of the wake, suggestingactive upwelling in the latter region. Vertical displacementof isotherms c) The vertical distribution of pressure is hydro- of the order of 60 m occurred near the center. His static. (This impijes that we confine attention to dis- observationswill be comparedwith the results of the turbances of fairly large wavdength.) presentwork. d) The traditional Coriolis approximations are Leipper's measurements are the most striking assumed and the Coriolis parameter f is constant. evidence of the phenomenon; however, there are e) Both molecular and turbulent lateral friction are additional observations. Fisher (1958) and Jordan neglected. (1964),using routine ship observation.saveraged over £) There is no exchange of mass or heat between the 1 day and 15 days, respectively,indicate that low sea atmosphere and the ocean. That is, evaporation, surface temperaturesare often found in the wake of precipitation, sensible and turbulent heat transfer, and intenseatmospheric cyclones. Hidaka and Akiba (1955) radiation exchange with the air are neglected. and Ichiye (1955) refer to occurrencesof cold surface It should be recognizedthat the variation of the waters near Japan following the passageof typhoons. Coriolis parameterf with latitude may produceasym- There are severalprocesses which may contribute to metrical motions aside from the modeling assumption the lowering of seasurface temperatures. Jordan (1964) that the responseto the external torque appliedby the estimated that, in the presenceof a deep mixed layer, wind stressis axially symmetric. However, the resmts the maximum seasurface temperature decreasewhich of the numerical models will be comparedwith actual can be accountedfor by processessuch as precipitation, evaporation, and sensibleand turbulent heat transfer east-westcross sections following hurricane Hilda and, in 72 hr is 2F. therefore, the approximation,f is constant, is retained. It is expected, a priori, that the ocean's initial The problem is to derive the changesin Pl, Pt, hl and dynamic and thermodynamic responseto a hurricane h, which result from tlie impressionof the stationary, would be to produce a well-mixed, isothermal surfaCe cyclonic storm on the syStem.The vertically-averaged layer which usually would lead to some cooling of the equations of motion and continuity for the upper and sea surface. However, it appears that only upwelling lower layers are: MARCH 1967 J J 0 'B R I E NAN D R O. REID 199
Upper layer between the two ocean layers. As an appro;\;mation, the density will be consideredto be a linear function of -+#1a.,1 a.,1 -+1VI 1 - - ( f temperature and salinity, i.e., at a,. ,. p=Pt-aT+bs, (9) -1 1Al+AI aPI (1'r~'~ =- -11_+ .., .,""'C "', (1) where T is temperature, s salinity and Po, a, bare Pt41 'I if' P1"1 positive constants.Here we neglectvariations of Powith pressure.If we let C representeither s or the specific 0171+#1-"--+8111 ( -+1VI )#1--(T,B_T,l) (J) enthalpy, c"T, then C should satisfy conservation at a,. ,. Plltl relations of the form aplh1 1 a a(p lC1h1) 1 a I-- (Plh1ulr)=O. (3) I --(PIC1h1ulr) = _Q.l 1 at ,. ar at r ar Lower layer ~~~~+~~(psCthtu'JY) =Q.I -au, +ura;-- aUI ( -;+VI f ) Vj - at rar ilt where Q.l is the downwardflux density of C at the interface, i.e., the transfer of salt or heat per unit time (4) from the upper to the lower layer. We assumethat the turbulent exchangeprocesses for heat and salt are identical and that (5) Q.l=B(C1-Ct), (11) where B is independentof C1 or Ct. Then if we take an ~~~(P"'2U21')=O. (6) appropriate linear combination of (10), it can be shown at ,. a,. that a 1 a -:-[Pl(Pl- ,.)h1]+- -[Pl(Pl- pO)h1uIT] at ,. a.. = -B(PI-P2), a 1 a {PI(PI- po)hl]+- -[PI (PI- po)h,ulf' ] at ,. a,.
