Effect of Baroclinicity on Double-Diffusive Interleaving

SEPTEMBER 1997 MAY AND KELLEY 1997

BRIAN D. MAY AND DAN E. KELLEY Department of Oceanography, Dalhousie University, Halifax, Nova Scotia, Canada (Manuscript received 8 August 1996, in ®nal form 27 February 1997)

ABSTRACT Although ocean fronts are often baroclinic, existing models of double-diffusive interleaving have ignored such baroclinic effects as velocity shear and horizontal density gradients. To determine the importance of these effects, the authors have formulated a linear instability analysis applicable to baroclinic fronts. Two limiting cases are considered: one for fronts with strong vertical and/or horizontal shear, the other for fronts with weak shear. In both limits, double-diffusive interleaving can be enhanced or suppressed by baroclinicity. Interleaving motion is enhanced if isopycnals rise toward the fresh side of the front. Conversely, interleaving is suppressed if isopycnals slope downward across the front. A signi®cant result is that the salinity gradient along isopycnals is not a good indicator of interleaving strength. As an example, the model is applied to a Mediterranean salt lens. The effect of baroclinicity is signi®cant: the predicted growth rates are increased by 35%±90%. The large-scale velocity and hydrographic ®elds indicate that Meddy Sharon lies somewhere between the high- and low-shear limits. Nevertheless, the model predictions agree reasonably well with the observed interleaving characteristics.

1. Introduction models of viscous/diffusive interleaving may also be invalid because they lack double-diffusive effects. In Inversions are often observed in vertical pro®les of this paper we address the ®rst of these de®ciencies; we temperature and salinity obtained near ocean fronts. The develop a new model of double-diffusive interleaving inversions typically have vertical scales of 10 to 100 m. for fronts that are both thermohaline and baroclinic. In the presence of horizontal temperature and salinity First, we review the dynamics and existing theories gradients, they are usually attributed to cross-front ``in- of double-diffusive interleaving (section 2). Then, we terleaving'' motions. Theoretical and laboratory studies formulate an instability analysis of in®nitely wide, bar- suggest that interleaving can arise from either of two oclinic, thermohaline fronts (section 3). In section 4, we mechanisms: present a new criterion for interleaving and examine its R In barotropic thermohaline fronts, unequal mixing of dependence on the isohaline and isopycnal slopes. Then heat and salt can drive double-diffusive interleaving we investigate properties of the fastest-growing modes, (Stern 1967; Ruddick and Turner 1979). separating the analysis into the high- and low-shear R In baroclinic fronts, unequal mixing of mass and mo- cases (section 5). To put the predictions into an ocean- mentum can drive viscous/diffusive interleaving ographic context, we apply them to a Mediterranean salt (McIntyre 1970; Calman 1977). lens (section 6). We conclude with a summary of our results and some new questions about the nature of dou- Ocean observations have provided evidence for double- ble-diffusive interleaving in ocean fronts (section 7). diffusive interleaving (Horne 1978; Joyce et al. 1978; Ruddick 1992). No conclusive evidence for viscous/ diffusive interleaving has been observed to date. 2. Dynamics of double-diffusive interleaving Many ocean fronts are both thermohaline and baroclinic, so they may have both types of interleaving (Rud- a. Basic mechanism dick 1992). However, existing theories of double-dif- Double-diffusive interleaving is thought to develop fusive interleaving may be inadequate for these fronts as an instability of thermohaline fronts. In the presence because the theories include neither horizontal density of horizontal gradients, vertically alternating lateral mo- gradients nor vertical shear. On the other hand, existing tions cause inversions in the pro®les of temperature and salinity (Fig. 1). These ¯uctuations lead to regions of enhanced double diffusion: salt ®ngering under warm, Corresponding author address: Mr. Brian D. May, Department of salty layers and diffusive convection under layers that Oceanography, Dalhousie University, Halifax, NS, B3H 4J1, Canada. are relatively cold and fresh (Turner 1973). E-mail: [email protected] Both forms of double diffusion generate an upgra-

᭧1997 American Meteorological Society

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FIG. 2. In this baroclinic front, the isopycnal slope increases the density gradient along the interleaving layers and thus opposes the interleaving motions.

FIG. 1. Side-view illustration of double-diffusive interleaving. The shaded arrows indicate lateral interleaving motions driven by the model of steady-state interleaving, McDougall (1985b) depth-varying double-diffusive density ¯ux (black vertical arrows). included both forms of double diffusion. If salt ®ngering is the dominant form of double diffusion (as shown), warm salty water rises as it crosses the front. These models apply to purely thermohaline fronts, that is, fronts with horizontal temperature and salinity gradients, but no horizontal gradients of density. There- dient (downward) density ¯ux. The convergence of this fore, the models may not apply to baroclinic fronts, density ¯ux drives interleaving motions. If the density which have nonzero horizontal density gradients as well ¯ux of salt ®ngering exceeds that of diffusive convec- as vertical shear. One model, by Kuzmina and Rodionov tion, water in the warm, salty layers becomes less dense (1992), considered the evolution of interleaving in bar- and, therefore, rises as it crosses the front. If diffusive oclinic fronts. The model included shear-dependent tur- convection dominates, water in the cold, fresh layers bulent mixing in addition to double diffusion and pre- should rise across the front. dicted that interleaving strength should decrease with increasing background shear. Unfortunately, the model b. Theoretical models of interleaving neglected advection of the background velocity ®eld, which could be important in some fronts. Also neglected Because our work is an extension of existing inter- was vertical advection of the background salinity ®eld. leaving models, we start with a brief review of the rel- This approximation is only valid when the density ratio evant models. Stern (1967) developed the ®rst model R␳ is very high, a situation in which double diffusion of interleaving, an instability analysis that predicted the is unlikely in the ®rst place. These assumptions are not initial growth of interleaving. The background front in made in the new model presented below. Stern's model was in®nitely wide with constant temperature and salinity gradients throughout. The instabilities predicted were driven by salt ®ngering; there- 3. Instability model fore, the layers sloped with warm salty water rising a. Introduction across the front. A number of models have followed Stern's (1967) In this paper, we consider two effects of baroclinicity approach, making re®nements along the way. Toole and that have not previously been investigated. Georgi (1981) added friction to Stern's model and pre- R In baroclinic fronts, background shear may distort the dicted an optimum vertical wavelength for the inter- interleaving layers. For example, twisting by vertical leaving layers. Niino (1986) extended the model to shear will alter the slope of the layers. Since the slope fronts of arbitrary width. His model matched Toole and is dynamically important, shear can be expected to Georgi's predictions for wide fronts, but also agreed affect the interleaving growth. with the energy-based model of Ruddick and Turner R The horizontal density gradient in baroclinic fronts (1979) for narrow fronts. alters the effect of the strati®cation on interleaving In Stern's model, salt ®ngering was assumed to be motions. For example, if the isopycnal slope opposes the only form of double diffusion and a constant dif- the slope of the interleaving layers, the density gra- fusivity was used to prescribe the vertical ¯uxes. Other dient along the layers is increased. Therefore, one parameterizations have been considered also. McDou- would expect the interleaving growth to be diminished gall (1985a) assumed the double-diffusive ¯uxes to be (Fig. 2). proportional to the salinity contrast between adjacent layers, rather than the gradient. More recently, Walsh In the model below, we include vertical shear and a and Ruddick (1995) included a nonconstant diffusivity. horizontal density gradient. We also include horizontal In addition, they mapped the analysis to the case driven shear, so the model applies not only to baroclinic fronts by diffusive convection, rather than salt ®ngering. In a but also to sheared barotropic fronts.

