Stability of a Cold Core Eddy in the Presence of Convection: Hydrostatic versus Nonhydrostatic Modeling

Transcription

1 MARCH 000 MOLEMAKER AND DIJKSTRA 475 Stability of a Cold Core Eddy in the Presene of Convetion: Hydrostati versus Nonhydrostati Modeling M. JEROEN MOLEMAKER* AND HENK A. DIJKSTRA Department of Physis and Astronomy, Institute for Marine and Atmospheri Researh, Utreht University, Utreht, Netherlands (Manusript reeived 8 Otober 1997, in final form 8 February 1999) ABSTRACT Geostrophi eddies in a stratified liquid are suseptible to barolini instabilities. In this paper, the authors onsider these instabilities when suh an eddy is simultaneously ooled homogeneously from above. As a linear stability analysis shows, the developing onvetion modifies the bakground stratifiation, the stability boundaries, and the patterns of the dominant modes. The oupling between the effets of onvetion and the largesale flow development of the eddy is studied through high-resolution numerial simulations, using both nonhydrostati and hydrostati models. In the latter models, several forms of onvetive adjustment are used to model onvetion. Both types of models onfirm the development of the dominant modes and indiate that their nonlinear interation leads to loalized intense onvetion. By omparing nonhydrostati and hydrostati simulations of the flow development arefully, it is shown that onvetive adjustment may lead to erroneous smallsale variability. A simple alternative formulation of onvetive adjustment is able to give a substantial improvement. 1. Introdution The transformation of surfae water into intermediate and deep water through deep onvetion appears to be a very important proess in the oean. It affets the strength of the thermohaline overturning irulation in the Atlanti and hene the meridional heat transport. By now, muh is known on the atual sales of onvetion, the physial proesses determining these sales, and the effets of onvetion on the large-sale flow development. Observations in the Greenland Sea (Shott et al. 1993), in the Labrador Sea (Gasard and Clarke 1983), and in the Mediterranean (Shott and Leaman 1991) indiate that deep onvetion ours only at speifi sites and is a very loalized proess both with respet to time and spae. Surfae ooling and possibly brine rejetion by sea ie formation indue vigorous onvetion in the form of plumes with a horizontal sale of O(1 km), whih are organized within larger-sale strutures of O(50 km). The mixing of heat and salt indued by the onvetion and the subsequent geostrophi ad- * Current affiliation: Institute of Geophysis and Planetary Physis, University of California, Los Angeles, Los Angeles, California. Corresponding author address: Dr. M. Jeroen Molemaker, Institution for Geophysis and Planetary Physis, University of California, Los Angeles, 405 Hilgard Avenue, Los Angeles, CA nmolem@atmos.ula.edu justment eventually leads to a large-sale modifiation of deeper water masses. This interation between smallsale onvetion and the larger geostrophi sales is a ruial problem in the parameterization of water mass transformation. Beause of the loalized nature of the onvetion proess, a prototype situation of study in the laboratory has been the development of onvetion in a stratified rotating layer of liquid due to a loalized negative surfae buoyany flux (Maxworthy and Narimousa 1994; Coates et al. 1995). In most of the experiments, the liquid is initially at rest and the surfae buoyany flux is onfined to a disk with a smaller extent than the total flow domain. During the first stages of flow development, a well-mixed layer grows downward through entrainment and a lateral buoyany gradient develops. The flow is not influened by rotation until it reahes the depth at whih the Rossby number based on the loal veloity and length sale beomes small enough (Coates et al. 1995; Coates and Ivey 1997). In a next stage, geostrophi adjustment on the sale of the ooling disk leads to a rim urrent along the edge of the onveting area. This rim urrent subsequently beomes unstable through barolini instability. The resulting vorties, having a horizontal sale of the loal internal Rossby deformation radius, spread away from the original onvetive region induing lateral transports of heat and salt. Eventually, a quasi-steady state may our in whih the energy loss through the surfae is balaned by the 000 Amerian Meteorologial Soiety