-B(Pl-PS). Note that th~ relationsarc obtained from principles of conservation of heat and salt j ilierefore, "they are independent of (3) and (6). The variable, P-Po, essentially correspondsto the quantity 0"1 commonly employedin oceanographicanalysis. In summa.ry,Eqs. (1)-(6), (12), (13) represent the general model with the pressure integrals in (1) and (4) determined by (8). The dependentvariables, UI, 1'1, hI, PI; U2, Vt, ht, Pt, are functions of the independent variables,. and t. The quantities, P.i, 'T,8,TIS, B, are to be prescribed. The functional fon11of the quantities. 1 I B B T,. , TI , T,. , TI ,mus tbe c hosen. A simple set of initial conditions is selected.We assumea state of rest and, in addition, each layer is initially of uniform deJ?th"and uniform density. Also, we stipulate that u, v, hand P have thcir initial values at r= (X) for eachlayer and u=v=o at r=O. This is a closedset of non-linear partial differential equations for which tllere are two possiblemethods of 200 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUJIE 24 solution: 1) linearized perturbation technique and 2) shearstresses in flUidsare found to dependquadratically the methodof characteristics(Freeman, 1951). Actually, on velocity, the interior stressr is definedas the entire set of equationsis quasi-linear,i.e., it involves "rl=pKqu only first powers of the derivatives of the dependent ",I=pKqrJ } , (16) variables. Since the method of characteristicsis best suited for quasi-linear systems with two independent whereK is a dimensionlessand constant coefficientand variables, it will be used in determining the solution. For the generalproblem involving eight dependent q= ..[;i"+:;;; . (17) variables there will exist eight characteristicpaths of integration in the r ,t-plane. Since the system of equa- We chooseK=CD, the atmospheric drag coefficient; O'Brien (1%5) presentsa physicalargument supporting tions is of h}1>erboliccharacter, the characteristicpaths this choice. are aJl real. Therefore, the equations of the specific model are: One can distinguish four different classesof behavior of the general system: a) baroclinic wave modes, au au j ah b) barotropic wave modes,c) vortex modes,d) mixing -+u ~- f+- -+ge--= (T,B_pKqu)/ph, (18) JIlodes.In genel:al,aJl of thesemodes can io-exist and at ar r ar they are coupleddue to the non-linearity of the system a. a. and through the influenceof rotation of the earth. The f+j-- (r.s-pKqv)fph, barotropic wave modes are characterizedby a very at7 large propagationalspeed Cbmpared with that of the baroclinic waVes.In the presenceof barotropic modes 0" 1 atIhr -+- -0. the numerical problem involves at least an order of at ,, M magnitude greater machine time as compared with certain spedaJcases in which the barotropic mode is The characteristic forms (O'Brien1 1965) of the filtered from the systemat the outset. above are: 3. The specific ocean model d -(u:i:2c) = (f+v/r)~f'r8Ih~ Kp/hTCU/r, (21) The generalset of equationsrepresents a formidable tit problem. Hence, a series of special problems with increasingcomplexity was defined. The present work along deals with the fust of these. In this sinlple model the d,./dt=u.