On Frontal Dynamics in Two Model Oceans

Transcription

1 OCTOBER 00 ELDEVIK 95 On Frontal Dynamics in Two Model Oceans TOR ELDEVIK* Department of Mathematics, University of Bergen, Bergen, Norway (Manuscript received July 000, in final form 0 April 00) ABSTRACT Vertically homogeneous variable-temperature layer models are often used to describe upper-ocean variability, the dynamics of jets and fronts included. Frontogenesis is known to have a preference for strong cyclonic shears. When a frontal wave winds up ageostrophically, one would expect intense cyclones and more diffuse anticyclones to be the result. This is characteristic of both atmospheric weather and the oceanic equivalent. The frontal dynamics of the variable-temperature layer model is here compared with that of the three-dimensional primitive equations, the origin of the layer model. Whereas the primitive equation numerical experiments produce dynamics according to the above, the cyclonic and anticyclonic shears and eddies are equally amplified in the corresponding variable-temperature layer experiments. It is suggested that thermal wind is basal to the desired frontal characteristics and that the layer description fails because of the way it has been defined to have none.. Introduction In the ocean there often exists a relatively well mixed upper layer separated from the underlying heavier water by a well-defined thermocline. This has motivated the use of the vertically homogeneous variable-temperature layer model for the description of upper-ocean dynamics (e.g., Fukamachi et al. 995; McCreary and Kundu 988; McCreary et al. 99; Ripa 993; Røed 996, 997; Røed and Shi 999; Shi and Røed 999; Young and Chen 995). In the widely used Miami Isopycnic Coordinate Ocean Model (MICOM), the upper layer is frequently chosen to be a variable-temperature layer as described by Bleck et al. (99). For a given computational cost, vertically homogeneous layer models can resolve smaller horizontal scales than three-dimensional primitive equations models, making the former amiable for studies of meso- and submesoscale ocean dynamics. This paper is a spinoff of Eldevik and Dysthe s (00) study of spiral eddies. Spiral eddies are submesoscale cyclones very frequently observed at the ocean surface (e.g., Munk et al. 000). Baroclinic instabilities are well known to generate (sub) mesoscale anomalies, an oceanic equivalent to the atmospheric frontal highs and lows constituting the weather at middle and high latitudes. Eldevik and Dysthe (00) model the spirals as * Current affiliation: Nansen Environmental and Remote Sensing Center, Bergen, Norway. Corresponding author address: Tor Eldevik, Nansen Environmental and Remote Sensing Center, Edvard Griegs vei 3a, N-5059 Bergen, Norway. tor.eldevik@nersc.no the nonlinear result of ageostrophic baroclinic instability. When a frontal wave winds up, intense cyclonic lows and more diffuse anticyclonic highs are the result. Thus spiral eddies are modeled as the sea surface signature of the bad weather below. Although the term ageostrophic baroclinic instability herein refers to an intrinsically ageostrophic process, the sought frontal dynamics should not be restricted to this. A cyclonic preference also evolves from traditional baroclinic instability when the frontal wave is allowed to grow beyond the quasigeostrophic approximation governing the onset of instability and becomes ageostrophic (e.g., Wang 993; Samelson and Chapman 995). The distinction between these two modes of baroclinic instability is discussed further by Barth (994). Because of the numerical efficiency of the variabletemperature layer model, and its ability to describe submesoscale frontal structures such as those observed off California and the Iberian Peninsula (cf. McCreary et al. 99; Røed and Shi 999), the numerical experiments of Eldevik and Dysthe (00) were originally planned to be performed in a variable-temperature layer model ocean. The preliminary results were discouraging with respect to spiral eddies. Contrary to the sought scenario, that is, oceanic bad weather generating cyclonic spirals, the unstable frontal wave emerging from a given initially geostrophic jet showed no preference for cyclones. Staggered rows of cyclonic and anticyclonic vortex pairs analogous to the Karman vortex street were the result. The layer model was accordingly discarded in favor of a three-dimensional primitive equations model. A comparison of frontal dynamics in these two model oceans is the scope of the present survey. 00 American Meteorological Society

