The inertial chimney: The near-inertial energy drainage from

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1 JOURNAL OF GEOPHYSCAL RESEARCH, VOL. 103, NO. C4, PAGES , APRL 15, 1998 The inertial chimney: The near-inertial energy drainage from the ocean surface to the deep layer Dong-Kyu Lee Department of Marine Sciences, Pusan National University, Pusan, Korea Peam P. Niiler Scripps nstitution of Oceanography, La Jolla, California Abstract. A three-dimensional primitive equation numerical model is used to study the behavior of near-inertial waves generated by surface wind stress on an fplane. The model is fixed depth with a rigid lid on the surface and is horizontally periodic. This study shows the behaviors of the wind-generated near-inertial waves for four different eddies: (1) a subtropical cold-core eddy, (2) a subtropical warm-core eddy, (3) a California subsurface warm-core eddy, and (4) the Gulf Stream warm-core ring. The mean secondary circulation generated by the wind-eddy interaction has magnitude to comparable to that of the Ekman current, and its characteristics are determined by the relative angle between wind and current. The propagation characteristics of near-inertial waves are quite different depending upon the sign of the relative vorticity. n a cyclonic eddy, near-inertial wave propagation is outward from the core of the eddy. The propagation direction in an anticyclonic eddy is downward and toward the core. Vertically propagating waves inside the eddy are trapped above a critical layer and slowly dissipated by parameterized viscous effects. The model shows that anticyclones efficiently drain near-inertial energy from the surface to the deep layer below the thermocline. 1. ntroduction used a nonhydrostatic two-dimensional numerical model to study the secondary circulation generated by the nonlinear in- The dispersion relation derived by Kunze [1985] for near- teraction of winds and ocean currents. According to their calinertial waves propagating in geostrophic shear reveals that culations, near-inertial energy was about 4 times larger in the the propagation behavior is affected by the spatial variation of negative vorticity side of the jet than in the positive vorticity the effective Coriolis frequency, j jj. = f+ (/2 where side. Near-inertial energy penetration was also deeper in the (= ov/ox-ou/oy is the relative vorticity. Using numerical ray- negative vorticity region. tracing, he showed trapping of near-inertial waves in regions This study is the extension of the two-dimensional numeriof negative vorticity. cal model of Lee et al. [ 1994] to three-dimensional structures. Kunze's study spurred observational [Kunze, 1986; Kunze A three-dimensional primitive equation numerical model is and Sanford, 1986] and numerical investigation of near- used to study the interaction between various eddies and a inertial wave propagation in the geostrophic shear. For exam- constant wind stress applied at the surface. Fine vertical (5 m) ple, Mied et al. [1986] observed the amplification of near- and horizontal (5 km) grids are used to model wind-eddy ininertial energy in the negative vorticity side of the front. Kun- teractions. Four experiments are performed to calculate these ze et al. [1995] found near-inertial energy maximum below interactions a subtropical warm eddy, a cold eddy, a Gulf the thermocline inside the Gulf Stream warm-core ring. Van Stream warm-core ring, and California subsurface eddies. Meurs [1996] found several tentative relationships between Vertical stratification used for all four cases is taken from obnear-inertial energy level and the sign of vorticity in the servations. The model is sufficiently deep (0 m) that bot- OCEAN STORMS experiment. Using a three-dimensional tom reflection is not an important process. primitive equation numerical model, Van Meurs [1996] showed that high near-inertial energy level at depth was found 2. Model in regions of strong relative vorticity gradient. Wang [1991] investigated near-inertial wave propagation in the frontal area This study uses a three-dimensional, hydrostatic, primitive using a two-dimensional numerical model of channel flow. He equation model. The model is 500 km wide in