2 476 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30 fluxes through the lateral boundaries of the onvetive region. Numerial simulations using nonhydrostati models (Jones and Marshall 1993; Klinger and Marshall 1995; Send and Marshall 1995; Visbek et al. 1996) have greatly ontributed to the understanding of these flows. Different flow regimes exist, depending on the magnitude of the rotation rate f, the initial stratifiation measured by the buoyany frequeny N, the surfae buoyany flux B 0, and the depth of the layer H. A key parameter for the small-sale flow development is the natural Rossby number Ro* B /(Hf 3/ 0 ), whih is the 1/ ratio of the harateristi vertial mixing timesale in the rotationally affeted regime and the geostrophi adjustment timesale. If Ro* is large, the developing onvetion remains essentially three-dimensional, while if Ro* is small, geostrophi adjustment is relatively fast and the onvetion is quasi two-dimensional (Klinger and Marshall 1995). In all of these studies, the initial horizontal sale of the onvetive region, that is, the radius of the ooling region, is presribed. In reality, the ooling by the atmosphere is not so stritly loalized and one would expet that, depending on the stratifiation, a muh larger region would overturn (Killworth 1983). To explain why the proess is so loalized it has been suggested that topography (Alverson and Owens 1996) or horizontal gradients in the bakground density field (Made et al. 1996) limits the sites of onvetive ativity. Doming of isopynal surfaes due to the large-sale yloni bakground flow has been observed in the Greenland Sea (Gasard and Clarke 1983) and in the Mediterranean (Shott and Leaman 1991). While this large-sale preonditioning of the density field may add to the ourrene of onvetion, it still does not explain its spatially loalized nature. The loalization of onvetion through the presene of old ore eddies was suggested by Johannessen et al. (1991) based on observations in the northern Greenland Sea. Typial horizontal sales of these eddies were 10 km and a typial eddy lifetime was about 0 30 days. The majority of these eddies rotated ylonially with orbital speeds of around 0 m s 1. In a reent study, Legg et al. (1998) studied a prototype problem of this loalization by studying numerially the flow development of a old ore eddy that was ooled homogeneously from above. Very loalized onvetion an indeed our due to the presene of suh an eddy. The struture of the eddy determines the initial stratifiation with a buoyany frequeny varying both as a funtion of distane to the eddy enter as well as with depth. The mixed layer deepening through onvetion is therefore inhomogeneous and leads to a restratifiation in the eddy region. After this restratifiation, smaller eddies develop along the edge of the original eddy through barolini instabilities, whih eventually leads to breakup into multiple eddies. The barolini instability proess is an essential feature in the large-sale flow development. The instability of an eddy has been studied in a laboratory experiment by Saunders (1973) and Griffiths and Linden (1981). In both studies it was found that the stability of the eddy ruially depends on the Burger number Bu (L /R 0 ), the square of the ratio of the internal Rossby deformation radius L NH/ f,and a harateristi horizontal length sale of the eddy, that is, the initial eddy radius R 0. A neessary ondition for barolini instability of a two-layer quasigeostrophi vortex is that Bu ¼ (Pedlosky 1985). Although their initial (standard) eddy is more ompliated, Legg et al. (1998) mention that this old ore eddy is stable to small perturbations in the absene of ooling. Cooling dereases L and therefore inreases the ratio R 0 /L sine R 0 is fixed, whih eventually gives onditions under whih the eddy beomes unstable. Legg et al. (1998) alulate the growth rate of the barolini perturbations from the numerial simulations and find that it ompares well with the lassial growth rates in the Eady problem (Eady 1949), whih sale as f N z being the meridional veloity of the initial eddy. However, they were not able to estimate the sale of the patterns of the instability and its dependene on the initial eddy size and strength. The barolini instability proess of the eddy is of major importane to the lateral exhange and therefore of the long time modifiation of the water masses involved. In this paper, we onsider the barolini instability problem of old-ore eddies in more detail, by solving the linear stability problem within the full 3D nonhydrostati model formulated in setion. In setion 3, the linear stability of geostrophi eddies is alulated. The effet of onvetion is modeled through its modifiation of the surfae stratifiation. We alulate the patterns and growth rates of the most unstable perturbations for different eddy sizes. Subsequently, a high resolution numerial simulation using a 3D nonhydrostati model is performed and the time and spae sales found are ompared to those predited by the linear stability analysis (setion 4). Reently, several studies have been arried out on shallow onvetion in an idealized oastal polynia (Gawarkiewiz and Chapman 1995; Chapman and Gawarkiewiz 1997). In setion 5, the same flow is simulated using a hydrostati model with a resolution that resolves the barolini eddy sale. The effets of onvetion on the mixing of buoyany is parameterized using onvetive adjustment. Considering the riteria developed in Marshall et al. (1997a), the large-sale flow development of the ooled geostrophi eddies is ertainly in the hydrostati regime. However, sine our linear stability results imply that the growth rate of the barolini instabilities is strongly modified by onvetion, one may

3 MARCH 000 MOLEMAKER AND DIJKSTRA 477 ask whether the orret large-sale flow development is obtained when onvetion is represented in a hydrostati model by onvetive adjustment. This representation issue of onvetion is studied in setion 5 and both the linear stability results and the high resolution nonhydrostati simulation serve as a referene ase for the results obtained with ourse resolution hydrostati simulations.. Formulation a. Model Consider a liquid with (eddy) visosity m and thermal diffusivity T in a rotating retangular domain of length L x, width L y, and height H. At the top of the liquid, a onstant negative buoyany flux B 0 is presribed through a onstant heat flux H, whih ools the layer homogeneously from above. With the veloity vetor u, the pressure p, the temperature T and the density, the governing equations (using the Boussinesq approximation) desribing the deviation from hydrostati equilibrium are given by u 1 u u f e3 u p (mu) t 0 g e (1a) u T 0 (1b) (1) T u T ( T T), (1d) t where f is the Coriolis parameter, is the thermal ompressibility, and e 3 the unit vetor in the vertial diretion. Both the visosity and the thermal diffusivity are assumed onstant and equal to. The governing equations are nondimensionalized using sales H, /H, 0 /H, H /, and 0 HB 0 /(g) for length, veloity, pressure, time, and density, respetively, suh that the total dimensional density * is alulated from the dimensionless by * 0 [1 HB 0 /(g)]. This leads to the following nondimensional equations: u u u Tae3 u pu e3ra t (a) u 0 (b) u, () t where the temperature has been eliminated using (1). The domain is assumed periodi in the horizontal. The top boundary is assumed to be stress free while the bottom satisfies no-slip onditions. The bottom boundary satisfies a no-flux ondition for the density while a onstant density flux is presribed at the top. The dimensionless boundary onditions at top and bottom are z 1: u 1; 0; 0; z z z w 0 (3a) z 0: 0; z u 0; 0; w 0. (3b) Apart from the two aspet ratios A x L x /H and A y L y /H of the basin, two other dimensionless parameters appear, the Taylor number Ta and the flux Rayleigh number Ra, whih are given by 4 4 HB0 Hf Ra, Ta. (4) 3 b. Cold ore eddies in geostrophi balane Our aim is to determine the influene of onvetion on the large-sale linear stability of old ore eddies. To define suh eddies, we first write the equations () in polar oordinates (r,, z) with radial veloity u, azimuthal veloity, and vertial veloity w; that is, u p u u Ta u t r r u (5a) r r u 1 p u Tau t r r u (5b) r r w p u w w Ra (5) t z 1 1 w (ru) 0 (5d) r r r z u (5e) t with u u w ; r r z 1 1 r r r r r z and with the boundary onditions (3) at top and bottom. Steady parallel flow solutions u 0, (r, z), w 0, (r, z) are not easily found analytially. Hene, a proedure is needed to find approximate parallel flow solutions. First, a bakground density field is defined by