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We purposefully limit consideration to simple (i.e., c. Perturbation equations of motion geostrophic) fronts in which the gradients of tempera- To model the initial growth of interleaving, we con- ture, salinity, and velocity are constant. Ocean fronts sider the evolution of perturbations to the base state. are generally more complex. Nevertheless, it is our hope The governing equations for the perturbations are lin- that the results of this model will apply, with the ap- earized and the ¯uid is assumed to be Boussinesq, so propriate choice of base-state gradients, to fronts in that which the gradients are nonconstant. 2uЈץpЈץuЈ 1ץuЈץ ϩ␷ÅϪf␷Јϩ ϪA ϭ0 (9) 2 zץ xץ y ␳oץ tץ b. Base state ␷Јץ␷Јץ -The relevant physical quantities are the velocity com ϩ␷ÅϩuЈ␷ÅϩwЈ␷Å yxzץtץ ponents (u, ␷, and w), pressure p, temperature T, salinity S, and density ␳. We use a linear equation of state 2␷ЈץpЈץ1 1 ϩfuЈϩ ϪA ϭ0 (10) 2 zץ yץ ␳ ϭ 1 ϩ ␤(S Ϫ Soo) Ϫ ␣(T Ϫ T ), (1) ␳o ␳o 2wЈץpЈץwЈg 1ץwЈץ where ␣ is the thermal expansion coef®cient, ␤ is the ϩ␷Åϩ␳Јϩ ϪA ϭ0 (11) z2ץ zץy ␳ ␳ץ tץ contraction coef®cient for salinity, and ␳o is the density oo wЈץ␷ЈץuЈץ at reference temperature To and salinity So. Following Stern (1967), we separate each of the variables into ϩϩ ϭ0 (12) x y z ץ ץ ץ -base-state (overbar) and perturbation (primed) compo ␳Јץ␳Јץ ,nents; for example ϩ␷ÅϩuЈ␳Åx yץ tץ (u(x, y, z, t) ϭ uÅ(x, z) ϩ uЈ(x, y, z, t). (2 2SЈץ The base state is a steady front and the perturbations are instabilities of the front. ϩwЈ␳ÅzfosϪ(1 Ϫ ␥ )␳␤K ϭ0 (13) z2ץ The base state we consider is an in®nitely wide front 2SЈץSЈץSЈץ with constant temperature and salinity gradients. The x ϩ␷ÅϩuЈSÅÅϩwЈSϪKϭ0. (14) z2ץyxzsץtץ axis is aligned across the front so that ÅÅÅ T(x, z) ϭ Toxϩ TxϩTz z (3) The underlined terms arise from the background veloc- SÅÅÅ(x, z) ϭ S ϩ SxϩSz. (4) ity ®eld and horizontal density gradient. These terms ox z have not been included in earlier theories. In our study, the horizontal gradients of temperature and An effective viscosity A (typically 10Ϫ5 to 10Ϫ3 m2 salinity are not assumed to be density compensating, sϪ1) is used to parameterize the momentum ¯uxes. As- Å that is, a nonzero horizontal density gradient ␳Åx ϭ ␳o(␤Sx suming salt ®ngering to be the dominant form of double Å Ϫ6 Ϫ4 2 Ϫ1 Ϫ ␣Tx) is allowed. The background strati®cation is as- diffusion, a diffusivity K (typically 10 to 10 m s ) Å s sumed to be appropriate for salt ®ngering (0 Ͻ ␤Sz Ͻ is assumed for the vertical salinity ¯ux F . The density Å Å Å s ␣Tz); analogy to the diffusive case (␤Sz Ͻ ␣Tz Ͻ 0) is ¯ux F␳ is then prescribed by straightforward. The ¯uid is stably strati®ed, so ␳Åz ϭ Å Å F␳ ϭ (1 Ϫ ␥f)␳o␤Fs, (15) ␳o(␤Sz Ϫ ␣Tz) Ͻ 0. The base state is geostrophic with where ␥f is a nondimensional ¯ux ratio appropriate for uÅ ϭ 0 (5) salt ®ngering (typically 0.5 to 0.9). In practice, the viscosity, diffusivity, and ¯ux ratio are likely to be de- ␷Å ϭ ␷ oxϩ ␷Å x ϩ ␷Å zz (6) pendent on parameters such as the density ratio R␳ (Kel- wÅϭ0, (7) ley 1990; Kunze 1994). However, in order to focus on the effects of baroclinicity, we assume these parameters where the horizontal shear ␷Åx and vertical shear ␷Åz are are constant. assumed to be constant. The vertical shear is related to Following Stern (1967), we assume that the solution the horizontal density gradient by the thermal wind re- can be expressed as a sum of time-dependent, spatially lationship harmonic modes. This permits the governing differential g equations to be reduced to simpler algebraic equations. f␷Å zxϭϪ ␳Å. (8) The assumed waveform is ␳o uЈϭuÃexp[i(kx ϩ ly ϩ mz) ϩ ␭t], (16) The horizontal shear is barotropic and, therefore, is independent of the strati®cation. Note that with constant where uÃ is the (complex) amplitude and k, l, and m are -shear, ٌ2␷Å ϭ 0, so friction does not play a role in the the (real) cross-front, alongfront, and vertical waven base-state dynamics. umbers, respectively. The growth rate ␭ may be real or