-:i:c, (22) lower layer is consideredto be infinitely deep which and permits it to be consideredof constant density and at rest for all r and t. This constraint automatically filters II -(vr+tJr') -".S /h-,.Kqv/h, (23) out the barotropic (external) modes of motion and at confinesattention to the baroclinic (internal) mode of response.We will not considerturbulent mixing between along the oceanlayers, i.e., PI is also constant. (The second d,./dt=u, (24) paper will considerthis additional feature.) where the baroclinic wave speedc is defmedas Numerical solutionswere obtainedfor r?,O, O~t<48 hr. The vertical velocity at the interf~ U1was deter. c=..f;gj;: (25) mined posteriorly using the integrated continuity Eqs. (22) and (24) defIne a family of characteristic equation all ah curves in the ,.,I-plane along which the in~gration of W-,- -+-- (14) (21) and (23) can be carried out using sjmple quadra- al Or ture techniques,thereby avoiding many of the unpre- dictable errors usually associatedwith finite difference This also follows as a kinematic conditiop it the methods of solving non-linear partial differential eq~- interface. tions in their original fonn. For conveniencea density contrast variable E is defined 4. Energyconservation .p,- PI - . (15) It ~ a~lutely essential"that any numerical tech- PI nique USed\0 solve the above special model conserve For normal oceanicsituations at mid-latitudes, E is of the sum of.tJie kinetic and potential energies.Conserva- the order of magnitude of 2Xlo-a. tion is used here in the sense that the total energy Consider the general set with 111=111= 0 and E= con- present in 'the system must equal the initial energy stant and suppressthe subscript1. Sinceturbulent plus the net gain (or loss) of energy to (or from) the MAKCB 1967 J. I. OIBRIEN AND R. 0 REID 201 system.Since it seemsimpractical to definea linearized However, the energy of the vortex mode can be trans- model to test the stability of the numerica.Iprocedure ferred to the energy of the wave mode through the for this highly non-linear problem, various energy coupling terDl, i.e., through the action of the Coriolis conservation relationships were derived and used to accelerationIv and the centripetal acceleration v/,.. ensure energy balance in the solutions. (The authors The latter must be negligible for large,.. Thus, near found theseenergy conservationtests to be indispen- the center of the storm both accelerationsmay transfer sablefor revealingprogramming errors.) vortex energy to wave energy and vice ve~ But, at We will definethe total energyE in a unit column by large radii, only the Coriolis accelerationis effective in the transfer of excessenergy from one mode to Eslfh+igEh2, (26) another. This coupling between the two modes effec- which has the cgs dimensions of ergs cm-tp-l. The tively prevents the forDlation of a bore in the solution total energyconsists of two parts: 1) the kinetic energy (O'Brien, 1965). if'" and 2) the potential energy due to stratification IgJII. The latter (except for the factor p) represents 5. Hurricane structure specifically the potential energy in excessof that for- a Let us consider a stationary, axially-symmetric homogeneouscolumn of density PI which has the same hurricane modeled after Hilda, 1964. We need to total massas the stratified column(per unit area).In the prescribea consistentair pressuredistribution and wind usual way, we can combineEqs. (18)-{20) and obtain stressdistribution. F'>llowingKajiura. (1956);we select aE 1 a the pressure distribution developed by Schloemer -+-~hw(geh+lf)]-flTrB+",.,B-Kt. (27) (1954). From a sample of nine hurricanes, Schloemer at ,. Or . found that a pressuredistribution,
