On the relevance of diapycnal mixing for the stability of frontal meanders *

The existence of an inverse relation between diapycnal (across-isopycnal) and horizontal oceanic diffusion is coherent with our physical intuition that natural turbulence in stratified fluids is anisotropic, i.e. when inhibited in one direction it will tend to align itself along a perpendicular plane. This physical argument was long ago used by very intuitive researchers such as Rossby (1936), Parr (1936) and Montgomery (1938), and has since been invoked in a number of works (e.g. Turner, 1973; Armi, 1979; Gregg, 1987). It is based not only on mass conservation (an ellipsoid has to become very elongated to contain the same mass as a sphere) but also on energetic arguments and observational data (e.g. Turner, 1973). The intrinsic relation between horizontal and diapycnal motions is well illustrated in a number of oceanographic examples, such as in the mechanism of double-diffusion and in quasi-geostrophic theory. In the former example the great horizontal coherence of relatively thin layers that have their origin in double diffusion is very striking (Schmitt, 1994). An important feature of the latter example is that horizontal geostrophic motions cannot maintain the geostrophic thermal wind balance. To maintain this SCI. MAR., 65 (Suppl. 1): 259-267 SCIENTIA MARINA 2001

The existence of an inverse relation between diapycnal (across-isopycnal) and horizontal oceanic d i i - sion is coherent with our physical intuition that natural turbulence in stratified fluids is anisotropic, i.e. whpg $Jjhi?pd iq ~n p Air=t;,ci, if s.~g tp,fid fe &$gr.itself alonp a perpendicular plane.This physical argument was long ago used by very intuitive researchers such as Rossby (1936), Parr (1936) and Montgomery (1938), and has since been invoked in a number of works (e.g.Tumer, 1973;Armi, 1979;Gregg, 1987).*Received June 20,2000. Accepted January 10,2001 It is based not only on mass conservation (an eiiipsoid has to become very elongated to contain the sarne mass as a sphere) but also on energetic arpnents and observational data (e.g.Turner, 1973).
The intrinsic relation between horizontal and diapycnal motions is well illustrated in a number of ~wwmp-yyhic ~X U I ~! P U , s x h as ii! &e m ~, ~h m i s m of double-difíusion and in quasi-geostrophic theory.In the former example the great horizontal coherente of relatively thin layers that have their oripin in double diffusion is very striking (Schmitt, 1994).An important feature of the latter example is that honzontal geost~ophic motions cannot maintain the geostrophic thermal wind balance.To maintain this 1 ., . .' .

