Instability and diapycnal momentum transport in a double diffusive, stratified shear layer

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1 Instability and diapycnal momentum transport in a double diffusive, stratified shear layer William D. Smyth and Satoshi Kimura College of Oceanic and Atmospheric Sciences Oregon State University, Corvallis OR Submitted to J. Phys. Oceanogr. November 5, 2005 Corresponding author. Telephone: (54) ; smyth@coas.oregonstate.edu Abstract We investigate the linear stability of a stratified shear layer in which stratification is generated by both temperature and salinity in a diffusively unstable configuration. We compute the effects of doubly diffusive stratification on inflectional shear instabilities and the effect of both shear and finite layer thickness on double diffusive instabilities. Momentum fluxes driven by salt sheets and diffusive convection cells are computed and parameterized in terms of Schmidt and Prantdl numbers.. Introduction Double diffusive instability exists throughout nearly half of the ocean interior (You, 2002). The instability often results in layering, which may occur through two mechanisms that have been seen as distinct, though their relationship is poorly understood. When temperature and salinity vary only in the vertical, the flow becomes organized into staircases of vigorously convecting layers separated by thin interfaces Radko (2003). When temperature and salinity vary in the horizontal, buoyancy fluxes may drive interleaving motions that have the potential to effect significant lateral transport between watermasses. Effects of shear (i.e. vertically sheared horizontal currents) may be important in both these scenarios. In the interleaving case, shear is clearly present between adjacent intruding layers (Mueller et al. 2005, in preparation). In a staircase, shear is not intrinsic, but the ubiquitous presence of gravity waves in the ocean means that such layers will be subjected to shear nonetheless (e.g. Kunze, 994). While the last few decades have seen many studies of double diffusive instability, the effects of shear on those instabilities have received relatively little attention. Linden (974) showed both theoretically and experimentally that a uniform shear favors convection rolls whose axes are parallel to the background flow. Analogous results have been found for ordinary convection in the presence of shear (Deardorff, 965). Thangham et al. (984) conducted detailed numerical studies of shear effects, but

2 excluded inflectional profiles. Kunze (990, 994) considered equilibration of finite amplitude salt fingers in shear, including the oceanically important case of inertially varying shear. Wells et al. (200) conducted lab experiments on salt fingers in a vertically periodic shear and showed how the strength of the salt fingers adjusted to create an equilibrium state. Other investigators have looked into the spontaneous generation of shear by divergent double diffusive fluxes (Paparella and Spiegel, 999; Stern, 969; Holyer, 98). The ability of double diffusive motions to flux horizontal momentum in the vertical has been addressed experimentally by Ruddick (985) and Ruddick et al. (989), and observationally by Padman (994). Here, we consider instability in a vertically localized layer where diffusively unstable stratification and shear coincide. The inflectional nature of the imposed shear yields the possibility of Kelvin-Helmholtz (KH) instability independent of double diffusion. We delineate regions of parameter space in which each class of instability is likely to dominate, and also consider the effects of double diffusive stratification on KH instability. We examine the effect that the finite vertical extent of the stratified layer has on the characteristics of double diffusive instability relative to the previously studied case of uniform gradients. One of the most useful outcomes that any study of ocean mixing can produce is a prediction of the fluxes due to the mixing process in question. Predictions of double diffusive fluxes are needed for the interpretation of observational data (e.g. Gregg and Sanford, 987; Padman and