An energy flux J and a.n energy supply S are de. P (r) -P o=e-Blr, (31) fined by P.-P. J.-(gth+lqi) } . (28) S-U'7'..8+1I'7',B gave an excellent fit to actual data. Here, P(r) is the pressureat radius r, R is the radius of maximwn winds, The teml Kqi is negativedefinite and representsthe - Po is the central pressure,and P. is pressureat the dissipationof kinetic energydue to transfer of momen- outer periphery of the hurricane. Eq. (31), when tum to the lower layer. Since S may be positive or differentiated, yields a radial pressuregradient which, negative, energy exchangewith the atmospheremay in turn, defines a gradient wind speed given f. Since be in either direction dependingon the sign and relative the surface wind is usually found to be ~ per cent magnitude of the products U'7'..8and 11'7',8. lower than the gradient wind V" the hurricane wind We recognizethat there are two distinct baroclinic speedV is taken to be 0.7 V,. In most mature hurricanes, mOdesinherent in the models and that it is possible the inflow angle a (the angle between the wind vector to write a separateenergy equation for eachmode. Let and the tangent to the local isobar) tends to be con- us define a vortex mode with associatedenergy 111'11stant in the outer portion of the storm. Near R, a tends and a radially-polarized wave mode with associated to decreaserapidly to zero in the eye. Therefore, we energy (tJII1I+1EgJsl).The partial energy equationsfor assumethat the componentsof the hurricane wind field eachmode are are given by a(jVh) 1 a V r= V.8 sin«. i-- -(lftlhv) (32) at ,. a,. V,= V.8cosa. }
--<1+-/,.)+_,.,8- Kqr, (29) where Vr is the radial colpponent,V, the tangential component and a. the maximum inftow angle at the exterior of the storm. The empirical function .8 is definedas t =uvh(f+,,/,.)+tlT,.8_Kqu'. (30) .8=" (33) 1+10/r Clearly the, sum of (29) and (30) yields the total energy equation (27). The term, """(f+"/,,), representsthe where in (33) only,. has the dimensionmiles. coupling mechanism between the two modes. This The componentsof the. turbulent shear stress, '"rs coupling deservesspecial comment. We recognizethat. and ""B, are calculatedby the relations in an atmospheric cyclone, T,8> T,.8 in ~tude. "'rB=PACDvv,.l Thus, a large portion of the energy supply from the (3-') atmospherewill initially dQminate the vortex mode. ",.B=PACDVV, 202 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUKE 24
Only the atmosphericdistributions of 1'rBand 1',B,as seenin Fig. 3, are neededfor the oceanmodel.
6. Numerical techniques The equations for the model, with the boundary conditions shown in Fig. 4, were solved on an mM 7094using a numericalform of the method of character- istics. Ordinarily, when this method is used to solve a :a systemof non-linear differential equations,the charac- ! teristics for the equations are constructed over the Q.C region of known initial data. By proceeding along these lines which lie in the coordinatespace, the solu- tions are determinedfor later times and new regionsof space. However, in this study a modification of the classicalmethod which employs evenly spacedincre-
0 50 100 150 200 2SO 300 350 r (Km) FIG. 2. Hurricane pressureprofile along any radia1coordinate line. where CD is the drag coefficientand PA is air density. The choice of CD for intense winds is controversial [e.g., Roll (1965)J. Here, the classical value, CD =2.SXIO""'1,is used. The values of all the ph}"Sicalconstants used to characterizethe hurricaneand oceanmodel are given in Table 1. The pressuredistribution is shown in Fig. 2. The hurricane wind and stressdistributions are shoWn in Fig. 3. A cOmparisonwith the actual wind field of Hilda is given by O'Brien (1965).For convenience,the value of 1'.8and 1',8follow (34) up to 100mi and then they are decreasedlinearly to zero at 200 mi. This permits us to designatethe forced region,0<,,<200 mi, as the Storm Region and the region, ,,> 200, the Free Region (see Fig. 4). This simplifies the numerical calculationsat the outer boundary of the Free Region.