x and y direcfound significant near-inertial energy accumulation in the tions and is 0 m deep. Boundary conditions are periodic in warm (negative vorticity) side of the front. Lee et al. [1994] horizontal directions, and the vertical velocity vanishes at the surface and bottom. Vertical mixing is Richardsonumber dependent with a maximum vertical eddy diffusivity of 0.01 Copyright 1998 by the American Geophysical Union. m2/s [Lee et al., 1994]. Grid resolution is 5 km in the horizontal, 5 m in the vertical near the surface, and 50 m near the Paper number 97JC bottom. The maximum initial Rossby number depends upon /98/97JC the speed and structure of the initial vortex, ranging from

2 7580 LEE AND NLER: NEAR-NERTAL ENERGY DRANAGE to 0.1. A description of the model, with these boundary and initial conditions, is presented below. nterested readers may consult the two-dimensional version of nondimensionalized momentum equations in the appendix of Lee et al. [1994]. Their scale analysis shows the scales resolved in this numerical model study Governing Equations The basic equations of the model use the Boussinesq approximation to the equation of motion on a rotating Cartesian plane. The governing equations are v v v v -Fu--+ v-- + w-- + ft =-- x y z + - (Av Ov) )z ' at at at at +u +v +w Po X poy + A hv2u + A hv2v (1) (2) P pg = -, (3) az u v w + + = 0, (4) 3y Ot x y z = KhV2T + 3- (Kv 3z ' (5) P = PO + P'= PO (1-aT), (6) where, in standard notation; (u, v, w) (x, y, z)components of velocity; f g Coriolis frequency acceleration of gravity, equal to 9.8 m/s 2 Po + p'(x, y, z,t) total density T(x,y,z,t) P(x,y,z) temperature thermal expansion coefficient, equal to 0.2x10-3 1/øC pressure. The magnitude of the constant horizontal turbulent viscosity term (A h = 10 m2/s) is determined not by physics but to avoid numerical generation of subgrid-scale waves. The vertical turbulent viscosity coefficient, Av, is given by A v = max[e v (1- Ri/Rc),Avo], R i = 0 if R i < 0 and E v = 10XAv0 and the vertical heat diffusion coefficient, Kv, is given by K v = max[d v (1- R i/r c), KvO ], R i = 0 if R i < 0 and D v = 10x KvO where R c is a critical Richardsonumber equal to 0.25 and Ri is the local Richardson number. The maximum vertical diffusivity of momentum, Ev, is 1 x 10-2 m2/s. We use the value from the recent estimate of the eddy viscosity by Chereskin [1995] which is between 0.9 x 10 '2 m2/s and 2 x 10 '2 m2/s in the Ekman layer, but our backgroun diffusivity becomes 10 times larger than values observed in layers at the base of warm-core rings by Kunze et al. [1995]. When we use their background viscosity, the instability develops, especially in the cyclonic eddy. Since one of the main objectives of our experiments is to study the difference between cyclonic and anticyclonic eddies, the background eddy viscosity, Avo, of 1 x 10-3 m2/s is chosen Boundary Conditions Periodic boundary conditions are used in the horizontal. The vertical boundary conditions are prescribed as u A = ' at z=o v az x v A v = 'y at z=o w=0 at z=0 and z=-h )T K =0 v )z at z=0 and z=-h u v=0 at z=-h Av = A v where H is the depth of the model domain. Setting w=0 at the surface is the "rigid lid" approximation. t filters out the surface gravitational waves that would require a very small time step to be resolved numerically nitial conditions The initial eddy velocity and density fields are obtained from the cyclogeostrophic balance without wind stress and frictional terms, a r r2 1 V O = V 0 x--exp (1- ) exp(91z) (7) VøJfa ' exp(91z' { [ T i = r(z)4 h' exp fa -;4) (8) where a is 33 km (one-third radius of the vortex), x = r cos fi, and y = r sin 6. V o is the tangential velocity, v r = 0 is the radial velocity of the vortex, and Vo is the maximum speed of the eddy. T(z) and 3, are basic stratification parameters from observation. The initial circulation field is cyclogeostrophically balanced and maintained against friction and diffusion by a body force introduced in the fluid. The body force is maintained throughouthe integration. The maximum value of this body force is less than 1/10,000 of the Coriolis force. nitial temperature and velocity fields are shown in Figure Numerical Technique The numerical scheme used to solve the governing equations is identical to that of Cox [1985] except for the grid scheme (staggered Arakawa C), the use of Cartesian coordinates, and the Richardson-number-dependent mixing in the mixed layer. External and internal modes are calculated separately. A forward time scheme is used every 51 time steps to eliminate time splitting of the solutions. Vertical and horizon-