2 96 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 3 The paper is organized as follows: the primitive equations model and the variable-temperature layer model are presented in section. If a mixing of momentum is introduced in the former model for the removal of thermal wind, the latter model will be the result. In section 3, simple manipulations of the equations governing the layer model suggest that the added mixing both removes baroclinicity in its basic form and dominates the vorticity dynamics. This seems to imply that no conservative potential vorticity can be found and that the expected dominance of cyclones is absent. Support for this is provided by the numerical experiments in section 4. The paper is summed up with concluding remarks in section 5.. The model oceans The governing equations of the two model oceans, the three-dimensional primitive equations and the twodimensional variable-temperature layer equations, are presented in this section. When thermal wind is taken out of the former model, the latter model is the result. a. The primitive equations model The nondiffusive, nondissipative primitive equations (PE) are the continuity equation 3 u3 0, () incompressibility D 0, () Dt the horizontal momentum equation under the Boussinesq approximation Du f k u p, (3) Dt 0 and the hydrostatic balance p 0 g. (4) z In the above, u 3 (u,, w) is the three-dimensional velocity vector, 3 is the three-dimensional gradient operator, D/Dt /t u 3 3 is the material derivative, denotes density with a reference value 0, f is the Coriolis parameter, p is pressure, and g is the gravitational constant. Quantities u and are the horizontal parts of u 3 and 3, respectively. The motion is restricted to an f plane. b. The variable-temperature layer model In the ocean there often exists a relatively well mixed upper layer separated from the underlying heavier water by a well-defined thermocline. This has motivated the use of the numerically efficient layer model presented below, as an alternative to the three-dimensional PE model. Let the depth of the mixed layer be h(x, y, t) (x, y, t) (x, y, t), where z is the position of the free surface, and z is the position of the thermocline. Any three-dimensional quantity of the upper layer,, can be written as the sum of the vertical average over the layer, (x, y, t) (x, y, z, t) dz, (5) h and the deviation from this, (with 0), (x, y, z, t) (x, y, t) (x, y, z, t). (6) 0 The integral # ( ) dz is replaced by # h ( ) dz in any average involving density anomalies under the Boussinesq approximation. Averaging the governing equations () (3) over the upper layer yields dh h u 0 (7) dt d (hu) (8) dt h du f k u p (huu), (9) dt 0 h where d/dt /t u is the material derivative with respect to the average motion. Density has been replaced by the relative density anomaly, ( 0 )/ 0, which can be regarded as a proxy for temperature. To simplify the discussion, we assume constant atmospheric pressure and a motionless layer below the thermocline with constant density 0. This is the so-called ½-layer formulation. The hydrostatic balance (4) then implies the horizontal pressure gradient force, p g(h) gz p, (0) 0 0 within the active layer. The average pressure force is p g(h) gh p. () The additional pressure, p 0 g # z dz, is caused by the vertical density structure. If the heating, [ (h u )]/h, and the forcing, p/ 0 [ (h uu)]/h, from the vertical structures on the mean quantities in Eqs. (8) and (9) are such that they may be represented by known functions of (x, y, t) and (h,, u) and their derivatives, say Q and F, the two-dimensional variable-temperature layer model due to Lavoie (97) follows:

3 OCTOBER 00 ELDEVIK 97 dh h u 0 () dt d Q (3) dt du f k u g(h) gh F. (4) dt When external heating or forcing is present, their integrated effects are included in Q and F. A source term can be introduced in the continuity equation () to account for compression or expansion of the layer. A simplified version of this model, to be introduced below [Eqs. (5) (7)], will be used for the comparisons with the PE model. c. The removal of thermal wind Horizontal density gradients imply vertically sheared flow through the second term of the pressure gradient force (0), that is, in the geostrophic approximation through thermal wind. Vertically sheared flow that is not exactly geostrophically balanced will, in general, introduce vertical structure to a horizontal stratification. This means that, unless mixing is included in the PE formulation () (4) to limit vertical inhomogeneities, the assumption that Q and F adequately represent the unresolved motion of Eqs. () (4) may be expected to be poor. Consider the following case: An initial flow with u and independent of z within the active layer is assumed. If the horizontal pressure gradient force (0) is replaced by its average () in the PE description () (4), the solution will remain vertically homogeneous as it evolves. In this case, and