TABLE 1. Values of physical constants and parameters. ~u - . 2.5Xl~ c atmospheric drag coefficient >- 980cm~ acceleration of gravity '0 - 100m initial depth of upper layer . 2.5Xl~ eddy diffiJsivity ... 1030mb air pressure outside the hurricane 930 mb central air pressure of the hurricane 2Omi radius of maximum wind 35 deg maximum inflow angle 2.0Xl~ initial density contrast 1.28Xl~ gm cm-t air density tor cm-1 water density of upper layer latitude of storm center 0 7 .~92X 10-S sec-1 a 50 100 150 200 250 300 350 anK:U1aT~d o! the Earth r (Km) lmi radial gn spaang . 18 min temporalgridspaang FIG. 3. Hurricane wind and stress distributions along any radius: f'DIUis the maximum radius of hurricane force winds. MARCH 1967 J J. O'BRIEN AND R 0 REID 203
For example, in Fig. 5, the value of 1IA, given r A, is FORCED REGION FREE REGION found by fitting one parabola to the known values of 11 at r _I, r "'-I, r. and a second pa.ra.bo~ to 1?e.known values of 11 at r_l, r., r--+l' Each qUadratIC 18 then evaluated at r A and the average result is used as 1IA. Now, given approximations of 11,v, II atr A, rB, rc, t .- / the integrated equations are solved using the Gauss- Seidel iteration method in which the initial guesses for the unknowns are their values found at the pre- vious time step. The criterion of convergence was that the residual in (35) and its counterpart for the negative "*\ characteristic were less than 10-' cm sec-1. Using this i~ criterion, the solutions at each grid Point were found :I c~. .o9ho in 3-8 cycles. Of course, after each cycle, new estiniates u.v.o, of 1IA, VA, etc., were determined at r A, rB and rc. 0 c. 200/ --- r- Special techniques are required at r=O and along the MIi.£S positive characteristic Co (see Fig. 4) at the exterior of FIG. 4. Schematic drawing of the coordinate space showing the storm. Along r=O, 11=11=0 and E=Eo,but 11varies. the OOundary conditions and defiDing the Forced Region and The new value of CO.4+1was found by solving the Free Region. counterpart of (35) for the negative characteristic and the" integrated form of dr/dt=1I-C for the two un- ments for the independent variables was used. The knowns, rB and CO,'+l-The values of 11,II, etc., needed at techniqueis attributed to Hartree (Fox, 1962). "negative" r for the double interpolation scheme were Consider the grid stencil shown in Fig. 5 which found using the principle of reflection on the velocities illustrates the point pattern and characteristicsused at the positive grid points. to determine the values of the dependent variables, At the exterior of the storm, we used an expanding u, tI, h, at an arbitrary grid point ["., t~lJ. Note that grid and integrated to the new v:alue of co. Actually, the characteristic lines may be curved and need not it was not necessary to calculate where Coexisted for a intersect the grid points since this is a non-linear particular time step, but just integrate until both problem. velocity components were zero. As shown in Fig. 5, the intersection of the ~tive characteristic, u+c, with the previous time level t. is designated".Ii; the intersection of the negative charac- 7. Results teristic, u-c, "B; and the intersection of the third The characteristic equations (21) through (24) were characteristic,u, "c. Hence,using the Trapezoidal Rule integrated numerically using the techniquesjust out- for the right-hand sidesof (21) through (24), we may lined. These computationsyielded values of 11,, and h write the integrated forms (shown for the positive at evenly spaced grid points in the Forced and Free characteristic)as Regions(Fig. 4) for O
'i 20I 100 200 100 400 500 . ' ' E .. '; " 100 200 400 $00 r (K..I
-10 FIG. 6. Radial velocity II as a function of, and ,; top~, 'i 0-24 hr; bottom diagram, 24-48 hr. Each curve illabeled l(hr) .- from 1-0. ,
-lIS j.. FIG. 8. Depth of upper layer h as a function of , and '; top ~ 0-24 hr; bottom diagram, 24-48 hr. Note that .4>0, but is shown as -.4 for illustrative purposes.