3 LEE AND N1LER: NEAR-NER L ENERGY DRANAGE 7581 T (C) o 500 T (C) m o _ (c) T (C) T(C) Figure 1. Vertical distributions of initial temperature along a zonal section with a vertical profile of temperature (øc) at the center of the eddy to the right for a subtropical cyclonic eddy, a subtropical anticyclonic eddy, (c) a Californi anticyclonic eddy, and (d) a Gulf Stream warm-core ring. tal diffusion terms are lagged one time step. A time step of 10 min for both external and internal modes is used in all simulations. The forced convective adjustment method is used to avoid unrealistic, unstable negative density gradients. 3. Results Four simulations are performed using four different cyclonic and anticyclonic eddies commonly found in the subtropical midocean (Figures la and lb) [Niiler and Hall, 1988], an anticyclonic subsurfac eddy observed near the California coast (figure l c) [Simpson and Lynn, 1990] and Gulf Stream warm core ring [Kunze et al., 1995]. The main objectives are to examine (1) the influence of an existing vorticity field on the near-inertial waves generated by a uniform wind, (2) the vertical and horizontal propagation patterns of near-inertial waves, (3) wave-eddy interaction for different eddies, and (4) the energy sink for the near-inertial waves propagating into the critical layer from the surface.

4 7582 LEE AND N1LER: NEAR-NERTAL ENERGY DRANAGE Vl o o o ooo - -0 i OO - 15oo o loo (c) (d) o o Figure 2. Vertical distributions of the vertical relative vorticity (10-5 l/s) for the subtropical cyclonic eddy, the subtropical anticyclonic eddy, (c) the California anticyclonic eddy, and (d) the Gulf Stream warm-core ring Experiment Settings The subtropical eddies used in this study have approximately a 100-km diameter and subtropical Pacific stratification from Levitus [1994]. Anticyclonic eddies found in the California coastal area subtropical Pacific have subsurface thermal structure: most of their density perturbation occurs between 350 and 1000 m; horizontally uniform thermal structure exists in the upper 200-m layer. The Gulf Stream warm core ring has a deep mixed layer and a larger diameter then subtropical anticyclones, but in this experiment we intentionally set all eddies to the same diameter and the same initial maximum speed. We tried to make our simulation as realistic as possible, but the Gulf Stream warm-core ring with realistic speed is not stable long enough to reach the statistical equilibrium. Thus we cannot analyze the dynamics and compare it with other simulations. nitial velocity contours and initial vertical stratification at the center of the eddy for each experiment are presented in Figure 1. The vertical components

5 ß LEE AND NLER: NEAR-NERTAL ENERGY DRANAGE 7583 i i i t X (Kin) X (Kin) Plate 1. nitial temperature (colored contours) and velocity (arrows) field for the subtropical cyclonic eddy and the subtropical anticyclonic eddy. 35O of the initial relative vorticity of subtropical eddies are shown in Figure 2. nitial velocity vectors and temperature fields on 3.2. Mean States the surface of subtropical cyclone and anticyclone are shown The mean surface horizontal velocities and temperature for in Plate 1. Only results of subtropical cyclone and anticyclone subtropical eddies are averaged from the fourth day, when are presented in most of the figures for the following reasons: near-inertial motions are fully set up throughout the water col- (1) the sign of the relative vorticity determines the propagation umn, to the eighth day (Plate 2). When mean states are comcharacteristics of near-inertial waves, and (2) the vertical dis- pared with initial states, an anticyclone (warm-cor eddy) distributions of near-inertial energy of the California subsurface plays more horizontal heat advection than does a cyclone eddy and the Gulf Stream warm-core ring are only signifi- (cold-core eddy) on the surface When cold water advects cantly different from the subtropical anticyclone. above warm, it becomes unstable and induce stronger vertical 350,,, 350,, 300, '.. /', :;.:...' 