Dillon, 987; Hebert, 988; Rudels et al., 999; Timmermans et al., 2003) and for models ranging from global simulations (Zhang et al., 999; Merryfield et al., 999) to small scale processes studies (e.g. Walsh and Ruddick, 998; Merryfield, 2000). In the linear (small amplitude) limit, fluxes grow exponentially. In reality, this exponential growth is arrested with the onset of nonlinear effects, which generally lead the flow to a turbulent state. Fluxes occurring in the turbulent state cannot be predicted from linear theory. The situation becomes more encouraging if one considers ratios of fluxes. As a linear eigenmode grows exponentially in time, any flux ratio will remain constant. One therefore knows the value of the flux ratio at the onset of nonlinear effects, and this provides a plausible estimate of the flux ratio that will pertain in fully nonlinear flow. For example, the ratio of thermal to saline buoyancy fluxes in exponentially growing salt fingers is a simple function of R ρ that agrees well with lab experiments and numerical simulations of the fully nonlinear state (e.g. Kunze, 2003). Here, we attempt to parameterize momentum fluxes in terms of the better understood buoyancy fluxes, using flux ratios in the form of Prandtl and Schmidt numbers. Results are in reasonable agreement with available laboratory and observational evidence, but provide a more comprehensive view of the parameter dependences of these flux ratios. Our primary focus is on the salt fingering regime, in which warm, salty fluid overlies cooler, fresher fluid. In the opposite scenario, linear theory predicts oscillatory diffusive convection. Diffusive convection as it occurs in the laboratory and the ocean is an intrinsically finite amplitude phenomenon and is poorly modeled by its linear counterpart. Nevertheless, the linear dynamics are not without interest and are included here for completeness. Our basic tool is linear stability analysis, whose application is described in detail in section 2. The main results are given in section 3, and conclusions are summarized in section Methodology a. The eigenvalue problem for linear normal modes We assume that the fluid is barotropic and Boussinesq, and that density is a linear function of temperature and salinity. The velocity field u(x, y, z, t) ={u, v, w} is measured in a nonrotating, Cartesian coordinate system {x, y, z}. The resulting field equations are: D u Dt = π +(b T + b S )ˆk + ν 2 u () u = 0 (2) Db T Dt = κ T 2 b T (3) Db S Dt = κ S 2 b S (4) 2

3 The variable π represents the pressure scaled by the uniform characteristic density ρ 0, and ˆk is the vertical unit vector. The variables b T and b S are buoyancy fields due to temperature and salinity, and ν, κ T and κ S are the kinematic viscosity and the thermal and haline diffusivities, respectively. Boundary conditions are periodic in the horizontal, flux free at z =0and z = H. The velocity, buoyancy and pressure terms are separated into two parts, a background profile and a perturbation: u(x, y, z, t) = U(z)î + ε u (x, y, z, t) (5) b T (x, y, z, t) = B T (z)+εb T (x, y, z, t) (6) b S (x, y, z, t) = B S (z)+εb S (x, y, z, t) (7) π(x, y, z, t) = Π(x, y, z, t)+επ (x, y, z, t) (8) We substitute (5-8) into (-4) then collect the O(ɛ) terms. The perturbations are assumed to take the normal mode form: w (x, y, z, t) =ŵ(z)e σt+i(kx+ly) + c.c. (9) (for example), where k and l are wavenumbers in streamwise and spanwise directions, respectively and σ is the growth rate of the perturbation. Substituting these perturbations into the O(ɛ) equations gives: σû = ŵu z ikˆπ + ν 2 û (0) σˆv = ilˆπ + ν 2ˆv () σŵ = ˆπ z + ˆb T + ˆb S + ν 2 ŵ (2) i(kû + lˆv)+ŵ z =0 (3) σˆb T = ŵb Tz ikuˆb T + κ T 2ˆbT (4) σˆb S = ŵb Sz ikuˆb S + κ S 2ˆbS, (5) where the subscript z indicates the derivative, 2 = d 2 /dz 2 k 2 and k 2 = k 2 + l 2. Eliminating ˆπ from (0-3), we obtain σ 2 ŵ = iku 2 ŵ + ikŵu zz + ν 4 ŵ k 2ˆbT k 2ˆbS. (6) Equations (4-6) form a closed, generalized eigenvalue problem whose eigenvalue is σ and whose eigenvector is the concatenation of {ŵ, ˆb T, ˆb S }. The horizontal velocities and the pressure may then be obtained separately if needed. The