L . & --_ L ! ---
~aiii~i~t.: iL 1s Iiecessary w riave ageosrropriic rnoíions, which include both vertical motions and diapycnal rnixing (Keyser and Shapiro, 1986;Vélez-Belchi and Tintore, 2001).One aspect to keep in mind hereafter is the difference between vertical and diapycnal motions.Because of the slope of isopycnais, vertical motions do not necessady imply changes in density and may involve both epipycnal (along-isopycnal) and diapycnal motions.It sould also be noted that, stnctly speaking, we should be concerned with dianeutral rather than diapycnai motions, since water parcels with constant composition moving adiabatically follow neutral surfaces rather than isopycnals (McDougall, 1987).The small difference between neutral surfaces and isopycnals in the upper thermocline of intense jets, however, allows us to use the isopycnic terminology hereafter (Pelegn and Csanady, 1994).Despite the above considerations, most instability anaiyses are systematically divided into horizontal (barotropic and baroclinic) and diapycnal analysis (mainly shear-induced and double-diffusion ), with no or little interaction between them.In particular, the unusual character of horizontal instability phenomena in western boundary currents, manifested by intense meanders, is the probable reason why the possibility of diapycnal mixing within these currents has so far received little attention (however, see Pelegrí and Csanady, 199 1, 1994;Rodríguez-Santana et al., 1999).
Furthermore, it may be argued that diapycnal instabiiities (of ikivin-~eimhoitz type) have temporal and spatial scales quite different from those of horizontal instabilities.The horizontal size and growing time of each individual Kelvin-Helmholtz set of disturbances, of the order of one hundred meters and just a few rninutes in the themodine, is certainly different from hose of horizontal instabilities in intense geophysical flows, of the order of one day and one hundred kilometers.The important point, however, is that Kelvin-Helmholtz type instabilities will only appear after frontogenesis has caused sufficient compression of the density field.This means that in oceanic (and atmospheric) jets there will be one single spatial and temporal relevant scale, the one associated with the intensification of t_S--mh&&g front_-system (tilnkg ~e f l k ~ pha-r of the horizontal instabilities, or meanderc, that develop in the flow (Keyser and Shapiro, 1986).This feature has been confirmed by recent analysis of Gulf Strearn data (Rodríguez-Santana et al., 1999; see also Pelegn' and Csanady, 1994).
Iñ .=Liswüik we exaiI~i,c 'he ~iiieiaC~üi, L & Y wecl ' diapycnal mixing and horizontal instability within meanders in oceanic jets (or waves in atrnophenc jets) in a very simple manner.Here we are not interested in obtaining the exact condition for instability, but in exarnining the relative size of the contribution due to diapycnal mixing.An important aspect of this work is how to estimate the intensity of rnixing.To this end, we use a two-dimensional isopycnic ocean model with the along-front velocity i n geostrophic balance, but allowing the existence of both frontogenesis and diapycnal mixing.The intensity of diapycnal mixing is calculated as the divergence of the vertical density Reynolds flux, which is determined from the vertical stratification and the vertical diffusion coefficient, the latter parameterised in terms of the ,pdient Richardson number.Hence, as the front evolves, the stratification and the gradient Richardson number change, leadin; to an intensfication of diapycnal mixing.To simulate the frontogenetical process that takes place between different phases of Gulf Strearn meanders, we use a deformation field that lasts 28 hours, in accordance with observations by Rodríguez-Santana et al. (1999).It should be clear that the same type of analysis could also be applicable to waves in atmosphenc jets, with adiabatic heating and clear air turbulence replacing diapycnal mixing.

HORIZONTAL INSTABILITY IN MEANDERS
r nere are severai reviews Uiar examine eirher Uie theones of instability in stratified shear fiow (Turner, 1973;Gregg, 1987) or the theories of barotropic and baroclinic instability (Pedlosky, 1979;Gill, 1982).The list of studies on the horizontal instability of meanders or vortices js also quite.impressive.
More recent studies on barotropic instability have been made by Leibovich and Stewmon (19831, Eckhoff (1984), Gent and McWilliams (1986) and Paldor (1999) for a review see Hopfinger and van Heijst (1993).In these papers the necessary condition for instability of barotropic vortices is that the radid &fivafive of pt-n!&d vo_rti&y mu-t c.hage sign somewhere within the fiuid domain.This result is consistent with results for parallel geophysical flows, which indicate that the necessary condition is the existence of a latitudinal inflection point of potential vorticity (e-g.Pedlosky, 1979).
While some of these studies indeed examine the interaction between barotropic and baroclinic insta-

LI
,r 26 ,r ;lo (6) bility, it seems surprising that little has been said about the effect of diapycnal mixing on radial insta- The radial component of the momentum equability.In this section we will investigate how large tion describes the behaviour of the azimuthal (tandiapycnal mixing has to be in order to become sig-gential) velocity, while the tangential component is nificant in the radial instability analysis.The coherent with the basic hypothesis of flow along 4 approach used here has severai irnportant simplifica-

Analysis in cylindricai coordinates
For our anaiysis we use the mass and momentum isopycnic equations in cylindrical coordinates.These equations apply equaily well for a vortex or a meander, the latter case being a portion of a vortex that is defined by its local curvature.We Taking the total time derivative of Equation ( 7), using Equation ( 8) and making use of the f-plane approximation leads to ---U, d2%- At this point we note that the last three terms in this equation depend on the perturbed variables and that, as we will see below, they are not modified by the presence of diapycnal rnixing.Since we are only interested in exarnining the relative contribution of diapycnal mixing, we will ignore them, i-e.we will examine the following equation:

Contribution due to diapycnal mimng
In the presence of diapycnal mixing the dimensional isopycnic equations in the tangential and radial directions become (e.g.Pelep' and Csanady, 1994): The density tendency is defined as wp=DplDt.l t is related to the diapycnal velocity, with units of distance over time, by w,=Jwpp, where J = M p is called diapycnal mixing greatly exceeds al1 other contnbutions, so that Equation ( 13) is approxirnated by Under this approximation the radial velocity of a water parcel will decrease in time (i.e.instability will be controlled by diapycnal mixing) whe'n the term on the nght-hand side of Equation ( 14) is negative.Except in steep anticyclonic meanders this term wiII be negative if wo<O, i.e. when the water parcel mixes towards the sea surface.
At this point we should discuss the meaning of the quantity within the fírst parenthesis of Equation (13).To do so let us consider the dimensional isopycnic vorticity equation in the presence of diapycnal terms: This equztien m2y iy t)~eli_tt~n h ~o~-dimensienU form as follows (dropping primes): where -10 obtain (iój we have used wp=dpidr-dpi& and awdar=ddt(dp/&-).Expanding the dependent variables in terms of the small parameter E and substitution in this last equation leads, to the lowest order: This result indicates that, to the lowest order, the absolute vonicity is given by l+v&+dwdar+ (w,lu,)(awddp) and it has to be conserved as a water parcel moves along the vortex.This suggests that if the motion is initially stable then the absolute vorticity cannot change, no matter how steep the vortex becomes, which is only possible if the t e m containing the density tendency changes accordingly.

MODELING DlAPYCNAL MIXING IN MEANDERS
In order to assess the importante of diapycnal mixing in the Gulf Stream, we make use of Rodríguez-Santana's (1997) model, ailowuig frontogenesis to take place during time scdes characteristic of Gulf Strearn meanders (Rodríguez-Santana e? al., 1999).For this purpose we use the original model formulation in a Cartesian coordinate system, leaving its analysis in cylindrical coordinates for a postenor study.

Model equations and methodology
The two-dimensiond model in isopycnic coordic ~n r i r ~ 2 ch-nb alnngrcelm g e ~~~~p h i ~ r-----balance, but allowing diapycnal mWng to regulate the separation between isopycnals through the mass conservation equation.The model is further simplified by assuming that both the Jacobian, J, and the baroclinic velocity, Yp are independent of the alongstteam coordinate.The uiitial frontal de@density field ( The component v, of the baroclinic velocity field, parallel to the front, is in geostrophic balance: where 4 is the Montgomery potential defined in the last section.The density tendency, wp=Dp/Dt, satisfies the mass conservation equation in isopycnic coordinates: with j E pJ.In a reference system moving with the deformation field (x ' , y, p), where x'=xe", Equation ( 20) reduces to: We integrate Equation ( 21

Model results
In order to illustrate the frontogenetical process we have produced Figure 1, which shows the evolution of selected isopycnals at three different times, t uidyycriai convergence vaiues of abvuí -i.ó X lo6 S".At t = 25 hours, the frontal system has compressed even further (Fig. 1c) and causes very high diapycnal shear.This creates a rather large central region with subcritical Ri values and very high wp values, of about 5 x lo-' kg m" S-' (Figs. 2c and 3c respectively).The concurrent diapycnal convergente in the central zone reaches maximum values of about -lo-' S-'.
To obtain a better perception of the last phases in the compression process, we present the evolurion of the Jacobian, the diapycnal shear, the gradient Richardson number, and the densiiy tendency at x = O km, from t = 25 hours to t = 27.2 hours (Fig. 4).We rnay appreciate how stratification increases or decreases at difFerent vertical positions, corresponding to the shrinking or stretching of the isopycnal layers.Such shrinking or stretching is caused by rhe distribution of diapycnal divergente or convergence, e.g. the isopycnals in the central region thicken because of diapycnal m a s convergence.The diapycnal convergenceJdivergence is driven by ver- In zones with high diapycnal convergente and divergente the vertical profile of the density is modified.In Figure 5 we present the evolution of the vertical density profile from t = 25 hours to t = 27.2 hours.Figure 5b clearly illustrates the fomation of a low stratified region at about 1026.8 kg m" and highly stratified regions at 1026.65 kg m-3 and 1026.95kg resemblinz a staircase type stnicture in the density profile (Peles' and Sangrá, 1998).