250,.,':j3"i,, o,,,../':... -,,...,.,x.:'.'-,,..,.,,,./',,......,"';..h :/:, [,.,,, :/:'i'.:i \,, :... ' ;'..." Jcm/s N,',o.... " O0 x (Kin) Plate 2. Mean temperature (colored contours) and velocity (arrows) field averaged over day 4 to day 8 for the subtropical cyclonic eddy and the subtropical anticyclonic eddy. The applied wind stress values are also drawn. Note that the velocity vectors outside the eddies are the same everywhere and are not drawn. 35O

6 7584 LEE AND NLER: NEAR-NERTAL ENERGY DRANAGE 350 O'! 1...,-'" a, x,,)......,' /, ;/ "' '/' ' ß ß 20o ß ß. -,--- -e--r',, --- " /'/' ß ß i-... '!-ø'if' ' '!... ''''/---t *..,, Nx, x..... ;-..,X x... i O Plate 3. The secondary circulation (arrows) resulted from nonlinear wind-eddy interaction. Temperature changes by the secondary circulation are drawn in color. mixing with downwelling than that induced by warm water advected above cold (Figure 4). The surface horizontal secondary circulation fields, which are departures from initial states and wind-generated Ekman currents, u = u -u i --tz E, are shown in Plate 3 along with temperature changes, T'=T-Ti-ATœ, where AT E is a temperature change outside the eddy. The Ekman current in our simulation is simply a mean current in an area away from the eddy where no wind- eddy interaction occurs and the Ekman velocities are derived from the far field in the numerical experiments. The magnitude of the secondary circulation generated by nonlinear inter- action between a wind stress of 1 dyn/cm 2 and the subtropical eddy is more than 10 cm/s and comparable to the 12-cm/s surface Ekman current. Maximum values are found in the center of the eddy. Directions of these circulations depend on the relative angle between wind and current. Larger cooling results from upwelling formed by the interaction between wind and current that have the same direction. The mean vertical velocities at the base of the mixed layer ,:::: ;f:?:-'... :-":-;.; ;?":' ½i;iii!?:3 '...:'::i:iii!"'½"- -" 5i' ii OO f v 2 :'""' : ':' ','"" ",'-!1 i t,-- - t J i i i x (Km) Figure 3. Mean vertical velocity (m/day) fields at 50-m depth for the subtropical cyclonic eddy and the subtropical anticyclonic eddy. The shaded regions are regions of upwelling.

7 LEE AND N]Y,ER: NEAR-NERTAL ENERGY DRANAGE , ½ , :... ;: :..:...-,.. :...-,: :;..: :½ ::!,...-:....,...,,..? o Y (Kin) Y (Kin) Figure 4. Vertical sections of mean vertical velocity (m/day) along the meridional centerline for the subtropical cyclonic eddy and the subtropical anticyclonic eddy. The shaded regions are regions of upwelling. are shown in Figure 3. As two-dimensional model results by Lee et al. [1994] indicate, the vertical velocity generated by wind-.eddy interaction depends upon the relative angle between wind stress and the existing current. Under the current flowing agains the wind, downwelling develops, while upwelling forms under a current in the same direction as the wind. The mean vertical velocities perpendicular to the wind direction are shown in Figure 4. The depth of secondary circulation cells is much deeper inside the eddies (over 300 m) than outside the eddies (about 30 m) Near-nertial Motion Rapid decay of inertial motions in the mixed layer within a few inertial periods was observed by D ;Asaro[ 1985]. D Asaro [1989] found that this decay was due to shortening of the north-south scales of the motion by the,b effect. Our numerical experimentshow that an efficient system for the draining of inertial energy can be set up withouthe,b effect by windeddy interaction. Two-dimensional simulations by Lee et al. [1994] showed that the pre.sence of both relative vorticity and nonlinear momentum transport ( w'dv'/dz ) both induced inertial wave leakage from the mixed layer. Their numerical expe ments also demonstrated that downward energy propagation caused rapid decay of the wind-generated near-inertial energy. Wang's [1991] channel flow model also showed the dependency of near-inertial wave propagation