4 478 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30 TABLE 1. Standard values of the dimensional and dimensionless parameters. Dimensional parameter H L x L y B 0 f H Dimensionless parameter A x A y Ra Ta e e A e Value 10 3 m m m 10 7 m s m s s 1 00Wm 10 4 K 1 Value b (z) e (z1), (6) where measures the vertial density gradient. This defines a dimensional buoyany frequeny N (z) [B 0 /]e (z1) with a maximum at the surfae. In Legg et al. (1998), a veloity field [0, (r, z), 0] is hosen orresponding to a yloni (old ore) eddy and the indued dynami density field is alulated using (5a) and (5) in the limit of zero frition. In this limit the density field satisfies the steady equation (5e). Our approah is slightly different, beause we want to hoose the density field a priori (and later on modify it by inluding effets of ooling) and alulate the resulting veloity field from this density field. In absene of ooling, the dynami density field of the eddy that is superposed on the bakground stratifiation is hosen as e (r, z) r (z1) Ae e e e e, (7) where A e is the amplitude of the eddy, e ontrols its horizontal sale, and e defines the vertial deay sale of the eddy. As in the basi state in Legg et al. (1998), also the total density field e b satisfies the steady equation (5e) in the limit of zero fition. To obtain the flow e (r, z), whih is in geostrophi balane with the density field, we use the thermal wind relation, negleting the ylostrophi term /r in (5a), to give z Ra e(r, z) dz. (8) Ta 0 r For the values of parameters as in Table 1, a vertial slie of the total density distribution for y A y / is plotted in Fig. 1a. This figure and following similar figures are plotted using retangular oordinates where x is given by x 0.5A x r os. The stratifiation due to the bakground density field is maximal near the surfae, with a maximum value orresponding to N FIG. 1. (a) Contour plot of the density field for a vertial slie at y 0.5A y. The ontour interval is 0.1, orresponding to a buoyany differene of 10 4 ms. (b) Similar plot of the meridional veloity field e os saled with its maximum value (0.4 m s 1 ). Solid (dotted) lines indiate positive (negative) veloities, the ontour interval is s. The presene of the geostrophi yloni eddy is learly seen by the doming of the isopynals near the enter of the domain, whih oinides with the enter of the eddy. The eddy redues the bakground density gradient. At the enter of the eddy, the resulting density gradient is nearly neutral. The geostrophi meridional veloity of the eddy is shown in Fig. 1b with amplitudes of O(10 1 ms 1 ). Our proedure of alulating the basi state enables us to take the effet of ooling into aount. Starting from the initial density field e b we determine the density profile that results if the layer is ooled for 4 hours using the presribed buoyany flux B m s 3. Here, it is assumed that ooling from the surfae leads to a mixed layer with a density that is equal to the density diretly underneath the mixed layer at every loation. The resulting density profile is shown in Fig. in whih the mixed layer is shaded. Sine the original stratifiation in the enter of the eddy is muh weaker than the far-field stratifiation, the resulting mixed layer is muh deeper in the eddy enter. Although any adaption of the density profile ours only FIG.. Contour plot of the density field, whih results after 4 hours of ooling of the density profile in Fig. a with the buoyany flux B 0. Sale and ontour intervals as in Fig. 1a.

5 MARCH 000 MOLEMAKER AND DIJKSTRA 479 in the upper part of the domain, it may be important sine it erodes the very strong stable density gradients near the surfae and therefore destabilizes the eddy. For the density profile, the geostrophi veloity field was realulated through the thermal wind relations. This veloity field hardly differs from e presented in Fig. 1b and is therefore not shown. The hoie of 4 hours as a period over whih to ool is rather arbitrary. It is hosen to be large enough for a surfae mixed layer to develop, but small enough to limit the modifiation of the original eddy to the removal of stable density gradients near the surfae. Subsequently, a homotopy parameter is introdued to deform the original basi state orresponding to 0 to the ooled basi state orresponding to 1 and a series of basi states is desribed by (r, z; ) (1 ) (r, z) (r, z) e (r, z; ) (1 )( (z) (r, z)) (r, z). b e (9a) (9b) These basi states will be used for a linear stability analysis in the next setion. The density field for 0 orresponds to that shown in Fig. 1, whereas that for 1 is shown in Fig.. The veloity field is formally only a steady solution of the invisid zonal momentum equations in the limit Ta (while keeping Ra/ Ta finite), that is, in the limit of fast rotation. As it is determined here, the veloity field is in exat geostrophi and hydrostati balane but it does not satisfy the invisid steady radial momentum equation sine the ylostrophi term is negleted. For the atual parameters used (Table 1), the ylostrophi term an be alulated from this solution and its maximum for 0 appears to be at most 10% of the geostrophi term. 3. Linear stability analysis In many studies, standard barolini instability theory (Eady 1949) has been applied to obtain the dominant sales assoiated with the growth of the barolini modes. However, this theory is based on a stability analysis of a zonal jet with onstant vertial shear and the nonparallel flow assoiated with a geostrophi eddy may have totally different stability properties. a. The eigenvalue problem and its solution To study the stability of the basi states given by (9), infinitesimally small perturbations are onsidered suh that (u,, w, p, )(r,, z, t) (0,,0,p, )(r, z) e tim (û, ˆ, ŵ, pˆ, ˆ )(r, z), (10) where ( û, ˆ,ŵ,pˆ, ˆ ) are (omplex) funtions, m is the azimuthal wavenumber of the perturbations and is the omplex growth fator. Substitution of (10) into the equations (5) and linearizing in the perturbation amplitude gives the eigenvalue problem ˆ pˆ û imû Taˆ r r r û ˆ û (11a) r r û impˆ ˆ û imˆ ŵ Taû r r z u r û ˆ (11b) r r pˆ ŵ imŵ ŵ Raˆ r z (11) 1 1 ŵ 0 (rû) imˆ r r r z (11d) ˆ û imˆ ŵ ˆ, (11e) r r z where 1 m r. r r r r z Together with the homogeneous boundary onditions (3) for the perturbation quantities and boundedness of all perturbation fields for r, the problem (11) is an ellipti eigenvalue problem with eigenvalue r i i and parameters (Ta, Ra, m) in addition to those parameters appearing in the basi state, suh as the eddy strength. To solve this ellipti eigenvalue numerially, first the domain [0, is transformed into the domain [0, 1] using the mapping r r, (1) 1 r where 0.1. The transformed problem is disretized using seond-order finite differenes and the (disretized) algebrai eigenvalue problem (11) an be written as Ax Bx, (13) where x is the disretized state-variable vetor and matrix A ontains the disretized right-hand side of (11). The diagonal matrix B is singular due to the ontinuity equation (11d) and the boundary onditions. The ode to alulate the eigenvalues and eigenvetors was verified by onsidering the lassial problem of the differentially heated rotating annulus (Hide and Mason 1975). In this problem, a rotating fluid between two onentri ylinders is subjeted to a radial temperature gradient T r. The parallel basi-state flow is suseptible to barolini instability and the relation between the azimuthal wavenumber of the most unstable barolini mode and the ontrol parameters Ta and T r is well