background profiles appearing in (0-6) are chosen to describe a standard stratified shear layer: U = u 0 tanh z h ; (7) B Tz = R ρb z0 R ρ sech2 z h ; (8) B Sz = B z0 z R ρ sech2 h. (9) Here, u 0 is the half velocity change across a layer of half depth h, B z0 is the maximum background buoyancy gradient (or squared buoyancy frequency), and R ρ = B Tz /B Sz is the density ratio. In the interleaving case, our assumption that the shear varies vertically on the same scale as the stratification is justified. If shear is imposed by an internal wave field, however, it is likely that the vertical scales will be very different. If thermohaline properties vary on a much smaller scale than velocity, there exists the potential for other classes of shear instability besides KH. If the shear field is aligned so that maximum shear coincides with a stratified interface, the result may be Holmboe instability (e.g. Smyth and Winters, 2003). On the other hand, if the shear maximum is located between adjacent stratified layers, Taylor instability may result (e.g. Lee and Caulfield, 200). These will be the subject of a separate publication. We nondimensionalize the problem using length scale h and time scale h 2 /κ T. This nondimensionalization introduces four new dimensionless parameters: the molecular Prandtl number Pr = ν/κ T, the diffusivity ratio τ = κ S /κ T, the Reynolds number Re = u 0 h/ν, and the Grashof number Gr = B z0 h 4 /νκ T. The equations needed for the analyses to follow become σû = ŵu z ikˆπ + Pr 2 û (20) σ 2 ŵ = iku 2 ŵ + ikŵu zz +Pr 4 ŵ k 2ˆbT k 2ˆbS. (2) σˆb T = ŵb Tz ikuˆb T + 2ˆbT (22) σˆb S = ŵb Sz ikuˆb S + τ 2ˆbS, (23) We use this term somewhat loosely, as the buoyancy gradient in Gr contains a contribution from a second scalar (salt) in addition to the one whose diffusivity appears in the denominator (temperature). 3

4 l with background profiles U = ReP r tanh z; (24) B Tz = R ρgrp r R ρ sech2 z; (25) B Sz = GrP r R ρ sech2 z. (26) In (20-26) and hereafter unless otherwise noted, all quantities are dimensionless. b. Parameter values The molecular Prandtl number and the diffusivity ratio have the values Pr =7and τ =0 2, appropriate for seawater. We also define the molecular Schmidt number, Sc = Pr/τ = 700. The Grashof number Gr is assumed to be positive, indicating stable stratification. Observations in a thermohaline staircase east of Barbados, summarized by Kunze (2003), suggest values in the range [0 8, ] (depending on how one relates h to the thickness of a double diffusive layer). Double diffusive instability requires R ρ > 0, and is strongest when R ρ is near unity. The Barbados observations yielded a typical value.6. Here, we consider a range of values from 0.9 to 00. The Reynolds number obtained from the Barbados observations is in the range Values used here will range from 0, in the limit of no shear, to 0 8. The latter may exist in thin, strongly sheared layers not resolved in measurements. Another parameter of interest is the bulk Richardson number, Ri b = Gr/(PrRe 2 ). This is the minimum value of the gradient Richardson number, occurring at z =0. Ri b < /4 is the standard criterion for KH instability (Miles, 96; Howard, 96). 3. Results We begin with an overview of the main classes of unstable modes that may occur in a diffusively unstable, sheared flow: salt fingers, salt sheets, KH instabilities, and diffusive convection. We discuss the physics of each and (a) k (b) k (c) k Figure : The real dispersion relation σ r(k, l) for three illustrative cases. (a) Pure salt fingering: Re = 0, Gr = 0 6, R ρ =.6. (b) Pure KH: Re = 300,Ri =0., R ρ = 00. (c) KH and salt sheets: Re = 00, Ri b =0., Gr = 7000 R ρ =.6. Bullets indicate points for which flux profiles are shown in figure 2 define Schmidt and Prandtl numbers with which to describe the momentum fluxes. In section 3.2, we look at the the effects of double diffusive stratification on KH instabilities. Background shear does not affect double diffusive instabilities directly (except to select their orientation), but the instabilities may affect the background flow by transporting momentum in the vertical. This process, and its parameterization, are