on the sign of the vorticity in the presence of ambient geostrophic flow. When we plot the time series of the current generated in our numerical experiments (not shown here), we find that most of the variation is a near-inertial fluctuation similar to the results of the two-dimensional experiments by Lee et al. [1994]. Thus the near-inertial energy is calculated from the high-passed (with cutoff frequency of 0.5 cycle per day) time series of the current from day 4 to day 9. Plate 4 shows the surface near-inertial energy along with relative vorticity (blue for the positive values and red for the negative). Minimum near-inertial energy holes are found in an area of the large surface vorticity gradient and upwelling. When another numerical simulation with the same initial conditions but different wind direction was performed (not shown here), it revealed that the energy minima holes are at upwelling regions and depend upon the relative angle between wind stress and current [Lee et al., 1994]. Energy levels in the energy minima are less than half the levels found outside of the eddy. Plate 5 shows near-inertial energy at the 200-m depth of maximum energy in the core of the subtropical anticyclone and at 200 m in the subtropi cyclone. The high inertial energy in the core of the anticyclone is striking. The vertical distributions of energy at the center of the eddy and outside the eddy are shown in Figure 5. For a subtropical anticyclone (Figure 5b) the subsurface maximum energy is found at 200-m depth and is more than half the energy found at the surface outside the eddy. The maximum energy for the Gulf Stream warmcore ring (Figure 5d) is found below the thermocline (300 m), and its depth is similar to one of three maxima observed by Kunze et al. [1995]. The California anticyclone's energy

8 7586 LEE AND NB_,ER: NEAR-NERTAL ENERGY DRANAGE Near-inertial Energy (cm/$) 2 (10-ss' Near-inertial Energy (cm/$) ; (10'Ss' O OO "' -750 " O - 0 Near-inertial Energy ( cm/s) ; (10'$s' Near-inertial Energy (½m/$) ½ (c) -25O (d) -5OO -5OO -750 e i i i i -0 Figure 5. The near-inertial energy (cm2/s 2) profiles at the center of the eddies (solid line with squares) and outside the eddies (dotted line with circles) for the subtropical cyclonic eddy, the subtropical anticyclonic eddy, (c) the California anticyclonic eddy, and (d) the Gulf Stream warm-core ring. The vertical vorticity (10-5 /s) at the center of the eddies for four cases is also drawn. maximum (Figure 5c) is found far below the thermocline, and it is by the stronger subsurface thermal structure (Figure l c). For all four cases the inertial energy is confined to a 50-mthick surface layer outside the eddy. The vertically averaged near-inertial energy for a subtropical cyclone and anticyclone is shown in Plate 6. n the core of the subtropical anticyclone the vertical integration of energy is about 3 times that outside the eddy. An energy minimum area encompasses the rim of the eddy. n the subtropical cyclone the maximum vertically averaged energy is found outside the eddy, and there is an energy minimum in the region of large vorticity gradient. Near-inertial wave phase propagation at 50- m depth are shown for the cyclone (Figure 6a) and anticyclone (Figure 6b). n the anticyclone, phase and energy propagate

9 LEE AND NLER: NEAR-NERTAL ENERGY DRANAGE Jl OO X(Km) Cyclonic Eddy Anticyclonic Eddy Figure 6. Space and time plot (x-t) of vertical velocity (m/day) of the subtropical cyclonic eddy and the subtropical anticyclonic eddy. toward the outside of the eddy. n the cyclonic eddy, phase and energy propagate toward the core of the eddy. Near-inertial waves in the core of the anticyclonic eddy then propagate downward and are trapped at a subsurface layer of 200-m depth Perturbation Kinetic Energy Budget To show the perturbation kinetic energy budget at the layer where near-inertial energy is trapped, the perturbation energy equations are derived and averaged over time and between 100-m and 300-m depth. The perturbation kinetic energy equation can be written as +v'2 ) 2 =A+P+B+Dv +DH where advection of perturbation energy A is - 1 t,2 _ w A=-.V and perturbation energy production P is z (9) (10) The horizontal dissipation of perturbation energy D H is and vertical dissipation Dv is D H = AH( 'V2t ') (12) D v =Av(u' 32u '+v' 32v'). (13) &2 3z 2 The constant background vertical diffusion coefficient, Av = lx10-3 m2/s, is used for the energy calculation because the Richardson number at the layer is always larger than Fi- nally, the work done by pressure B is B = a'.vp' (]4) When these terms are integrated over the horizontal, the horizontal advection term in (10) becomes zero because of the periodic boundary condition. Thus only vertical advection remains, (15), =,7.,7. v + u'v'. V + u w - -z. + v W. z (11) and (14) becomes