u u, and p, Q, and F vanish. Thus the equations of motion () (4) give the variable-temperature layer formulation dh h u 0 (5) dt d 0 (6) dt du f k u g(h) gh, (7) dt hereafter referred to as the VTL model. This set of equations is commonly considered (e.g., Fukamachi et al. 995) to be the variable-temperature model corresponding to the nondiffusive, nondissipative primitive equations () (4), the ½-layer approach taken into account. These are the two model oceans to be compared. Consistency of Eq. (7) with the underlying PE model requires the addition of the term g(z h/) (8) to the right-hand side of Eq. (3). Following McCreary and Kundu (988), this can be interpreted as a vertical mixing of horizontal momentum. It is an ad hoc argument, as pointed out by Ripa (999), but the introduction of is convenient for the comparison of the two models. The mixing may be written as the sum th bt, where th gz (9) counterbalances thermal wind, and bt gh (0) is a barotropic part that assures 0, and therefore the VTL to be conservative. The causality between horizontal stratification and vertical shear is a key element in understanding baroclinic flows and instabilities (e.g., Pedlosky 987, p. 503). As the homogeneity of each water column in the VTL is assured by the added mixing, thermal wind is lost. This fundamental difference between the two conservative model oceans at hand, the PE () (4) and the VTL (5) (7), has been acknowledged since the latter formulation was first suggested by Lavoie (97). Still both models are used to describe baroclinic instabilities in the upper ocean; see, for example, Fukamachi et al. (995). 3. The oceanic weather Baroclinic dynamics is known to generate meso- and submesoscale eddies in the ocean, oceanic weather that is. Simple manipulations of the governing equations suggest that the VTL model lacks some of the weather s key features. a. Vorticity equations and frontal instability The vorticity equation following from the PE () (4) is D ( f k) 3 u 3 B, () Dt where (,, ) 3 u and B g k are the vorticity and the baroclinicity vector, respectively, as the Boussinesq approximation and hydrostatic balance have been assumed. The baroclinicity generates horizontal vorticity only. The VTL model (5) (7) implies the equation d ( f ) u g(hx y hy x) () dt for the vorticity x u y. Only vertical vorticity is present since u is z independent. Instability caused by the vorticity generated through the rightmost term in Eq. () has been termed frontal instability by McCreary et al. (99) because of its dependence on the horizontal density gradient. This definition is used in the remaining part of the paper. Frontal instability has been shown to be a major component in the generation of meso- and submesoscale eddies in the ½-

4 98 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 3 layer variable-temperature layer models of McCreary et al. (99), Fukamachi et al. (995), Røed (996), Shi and Røed (999), and Røed and Shi (999). In all the papers but the first, frontal instability is described to be a type of baroclinic instability as it is related to the horizontal density gradient. Below, we show that this relation can be expressed through the mixing. As shown in the previous section, the simplified ½layer PE model, with thermal wind removed by the mixing (8), is the VTL model. The vorticity equation (), with the curl of added to its right-hand side, D k ( f )k u B 3, (3) Dt is then the the same as Eq. (). Using the expressions (9) and (0), it is readily found that the two components of 3 are B (4) 3 th bt g(hx y hy x)k. (5) From a basic point of view, that is, B as the source of baroclinic instabilities, the VTL may thus be argued to have none. The part of the mixing responsible for removing thermal wind also cancels the effect of the baroclinicity vector [Eq. (4)]. This could be expected as the vertical shear of thermal wind is prescribed by B. Still, frontal instability can be described as baroclinic in the sense that it allows for the conversion of energy associated with the VTL density structure to eddy kinetic energy (cf. Fukamachi et al. 995). Equation (5) shows that frontal instability can be interpreted to take place through the implicit mixing. b. Conservation of potential vorticity and cyclonic preference Going back to the classical papers of Charney (947) and Eady (949), frontogenesis is understood to take place through baroclinic instability. Their linear stability analyses, performed within a quasigeostrophic (QG) framework, predict instability to take place on a scale consistent with the synoptic weather. Quasigeostrophy requires that the characteristic timescale of the dynamics is much larger than f, or equivalently that the internal radius of deformation is much larger