There is a surprising amount of fine structure in the velocities which would require lengthy description. However, there are a few noteworthy.features : a) The initial radial velocity u is neg~tive since ,.,.8<0 for all 1'. b) The initial tangential velocity v is positive since ,.,8>0 for all 1'. c) After approximately 3 hr, the radial velocity changessign near the center due to the action of the terms, v/1' andfv. d) A series of radial dispersive waves forms near 30 km and propagates outward. (Two are easily distinguishable.) e) Tbe tangential velocity becomesapproximately stationary after 24 hr inside 20 km. f) A narrow, stationary anticyclonic vortex forms beneath the eye of the storm. g) After 12 hr, the profiles of v mirror the profile of T,8. h) A permanentanticyclonic vortex forms at 320km
-30" (the border between the Forced and Free Regions.) FIG. 7. Tangential velocity. as a fUnction of , and I; top diagram, This is an artificial feature introduced by the dis- 0-24 hr; bottom diagram, 24-48 hr. tribution of ,.8 (i.e., the rather sudden cutoff at 200
~ MUCH 1967 J J O'BRIEN AND R O. REID 205 mi) and the monotonic decreasing nature of 11along the majority of the later profiles. The strong upwelling createdby the velocity diverg- enceis shown in Fig. 8. Hete, for illustrative purposes, the profiles of II are.shown as if eachprofile represented the configurationof the bottom of the layer which was initially at 100m everywhere.The noteworthy features are: a) The initial response(11<0) of the system causes a thickening (downwelling)in the center of the system. b) When 11becomes positive near the center, the water is swept from the center and 11decreases rapidly within ,.< 80 km. c) The maximum upwelling occursat awroximately 30 km, which is an expectedresponse to the maximum value of T8. d) During the second 24 hr, the rate of upwellipg diminishes.This evidenceof numerical stability is an indication of a degreeof physical reality in the model. We may imagine that the cold densesea water of the lower layer rosein the area of upwelling to replacethe lighter surface water which was removed from the central region. The displacedwarmer waters accounted for the thickening of the upper layer between 80 and 300 km. The vertical velocity w was determined from the values of 11and II using a finite differenceform of Eq. (14). The profiles of w (Fig. 9) show that:. When we compare Leipper's Figs. 12b and 12e we a) Initially, w is negativenear the center. find that considerableupwelling is evident. There is a b) Mter 4 hr, w becomespositive near the center and shallow mixed layer less than 25 m deep along the reachesa maximum of 0.25 cm gec-l after 7 hr. hurricane path and a deeper mixed layer (60-80 m) c) As h approacheszero near the center,w diminishes at the edgesof the section.The upwelled area extends 100-150kIn on eachside of the ,hurricanepath. In his rapidly. Fig. 14b the Sigma-Isurfaces show a similar stnlcture, 8. Comparison with observations i.e., strong evidence of upwelling along the hUJrlcane track, a shallow mixed layer in the center, and a The only observationswith which we can comparethe deepermixed layer at the edgesof the section.In this presentwork are thoseof Leipper (1967).However, first figure there is evidence that the strongest.upwelling it is necessaryto understandto what degreethe model is not exactly in the center but slightly displaced, ocean approximates the real ocean. It is recognized e.g., considerthe lines 0',= 26. that the density and the velocity of the real oceanvary Let us recall someof the results (say Fig. 8) of the continuouslywith depth. In the presentmodel, we have present work which are similar to the structure along approximatedthe density structure beforethe hurricane section CC': with two uniform layers.The equationsof motion have been integrated with respect to depth. Hence, the a) On the first day, the upwelling is confined to the velocity components are considered as the depth- region of hurricane force winds. averagedvelocity componentsfor the layer. A priori, b) Upwelling producesa shallow mixed layer in .the we have assumedthat the initial dynan1icresponse of center and a deeper mited layer outside the center. the real oceanwould be to stir the upper portion of the c) The maximum upwelling is slightly