Unauthenticated | Downloaded 09/26/21 10:58 PM UTC 2000 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27 complex, corresponding to stationary or traveling-wave with strong baroclinic and/or barotropic shear. The low- solutions. We will often describe the interleaving layers shear limit applies to fronts in which the shears (and in terms of their vertical wavenumber (m), cross-front therefore the horizontal density gradient) are weak. Note slope (Ϫk/m), and alongfront slope (Ϫl/m). that the criterion for applying the high- and low-shear limits depends on characteristics of the solution (i.e., the growth rate ␭). d. Effect of background shear In baroclinic fronts and sheared barotropic fronts, ad- e. High-shear limit -y) can disץ/uЈץvection by the background ¯ow (e.g., ␷Å tort the interleaving layers. For the ¯ow given by (5)± The high-shear limit is obtained by setting the along- (7), the WKB equations describing this distortion (i.e., front wavenumber to zero. This implies that the inter- the evolution of the wavenumbers in time) are leaving layers do not slope along the front. Using (16) with l ϭ 0, the governing equations (9)±(14) become dk k ϭϪl␷Å (17) 2 dt x ␭uÃ Ϫ f␷Ã ϩ ipÃϩAm uÃ ϭ 0 (19) ␳o dm g␳Å ϭϪl␷Åz (18) x 2 dt ␭␷Ã ϩ ( f ϩ ␷Å x)uÃ Ϫ wÃ ϩ Am ␷Ã ϭ 0 (20) f␳o (Kunze 1990). If the alongfront wavenumber l is non- gm zero, the cross-front and vertical wavenumbers will ␭wÃϩ␳ÃϩipÃϩAm2 wÃ ϭ 0 (21) grow in time, ultimately to in®nity. ␳␳oo We concentrate on two limiting cases: ikuÃ ϩ imwÃ ϭ 0 (22) R High-shear limit. If the rate of vertical or horizontal ␭␳Ã ϩ ␳Å uÃ ϩ ␳Å wÃ ϩ (1 Ϫ ␥ )␳␤KmS2 Ã ϭ0 (23) shear is large compared to the growth rate ␭, the in- xz fos terleaving layers will be tilted out of their unstable ÃÅ Å2 Ã ␭SϩSuxzÃϩSwÃ ϩKmS sϭ0, (24) range before signi®cant growth can occur, unless the where (8) was used to substitute Å gÅ /(f ). Note alongfront wavenumber is zero. Therefore, the high- ␷z ϭ Ϫ ␳x ␳o shear modes are derived by setting l ϭ 0. that (22) gives wÃ /uÃ ϭϪk/m, so the cross-front inter- R Low-shear limit. If the shear rate is small compared leaving ¯ow is along the layers. Additionally, there is to the growth rate, growth to ®nite-amplitude can oc- nonzero perturbation velocity along the front (␷Ã ), given cur before signi®cant tilting takes place. In this case, by (20). a nonzero alongfront wavenumber is allowed. Eliminating the perturbation amplitudes (uÃ, ␷Ã, wÃ, pÃ, ␳Ã, and SÃ) from (19)±(24) yields a single equation for Accordingly, we split the analysis into the high- and the growth rate as a function of the cross-front and low-shear cases. The high-shear limit applies to fronts vertical wavenumbers. This is

kk22 k 2 gkgkk 1ϩ ␭4ϩ1ϩ (2A ϩ K )m23␭ ϩ 1 ϩ (A ϩ 2K )Am4ϩ f ( f ϩ ␷Å ) ϩ ␳Å ϩ ␳Å Ϫ ␳Å ␭ 2 mm22ssxxxz m 2 ␳m␳mm ΂΃΂΃[] ΂΃oo ΂ ΃ kgkgkkkk2 ϩ1ϩAm24ϩf(fϩ␷Å)ϩ ␳Åϩ(1 ϩ A/K ) ␳Å Ϫ ␳Å Ϫ (1 Ϫ ␥ )g␤ SÅÅϪ SKm2␭ m2 xx␳m sxzfxzs␳mm mm [΂΃oo ΂ ΃ ΂ ΃] gk k k k ϩ ␳ÅϪ␳ÅϪ(1 Ϫ ␥ )g␤ SÅÅϪ SAKm4ϭ0. (25) ␳mmxz f mm xzs []o΂΃ ΂΃

ÅÅ The terms␳ÅxzxzϪ(k/m)␳Åand S Ϫ (k/m)S are the For any set of input parameters, the growth-rate poly- cross-front gradients of density and salinity along the nomial has four roots, each representing an eigenvalue interleaving layers; they arise from advection of the of the dynamical system. The sign of the real part of base-state ®elds by the perturbation ¯ow. The combi- the growth rate indicates whether the associated solution nation f (f ϩ ␷Åx) ϩ (g/␳ox)(k/m)␳Å arises from planetary is growing (positive), decaying (negative), or neutrally rotation and background shear. stable (zero).

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f. Low-shear limit g␳Å x l ( f ϩ ␷Å x)uÃ Ϫ wÃ ϩ ipÃϭ0 (27) Before continuing, we derive the corresponding poly- f␳␳oo nomial for the low-shear limit. We assume a priori that gm the rate of shear deformation is small. Therefore, we 2 allow a nonzero alongfront wavenumber, or equivalently ␭wÃϩ␳ÃϩipÃϩAm wÃ ϭ 0 (28) ␳␳oo a nonzero alongfront layer slope. When applying the model, care must be taken to ensure that the shear rate ikuÃ ϩ imwÃ ϭ 0 (29) is actually small compared to the predicted growth rate. 2 Ã McDougall (1985a) showed that the alongfront layer ␭␳Ã ϩ ␳Å xzuÃ ϩ ␳Å wÃ ϩ (1 Ϫ ␥ fos)␳␤KmSϭ0 (30) slope allows the fastest-growing modes to reach a bal- ÃÅ Å2 Ã ance with no alongfront interleaving ¯ow. Setting ␷Јϭ ␭SϩSuxzÃϩSwÃ ϩKmS sϭ0. (31) -y), the equaץ/uЈץand dropping the shear terms (e.g., ␷Å 0 tions of motion (9)±(14) become Again, wÃ /uÃ ϭϪk/m, so the perturbation ¯ow is along k ␭uÃ ϩ ipÃϩAm2 uÃ ϭ 0 (26) the interleaving layers. ␳o Manipulation of (26) and (28)±(31) yields

kk22 kgkk 2 1ϩ ␭3ϩ1ϩ (AϩK)m22␭ϩ1ϩ AK m4ϩ ␳Å Ϫ ␳Å ␭ mm22ssxz m 2␳mm ΂΃΂΃[] ΂΃o ΂ ΃ gk k k k ϩ ␳ÅϪ␳ÅϪ(1 Ϫ ␥ )g␤ SÅÅϪ SKm2ϭ0. (32) ␳mmxz f mm xzs []o΂΃ ΂΃