than the inertial length scale. It is well known that the strong horizontal velocity shear associated with fronts is predominantly cyclonic. This preference is not present in the QG formulation because of the restrictions mentioned above. The ageostrophic asymmetry between cyclonic and anticyclonic shear is often explained using the conservation of the Ertel potential vorticity (PV) DQ 0, (6) Dt where Q ( f k) 3. (7) 0 The conservation applies to flow governed by Eqs. () (4). Assume in the following that the horizontal vector product is nonnegative. This is not a severe restriction. If A is the ageostrophic vorticity, then g /(0 f ) A from thermal wind. A negative vector product therefore requires the flow not only to be ageostrophic, but to the extreme extent that the ageostrophic horizontal vorticity antiparallel to the geostrophic is larger than the geostrophic. An example is flow close to cyclostrophy where distinguishing between cyclones and anticyclones is void of meaning (cf. Cushman-Roisin 994, p. 5). Prior to any frontogenesis, Q ( f/ 0 )/z 0is a good approximation. It can therefore be assumed that the (conserved) PV is nonnegative as well. From Eq. (7) it then follows, assuming static stability, f ( Q) 0, (8) z which gives a lower bound ( f ), but no upper bound for the vorticity. Large horizontal density gradients are thus expected to produce large cyclonic shears. This is experienced in our everyday lives (particularly in Bergen): Cyclonic lows dominate the baroclinic flow that constitutes the synoptic weather. See also Gill (98, p. 575) for a two-dimensional argument similar to the above. As shown by Eldevik and Dysthe (00), the socalled spiral eddies, submesoscale cyclones observed at the ocean surface, may possibly be described as oceanic bad weather. A dominance of cyclones is also found by Wang (993) and Samelson and Chapman (995) in their numerical studies of longer frontal waves. All three papers are based on PE ocean models. The atmospheric literature is of course rich in papers demonstrating cyclogenesis, with Garnier et al. (998) as a recent example. 0 c. No conservation of potential vorticity and no cyclonic preference? It does not seem possible to find a conserved PV for the VTL model (cf. Ripa 993, 995). Potential vorticity as defined by (7) is identically zero in a vertically homogeneous layer. Thus no vorticity characteristics can be deduced from its trivial conservation. Equation () implies the evolution dq g(hx y hy x) (9) dt h for the PV conserved in a layer of constant density, f q, (30) h

5 OCTOBER 00 ELDEVIK 99 which is a correct approximation of the PE evolution of ( f )/h to the (zeroth) vertical order of the VTL (Ripa 993, 995). Using Eq. (5), this PV-like equation can be rewritten dq ( bt) k. (3) dt h Conservation of q fails because of the very same curl that characterizes the VTL frontal instability. A preference for cyclones is not evident. This might have been expected from the previous discussion. The cyclone/ anticyclone asymmetry inferred was based on the conservation of the Ertel PV and a nonnegligible thermal wind. Neither applies to a VTL. In the numerical VTL experiment described in the next section, cyclones and anticyclones are just as intense in the evolving flow. Small-scale eddies generated in similar experiments, for example, Shi and Røed (999), also display this lack of asymmetry. 4. The numerical experiments Two idealized, numerical experiments were set up to compare the oceanic weather generated in the two model oceans, () (4) and (5) (7). The two models were perturbed from equivalent initial conditions. The results are displayed and compared in light of the previous discussion. a. The initial configuration The initial PE configuration, taken from Fukamachi et al. (995), is an upwelling stratification with a rather weak horizontal density gradient and a corresponding geostrophic current jet: [ ] y fu (y, z) Y(y) dy gh 0 z H/ H (3) z H u(y, z) UY(y) H (33) for z H, where [ ] y cos, y L Y(y) L (34) 0, y L gives the cross-flow variation. In the above, U is the maximum velocity found at the surface centerline y 0, H is the depth of the active layer, L is the half-width of the jet, is the density difference over the layer, and is the density at the center of the layer, (y, z) (0, H/). Numerical values are displayed in Table. For z H, the fluid is assumed motionless and with TABLE. U (m s ) H (m) The dimensional parameters of the initial flow. L (km) (kg m 3 ) ( t ) 0 ( t ) f (s ) constant density 0. The current jet is symmetric about y 0 with equal amounts of cyclonic and anticyclonic shear. This geostrophic flow is not intended to represent any specific location but should be representative for a rather wide, midlatitude, buoyant