DISCUSSION
Une resuit derived irom the first part of this paper is that the absolute vorticity of a water pace1 is, to lowest order, conserved as it moves along a vortex.

This imposes ( ~$ ~) ( a v l a p ) + a ~& + ~/ r + f = f , ,
with f o be& the planetary vorticity of the straight unpermrbed flow.If the changes in f are small, such as when the Gulf Stream flows eastward, we obtain that diapycnai mixing will become relevant when the absolute value of (wfi)(¿h@p) is of the same order as avlaptvlr.This result is coherent with the one aerived following Equation ( 13), on the relative size that diapycnal mixing has to have to become important in controlling the growth of radial velocities in vortices.
Satellite pictures in Lee and Atkinson (1983) allnw -.ES ?n e s t i m t ~ SI EI P nf the ahcm nnmkrs f a meanders near CharIeston Bump.For anticyclonic meanders, for exarnple, using a tangential velocity of 1.5 m/s and a radius of 35 km, we obtain that both dv/dr and -f/2 are about -4 x lQ5 S-'.The above condition, however, applies for epipycnal gradients The solid, dotted and dashed lines correspond to r = 25,27.1 and 27.2 pectivel y.
that cannot be directly estimated from usual density and velocity cross-sections.Using Pelegn and Csanady's (1994) isopycnic cross-sections we rnay estimate the epipycnal velocity gradient to be probably much larger (smaller in absolute value), about -10" S-'.Hence, a gross condition for diapycnai mixing to become important in the developrnent of horizontal instabilities is that (wdu)(av/dp) has to be of the sarne order as the total vorticity, Le. of the order lo4 S-'.
A second result from the first p m of the paper comes after Equation ( 14), and may be used to set a limit on the characteristic time scale over which strong diapycnal mixing must act in order for the radial velocity to decrease.This time scale is given by or, using the above result, by In western boundary currents this time scale is of the order of lo4 S, hours to days, and coincides with the characteristic time for the development of meanden in oceanic jets.
In the second part of the paper we have proposed a very simple model for a frontal system characteristic of intense ocean currents, such as those associated with the Gulf Stream.Using this model we have simulated a frontogenetical process that develops at a rate, and during a rime, similar to hose that characterise Gulf Stream meanders (Rodríguez-Santana, et al., 1999;Pelegrí and Csanady, 1994).In our simulation we find that the density tendency reaches values of rt 2 x 10" kg m-3 S-' and that diapycncri shear reaches a maximum value of 7 m4 kg-' S-'.Even if these figures are one order of magnitude too large, and using radial velocities ranging between 10-l of 10" m S-', this implies that (wfi)(av/ap) reaches values larger than the total vorticity, of the order 10A S".This supports the idea that the intensity of diapycnal mixing during frontogenesis may indeed be high enouph to control the development of barotropic instabilities.
One important limitation in our study is due to the severd approximations used in the instability analysis.Another limitation is that the stability condition we have obtained is characteristic of barotropic type instabilities, while diapycnal mixing results from increased diapycnal shear (change in alongstrearn velocity with density) in some phases of meanders, which is a baroclínic process.Nevertheless, we believe that our results indeed endorse the potential role played by diapycnal mixing in the inhibition or generation of horizontal instabilities in strong oceanic currents.This agrees with the simple idea uiat diapycnal mixing, if intense enough, will redistnbute momentum and vorticity in such a way as to significantly infiuence the development of horizontal instabilities.
contours.The O(&) equations are: tions and is aimed only at providing a tool for the identification of the relative irn?ortance of diapycnai duo - assume a basic flow model with streamlines flowing aiong Iines of constant Montgomery potential, 4 = plp + gz.-h t us neglect al1 frictional terms and temporarily also ignore diapycnal mixing.The horizontal momentum equations in acceleration form (which have already made use of the isentropic continuity equation) are: whprp f is C ~f i ~l i ~ pazmeter, zqd ~7 & ~d 2 & T I &e tangential and radial velocities respectively.Let us scale the variables as u=Uu', v=VvJ, t= (l/B(V/U)t', r=(V/nr', +V@' , and further let U ' = & be a smail quantity.Notice that in isopycnic coordinates, in the absence of diapycnal rnixing, the density p is constant.The non-dimensional equations become (dropping primes): Let us now expand al1 dependent variables in terms of the small parameter E, e.g.v = v o + ~, + 2 v 2 + O($).The O(1) equations are: dt" Jar&iznin.i! s t ~n & forJac&izn onf !he DIAPYCNAL AND HORIZONTAL INSTABILiTiB 261transfonnation from Cartesian to isopycnic coordinates.Following the above procedure, with the additional scaling of z = D z', p =p, p', and wp = ~f p , wp', leads to (dropping prirnes and with w, as the order zero density tendency):One possible interpretation of this last equation is that instability occurs when (1 +vdri~vd~r+(wdu,)~v~~p)(l+2vdr)<O.Another possible approach is to consider the case in which -- Fig. la), is &ven with a .externally imposed horizontai deformation fieid ( H o s h s , 1982), which results in the compression of the frontal systern (Fig. lb-c).The velocity field is the result of a deformation velocity field, 3' ,= (u, v, O) , and a baroclinic velocity field, Yd= (O, Y, w>.The components of the deformation velocity field are u,=-yx in the direction normai to the front and vd=yy in the direction parallel to the front, ybeing a constant.This definition produces a non-divergent deformation field: . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . .l . . . . . . . . . .-Depth distribution of the isopycnals between 1026.2 and 1027.7 kg m-', every 0.1 kg m-3, at times (a) O hours, (b) 19 hours, and (c) 25 hours.