6 480 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30 known from experiments (Hide and Mason 1975) and theory. This relation [Fig. 7 in Hide and Mason (1975)] ould be reprodued very well with our ode. It turned out that for the alulation of the barolini stability of the eddies that are onsidered, a resolution of 64 3 grid points in the (r, z) plane proved to be adequate. b. Results The parameters for the standard ase are shown in Table 1 and represent a typial situation in the Greenland Sea (Shott et al. 1993). Of ourse, the bakground eddy diffusivity is quite unsure and hosen as small as possible and still be able to perform the high-resolution numerial simulations in the next setion. To be ompatible with these simulations, the same value is onsidered in the linear stability analysis. In Fig. 3, the neutral urve in the (m, ) plane is shown in panel (a) whereas the angular frequenies i are presented in panel (b). For 0, whih orresponds to the basi state unmodified through ooling at the surfae, the eddy is linearly stable. At 0.503, the basi state beomes unstable to a disturbane having an azimuthal wavenumber m 4. The angular frequenies are negative, indiating yloni propagation of disturbanes, that is, in the same diretion as the basi state. The angular frequenies inrease linearly with azimuthal mode number and orrespond with a propagation veloity that is nearly onstant for every mode number. A horizontal slie of the vertial veloity of the ritial m 4 mode is shown in Fig. 4a. The orresponding density perturbation of the most unstable mode in Fig. 4b has a similar struture as the vertial veloity pattern but it is slightly displaed. Note that these plots only show a snapshot of a propagating mode at a partiular phase of the osillation. In regions of positive vertial veloity, heat is transported downward leading to ooler surfae water and hene a positive density perturbation. This density perturbation ours slightly downstream from loations where positive density anomalies were at an earlier time and hene the perturbation propagates downstream. The slies at y/a y 0.5 of both fields indiate the strong loalization of the perturbation in the region of maximum vertial shear of the basi state near R 0 1/ e (at x 0.5A x R 0 in the figure), with larger amplitude at the enter side of this loation. The loation of maximal shear of the basi state defines the (dimensionless) wavelength and meridional phase speed V of the perturbations, whih are given by R 0 /m and V R 0 i /m. The phase speed of perturbations is slightly higher than the maximum azimuthal veloity of the eddy, in the downstream diretion, and its magnitude is about 0 m s 1. This is larger than the vertially averaged veloity of the basi state, whih is the propagation veloity in the standard Eady problem. To ompare more losely to the Eady problem, the dependene of the growth fators on the sale of the eddy, determined by e, is shown in Fig. 5a for FIG. 3. (a) Neutral urve in the (m, ) plane where m is the azimuthal mode number and ontrols the shape of the basi state. (b) Angular frequeny of the neutral modes in (a). 1. The ase onsidered above (with e 0.04) is shown as the solid urve and shows a maximum growth rate at m 4. Clearly, the mode number m of maximum growth inreases with inreasing eddy size. For e 0.01, the maximum growth rate ours for m* 14 while for e 0.05 it ours for m* 3. Smaller eddies have in this ase larger growth rates of perturbations, whih is not surprising sine the gradients from whih the perturbations feed are stronger. The range of azimuthal sales of the perturbations shows a utoff for large m. However, as is evident from Fig. 5 for large m the eddy may beome suseptible to other types of instabilities, suh as shear instabilities. For example, the seond peak in growth rate at m 9 for e 0.0 is assoiated with a vertial shear instability. This an be