geostrophic current. Eldevik and Dysthe (00) used the same initial conditions in their numerical reference experiment. When the flow was perturbed, frontogenesis through ageostrophic baroclinic instabilities was the result. Strictly speaking, there was also a barotropic component present. Its contribution was found to be negligible with regard to the classification of the instability. This is also the case for the barotropic component in the linear stability analysis of Fukamachi et al. (995) referred to below. As the frontal wave evolved, it wound into cyclonic spiral eddies. The vertical averages over the layer, [ ] y fu (y) Y(y) dy (35) gh 0 u(y) UY(y), (36) constitutes the initial geostrophic flow in the VTL model (5) (7). An interesting point is that geostrophic flows as described by the PE model will in general not average to geostrophic flows in the VTL model. Consistency requires the additional pressure gradient force in Eq. (), p, to be negligible. The initial stratification and jet are shown in Fig. for the two models. Fukamachi et al. (995) studied the linear instability of both models for the given initial flow. For the values in Table, the evolution governed by the PE predicts the wavelength of the most unstable wave growing along the jet to be about 0 km. In the absence of friction, 0 gives maximum growth for the VTL. Agreement with regard to the preferred wavelength is assured by the addition of harmonic diffusion and dissipation, with an eddy coefficient O(0 m s ), to the perturbed VTL equations. This amount of friction was found to have essentially no impact on the linear PE flow. The growth rate of the PE wave was estimated to 0.5 f, about twice the growth found for the VTL wave (diffusion/dissipation included). In spite of the aforementioned differences, using additional energy analyses Fukamachi et al. (995) found that both models describe ageostrophic baroclinic instability. The difference in growth rates between the two regimes were understood to be resulting from the different vertical structures of the waves rather than a fundamental change in dynamics.

6 90 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 3 b. The numerical models The primitive equations, -coordinate numerical model of Berntsen et al. (996) was used for the experiment solving the equations of motion () (4), evolving from the perturbed initial conditions (3) (33). The three-dimensional results, and details on the implementation, are given in Eldevik and Dysthe (00). The fluid below the thermocline was practically motionless during the experiment. Thus an active layer is well defined. The average quantities h,, and u are calculated to make a comparison with the solutions found by the VTL model. The VTL equations (5) (7), perturbed from the initial conditions (35) (36), were solved by the fourthorder accurate reduced-gravity model described by Sanderson and Brassington (998). The inclusion of horizontal density gradients in their homogeneous layer model is numerically trivial. As the CFL criterion allows for explicit time stepping for the temporal resolution (t) chosen herein, this was used instead of the original implicit scheme. The computational domain for both models is a channel periodic, with period 0.8 km in the x direction, and with walls at y 50 km. For the PE solver, there is a flat bottom at z H 00 m. Thus the coordinate description is basically Cartesian. The values used [those of Eldevik and Dysthe (00)] are such that coincide with the most unstable wavelength predicted by Fukamachi et al. (995) and such that the solid boundaries have no influence on the evolving dynamics. Harmonic diffusion and dissipation, 0 m s, was prescribed to the VTL solver in agreement with the above discussion. The same temporal and uniform horizontal resolution was used for both experiments: t 60 s and 3 cells to the length of the channel. The threedimensional model had 4 uniformly spaced layers. For the chosen resolution, the VTL solver required 80 times less CPU time than the PE solver. FIG.. The initial geostrophic solution in (a) the PE and (b) the VTL ocean models. Solid lines are isopycnals, and broken lines are isolines of velocity. Units are m s and t, respectively. c. The bad weather generated A useful quantity in describing and comparing the instabilities taking place in the two models is the eddy kinetic energy (EKE). Let be a VTL quantity and let the two along-channel averages, # 0 dx/ and h/h, be defined. Then, following Røed (997), a measure of the EKE in the computational domain is 0 EKE h(u* * ) dy, (37) where u* u ũ is the eddy velocity. The evolution of the EKE in the two models, perturbed from the given geostrophic flows, is seen in Fig.. One day is denoted by the unit d. Time t 