In this case, the interleaving dynamics are independent Both requirements depend on the solutions (i.e., l/m and of the y-momentum equation, so the growth-rate poly- ␭), so they must be checked after solving the growth- nomial is a cubic. With ␳Åx ϭ 0, (32) reduces to the rate polynomial. In section 5, we investigate fastest- model of Toole and Georgi (1981). Our low-shear limit growing modes and give some guidelines as to when is an extension of their model to baroclinic fronts. the low-shear limit applies. Equation (27) prescribes the alongfront slope required to maintain ␷Јϭ0. Combining (26)±(27) and (29), we 4. Criterion for growth of double-diffusive ®nd that the slope is interleaving lkg␳Åk/m a. Modes of instability ϭfϩ␷Åϩx . (33) x 2 To expose instabilities of the front, we consider only mmf΂΃␳␭o ϩAm solutions with positive ␭. As Stern (1967) noted, we are The predicted alongfront slope can be much larger than guaranteed at least one positive root if the term inde- the cross-front slope, indicating that the interleaving pendent of ␭ in the growth-rate polynomial [(25) or (32)] motions are strongly affected by rotation. is negative. This is the criterion for double-diffusive The WKB equations (17)±(18) show that the layers interleaving. will be distorted by background shear if they slope along In baroclinic fronts, there are additional possibilities the front. The low-shear polynomial (32) is only valid for growth. If the ␭ coef®cient is negative, there are if the rate of this distortion is small compared to the generally two roots of the growth-rate polynomial with growth rate (␭). positive real parts. These roots correspond to the vis- R The rate at which the wave vector will be rotated by cous/diffusive instability predicted by McIntyre (1970). horizontal (i.e., barotropic) shear is Å . Therefore, A third instability, ``slant-wise convection'' (Emanuel ͦ␷xͦ 2 application of the low-shear limit requires 1994), can occur if the ␭ coef®cient is negative. This form of instability cannot occur unless a modi®ed Rich- 2 2 ͦ␷Åxͦ K ␭. (34) ardson number (N /␷Åz )(1 ϩ ␷Åx/f) is less than one. R The rate of distortion by vertical (i.e., baroclinic) shear b. Criterion for double-diffusive interleaving is ͦ(l/m)␷Åzͦ. Application of the low-shear limit requires The focus of our investigation is double-diffusive in- llg␳Å x terleaving. Dividing (25) by gAK m4 and (32) by gK m2, ␷Åz ϭ K␭. (35) s s ΗΗΗmmf␳o Η we see that the term independent of ␭ is negative when

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FIG. 3. Range of interleaving slopes (shaded) when the isopycnals

are ¯at (i.e., ␳Åx ϭ 0).

1 kk kkÅÅ ␳ÅxzϪ␳ÅϪ(1 Ϫ ␥ f)␤ S xzϪ S Ͻ 0. ␳omm΂΃ mm ΂΃ (36) This is the criterion for double-diffusive interleaving. It holds in both the high- and low-shear limits. ∈ The ®rst term in (36) is negative if k/m (0,␳Åxz/␳Å ), FIG. 4. Range of interleaving slopes (shaded) when the isopycnal Å Å that is, if the layer slope (Ϫk/m) lies between zero and and isohaline slopes are of opposite sign [i.e., (␳Åxz/␳Å ) /(Sx/Sz) Ͻ 0]: (a) the isopycnal slope (Ϫ␳Å xz/␳Å ). This range of slopes de- small isopycnal slope and (b) large isopycnal slope. ®nes the baroclinic wedge of instability; when the layer slope is in this range, baroclinicity enhances the inter- Å leaving motion. With Sz Ͼ 0, the second term in (36) ∈ Å Å tation of (38) is straightforward. The maximum inter- is negative if k/m (0, Sx/Sz), that is, if the layer slope Å leaving slope always lies between zero and the isohaline (Ϫk/m) lies between zero and the isohaline slope (ϪSx/ Å slope, so the interleaving layers slope upward into fresh- Sz). We refer to this range of slopes as the double-diffusive wedge of instability; in this range, double dif- er water (Fig. 3). Due to the strati®cation, the limit for fusion provides the driving mechanism for interleaving growth is less than the isohaline slope by the factor ⑀z/(⑀z motions. ϩ 1). The instability criterion (36) depends on both the density and salinity distributions. It is satis®ed if d. Opposing isopycnal and isohaline slopes k (1 Ϫ ␥ )␤SÅ Ϫ ␳Å /␳ ∈ fx xo If the isopycnal slope opposes the isohaline slope [i.e., 0,Å . (37) m΂΃(1 Ϫ ␥ fz)␤S Ϫ ␳Å zo/␳ (␳Å /␳Å )/(SÅ /SÅ ) Ͻ 0], the double-diffusive and baroclinic xz x z In this range, the combined effects of density advection wedges of instability do not overlap (Fig. 4). The range and diffusion are destabilizing. The maximum unstable of interleaving slopes depends on the relative magni- Å interleaving slope is the cross-front slope along which tudes of␳Åx and Sx. the ratio of the density and salinity gradients (⌬␳/␳ )/ o R If the horizontal salinity gradient dominates [i.e., (1 (␤⌬S) equals the ¯ux ratio (1 Ϫ ␥f). Å Ϫ ␥f)␤ͦSxͦ Ͼ ͦͦ␳Åx /␳o], the maximum interleaving slope Introducing the nondimensional quantity ⑀z ϵϪ(1 Ϫ Å lies between zero and the isohaline slope (Fig. 4a). ␥f)␤Sz/(␳Åz /␳o) ϭ (1 Ϫ ␥f)/(R␳ Ϫ 1) (Toole and Georgi 1981), we rewrite (37) as Therefore, the instabilities lie in the double-diffusive wedge of instability. ÅÅ R If, on the other hand, the density gradient dominates k ⑀zxS/S zϩ␳Å x/␳Å z ∈0, . (38) [i.e., ␳Å /␳Ͼ(1 Ϫ ␥ )␤ SÅ ], the maximum interleav- m ⑀ϩ1 ͦͦx o f ͦ xͦ ΂΃z ing slope lies between zero and the isopycnal slope This shows that the maximum layer slope is a weighted (Fig. 4b). The instabilities lie in the baroclinic wedge average of the isohaline and isopycnal slopes. In the of instability. ocean, they are typically 10Ϫ4 to 10Ϫ2, so the interleaving slopes are roughly this size. The weight ⑀z is positive The second case is a form of baroclinic instability that Å under the assumption Sz Ͼ 0 and is typically in the range has not been predicted before. Unlike McIntyre's (1970) 0.2 to 1. viscous instability, the instability criterion does not depend on the relative rates of viscosity and diffusion. Instead, it depends on the inability of double-diffusive c. Flat isopycnals ¯uxes to completely remove ¯uctuations in the density

When the isopycnals are ¯at (i.e., ␳Åx ϭ 0), interpre- ®eld.