off center. ocean.This stirring will producea well-mixedlayer and will usually lead to a temperaturedrop at the surface. Although section CC' doesnot conclusivelyshow an In the simple model describedhere, it is assumed,as a area of downwelling between 80-300 km of the center, first approrlmationi that only the upper 100 m of the as found in the present results, there is someevidence ocean will then respondto the hurricane's wind field. that the downwelling does occur. By c<»nparingthe It is our intent to comparethe results of the theoreti- ocean configuration in the eastern Gulf of Mexico cal model with the actual observationsin order to lend prior to and following Betsy, 1965,Leipper finds that someconfidence to the results. . To be published. 206 JOURNAL OF.THE ATMOSPHERIC SCIENCES VOLUKE 24 the distribution of oxygen indicates an area of down- of heat and moisture with the atmospherecould be welling adjacent to the central upwelling region. The studied. There appears to be ample extensions for upwelling found after Betsy was not as marked as that future work, however, the pertinent and detailed after Hilda. This is attributed to the fact that Betsy observations needed to confirm the theoretical work moved rapidly through the easternGulf of Mexico and will be difficult to obtain. passedover a large oceaneddy current in the region of observations. 10. Summary
9. Comments on previous theoretical work It has been demonstrated that the simple model approximates the structure of the intense upwelling Hidaka and Akiba (1955) studied the steady-state found a.fter hurricane Hilda to a reasonabledegree. upwelling due to a stationary cyclonewith a maximum The active upwelling is confinedto the central portion wind stress of 1 dyne cm-l. Clearly, it is not entirely of the Forced Region. The initial responseof the ocean permissible to compare the present transient models results in a thickening of the upper layer at the center. with a steady-statemodel. However, we can make a However, a.fter several hours significant velocity few comparisons.First, in the presentproblem, we find divergencedevelops and the wann central waters are approximatest~dy-state velocities only after one day displaced,which permits strong upwelling in the center and within the radius of maximum wind stress.Second, of the system. Hidaka and Akiba find a maximum vertical velocity One may logically inquire into how permanent will of 2.4X 10-&cm sec-1.If we increasetheir maximum be the obserVeddensity configuration created by the vertical velocity (or wind stress) 50-fold, we obtain a intense upwelling. In nature, we can assumethat, in value of w comparableto that observedin the present the absenceof a moderately-strong,externally-driven work. Third, they find that the upwelling is confinedto oceancirculation, the kinetic energy remaining in the within twice the radius of maximum wind stress,-a upper layer of the real oceanwill be quickly dissipated. result similar to both the present work and Leipper's This will leave a quasi-permanent upwelled region observations. which may be identifiahle for several weeks after the Ichiye (1955) solves a transient problem similar to hurricane'spassage. the presentwork. However,for the purposeof evaluat- ing the vertical velocity via the vorticity equation, he Acknowledgments.We wish to 'thank Prof. Dale F. assumesthat the current fluxesare given by the Ekman Leipper for his continuous encouragementand advice. relations, The Texas A&M University Data ProcessingCenter provided the extensivecomputer time required fQr this uh=T,8ffp, "h=-T..8/fp. (37) research.The Officeof Naval Research,Contract Nonr- This approximation is not supported by the present 2119(04), provided data, staff and clerical assistance. work. The right-hand side is constantin both problems, One of us (]. O'B.) was supported by National Aero- but in the presentproblem the left-hand sideis definitely nautics and Space Adminstration and Graduate not constant (not illustrated). In fact, the radial CollegeFellowships during the completionof this work current flux even differs in sign at thoseplaces where u which is containedin part in his Ph.D. dissertation.The is negative, e.g., during the first few hours in each of National Centerfor AtmosphericResearch provided the the presentmodels. support for preparing the final publication copy of this On the positive side, Ichiye finds that the mmmum paper. 