the subjects of section 3.3. We then give an equivalent (though less detailed) discussion for diffusive convection in section 3.4. In 3.5, we delineate the regions of parameter space in which each of these classes of instabilities is expected to dominate. a. Overview of instabilities In the absence of shear, double diffusive instability creates convection rolls whose orientation is not determined, i.e. the growth rate depends on the magnitude of the wave vector but not on its direction (figure a). In the absence of double diffusive instability, inflectional shear may create dynamical instability, which for this flow configuration is KH instability. Unlike double diffusive instability, KH instability is strongly directional, with wavevectors oriented parallel to the background shear flow growing most rapidly. These modes have l =0, and we refer to them as transverse. Of these, the mode with wavevector near one half (or wave length about 7 times the thickness of the shear layer) grow the fastest (figure b). Double diffusive instability becomes directional in the presence of shear, with wavevectors oriented at right an- 4

5 z 0 (a) b w (b) b w (c) u w (d) b w (e) u w (f) b w (g) u w fingers growing on uniform background gradients (e.g. Baines and Gill, 969), motions are concentrated in a thin layer where background gradients are largest. (Note that the vertical extent included in figure 2a is much less than that of the domain.) The amplitudes of the fluxes are arbitrary in the linear regime, but their ratio is not. The flux ratio γ sf = b T w /b S w (27) Figure 2: Vertical fluxes of buoyancy and velocity for the three illustrative cases shown by bullets in figure, plus an example of sheared diffusive convection. (a) Pure salt fingering: Re =0, Gr =0 6, R ρ =.6, k =22.36, l = (b,c) Pure KH: Re = 300,Ri b =0., R ρ = 00, k =0.47, l =0. (d,e) Salt sheets: Re = 00, Ri b =0., Gr = 7000 R ρ =.6, k =0, l =9.5. (f,g) Sheared diffusive convection: Re = 00, Ri b =0., Gr = 7000 R ρ =0.96, k =0, l =9.5. (a,b,d,f) buoyancy fluxes: total (solid), thermal (dashed), saline (dotted). All buoyancy fluxes have been divided by 000 for plotting convenience. (c,e,g) velocity flux. gles to the mean flow growing fastest. These modes have k =0, and they are referred to here as longitudinal. In accordance with Squire s theorem, longitudinal modes are unaffected by the background shear, a property we will explore in detail in the next section. Figure c shows an example in which both salt fingers and KH instability are present. The KH modes are strongest near k = 0.45,l =0. Salt fingers, instead of being nondirectional as in figure a, are strongly damped except near k =0. This monochromatic character causes the instability to take the form of convections rolls oriented parallel to the background flow, or salt sheets. Diffusive convection cases are not included in figure because their dispersion relations are qualitiatively similar to the salt sheet case (figure c). The difference is that, rather than being stationary, modes are oscillatory with frequencies often far in excess of the growth rate. Salt fingers grow because the positive buoyancy flux due to salinity advection outweighs the negative buoyancy flux due to thermal advection (figure 2a). Unlike salt has the value 0.59 for this example. This value agrees well with measurements of salt fingers in the lab and numerical experiments (Kunze, 2003) in the ocean (Schmitt, 2003), despite having been derived from linear theory. KH instability grows because the momentum flux (figure 2c) acts to reduce the background shear and thus draws energy from the mean flow into the perturbation. Because the stratification is almost purely thermal in this example, the same is true of the buoyancy flux (figure 2b). In fact, the flux ratio for this case is 93. This is similar to the value of the density ratio R ρ = 00, indicating that the difference in the diffusivities of the two scalars plays no significant role in the dynamics, consistent with the fact that mechanism of KH instability is essentially inviscid. Salt sheets, like salt fingers, grow because of the dominance of the positive buoyancy