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FIG. 6. Contour plot of the high-shear growth rate [␭ in (25)] as a function of the cross-front layer slope (Ϫk/m) and vertical wave- FIG. 5. Range of interleaving slopes (shaded) when the isopycnal number (m). The input parameters were taken from Table 1. The Å Å Å Å and isohaline slopes have the same sign [i.e., (␳Åxz/␳Å ) /(Sx/Sz) Ͼ 0]: (a) isopycnal slope (Ϫ␳Åxz/␳Å ), isohaline slope (ϪSx/Sz), and maximum un- small isopycnal slope and (b) large isopycnal slope. stable slope [given by (37)] are indicated. The shaded area illustrates the range in which ␭ is greater than 90% of its maximum value (e- folding period of 11 days). e. Isopycnal and isohaline slopes of same sign If the isopycnal and isohaline slopes have the same Å Å sign [i.e., (␳Åxz/␳Å )/(Sx/Sz) Ͼ 0], the double-diffusive and wavelength of the fastest-growing mode is roughly the baroclinic wedges of instability overlap (Fig. 5). Again, Ekman scale 2␲(A/ͦ f ͦ)1/2. the range of interleaving slopes depends on the relative To illustrate the effect of baroclinicity on the fastest- Å magnitudes of␳Åx and Sx. growing solutions, the optimum cross-front slope, ver- R If the isohaline slope exceeds the isopycnal slope (i.e., tical wavenumber, and growth rate have been plotted as Å Å ͦSx/Sz ͦ Ͼ ͦͦ␳Åxz/␳Å ); the limit for growing instabilities functions of isopycnal slope (Fig. 7). The cross-front extends beyond the slope of the isopycnals (Fig. 5a). slope increases linearly with isopycnal slope and the R If, on the other hand, the isohalines slope less steeply growth rate has a quadratic dependence. Interleaving is Å Å than the isopycnals (i.e., ͦSx/Szͦ Ͻ ͦ␳Åxz/␳Å ͦ), the limit for enhanced if the isopycnal slope is positive (i.e., if the growing instabilities extends beyond the isohaline isopycnals rise toward the fresh side of the front). Note slope (Fig. 5b). that the slope and growth rate of the fastest-growing Å Depending on the slope of the interleaving layers rel- mode go to zero when (1 Ϫ ␥f)␤Sx ϭ ␳Åx/␳o. The optimum ative to the isopycnal and isohaline surfaces, interleav- vertical wavelength is affected little by baroclinicity. ing motions may be driven by double diffusion, baro- Numerical maximization of the growth rate in (25) is clinicity, or both. straightforward. However, we must make a number of approximations in the growth-rate polynomial to obtain analytical expressions for the fastest-growing mode. 5. Fastest-growing modes a. High-shear limit R In the ocean, the isohaline and isopycnal slopes are In the high-shear limit, (25) is used to calculate the usually much smaller than one, so the layer slopes are 2 2 growth rate (␭). For a given base state and ¯ux param- also very small. We can approximate 1 ϩ k /m ഠ 1. eterization, the growth rate is a function of the cross- R When the slope of the interleaving layers is small front slope (Ϫk/m) and vertical wavenumber (m) (Fig. compared to the frequency ratio f/N, the polynomial 6). It is positive within the range of slopes prescribed can be simpli®ed even further. We will show that ␭ 2 2 by (37) for all values of vertical wavenumber. K Ksm and ␭ K Am in this case. The values of k and m that maximize ␭ are of par- ticular interest. These are the properties one might ex- Because these approximations depend on characteristics pect to ®nd when a set of random initial perturbations of the solutions, it is necessary to verify a posteriori has grown to ®nite amplitude. In the high-shear limit, that they hold. 2 2 2 2 the cross-front slope of the fastest-growing mode is With 1 ϩ k /m ഠ 1, ␭ K Ksm , and ␭ K Am , (25) roughly half the maximum unstable slope. The vertical reduces to the much simpler relation

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FIG. 8. As in Fig. 6 but for the low-shear limit [␭ given by (32)]. The input parameters were taken from Table 1, except the horizontal density gradient, which was reduced by 50% to ensure satisfaction of (35) near the maximum. The e-folding period of the fastest-growing mode is roughly 3 days.

FIG. 7. Optimum cross-front slope (Ϫk/m), vertical wavenumber imate solutions (40)±(42) are quite close to the numer- (m), and growth rate (␭) for the high-shear limit vs isopycnal slope. The input parameters were taken from Table 1. The observed iso- ical results. pycnal slope is indicated. To illustrate dependence on the Prandtl number (A/K ), the calculations were performed with A ϭ K (solid s s b. Low-shear limit lines) and A ϭ 10Ks (dashed lines), holding Ks constant. The dotted lines indicate approximate solutions given by (40)±(42). The vertical In the low-shear limit, the growth of interleaving is wavenumber axis was scaled to take into account the theoretical Prandtl-number dependence. prescribed by (32). As in the high-shear limit, the rate of growth is positive within the range of cross-front slopes speci®ed by (37) (Fig. 8). Compared to the high- shear case, the maximum is shifted to smaller cross- front slope and vertical wavenumber. The optimum gk k [Am24ϩf(fϩ␷Å)]␭ ϩ ␳Å Ϫ ␳Å growth rate is increased. xxz␳mm [o΂΃ The dependence of the fastest-growing mode on the isopycnal slope is illustrated in Fig. 9. At low Prandtl kk number (A ϭ K ), the cross-front interleaving slope in- Ϫ(1 Ϫ ␥ )g␤ SÅÅϪ SAm2ϭ0. (39) s fxzmm΂΃] creases with isopycnal slope and the vertical wavenumber is roughly constant. The dependence at higher Maximizing ␭ with respect to k and m gives Prandtl number (A ϭ 10Ks) is signi®cantly different. kg(1 Ϫ ␥ )␤SÅ Ϫ ␳Å /␳ The cross-front slope is roughly constant and the vertical ϭ fx xo (40) wavenumber decreases as the isopycnal slope is in- 2 m 2N (1 ϩ ⑀z) creased. In both cases, the growth rate increases with f increasing isopycnal slope. This implies that interleav- 2 ͦ ͦ 1/2 m ϭ (1 ϩ ␷Å x/ f ) (41) ing is enhanced if the isopycnals slope upward toward A the fresh side of the front. g22[(1 Ϫ ␥ )␤SÅϪ ␳Å /␳ ] In Fig. 9, and we only plotted solutions that grow ␭ϭ fx xo, (42) 21/2faster than the rate of distortion by vertical shear [i.e., 8ͦfͦN(1 ϩ ⑀zx)(1 ϩ ␷Å / f ) ␭ Ͼ ͦ(l/m)␷Åzͦ]. The ®gure shows that the low-shear limit 2 Å where N ϭ Ϫg␳Åzz/␳o and ␧z ϭϪ(1 Ϫ ␥f)␤Sz/(␳Å /␳o) ϭ is restricted to fronts with small isopycnal slope. The

(1 Ϫ ␥f)/(R␳ Ϫ 1). observed isopycnal slope is near the upper bound for 2 2 It is easily shown that ␭ K Ksm and ␭ K Am , application of the low-shear polynomial. 2 2 2 2 2 2 2 2 provided k /m K f /N and k /m K (A/Ks)f /N . In the As in the high-shear limit, the growth-rate polynomial example, f/N ഠ 0.03, so these approximations are valid (32) must be approximated to obtain analytical expres- over most of the illustrated range (Fig. 7). The approx- sions for the fastest-growing mode.