111,induced by a stationary cyclone w;ith maximUm wind speedof 29.7m sec-l, is 0.5 cm see-I. In addition, List of Symbols the upwelling is confined to within twice the radius of cJ, b positive constantsdefined by Eq. (9) mmmum wind stress.Both of theseresults agreewith B mixing coefficientdefined by Eq~ (11) the presentresearch. c wave speed,defined by Eq. (25) An obvious extension of the present work is the c. initial wave speed investigationof an axially-symmetriccyclone moving at CD atm~pheric drag coefficient a constant velocity U. Following Kajiura (1956), a C, C1,C. salinity or specific enthalpy defined by choiceof independentvariables would be ,1;- Ut and y, Eq. (10) where ,1;and y are Cart~ian coordinates.The moving Cp specific heat of sea water at constant hurricane problem is fundamentally distinct from the pressure stationary model and the method of analysisby charac- E total energy in a unit column, Eq. (26) teristics ma.ybe of limited value. f Coriolis parameter There are other immediately feasible extensionsof F right-hand side of (21), see (35) the present study. Future work might include con- g accelerationof gravity sideration of a finite lower layer, multiple layers, h. h1.h2 thicknessof an oceanlayer bottom topography, etc. Also, the physical exchange h. initial thicknessof upper layer MA:RCB 1967 ]. 1. O'BRIEN AND R 0 REID 207
J energyflux density definedby Eq. (28) PA air density K internal momentum e.'tchangecoefficient, p, PI, Pt, p, sea densities Eq. (16) " Sigma-t PJ P.t pressureat radiusr in hurricane T8, T,.8,T,8 wind stressmagnitude and componentsat P. central pressureof hurricane seasurface P. pressureoutside of hurricane ,.1, .,.,.1,T,I !Qagnitude and components of internal Ph P, oceanpressure , shearingstress at the interface f magnitude of velocity, Eq. (17) T,.-. .,.,s shearingstress componentsat the sea bed Qc' flux density of C between ocean layers, ~ latitude of storm center Eq. (11) I) angular speedof the Earth r independent radial coordinate, increasing outward r.t, rB, 'c intersectionsat " in Fig. 5 REFERENCES r maz maximum radius of hurricane force winds FISher,E. L., 1958:Hurricanes and the sea-surfacetemperature R radius of maximum wind field. J. J{IIeor., 15,328-333. $ salinity Fox.L., 1962:N~ s~ of0rdiIIIJry - ParlialDiffer- Mal~. London, Addison-Wesley,339-342. S energysupply from atmospheredefined by Freeman, J. C., 1951: The solution of non-linear meteorological Eq. (28) problems by the method of characteriStics. Comp",..- of 1 independenttemporal coordinate J{IIeorology, B~ton, Amer. Meteor. Soc., 421-433. '" "-1, "-+-1arbitrary time levels usedin Fig. 5 ffidaka, K., and Y. Akiba, 1955: Upwelling induced by a circular T temperature wind system. RecordsDee_g. Works Jap_, 2,7-18. -, -1, -, radial velocity in ocean Ichiye, T., 1955: On the variation of oceanic circulation (Y). Geophjs. Mag., 26,283-342. U translational speedof hurricane Jordan, C. L., 1964: On the influence of tropical cyclones on the v, Vi, v, tangential velocity in ocean sea surface temperature field. Proc. Symp. Trop. MIIeor., V, gradient wind in hurricane New Zealand Meteor. Service. Wellington, 61~2. V, V r, V, magnitudeand componentsof surfacewind Kajiura, K., 1956: A forced wave caused by atm~pheric dis- speedat 10 m aboveocean turbances in deep water. Tech. Report 133-1, A&M College fD vertical velocity in ocean of Texas, Ref. 56-26T, 32 pp. Leipper, D. F., 1967: Observed ocean C9"ditions and hurricane J independentvertical coordinate,increasing Hilda, 1964. J. AlIIIos. Sci., 24, 182-196. upward O'Brien, J. J.. 1965: The non-linear response of a two-layer, cI inflow angle in hurricane baroclinic ocean to a stationary, axially-symmetric hurricane. cI. maximum inflow angle Tech. Report. Ref. 65-34T, Texas A&M University, 99 pp. .8 empirical function definedby Eq. (32) Roll, H. V, 1965: Pilysk$ oflhe MariMAImosphere. New York,' Ar, l1.1 grid intervals Academic Press, 152-100. Schloemer, R. W.. 1954: Analysis and synthesis of hurricane wind e density contrast definedby Eq. (15) patterns over Lake Okeechobee,Florida. Hydrometeorologi- 8 independentazimuth coordinate caI Rept., No. 31, U. S. Weather Bureau, 49 pp.