flux due to salinity (figure 2d). For this example, γ s =0.55. In addition, convective motions within the salt sheets tilt against the shear, so that the momentum flux (figure 2e) draws energy from the mean flow to the perturbation. This energy transfer, however, amplifies only motions in the x direction, and thus does not supply energy to the salt sheets per se. The momentum and buoyancy fluxes may be expressed in terms of effective diffusivities via the usual fluxgradient formalism. The ratio of momentum to thermal diffusivity, here called the salt sheet Prandtl number Pr ss has the value 0.9, while the Schmidt number Sc ss, the ratio of momentum to saline diffusivity, is Diffusive convection is an oscillatory instability driven by the dominance of the positive thermal buoyancy flux over the negative saline buoyancy flux (figure 2f). The finite amplitude expression of diffusive convection is a non-oscillatory convective layering, suggesting a qualitative change in the physics between the small and large amplitude regimes, so that results from linear theory must be interpreted with caution. The flux ratio is defined op- 5

6 R ρ k Figure 3: Effects of double diffusive stratification on KH instability. Shaded contours indicate log 0σ r as a function of k and R ρ for modes l =0, Re = 00, Ri b =0.0, and Gr = The white contour shows the wavenumber of the fastest growing mode for each R ρ. Modes are stationary except inside the black contour, where oscillatory instability occurs. positely to that for salt fingers: when damped by shear, it overwhelms shear instability. Of course, in this case the longitudinal salt sheet instability, being free of the damping effect of shear, would be the dominant mode. An example is given in figure 3, which shows growth rate as a function of R ρ and k for purely transverse modes with shear strong enough that the Richardson number is subcritical. The black contour signifies σ i =0; modes enclosed by (i.e. to the right of) that contour are oscillatory. The white contour indicates the fastest-growing mode (FGM) at each R ρ. For R ρ >.2, the FGM is stationary and has wavenumber near 0.45, as is characteristic of KH instability. To the right of the white contour is a secondary maximum in σ r, where salt fingering instabilities are damped and rendered oscillatory by the shear. For <R ρ <.2, these oscillatory salt sheets are the dominant instability. For R ρ <, the dominant instability is diffusive convection, which is oscillatory even in the absence of shear. For 0.9 <R ρ <.2, the double diffusive instabilities represent the FGM. R ρ < 0.9, KH modes are again dominant and are not significantly affected by the double diffusive stratification. γ dc = b S w /b T w, (28) and has the value 0.2 for this example. Like salt sheets, sheared diffusive convection generates a vertical transport of momentum (figure 2f) via longitudinal convection cells. Unlike in the salt sheet case, momentum diffusivities are larger than scalar diffusivities. For the case shown in figure 2e,f, the Prandtl number of the diffusive convection is Pr dc =.46 and the Schmidt number is Sc dc =7.07. In the rest of this section, we will explore in greater detail the the relationships between sheared, doubly diffusive flow profiles and the instabilities they generate. b. How are KH billows affected by diffusively unstable stratification? For most oceanically relevant parameter values, diffusively unstable stratification has negligible effect on KH instability. An exception occurs where R ρ. There, double diffusive instability becomes so strong that, even c. How do salt sheets interact with shear? As illustrated in figure c, shear suppresses the salt fingering instability whenever the convective motions have a component in the direction of the mean flow, i.e. whenever k 0. When k =0, the mean flow U no longer appears in (2-23), so that the growth rate and the eigenfunctions ŵ, ˆb T and ˆb S are just as they would be in the absence of a mean shear (cf. Linden, 974). In contrast, U does not vanish from the streamwise momentum equation (20), which becomes (σ Pr 2 )û = ŵu z. (29) The vertical motion of the salt sheets advects the background shear to force a perturbation horizontal velocity. While this process does not affect the salt fingers (at lowest order), it may have an important effect on the mean flow via the Reynolds stress u w = 2(ŵ û) r e 2σrt,in which the dagger and the subscript r represent the complex conjugate and the real part, respectively. 6