10 7588 LEE AND N]XLER: NEAR-NERTAL ENERGY DRANAGE Energy Flux (10-8m3/s3) Energy Flux (10-8m3/s3) D v -loo -200 Y -300 B OO Kinetic Energy (cm2/s 2) Kinetic Energy (cm2/s 2) Figure 7. The vertically integrated energy flux profiles at the center of the cyclonic eddy and the anticyclonic eddy. Shading denotes the perturbation kinetic energy. 4 above 100-m depth is larger than the pressure-work. The production of perturbation kinetic energy is significant for the where angle brackets mean horizontal average. Energy fluxes anticyclonic eddy. As there is no balance in the fluxes, there is are calculated by using model history data from day 4 to day 9, a continual readjustment of the energy level during the 6 days which were saved every 10 time steps and at every other grid. of integration. The kinetic energy of the whole model domain reaches the Horizontal mean energy fluxes in the layer from 100-m to statistically steady state, but the kinetic energy fluxes do not 300-m depth in the subtropical cyclone and anticyclone are balance when they are calculated at a particular location or presented in Figure 8. n the anticyclonic eddy the perturbawithin a certain depth range because the rate of change of lo- tion kinetic energy is dissipated mostly by vertical diffusion. cal energy density is also locally important. n the cyclonic eddy the perturbation energy production is The vertically integrated P, B, and Dv at the center of the negative; i.e., there is a small perturbation-to-mean energy eddy are shown in Figure 7 with vertical distribution of the flux. The anticyclonic eddy's perturbation energy production near-inertial kinetic energy. At the center of cyclonic eddies from mean flow is larger than the pressure work. The amplifithe pressure work is a dominant flux from the surface to 100 cation of inertial energy in the center of the negative vorticity m. The vertical dissipation much smaller than the pressure- eddy is caused by the increased energy flux from the mean to work at all depths. On the contrary, in the case of the anticy- the near-inertial wave as well as by the pressure-work below clonic eddy the pressure work has a large influx below the the critical layer. maximum near-inertial energy layer and an outflux above the Observations in the Gulf Stream warm-core rings by Kunze maximum near-inertial energy layer. The vertical dissipation et al. [ 1995] showed that most of the vertical transport of the

11 LEE AND N1LER: NEAR-NERTAL ENERGY DRANAGE '!! E Plate 4. Mea near-inertial energy (cm2/s 2) and vorticity (10-5 l/s) at the surface for the subtropical cy- clonic eddy and the subtropical anticyclonic eddy. The blue areas are the positive vorticity regions, and the red areas are the negative vorticity regions. near-inertial energy by the pressure was balanced by turbulent dissipation except above 350-m depth, where the turbulence dissipation much larger than the pressure work. Most of the perturbation kinetic energy production in our experiments occurs above the critical layer (the layer of the maximum perturbation kinetic energy) and by the vertical shear of the horizontal mean current, 3z 3z which is neglected by Kunze et al.'s [1995] model approximation The conditions of their observations are not well represented by our numerical experiments. We are discussing the evolution of a single event where there is a readjustment of energy at all levels, while in the real ocean the influence of ), 350 t t l_ 1 F' Lo_.[- O--L_ 1._, 0 Fo Plate 5. Meanear-inertial energy (cm2/s 2) and vorticity (10-5 /s) at the depth of subsurface energy maxi- mum for the subtropical cyclonic eddy and the subtropical anticyclonic eddy. The colors are the same as in Plate