0 refers to the onset of exponential growth in the respective models. The two curves are quite similar, the : ratio of the timescales taken into account. Thus comparisons between the model evolutions will be done using that ratio of scales. Both curves describe linear instability in agreement with Fukamachi et al. (995) for the first few days. The PE wave has the growth rate d 0.4 f, twice that of the VTL wave. Then nonlinearity sets in to weaken the growth. Both waves reach their energetic peak in the displayed time interval. Eldevik and Dysthe (00) performed a variety of numerical PE experiments, including the one described here. The incipient linear growths were all consistent with the PE findings of Fukamachi et al. (995) who explicitly prescribe the lower layer as motionless. This agreement supports the previous statement that the lower layer in the present PE experiment can be considered passive. Further, Eldevik and Dysthe (00) find that

7 OCTOBER 00 ELDEVIK 9 different (Figs. 4d f), as expected from the discussion of the previous section. Staggered rows of cyclonic and anticyclonic vortex pairs analogous to the Karman vortex street are the result. The pairs are hereafter referred to as dipoles. There is no asymmetry. A possible explanation is provided by the following argument: The vorticity generation d/dt, given by Eq. (), can be separated into the contribution from convergence, FIG.. The EKE (ln scale) of the instability in the PE (broken line) and the VTL model (broken-dotted line). Note that the timescale of the former (given at the base) is one-half that of the latter (given at the top). The solid line is the tangent with slope.5 d and.5 d for the two timescales. their experiments display dynamics akin to the surfacetrapped ageostrophic baroclinic instability described by Barth (994) in his PE numerical analysis of a continuously stratified ocean. Barth s model ocean also has a deeper, Eady-like, baroclinic response. This deep mode is excluded from our cyclic channel as it has a wavelength of roughly 00 km. The onset of instability and the cyclone generated in the PE experiment are shown in Fig. 3 for the relative vorticity x u y. Note that this quantity is not identical to the average of the three-dimensional vertical vorticity. It is used instead of for a straightforward comparison with the vorticity produced in the VTL model. Still, the expected cyclone/anticyclone asymmetry of ageostrophic three-dimensional flow should be (and is indeed) present. The development of x u y in the VTL experiment for the equivalent period of time is displayed in Fig. 4. The two experiments are very similar during linear instability (cf. Figs. 3a c, 4a c), but the emerging front in Fig. 3c does show a cyclonic preference not present in Fig. 4c. Clear differences are observed as the waves wind up to produce oceanic weather in Figs. 3d and 4d. Then the PE simulation produces a distinct submesoscale cyclone with a superf core. The asymmetry between cyclonic and anticyclonic shears predicted from the conservation of potential vorticity, is pronounced. Eldevik and Dysthe (00) give a detailed diagnosis of this essentially conservative vorticity dynamics. Figures 3a f qualitatively reproduce the surface vorticity displayed in Figs. 3a f of Eldevik and Dysthe (00), but their snapshots are more dramatic as the three-dimensional flow within the active layer is intensified toward the surface. The nonlinear vortex dynamics of the VTL is quite def conv ( f ) u, (38) and the contribution from frontal instability, def FI ( ) k. (39) bt It is easily shown that the circulation in the computational domain is constant in time, and with our initial conditions (x, y, t) dx dy (x, y, 0) dx dy 0. (40) Any (anti) cyclonic dominance in the evolving flow must be caused by convergence compressing the (anti) cyclonic region. Vorticity generation caused by FI alone should amplify the magnitude of vorticity in both regions equally. An estimate of the two effects can be made on the basis of the linear instability. The alongflow structure and growth for any eddy quantity are then given by exp (ikx t), where k / is the wavenumber. Writing the convergence as u h dh/dt, the linear ap- proximations are h conv f (4) H FI gk h, (4) where h is a typical depth anomaly growing up over the time and is the initial density difference in the y direction over the distance k. For the given k, 0 4 (cf. Fig. b). This implies the ratio conv f 0.. (43) FI ghk The amplification of vorticity, at least in the early stages, is therefore expected to be due to frontal instability, and no preference for either vorticity can be expected. This is confirmed in Fig. 5, showing the generation of cyclonic and anticyclonic vorticity in the experiment, averaged over areas of respective vorticity from the two contributions. The effect of convergence is negligible in the displayed period of time. This provides an explanation for the symmetric evolution of vorticity causing the characteristic dipole structures in the VTL experiment (cf. Figs. 4d f). The dissipation added to the VTL was ignored in the above discussion. The linear