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TABLE 1. Base-state properties for Meddy Sharon. Quantity Value g 9.8msϪ2 f 7.7 ϫ 10Ϫ5 sϪ1 ␤ 7.5 ϫ 10Ϫ4 psuϪ1 Å Ϫ5 Ϫ1 Sx Ϫ2.9(Ϯ0.4) ϫ 10 psu m Å Ϫ3 Ϫ1 Sz 1.23(Ϯ0.03) ϫ 10 psu m Ϫ3 ␳o 1032 kg m Ϫ6 Ϫ4 ␳Åx 2.3(Ϯ1.0) ϫ 10 kg m Ϫ4 Ϫ4 ␳Åz Ϫ7.52(Ϯ0.07) ϫ 10 kg m Ϫ5 2 Ϫ1 Ks 1.2 to 7.5 (ϫ10 m s ) A 0.3 to 4.0 (ϫ10Ϫ4 m2 sϪ1)

␥f 0.7(Ϯ0.1) ␨ 0.6(Ϯ0.2) ϫ 10Ϫ5 sϪ1) ␩ 1.1(Ϯ0.4) ϫ 10Ϫ5 sϪ1

solution (␳Åx ϭ 0) matches existing theories, which do not include baroclinicity (Toole and Georgi 1981; Mc- Dougall 1985a). To determine the ®rst-order effect of baroclinicity, we maximize ␭ with respect to k and m, retaining terms of Å order (␳Åx /␳o)/(␤Sx). This yields kg (1 Ϫ ␥ )␤SÅ ϭ fx (44) 21/2 FIG. 9. As in Fig. 7 but for the low-shear limit [␭ given by (32)]. m2N1ϩ⑀zzϩ(1 ϩ ⑀ ) The observed isopycnal slope is indicated. Only solutions satisfying g(1 Ϫ ␥ )␤SÅ␳Å the low-shear criterion (35) are plotted. Fastest-growing modes were m2ϭϩfx x(45) calculated with A ϭ K (solid lines) and A ϭ 10K (dashed lines), 1/2 1/2 s s 2N(AKsz)1ΗΗϩ⑀ϩ(1 ϩ ⑀ zo) ␳ holding Ks constant. The dotted lines indicate approximate solutions given by (44)±(46). gK1/2 (1 Ϫ ␥ )␤SÅ ␳Å ␭ϭϪsfxx. (46) 1/2 2NA΂΃ΗΗ1ϩ(1 ϩ ⑀zo) ␳ R Assuming the isohaline and isopycnal slopes to be much smaller than one, we set 1 ϩ k2/m2 ഠ 1. From these solutions, it is easy to show that ␭ K Am2, R Following Toole and Georgi (1981) and McDougall provided A/Ks k 1. Comparison of (44)±(46) with the (1985a), we consider a limit in which the Prandtl num- numerical results (at high Prandtl number) yields good ber is large (i.e., A/Ks k 1). We will show that ␭ K agreement (Fig. 9). Am2 in this case. Using the approximate solutions (44)±(46) we can The Prandtl number is not necessarily large in the ocean derive rough guidelines for determining when the low-shear limit should apply. From (34), the require- (A/Ks ഠ 4 in Table 1). Nevertheless, the second approximation is useful in that it allows us to derive an- ment for the horizontal (i.e., barotropic) shear is alytical solutions and to compare these solutions with 1/2 gK (1 Ϫ ␥ )␤SÅ ␳Å earlier theories. ␷ÅKsfxxϪ. (47) ͦxͦ 1/2 With 1 ϩ k2/m2 ഠ 1 and ␭ K Am2, the low-shear 2NA΂΃ΗΗ1ϩ(1 ϩ ⑀zo) ␳ polynomial (32) becomes This condition should be satis®ed in order to apply the gk k low-shear limit. Am22␭ ϩ AK m 4Ϫ ␳Å Ϫ ␳Å ␭ sxz␳mm The required alongfront slope is given by (33) and is []o΂΃ roughly

gk k k k 1/2 2 ϩ ␳ÅϪ␳ÅϪ(1 Ϫ ␥ )g␤ SÅÅϪ SKm l ͦ f ͦ Ks xz f xzs . (48) ␳omm΂΃ mm ΂΃ ഠ []ΗΗmNA΂΃ ϭ0. (43) Therefore, the requirement on the vertical (i.e., baro- Despite these simpli®cations, maximization of (43) re- clinic) shear (35) becomes mains a challenge. Keeping in mind that the low-shear limit is applicable at small isopycnal slopes, we tackle ␳Å 1(1Ϫ␥)␤SÅ the problem as a perturbation analysis, using the ratio xfxK. (49) Å ␳ 21ϩ(1 ϩ ⑀ )1/2 (/␳Åx ␳o)/(␤Sx) as an ordering parameter. The zeroth-order ΗΗoz Η Η

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Application of the low-shear limit is restricted to weakly baroclinic fronts, that is, fronts in which the horizontal density gradient␳Åx /␳o is much smaller than the hori- Å zontal salinity gradient ␤Sx.

