SEPTEMBER 1997 MAY AND KELLEY 1997 Effect of Baroclinicity on Double-Diffusive Interleaving 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 baro- clinic, 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- q1997 American Meteorological Society Unauthenticated | Downloaded 09/26/21 10:58 PM UTC 1998 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27 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 Rr 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 tem- perature and salinity gradients throughout. The insta- bilities 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. Unauthenticated | Downloaded 09/26/21 10:58 PM UTC SEPTEMBER 1997 MAY AND KELLEY 1999 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. ]u9]u9 1]p9]2u9 1yÅ2fy91 2A 50 (9) 2 b. Base state ]t ]y ro ]x ]z The relevant physical quantities are the velocity com- ]y9]y9 1yÅ1u9yÅ1w9yÅ ponents (u, y, and w), pressure p, temperature T, salinity ]t]yxz S, and density r. We use a linear equation of state 1]p9]2y9 1 1fu91 2A 50 (10) 2 r 5 1 1 b(S 2 Soo) 2 a(T 2 T ), (1) ro ]y ]z ro ]w9]w9g 1]p9]2w9 where a is the thermal expansion coef®cient, b is the 1yÅ1r91 2A 50 (11) ]t ]y r r]z ]z2 contraction coef®cient for salinity, and ro is the density oo at reference temperature To and salinity So.