8 9 JOURNAL OF PHYSICAL OCEANOGRAPHY FIG. 3. The onset of instability and the generation of eddies as seen in the relative vorticity y x u y for the PE case. The gray scale is related to the planetary vorticity f. Note that its range varies. The spatial resolution of the figures is that of the experiment. The periodic domain is shown twice. VOLUME 3

9 OCTOBER 00 ELDEVIK 93 FIG. 4. As in Fig. 3 but for the VTL case. approximation n¹ v ; n (U/)/L 3 ø 0 4 f should justify that approach (cf. Fig. 5). The present VTL experiment did not produce velocity shears larger than f. The vorticity generated is substantial, but still notably weaker than that of the PE experiment. The reason for this is not clear. One can only speculate that the prescribed eddy diffusion and dissipation cause the VTL instability to peak at a less energetic level (see Fig. ). An additional experiment was performed where the initial maximum speed was

10 94 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 3 FIG. 5. The production d/dt in the VTL experiment in units of 0 3 f. Solid (broken) lines are the averages over the cyclonic (anticyclonic) region. The thick lines are the vorticity generated by frontal instability, and the thin lines the vorticity generated by convergence. The anticyclonic values have been multiplied by. tripled, 0.5U 0.45 m s. Figure 6 is a snapshot from that experiment. The vorticity is still characterized by dipole structure (and no asymmetry), and it clearly illustrates that anticyclones with f are not inhibited in the VTL model. A complimentary comparison is the ongoing survey of Ådlandsvik et al. (00). They use the PE model Regional Ocean Model System (ROMS: documented online at index.php) and MICOM, with a VTL on top, to model the instability and subduction at the Arctic Front in the Nordic seas. In twin experiments, the frontal dynamics displayed by the two model oceans differs in consistence with the above. The discussion of the present experiments has been limited to the vorticity for the purpose of simply illustrating the arguments put forward in the previous section. The PE (VTL) model displayed cyclonic (dipole) structures in h and as well. FIG. 6. The vorticity at t d for the experiment with a fast jet. d. On VTL validity and improvement The VTL model may be interpreted as the lowestorder polynomial approximation to the vertical structure of the PE model. Ripa (999) states that a condition for its validity with respect to baroclinic instability is Re k Ri. The quantities R i and R e, respectively, are the two internal radii of deformation associated with the stratification of the upper layer and the density difference across the thermocline. They are 4 and 0 km in the present experiment. Thus Ripa s criterion is met for the wavelength of the original instability (and the diameter of the frontal eddies), but it is not met for the finer scales of the developed flow. The latter may cause inconsistencies. However, the discussion herein suggests a more generic explanation for the discrepancies displayed by the VTL frontal dynamics, that is, the symmetric nature of its frontal instability. It should be noted that approximations of the next polynomial order (i.e., models that partially include thermal wind in the VTL) do exist. Young (994) and Ripa (999) have developed subinertial, quasigeostrophic-like versions. This restriction on timescales is not suitable in the present context. A highly relevant model, without the subinertial assumption, is that of Ripa (995). To our knowledge, it is yet to be implemented as a nonlinear numerical ocean model. The main problem is a seeming lack of a continuity-like equation for the vertical structure quantities. Hence implementing a numerical advection scheme for these quantities is difficult (impossible?). 5. Concluding remarks Frontogenesis is known to have a preference for strong cyclonic shears due to ageostrophy. When a frontal wave winds up, intense cyclones and more diffuse anticyclones will be the result. This is characteristic of both the atmospheric and the oceanic weather. Several authors have found the variable-temperature layer (VTL) model (5) (7) to be suitable for describing oceanic frontal dynamics. The present investigation suggests that the VTL dynamics do not capture the above characteristics of the weather. The addition of the mixing (8) to the primitive equations (PE) model () (4) removes thermal wind to give the slablike motion of the active layer described by the VTL model. This mixing appears to dominate the VTL vorticity dynamics. For potential vorticity as defined by (30), conservation is broken during frontal instability, and no lower bound for the vorticity can be stated. This is illustrated by the numerical experiment of section 4, cf. Figs. 4, 5, and 6. Frontal instability is found to amplify