7 MARCH 000 MOLEMAKER AND DIJKSTRA 481 FIG. 4. (a) Vertial veloity distribution of the most unstable perturbation at z (b) Density distribution of the most unstable perturbation at z () Vertial veloity distribution of the most unstable perturbation at y/a y 0.5. (d) Density distribution of the most unstable perturbation at y/a y 0.5. seen from the struture of the mode (not shown) that is markedly different from those shown in Fig. 4. This mode does not have the tail-like strutures as in Fig. 4 and its vertial struture reahes muh deeper, ontaining extra nodes. A omplete investigation of all possible instabilities on the basi-state eddy would require an extensive investigation that is outside the sope of this paper. Different ooling times to alulate the basi state for 0 are not expeted to hange the result signifiantly as long as they are short enough to limit the effets of ooling to the density gradients in the surfae layer. The only obvious differene is that for shorter ooling times the eddy will be more stable, but the same modes are expeted to dominate. Eady theory indiates a maximum growth rate for perturbations that have a length-sale about four times the deformation radius L. In Fig. 6, the value of m* is plotted as a funtion of R 0, showing a near-linear relationship. For eah value of e, the azimuthal wavenumber adjusts to fit a number of wavelengths of sale L on the irumferene R 0 at the radius of maximum shear. The proportionality onstant between m* and R 0 an be omputed from the slope in Fig. 6. It follows that, measured at the radius of maximum shear, the most unstable perturbations have a wavelength of about 5.5 km for the eddy sales onsidered. It is well known that the spatial sale of the fastest growing disturbane sales with the Rossby deformation radius L. However, in this situation, it is not an easy task to define an appropriate L sine it varies strongly over the field. A loal density gradient is likely not representative for the whole domain in whih the perturbations grow, although it provides small values of L when onvetion erodes the surfae stratifiation. A nonloal estimate of L an, for example, be obtained through the density differene g between surfae and bottom. This leads to an estimate of L of

8 48 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30 FIG. 5. Growth fator r of the fastest growing modes as a funtion of azimuthal mode number m for 1.0 (all other parameters as in Table 1). Different urves are for different eddy sale parameters e and the solid urve is for e FIG. 6. Most unstable azimuthal wavenumber m* as a funtion of the radius of maximal shear R 0 of the eddy. gh g(sur bot)/ 0)H L, (14) f f whih inreases from 3. km at the enter of the eddy to 7.1 km of the bakground stratifiation. Another alternative is to estimate L from the (surfae) density differene between the enter of the eddy and the bakground field. This would lead to an estimate of the Rossby deformation radius of 6.4 km. Both estimates give a value of L that is too large to omply with standard Eady theory. Another possibility is to use a volume averaged value for the buoyany frequeny N ave, leading to an integral sale for L. We may use Fig. 4 to determine the volume where the perturbation has substantial amplitude and average the buoyany frequeny N over that volume to obtain N ave. The boundaries of suh a volume are by no means learly defined, whereas the resulting values of N ave depend strongly on these. However, if we hoose the boundaries of this volume as x/a x [0.5, 0.65], z [0.8, 1] a value of N ave s 1 results, orresponding to a muh smaller value of L.1 km. This estimate of L is muh loser to the value expeted from standard Eady theory. However, as it is not lear whether the behavior of the growth rates of the perturbations is similar to that in Eady theory, the partiular relevant L remains unlear. 4. Numerial simulations: Nonhydrostati model As a next step toward understanding the impat of onvetion on the large-sale flow development, a highresolution simulation was performed using the nonhydrostati model as desribed in the appendix. The domain is a 3 3 1km 3 retangular box and the initial onditions are exatly the basi state (9) for 0. As a first hek on the linear stability results, a simulation was performed at a resolution , orresponding to x y 15 m, z 30 m, of the flow development of the eddy without surfae ooling. The flow deays to zero in this ase, and the eddy is stable, just as predited by the linear stability results for 0. It appears that the density gradients near the surfae are strongly stabilizing the initial state. Next, a high resolution with x y 80 m, z 30mis used ( grid points) using the parameters as in Table 1. The value of the parameter Ro* (Ra/ (Ta 3/ ) 1/ 0.3, whih ombined with Ra 10 8 shows that the simulation is in the 3D turbulent onvetion regime (Klinger and Marshall 1995). The governing equations were integrated over 0.1 dimensionless units in time orresponding to approximately 1 days. a. Flow development This simulation differs from that in Legg et al. (1998) in the initial onditions and parameters, but the flow development obviously shows similar features. Due to the ooling at the surfae, a thermal boundary layer forms, whih very soon beomes unstable to diret buoyany driven instabilities. The depth of penetrative onvetive ativity is diretly related to the initial stratifiation. Away from the eddy the onvetive layer remains limited to the upper 100 m, whereas in the enter of the eddy, onvetion reahes to muh greater depth.

9 MARCH 000 MOLEMAKER AND DIJKSTRA 483 FIG. 7a,b. Horizontal slie at z 0.95 of different fields after three days of ooling. (a) Vertial veloity (dark olors downward, bright olors upward). (b) Temperature (bright olors warm and light, dark olors old and dense). FIG. 7,d. Horizontal slie at z 0.95 of different fields after six days of ooling. () Vertial veloity (dark olors downward, bright olors upward). (d) Temperature (bright olors warm and light, dark olors old and dense). After 3 days the onvetion is ative within a radius of about 6 km as an be seen from a slie of the vertial veloity just below the surfae (z 0.95) in Fig. 7a. A dominant feature in the vertial veloity field are the spiral strutures along whih downward veloities have been organized. The width of these spirals is about 100 m and the distane between the arms is a few kilometers. Sine the layer is ooled from above, temperature fields (saled by HB 0 /g) will be shown of the simulations. In the temperature field after 3 days of ooling (Fig. 7b) the small-sale details as in the vertial veloity field an be observed but are superposed on the initial temperature distribution. In Figs. 7,d the vertial veloity and temperature fields are shown after 6 days of ooling. The temperature (Fig. 7d) shows two onvetive pathes with a horizontal dimension of about 4 km moving outward of the original onvetive region. The veloity field shows again many small-sale onvetive elements (Fig. 7), but these are organized into larger-sale features that oinide with the ooler pathes in the temperature. At a later stage in the evolution both pathes are found to move outwards from the onvetive region. The signal of one of these pathes passing the point x/a x y/a y 0.75, z 0.9 an be seen in the time series of the azimuthal veloity (solid) and vertial veloity w (dotted) in Fig. 8 near t 10 h. This figure gives also an impression of the veloity sales during the evolu-