12 ß 7590 LEE AND N ER: NEAR-NERTAL ENERGY DRANAGE i t ß i [ i i Plate 6. Vertically integrated perturbation kinetic energy for the subtropical cyclonic eddy and the subtropical anticyclonic eddy. many events in eddies was calculated by Kunze et al. [1995]. Their field observations are not capable of evaluating the evolution of one wave packet as we have done here. A secondary circulation is generated in the eddies by the nonlinear wind-eddy interaction, and its amplitude is comparable to the Ekman current of 12 cm/s. Upwelling forms under the eddy's downwind current, and downwelling forms under 4. Summary and Discussion the upwind current. The eddy current perpendicular to the wind does not interact with the wind as much as the current The numerical simulations presented in this paper are an parallel to the wind, consistent with the two-dimensional reextension of the nonhydrostatic two-dimensional numerical sults of Lee et al. [ 1994] and with the quasi-geostrophic threemodel by Lee et al. [1994] to a hydrostatic three-dimensional dimensional numerical model by Klein and Hua [1988]. The numerical model of non-linear interactions between wind- penetration depth of the secondary circulation is about 300 m, much deeper than the 30-m Ekman depth. After a few inertial oscillations the distribution of near- stress and various eddies in the ocean. Kunze [ 1985] and Kunze et al.'s [1995] extensive theoretical works are almost entirely related to linear perturbation analyses around a mean flow whose spatial scale is larger than the wave scale, or conditions where WKBJ or ray tracing methodology applies. Here the calculationshow that the interactions among the eddy and the Ekman layer are fully nonlinear, and perturbation methods would fail in their first-order physics. Ray tracing methods are valuable indicators of what might happened to nearly linear wave trains. When these methods blow up, or fail, one should look for higher-order, or different, physics to apply. For example, internal gravity waves do not interact with the mean flow to extract energy to the first-order WKBJ approximation, as has been shown by Kunze [ 1986] and many others. The WKBJ approximation or ray theory must be carried out to third order to see the mean flow and wave interactions we computed with the numerical model (for example, see a thirdorder WKBJ calculation compared with a numerical model, in the work of Stabeno [1982]). Furthermore, the generating region of the upper ocean in our solutions is nonlinear, and the regions of wave and mean flow interactions are nonlinear, so for the purpose of calculating the interaction of the mean flow with the perturbations, which is the interest in this paper, Kunze's theoretical applications of ray tracing would not be applicable. inertial energy is not uniform. An energy minimum appears in the surface layer in the region of a large relative vorticity gradient caused by anomalous horizontal propagation of the nearinertial waves from this region. Near-inertial energy in the anticyclonic eddy also propagates downward and becomes trapped in the core. Near-inertial waves are trapped at what appears to be a critical layer where near-inertial energy reaches its maximum. On the other hand, in the cyclonic eddy, near-inertial energy propagates freely in the horizontally away from the eddy. n the latter case, near-inertial energy is present only in the surface layer. These different propagation behaviors dependent upon the relative vorticity are consistent with Kunze's [ 1985] theory. Kunze [ 1986] and Kunze et al. [ 1995] also observed downward propagation of near-inertial waves inside Gulf Stream warm-core rings. n the ocean we expect that an "inertial chimney" is set up in closed regions of negative relative vorticity. n the chimney the near-inertial energy not only propagates to the layer below the thermocline but also draws energy from the mean field in the subsurface layer. The near-inertial wave energy production near the energy maximum at depth is balanced by vertical transport by the pressure field and by viscous dissipation due to vertical diffusion.