FIG. 3 .
FIG. 3. -Density dismbution of w at times (a) O hours, (b) 19 hours and (c) 25 hours.Negative valuesPare represented with dashed lines s d p i f i v e v & ~ wifi snbd h ~.
2ó4 A. RODR~GUEZ-SANTANA er al. = 0, 19 and 25 hours.Figures 2 and 3 illustrate the concurrent gradient Richardson number, Ri, and densíty tendency, wp fields, respectively.The initial conditions correspond to rather low diapycnal shear, and have minimum Ri values of about 0.6 and maximum wp values of about 1.2 x kg m" S-'.At t = 19 hours, the frontal system is quite compressed (Fig. lb) and causes an increase in diapycnal shear near the central zone.This leads to low Ri values (Fig. 2b), subcritical (< 0.25) near x = O and p = 1026.7 kg m-j.This in tum produces relatively large density tendencies, of about 2 x lo-' kg m-' S-' , in the central subcritical zone (Fig. 3b), and concur- flux, which uitirnateiy depends both on the vertical stratífícation and the diapycnal shear.In the cena-al region, for exarnple, both the stratification and the diapycnal shear initially attain their maximum values, causing maximum positive and negative density tendencies around it and inducing ihe observed mass convergente.At r = 27.2 hours ihe density tendency attains positive and negative maxirna of absolute value +2 DIAPYCNAL AND HORIZOhTAL INSTABILITIES 265 (a) Density profile at x = O.(b) Detail of the area of interest.hours res xlO" kg m' : S".At this time the diapycnal shear reaches a maximum value of 7 m4 kg-' S-'.