10 484 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30 FIG. 8. Time evolution of the azimuthal veloity (solid) and vertial veloity w (dotted) at the point x 0.75A x, y 0.75A y. tion. The azimuthal veloity osillates initially around a mean value of ms 1, whih orresponds to the veloity field of the presribed eddy, up to about 3 days. The timesale of these osillations orresponds to onvetive plumes with a size of several 100 m that are adveted by the bakground veloity field. For t 60 h the veloity beomes more irregular orresponding to the breakup of the original eddy into different onvetive pathes. This breakup an also be seen in the vertial veloity showing large amplitude osillations (the maxima orrespond to 10 ms 1 ) around that time, whih subsequently beome irregular. In the initial stages of the flow development, the sales of motion orrespond to those derived earlier in the numerial simulations where loalized ooling over a disk within the flow domain was applied (Jones and Marshall 1993). By alulating Ta based on the depth of the unstable thermal boundary layer, using a depth of 0.1H, we find a value of Ta 00. This value is suffiiently small to neglet the effet of rotation during the initial stage of onvetion. This an also be seen from the orresponding timesale of the onvetive motions in that layer (Deardorff 1980); t ( /B 0 ) 1/3 or t s, whih means that t f 1 and rotation is not (yet) important. The resulting sales for (nonrotating) onvetion are l onv 100 m, u onv (l onv B 0 ) 1/3 10 ms 1.Att f 1, rotational effets beome important and the sales l rot (B 0 / f 3 ) 1/, u rot (B 0 / f) 1/ are expeted (Maxworthy and Narimousa 1994; Fernando et al. 1991; Jones and Marshall 1993). In the present simulation, l rot 150 m and u rot 10 m s 1 and, beause the differene between rotationally and nonrotationally dominated sales is small, a transition between both regimes annot be learly distinguished in Fig. 8. In agreement with the results of the linear stability alulation in the previous setion the m 4 mode is learly dominating the results of the numerial simulation, where it an be identified by the four spirals in the vertial veloity (e.g., Fig. 7a). As argued in the previous setion, the exat value of the ooling time to alulate the basi states is not important as long as it is small. Apparently, the effet of the onvetion on the barolini instability has been modeled well by the parameterization of the onvetion through the parameter ; that is, only the erosion of the surfae stratifiation is important. The reason may be that the large-sale dynamial effets of the small-sale veloities that result from onvetion are small. In the loalized ooling ase the density anomaly aused by onvetion, say measured by a redued gravitg, indues a horizontal density gradient and sub- sequently a rim urrent that is suseptible to barolini instabilities. However, in our ase, a horizontal dynamial density gradient is already present through the geostrophi eddy. The g e due to the eddy an be diretly obtained from the initial onditions g e g, where an be estimated by the initial differene between the density in the enter of the eddy and that of the bakground density field. An estimate of g is not so easy to define in our ase. Sine the ooling is not loalized over the eddy, previously used estimates that onsider a balane between loalized ooling and horizontal transport by the onvetive pathes (Visbek et al. 1996) are not valid here. Any additional density differene to g e will only be due to horizontal mixing sine the ooling is homogeneous over the whole surfae. The effet of horizontal mixing will be small, at least up to the time where the large-sale perturbations are not yet fully developed. However, an upper bound is ertainly given by the estimate based on the near steady state thermal balane from Visbek et al. (1996), g N(B 0 R 0 ) 1/3. If we base g e on the surfae values, then the ratio g/ ge for 0 is given by 1/3 g N(B0R) 0, (15) g AHB e e 0 and its value will determine whether onvetion will influene the mean irulation of the eddy or not. In the present simulation, g/ ge and onsequently the rim urrent indued by onvetion is negligible ompared to the veloity field of the original eddy. This shows that the dominating unstable modes are determined by the initial geostrophi eddy and not by the strength of the surfae buoyany flux. b. Spetral analysis of the flow patterns As an be observed in the high-resolution nonhydrostati simulation there is muh energy both in the large sales (barolini instability) and the small sales (onvetion). To look at the energy-ontaining sales in more detail, two-dimensional disrete Fourier spetra were alulated for the spatial pattern of quantities in