13 LEE AND N]T.ER: NEAR-NER L ENERGY DRANAGE 7591 D v 2.4 D v 7.0 D H P DH B 6.6 B 4.3 Figure 8. The energy flux (10-7 m3/s 3) diagram at the critical layer ( m) for the subtropical cy- clonic eddy and the subtropical anticyclonic eddy. The vertical diffusion coefficient for momentum used in Kunze, E., The mean and near-inertial velocity fields in a warm-core this experiment is relatively large (1 x 10-3 m2/s 2) in compari- ring, J. Phys. Oceanogr., 16, , son with the value measured in the thermocline in the real Kunze, E., T. B. Sanford, Near-inertial wave interactions with mean flow and bottom topography near Caryn Seamount, J. Phys. ocean; our numerical experiments produce some of the essen- Oceanogr., 16, , tial features of the generation of secondary circulation by Kunze, E., R. W. Schmitt and J. M. Tooles, The energy balance in a wind-eddy interactions and the propagation characteristics of warm-core ring's near-inertial critical layer, J. Phys. Oceanogr., 25, , near-inertial energy in the presence of geostrophic eddies that Lee, D-K., P. P. Niiler, S. Piaseek, and A. Warn-Varnas, Wind-driven are not dependent upon the size of the coefficient. Because in- secondary circulation in ocean mesoscale, J. Mar. Res., 52, 371- ertial energy shear is large in the core of an anticyclonic eddy, 396, observations of vertical mixing in such an eddy might indeed Levitus, S., World ocean atlas, Natl. Oceanic and Atmos. Admin., U. also find large values of vertical diffusivity [Lueck and Os- S. Dep. of Comm., Washington D.C., Lueek, R., and T. Osborn, The dissipation of kinetic energy in a born, 1986; Kunze et al., 1995]. warm-core ring, J. Geophys. Res. 91 (C1), , Mied, R. P., C. Y. Shen, C. L. Trump, and G J. Lindemann, nternal- Acknowledgments. This study was funded by the Pusan Na- inertial waves in a Sargasso Sea Front, J. Phys. Oceanogr., 16, tional University and the Office of Naval Research. We would like to , thank Eric Kunze for his valuable comments and suggestions. Niiler, P. P., and M. Hall, Low-frequency eddy variability at 28 ø N, 152 ø W in the eastern North Pacific subtropical gyre, or. Phys. References Oceanogr., 18, , Simpson, J. J., and R. J. Lynn, A mesoscale eddy dipole in the offsho- Chereskin, T. K., Direct evidence for an Ekman balance in the Cali- re California current, or. Geophys Res., 95 (C8), , fornia Current, J. Geophys. Res., 1 O0 (C9), 18,261-18,269, Cox, M., An eddy resolving numerical model of the ventilated ther- Stabeno, P.J., The reflection, transmission and scattering of internal mocline, or. Phys. Oceanogr., 15, , waves at ocean fronts, Ph.D. thesis, Oreg. State Univ., Corvallis, D'Asaro, E. A., Upper ocean temperature structure, inertial currents and Richardsonumbers observeduring strong meteorological Van Meurs, P. V., The importance of spatial variability on the decay forcing, o r. Phys. Oceanogr., 15, , of near-inertial mixed-layer currents: theory, observation and D'Asaro, E. A., The decay of wind-forced mixed layer inertial oscil- modeling, Ph.D. thesis, Univ. of Calif. at San Diego, La Jolla, lations, or. Geophys. Res., 94 (C2), , D'Asaro, E. A., C. C. Erikson, M.D. Levine, P. P. Niiler, and P. Van Wang, D.-P., Generation and propagation of inertial waves in the Meurs, Upper-ocean inertial currents forced by a strong storm,, subtropical front, or. Mar. Res., 49, , data and comparison with linear theory, J. Phys. Oceanogr., 25, , Klein, P., and B. L. Hua, Mesoscale heterogeneity of the wind-driven D.-K. Lee, Department of Marine Sciences, Pusan National Unimixed layer: nfluence of a quasigeostrophic flow, or. Mar. Res., 46, versity, Pusan, , Korea. ( lee po.ocean.pusan.ac.kr) , P. P. Niiler, Scripps nstitution of Oceanography, La Jolla, CA Koblinsky, C. J., P. P. Niiler, and W. J. Schmitz, Observations of ( Pniiler ucsd.edu) wind-forcedeep ocean currents in the North Pacific, or. Geophys. Res., 94 (C8), , Kunze, E., Near-inertial wave propagation in geostrophic shear, J. (Received September 12, 1996; revised September 8, 1997; Phys. Oceanogr., 14, , accepted November 4, 1997.)