11 OCTOBER 00 ELDEVIK 95 both positive and negative shears equally, and the front eventually winds up to produce a dipole as opposed to the cyclone of the PE experiment (Fig. 3). Figure 6 shows that anticyclones with f are not inhibited in the VTL model. Acknowledgments. The author thanks J. Berntsen and B. G. Sanderson for helping to set up the numerical experiments, and G. Alendal, K. B. Dysthe, and the reviewers for their many helpful comments in the preparation of this manuscript. Part of this work was done while the author was visiting the School of Mathematics, University of New South Wales, Sydney, Australia, August 998 to January 999, at the kind invitation of M. A. Banner. The stay was funded by The Research Council of Norway, which also supported this work through a grant of computing time (Programme for Supercomputing) and funded the revision of the manuscript through the NOClim Project (3985/70). REFERENCES Ådlandsvik, B., X. B. Shi, W. P. Budgell, and L. P. Røed, 00: Modelling the Arctic front in the Nordic Seas, preliminary results from an idealized test case. Norwegian Ocean Climate Project Tech. Rep. No., Geophysical Institute, University of Bergen, Barth, J. A., 994: Short-wavelength instabilities on coastal jets and fronts. J. Geophys. Res., 99 (C8), Berntsen, J., M. D. Skogen, and T. O. Espelid, 996: Description of a sigma-coordinate ocean model. Institute of Marine Research, Bergen, Norway, Tech. Rep. Fisken og havet, 33 pp. Bleck, R., C. Rooth, D. Hu, and L. T. Smith, 99: Salinity-driven thermocline transients in a wind- and thermocline-forced isopycnic coordinate model of the North Atlantic. J. Phys. Oceanogr.,, Charney, J. G., 947: The dynamics of long waves in a baroclinic westerly current. J. Meteor., 4, Cushman-Roisin, B., 994: Introduction to Geophysical Fluid Dynamics. Prentice-Hall, 30 pp. Eady, E. T., 949: Long waves and cyclone waves. Tellus,, Eldevik, T., and K. B. Dysthe, 00: Spiral eddies. J. Phys. Oceanogr., 3, Fukamachi, Y., J. P. McCreary, and J. A. Proehl, 995: Instability of density fronts in layer and continuously stratified models. J. Geophys. Res., 00 (C), Garnier, E., O. Métais, and M. Lesieur, 998: Synoptic and frontalcyclone scale instabilities in baroclinic jet flows. J. Atmos. Sci., 55, Gill, A. E., 98: Atmosphere Ocean Dynamics. Academic Press, 66 pp. Lavoie, R. L., 97: A mesoscale numerical model of lake-effect storms. J. Atmos. Sci., 9, McCreary, J. P., and P. K. Kundu, 988: A numerical investigation of the Somali Current during the Southwest Monsoon. J. Mar. Res., 47, 5 58., Y. Fukamachi, and P. K. Kundu, 99: A numerical investigation of jets and eddies near an eastern ocean boundary. J. Geophys. Res., 96 (C), Munk, W., L. Armi, K. Fischer, and F. Zachariasen, 000: Spirals on the sea. Proc. Roy. Soc. London, 456A, Pedlosky, J., 987: Geophysical Fluid Dynamics. d ed., Springer- Verlag, 70 pp. Ripa, P., 993: Conservation-laws for primitive equations models with inhomogeneous layers. Geophys. Astrophys. Fluid Dyn., 70, 85., 995: On improving a one-layer ocean model with thermodynamics. J. Fluid Mech., 303, 69 0., 999: On the validity of layered models of ocean dynamics and thermodynamics with reduced vertical resolution. Dyn. Atmos. Oceans, 9, 40. Røed, L. P., 996: Modelling mesoscale features in the ocean. Waves and Nonlinear Processes in Hydrodynamics, J. Grue, B. Gjevik, and J. E. Weber, Eds., Kluwer Academic, , 997: Energy diagnostics in a ½-layer, nonisopycnic model. J. Phys. Oceanogr., 7, , and X. B. Shi, 999: A numerical study of the dynamics and energetics of cool filaments, jets and eddies off the Iberian Peninsula. J. Geophys. Res., 04 (C), Samelson, R. M., and D. C. Chapman, 995: Evolution of the instability of a mixed-layer front. J. Geophys. Res., 00 (C4), Sanderson, B., and G. Brassington, 998: Accuracy in the context of a control-volume model. Atmos. Ocean, 36, Shi, X. B., and L. P. Røed, 999: Frontal instabilities in a two-layer, primitive equation ocean model. J. Phys. Oceanogr., 9, Wang, D.-P., 993: Model of frontogenesis: Subduction and upwelling. J. Mar. Res., 5, Young, W. R., 994: The subinertial mixed layer approximation. J. Phys. Oceanogr., 4, 8 86., and L. Chen, 995: Baroclinic instability and thermohaline alignment in the mixed layer. J. Phys. Oceanogr., 5,