11 MARCH 000 MOLEMAKER AND DIJKSTRA 485 TABLE. Spetral distribution of the vertial heat flux for the different simulations onsidered. The entry (wt) b is the barolini omponent, defined as the ontribution to the total heat flux wt for k 40. The quantity (wt) is the onvetive omponent, defined as the ontribution to wt for k 40 and (wt) b (wt) wt. IM and CA are different types of onvetive adjustment as explained in the text. n m Type of adjustment wt wt b wt nonhydrostati IM, 0 IM, 0 CA IM, 1 IM, FIG. 9. Analysis of the spatial patterns in the high-resolution simulation at t* 6 days and a horizontal slie at z (a) Graysale plot of the two-dimensional ospetrum of the vertial veloity and the temperature Qˆ as a funtion of k x (N/ k x N/) and k y (M/ k y M/). Light (dark) olors indiate large (small) values. (b) Qˆ as a funtion of the absolute wave number k as obtained from the two-dimensional ospetrum. a horizontal plane just below the surfae. More speifi, for any quantity F the (omplex) disrete Fourier transform Fˆ is defined as ˆF k x,k y [ ] N M x mk n,m NM n1 m1 N M 1 nk y F exp i, (16) where k x N/,,1,0,1,,N/, k y M/,,1,0,1,,M/ and the F n,m are the values of F at the grid points. An important quantity to onsider in the problem is the vertial onvetive heat flux Q wt. All the presented spetra are therefore hosen to be ospetra of vertial veloity w and temperature T. The ospetrum Qˆ is defined as Qˆ Re(ŵ)Re(Tˆ ) Im(ŵ)Im(Tˆ ), (17) with Re and Im indiating real and imaginary part and ŵ and Tˆ defined as in (16). It follows that Qˆ is real and the sum over the (disrete) wavenumbers of the ospetrum Qˆ equals the integrated vertial heat flux Q (Stull 1988). At t* 6 days, a gray shade plot of ˆQ k,k is shown x y in Fig. 9a for a slie at z 0.95, whih orresponds to the pattern in Figs. 7,d. The bright area enlosing the enter shows the energy ontaining large sales. Muh energy is still ontained in the initial eddy, orresponding to k x k y 0, other large sales arise through its barolini development. Enlosing this enter area is a band of relatively small amplitude followed by a broad small-sale band orresponding to the onvetive sales of O(10 m). The orresponding spatial sale of a speifi wavenumber an be found from L/ k, k ( kx ky ) 1/, where L 3 km is the horizontal length of the domain. The dominant spatial sales an be seen more easily in a plot of Qˆ as a funtion of k (Fig. 9b). Two bands of high energy ontaining sales are again seen, whih an be roughly divided into a large-sale band 0 k 40 and a small-sale band 40 k 00. The two spetral bands in Fig. 9b motivate one to ompute a band averaged spetral amplitude of the onvetive heat flux. These values are given as (wt) b, for the large sales (k 40) and (wt) for the small sales (k 40) in the first row of Table for the nonhydrostati high-resolution simulation. Most of the energy (about 90%) is ontained in the small sales. We will use these band averages in the subsequent setion to study the issue of representation of onvetion in hydrostati models. 5. Convetion in hydrostati models Having established the basi harateristis of the flow development as observed in the high resolution

12 486 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30 nonhydrostati simulation, we now investigate how these are represented in hydrostati models. A hydrostati version of the ode an be easily developed from the nonhydrostati ode and, sine the approah differs from that in Marshall et al. (1997a), more details are presented in the appendix. a. Convetive adjustment In the nonhydrostati model, onvetion is expliitly resolved and no adaptations are needed when the stratifiation beomes loally (statially) unstable. In hydrostati models, the effets of onvetion have to be parameterized and several ad ho proedures, generally referred to as onvetive adjustment, are used. In all these proedures, the temperature (and salinity) fields are loally adjusted in suh a way that a stable stratifiation is ahieved. In the first proedure, whih we will indiate by lassial adjustment (CA), the temperature is expliitly mixed in adjaent vertial levels of the water olumn if the density stratifiation is unstable. This proedure has to be repeated a number of times at eah time step in the evolution of the flow, as an iteration toward omplete removal of stati instabilities (Cox 1984). A variation of this tehnique is suggested by Marotzke (1991) and others (Yin and Sarahik 1994; Rahmstorf 1995) in whih groups of levels in the water olumn are treated as one onvetive region. The latter proedure was implemented sine it is guaranteed that the liquid is stably stratified after the proedure terminates. This ours within one time step of the model, and hene it is assumed that the timesale of onvetive mixing is muh smaller than the time step of the numerial model. Furthermore, it is assumed that the onvetion only mixes quantities vertially; no horizontal mixing is involved. The other proedure of onvetive adjustment is indiated by impliit mixing (IM) and assumes that the effet of onvetion on a subgrid sale an be modeled by a large vertial diffusion oeffiient for the traers (Cox 1984). Hene, the vertial mixing is parameterized as z with the mixing oeffients given by T K (18) z K K, (19) where is the bakground value (as in Table 1) and K models the subgrid-sale onvetion as an additional diffusive proess. The value of K is large in areas with unstable stratifiation and zero otherwise. A value of K an be estimated from a straightforward mixing length argument using the appropriate veloity U and length sales L for onvetion. In the rotationally ontrolled regime, one would have U B 0 / f and a length FIG. 10. (a) Two-dimensional ospetrum Qˆ obtained from a horizontal slie at z 0.95 for the hydrostati model after six days of ooling. In these results, 0 and a grid is used. (b) As (a) but with a grid. sale L B 0 / f 3. This saling leads to a mixing oeffiient of B K UL 0, (0) f whih gives values of K O(10 m s 1 ), a value about 100 times larger than the bakground diffusivity of 0.1 m s 1. For all subsequent alulations where IM is used a value of K 10 m s 1 is hosen. Instead of this estimate for the mixing length sale one ould also use the depth of the mixed layer as suggested in Klinger et al. (1996) but this does not lead to an order of magnitude differene.