6. Application to Meddy Sharon a. Overview In order to establish an oceanographic context for our model, we applied it to a previously studied front in which double-diffusive interleaving has been observed. Meddy Sharon, a lens of Mediterranean Water observed in the eastern North Atlantic (Armi et al. 1989), is a good test site for several reasons. During the period of observation, its heat and salt characteristics were grad- ually eroded away by lateral interleaving (Hebert et al. 1990). The doming of isopycnals associated with the meddy indicates that it caused a density front as well as a front in temperature and salinity. We should note that the meddy example may not represent a typical ocean regime because it was a region of particularly high gradients and strong shears. However, it does make a good test case for our theory. b. Base state The lower part of the meddy was strati®ed appropri- ately for salt ®ngering and has been most intensively studied, so we focus on that region. The quantities used in this analysis are listed in Table 1. The seven ``tow-yo'' pro®les considered by Ruddick (1992) were reanalyzed to estimate the horizontal and vertical gradients of salinity and density. The salt dif- FIG. 10. Cross-front layer slope, vertical wavelength, and e-folding fusivity Ks was calculated following Ruddick and Hebert (1988), but the uncertainty in the observed vertical period of the fastest-growing modes, for the high- and low-shear limits. The calculations were made using the observed cross-front wavelength was included in the calculation. We used density gradient (␳Åx ± 0) and, for comparison, setting the cross-front microstructure measurements (Oakey 1988) together density gradient to zero (␳Åx ϭ 0). In the ®rst two panels, the shaded with the observed velocity ®eld (Armi et al. 1989) to regions indicate the observed slope and wavelength. In the ®nal panel, estimate a vertical eddy viscosity appropriate to the the rate of distortion by background shear is shaded. 2 scales of interest (A ഠ ␧/␷Åz ). The value obtained, 0.3 to 4.0 (ϫ10Ϫ4 m2 sϪ1), is consistent with other estimates of viscosity in double-diffusive systems (Schmitt et al. c. Model predictions 1986; Padman 1994) and with Ruddick et al. (1989), It is not clear a priori whether the high- or low-shear who suggested that the Prandtl number should be O(1). limit applies to Meddy Sharon. Therefore, we estimate Our estimate of the salt-®nger ¯ux ratio (␥f) is taken the optimum cross-front slope (Ϫk/m), vertical wave- from McDougall and Ruddick (1992), who reviewed length (2␲/m), and e-folding period (1/␭) for both cases published estimates and concluded that the most prob- (Fig. 10). To illustrate the effect of baroclinicity, the able range for oceanic salt ®ngering is 0.6 to 0.8. calculations were performed using the observed hori- Estimates of vorticity ␨ and strain ␩ were obtained zontal density gradient (␳Åx ± 0) and, for comparison, from Hebert (1988). Because the meddy was a circular setting the horizontal density gradient to zero (␳Åx ϭ 0). vortex, we cannot simply substitute a single value for In the high-shear limit, numerical maximization of Ϫ3 ␷Åx in our equations. Where ␷Åx appears in the form 1 ϩ (25) gives a predicted cross-front slope of 4 ϫ 10 , ␷Åx/f, the vorticity is the relevant quantity (i.e., we sub- vertical wavelength of 8 m, and e-folding period of 11 stitute 1 ϩ ␨/f ). In (34) and (47), however, the strain days (Fig. 10). The predicted layer slope roughly match- Ϫ3 dictates the rate of distortion, so we substitute ␷Å x ϭ ␩. es the observed isopycnal slope of 3(Ϯ1) ϫ 10 . This

Unauthenticated | Downloaded 09/26/21 10:58 PM UTC SEPTEMBER 1997 MAY AND KELLEY 2007 suggests that the layers should slope upward relative to interleaving wavelengths to be 35(Ϯ15) m. However, geopotential surfaces, but should not necessarily slope there appears to be an additional peak in the range relative to the isopycnals. Including the horizontal den- 15(Ϯ5) m. This could be an interleaving mode, but it sity gradient increased the predictions of cross-front might be an artifact of the processing (i.e., a harmonic slope and growth rate by 35% and 45%, respectively. of the main peak). Taking a cautious approach, we ex- However, the predicted vertical wavelength was un- tend our range to include this second maximum (Fig. changed. 10). The low-shear prediction agrees well with the ob- In the low-shear limit, maximizing (32) gives a pre- served wavelength. The high-shear prediction falls out- dicted cross-front slope of 2.5 ϫ 10Ϫ3, vertical wave- side the observed range, but the uncertainties overlap. length of 25 m, and growth rate of 2.5 days (Fig. 10). This suggests that the high-shear limit may not apply, The growth predicted in the low-shear limit is signi®- but it is insuf®cient evidence to reject that limit outright. cantly faster than the high-shear case. Including the hor- From the analysis above, it is not clear which limit izontal density gradient increased the predicted cross- applies best to Meddy Sharon. In fact, the observed front slope and growth rate by 30% and 90% respec- velocity ®elds suggest that the shear is intermediate, tively. The effect on the vertical wavelength was small- rather than ``high'' or ``low.'' This may be consistent er: an increase of roughly 15%. with the rough overlap between observed values and The alongfront slope required in the low-shear limit the predictions of each limiting case. is 2(Ϯ1) ϫ 10Ϫ2, calculated from (33). With this alongfront slope, the relative rate of deformation by hori- 7. Discussion zontal shear (i.e., ͦ␷Åxͦ/␭)is2.5Ϯ1.5 from (34). The rate of deformation by vertical shear [i.e., ͦ(l/m)␷Åzͦ/␭]is In this paper, we have investigated the behavior of 1.1 Ϯ 0.6 from (35). The shear rates and the predicted double-diffusive interleaving in baroclinic fronts. An growth rate are of similar magnitude (Fig. 10). There- important outcome of the study is the prediction that fore, it is not clear that the shear is either ``high'' or interleaving layers will not slope in the alongfront di- ``low.'' Perhaps the observed interleaving re¯ects an rection if the background vertical or horizontal shear is intermediate case. strong. Comparing our high- and low-shear solutions, we have shown that the growth rate is signi®cantly d. Observed values smaller in the high-shear case (Fig. 10). This suggests that double-diffusive interleaving may be diminished in Ruddick (1992) estimated the density and salinity gra- fronts with strong baroclinic or barotropic shear. dients along a number of interleaving layers. In the low- In both the high- and low-shear limits, the strength er part of the meddy, he found the ratio of the along- of interleaving (i.e., the growth rate) is a function of layer gradients to be ⌬␳/⌬S ϭ 0.07(Ϯ 0.03) kg mϪ3 the baroclinicity. Interleaving can be enhanced or sup- psuϪ1. The cross-front and alongfront slopes of the lay- pressed, depending on the sign of the isopycnal slope ers are related to the observed ratio by (Figs. 7 and 9). Interleaving is enhanced if the isopyc- ⌬␳␳ÅϪ[k/mϩ(l/m)tan␾]␳Å nals rise toward the fresh side of the front. Conversely, ϭxz, (50) if the isopycnals slope downward across the front, the SSÅÅ[k/m(l/m)tan ]S ⌬xzϪ ϩ ␾ growth of interleaving is suppressed. where ␾ is the angle of the ship's path relative to the Previous studies have suggested that the driving force front. Using (50), we estimate the slope of the layers for double-diffusive interleaving is the cross-front gra- along the transect Ϫ[k/m ϩ (l/m)tan␾] ϭ 5.1(Ϯ1.5) ϫ dient of salinity along isopycnal surfaces (McDougall 10Ϫ3.If␾were zero, one could easily obtain the cross- 1985a; Ruddick 1992). While this holds in the absence front slope from this value. However, it is unlikely the of a horizontal density gradient, it is not true in baro- transect was oriented directly across the front, so care clinic fronts. We have shown that interleaving is en- must be taken while interpreting the observations. hanced if the isopycnals rise toward the fresh side of Å In the high-shear limit, the interleaving layers do not the front (Figs. 7 and 9). However, with Sz Ͼ 0, the slope along the front. Therefore, to evaluate the high- cross-front gradient of salinity along isopycnals is di- shear prediction, we set l/m ϭ 0, which gives Ϫk/m ϭ minished in this case. Therefore, we conclude that the 5.1(Ϯ1.5) ϫ 10Ϫ3. Comparing this value with the high- salinity gradient along isopycnals is not a good indicator shear prediction yields good agreement (Fig. 10). of expected interleaving strength. In the low-shear limit, the predicted alongfront slope The slope of interleaving layers relative to isopycnals (Ϫl/m) is roughly 8 times the cross-front slope (Ϫk/m). is of interest because it affects directly the amount of Taking this into account, and allowing a reasonable un- diapycnal mixing generated by interleaving motions. It certainty in the orientation of the transect (␾ ϭϮ10Њ), is often assumed that warm, salty ¯uid should rise rel- we are unable to estimate the cross-front slope or along- ative to isopycnals if salt ®ngering is the driving mech- front slope from the observed value. anism. We have shown that this is not the case (Figs. From the spectrum shown in Ruddick and Hebert 5 and 6). It is the layer slope relative to geopotential (1988), we estimate the most likely range of observed (i.e., horizontal) surfaces that determines the buoyancy