13 MARCH 000 MOLEMAKER AND DIJKSTRA 487 FIG. 10. (Continued) () Vertial veloity field orresponding with (a). (d) Vertial veloity field orresponding with (b). (e) Temperature field orresponding with (a). (f) Temperature field orresponding with (b). b. Results The total heat flux Q is defined as T Q wt K. (1) z In the nonhydrostati model, the range of sales is large and, in partiular, the small sales ontribute to the advetive term wt, whereas the diffusive term K T/z in (1) is relatively small (K equals ). The basi idea of the adjustment shemes in hydrostati models is that the ontribution of the large sales to the heat flux Q should be represented well, whereas the small-sale ontribution to wt is parameterized by the onvetive adjustment as in (18). In other words, there should be no smallsale amplitude in the expliitly resolved part of wt in hydrostati models that employ a form of onvetive adjustment. The ontribution of the smaller onvetion sales must be represented by the relatively large term K In T/z. Fig. 10a, the ospetrum of Qˆ is shown for the IM sheme (18) using a horizontal resolution of In Table, the orresponding band amplitudes are given in the seond row. Although the ontribution of the small sales redues to 30% of the total, it ertainly does not vanish as would be desired. In fat, Fig. 10a

14 488 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30 shows erroneous ativity in the small sales. The ativity inreases toward the smallest resolved sales, indiating a soure at the grid sale that most likely has a numerial origin. In Fig. 10, these small-sale features an be observed in the vertial veloity pattern. As a onsequene of the presene of these erroneous sales, the breakup of the eddy is quite different from the referene ase, as an be seen for the temperature field in Fig. 10e. For a horizontal resolution of 18 18, the spetrum (Fig. 10b), vertial veloity (Fig. 10d), and band amplitudes (the third row of Table ) show similar behavior. Here again, erroneous small-sale ativity is present, although the total amount is less, due to the fat that the highest wavenumber present at that resolution is only k 64. This is similar to the results for CA (fourth row of Table ) of whih the spatial fields and spetrum are not shown. The origin of this small-sale ativity is most likely the ourrene of nonresolved boundary layers in areas where onvetion ours. The essential diffiulty an be illustrated by a simple reasoning that only assumes diffusive proesses. Consider the timesale of vertial exhange that is assoiated with the onvetive adjustment to be t mix. Within this time, horizontal temperature differenes are reated on the smallest resolved (grid) sales. These horizontal gradients will lead to diffusive boundary layers of whih the thikness an be readily estimated from lassial diffusion theory (Carslaw and Jeager 1959); that is, (t mix K h ) 1/, where K h is the horizontal diffusivity. The boundary layer thikness should be resolved by the horizontal grid spaing; otherwise they lead to spurious ativity on the smallest sales. An estimate for the ondition on the minimal horizontal resolution is x (t mix K h ) 1/. () If this boundary layer is not resolved, large numerial errors may result, whih an ompletely destroy the auray of the solution and have other undesirable, nonphysial effets. Another possible ause for the small-sale ativity is hydrostati overturning, as disussed in Marshall et al. (1997b). Hydrostati models are not stable in the presene of statially unstable gradients but exhibit a type of onvetion. However, sine in onvetive onditions the hydrostati assumption is not valid, it is not lear what value an be attributed to this hydrostati onvetion. If a large vertial diffusivity is used in the onvetive adjustment proedure (IM), the timesale of vertial exhange and onsequently the timesale in whih the horizontal boundary layers are formed an be found using the onvetive mixing oeffiient K. The ver- tial exhange timesale is now given by t D /, where D is a vertial sale over whih mixing takes plae. Consequently, Eq. () an be written as a ondition for the horizontal diffusivity K FIG. 11. (a) Two-dimensional ospetrum Qˆ obtained from a horizontal slie at z 0.95 for the hydrostati model after 6 days of ooling. In these results, 1 and a grid is used. (b) As (a) but with a grid. x K K h. (3) D On the other hand, if an algorithm is used that ompletely removes unstable vertial gradients within one time step (i.e., CA), the timesale on whih the horizontal boundary layers are reated is equal to the time step of the numerial sheme t. Equation () an now be written as a ondition for the time step x t. (4) K h From (4) it follows immediately that numerial problems are expeted if the time step t is dereased, whih

15 MARCH 000 MOLEMAKER AND DIJKSTRA 489 FIG. 11. (Continued) () Vertial veloity field orresponding with (a). (d) Vertial veloity field orresponding with (b). (e) Temperature field orresponding with (a). (f) Temperature field orresponding with (b). is [similar to the grid-sale instability in Cessi (1996)] a very undesirable quality in any numerial integration.. Modified adjustment sheme To test whether this simple piture is indeed responsible for the small-sale ativity, as seen in the twodimensional spetra for the hydrostati model, several simulations were performed in whih additional horizontal diffusion was added loally in areas where onvetive adjustment ourred. The IM proedure was used; in this ase, the Laplaian diffusion terms are written as h (Khh) K (5) z z with h (/x, /y). The mixing oeffiients are given by K h Kh; K K, (6) where the supersript again refers to values due to onvetive adjustment.

16 490 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30 FIG. 1. (a) Two-dimensional ospetrum Qˆ obtained from a horizontal slie at z 0.95 for the hydrostati model after 6 days of ooling. In these results, 4 and a grid is used. (b) Vertial veloity field orresponding with (a). () Temperature field orresponding with (a). In the simulations we take K h K with K 10 m s 1 and 1. This is believed to be physially realisti in a high-resolution hydrostati model and also guarantees the absene of spurious numerial behavior, as desribed above, if the value of is suh that Eq. (3) is satisfied. For the oarsest resolution (x 40 m) indeed (3) is satisfied with 1. From a physial point of view, the hoie 1 also makes sense sine, for three-dimensional onvetion, horizontal and vertial sales of onvetive elements that have to be parameterized are the same. Consequently horizontal and vertial mixing that result from this onvetive ativity should also be the same. In Figs. 11a,b, the ospetra Qˆ are shown for equal vertial and horizontal diffusivities ( 1) for both and resolution. In both ases the small-sale ativity has dereased signifiantly, in orrespondene with the patterns that are shown in Figs. 11 f. From Table, whih shows the band amplitudes in this ase, it may be observed that the onvetive heat flux indued by the barolini sales (k 40) has roughly the same amplitude as the orresponding heat flux in the referene run with the nonhydrostati model. With 4 and a resolution of the small-sale energy is dereased enormously (Fig. 1a), but the breakup of the eddy is now