Unauthenticated | Downloaded 09/26/21 10:58 PM UTC 2008 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27 force along a layer. If salt ®ngering is the driving mech- housie University, 187 pp. [Available from Dalhousie Univer- anism, warm salty ¯uid rises relative to surfaces of con- sity, Halifax, NS, B3H 4J1 Canada.] , N. Oakey, and B. Ruddick, 1990: Evolution of a Mediterranean stant geopotential, but does not necessarily rise relative salt lens: Scalar properties. J. Phys. Oceanogr., 20, 1468±1483. to surfaces of constant density. Horne, E. P. W., 1978: Interleaving at the subsurface front in the slope Our analysis of Meddy Sharon highlighted a geo- water off Nova Scotia. J. Geophys. Res., 83, 3659±3671. Joyce, T. M., W. Zenk, and J. M. Toole, 1978: The anatomy of the metrical aspect of interleaving that must be taken into Antarctic Polar Front in the Drake Passage. J. Geophys. Res., account when surveying ocean fronts. Because inter- 83, 6093±6113. leaving layers may slope to a much greater extent along Kelley, D. E., 1990: Fluxes through diffusive staircases: A new for- fronts than across them, it is insuf®cient to survey only mulation. J. Geophys. Res., 95, 3365±3371. Kunze, E., 1990: The evolution of salt ®ngers in inertial wave shear. in the cross-front direction. Estimates of cross-front J. Mar. Res., 48, 471±504. slope can easily be contaminated by the alongfront slope , 1994: A proposed ¯ux constraint for salt ®ngers in shear. J. if the transect is not directly across the front. A mini- Mar. Res., 52, 999±1016. mum of two transects (one across and one along the Kuzmina, N. P., and V. B. Rodionov, 1992: In¯uence of baroclinicity on formation of thermohaline intrusions in ocean frontal zones. front) are needed to adequately sample the three-di- Izv. Atmos. Oceanic Phys., 28, 804±810. mensional structure of ocean interleaving. McDougall, T. J., 1985a: Double-diffusive interleaving. Part I: Linear In this study, we have investigated two limiting cases stability analysis. J. Phys. Oceanogr., 15, 1532±1541. of double-diffusive interleaving. One limit is appropri- , 1985b: Double-diffusive interleaving. Part II: Finite amplitude, steady state interleaving. J. Phys. Oceanogr., 15, 1542±1556. ate when the shear rate is high; the other when the , and B. R. Ruddick, 1992: The use of ocean microstructure to background shear is low. This raises the question of quantify both turbulent mixing and salt-®ngering. Deep-Sea Res., how interleaving develops when the shear rate and the 39, 1931±1952. growth rate are roughly equal (e.g., Meddy Sharon). Do McIntyre, M. E., 1970: Diffusive destabilization of the baroclinic circular vortex. Geophys. Fluid Dyn., 1, 19±57. the interleaving layers simply remain unsloped along Niino, H., 1986: A linear stability theory of double-diffusive hori- the front? Or, do they develop with the optimum along- zontal intrusions in a temperature-salinity front. J. Fluid Mech., front slope and then ¯atten over time, twisting in the 171, 71±100. background shear? In the latter case, could shear provide Oakey, N. S., 1988: Estimates of mixing inferred from temperature and velocity microstructure. Small-Scale Turbulence and Mixing a limiting mechanism, choking the growth of interleav- in the Ocean, J. C. J. Nihoul, and B. M. Jamart, Eds., Elsevier ing at ®nite amplitude? Science, 239±247. Padman, L., 1994: Momentum ¯uxes through sheared oceanic thermohaline steps. J. Geophys. Res., 99, 10 799±10 806. Acknowledgments. We thank David Hebert, Barry Ruddick, B., 1992: Intrusive mixing in a Mediterranean salt lensÐ Ruddick, and David Walsh for many interesting dis- Intrusion slopes and dynamical mechanisms. J. Phys. Oceanogr., cussions on the dynamics of interleaving. We also thank 22, 1274±1285. , and J. S. Turner, 1979: The vertical length scale of double- the reviewers who provided many excellent suggestions diffusive intrusions. Deep-Sea Res., 26A, 903±913. for improvement of the manuscript. This work was sup- , and D. Hebert, 1988: The mixing of Meddy ``Sharon.'' Small- ported by grants and scholarships from the Natural Sci- Scale Turbulence and Mixing in the Ocean, J. C. J. Nihoul, and ences and Engineering Research Council of Canada and B. M. Jamart, Eds., Elsevier Science, 249±262. , R. W. Grif®ths, and G. Symonds, 1989: Frictional stress at a the Izaak Walton Killam Foundation. sheared double-diffusive interface. J. Geophys. Res., 94, 18 161± 18 173. Schmitt, R. W., R. G. Lueck, and T. M. Joyce, 1986: Fine- and mi- REFERENCES crostructure at the edge of a warm-core ring. Deep-Sea Res., 33, 1665±1689. Stern, M. E., 1967: Lateral mixing of water masses. Deep-Sea Res., Armi, L., D. Hebert, N. Oakey, J. F. Price, P. L. Richardson, H. T. 14, 747±753. Rossby, and B. 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