7 To gain some initial insight into how this salt sheet - mean flow interaction might operate, consider the simple case in which the aspect ratio of the salt sheets is such that the vertical derivative in 2 is negligible relative to the horizontal derivatives. This is the tall fingers (hereafter TF) approximation first used by Stern (975). When k = 0, we may then write 2 l 2. Validity of this approximation requires that the vertical scale over which salt sheet properties vary is much larger than 2π/l. We express that vertical scale in dimensional terms as ch, where c is a dimensionless scaling factor. Assuming that salt sheets can be vertically uniform only inasmuch as the background flow is, we expect that c will be at most O() and probably smaller. The condition for validity of the TF approximation is then, in nondimensional terms, c 2π/l, orl 2π/c. Consistent with the above assumptions, we replace the background buoyancy gradients B Sz and B Tz, and velocity gradient U z with constants B Tz0 = R ρ GrP r/(r ρ ), B Sz0 = GrP r/(r ρ ) and U z0 = ReP r, equal to the gradients evaluated at z =0. With these assumptions, the eigenvalue problem (2-23) reduces to a cubic polynomial for the growth rate (Stern, 975): (σ + τl 2 )(σ + Prl 2 )(σ + l 2 ) + GrP rσ +(R ρ τ ) GrP r R ρ l2 =0. (30) While the TF approximation simplifies the problem considerably, it still admits no closed form solution simple enough to be of practical use. To make further progress, one may make the additional assumption that the growth rate is small in comparison to the viscous decay rate Prl 2, but large in comparison to the salinity decay rate τl 2. (This formulation assumes that growth rates are purely real.) We refer to this as the viscous control (hereafter VC) approximation. With it, the cubic is reduced to a quadratic, which is easily solved to find the growth rate and wavenumber of the fastest-growing instability: ( ) /2 σ = Gr /2 Rρ f(r ρ ); f(r ρ )= (3) R ρ and l = Gr /4. (32) The assumption l 2π/c becomes Gr (2π/c) 4 = 559/c 4. Given the uncertainty of c, all we can say is that Gr must be greater than O(0 3 ), possibly by several orders of magnitude, for these approximations to be valid; however, that condition is frequently satisfied in the ocean. The condition for the analytical solution (3,32) to be valid, τl 2 σ Prl 2, becomes τ f(r ρ ) Pr. (33) Common oceanic values of R ρ at which salt fingering is observed range between.25, for which f =.24, and 3, for which f =0.22. With τ =0.0 and Pr =7, these values are within the range specified in (33), so we may reasonably expect that the analytical solution (3,32) will be relevant. In limited regions, such as certain sublayers of a thermohaline interleaving complex, R ρ may be less than.25, and the resulting intense instability may not be well described by (3-33). The TF approximation allows explicit solution of the buoyancy equations (22,23) for longitudinal modes: ˆbT = ŵb Tz0 σ + l 2 ; ˆbS = ŵb Sz0 σ + τl 2. (34) The flux ratio (27) thus becomes γ sf = σ + τl2 σ + l 2 R ρ, (35) where once again the growth rate is assumed to be real. (These results will be generalized to include complex growth rates in the next subsection.) We cannot evaluate γ sf explicitly without knowledge of σ and l, but we can deduce the bounds R ρ τ γ sf <R ρ. With the additional assumption (33), we can substitute from (3,32) to obtain γ sf = R /2 ρ [R /2 ρ (R ρ ) /2 ], (36) (e.g. Kunze, 2003). Results quoted so far are independent of the mean flow. We turn now to the momentum flux driven by tall salt sheets aligned with the mean flow so that k = 0. The streamwise momentum equation (29) may be solved and multiplied by ŵ to obtain: u w = 2 ŵ 2 U z0 σ + Prl 2 e2σrt. (37) This specific momentum flux may be expressed in terms of a salt sheet viscosity ν ss that is independent of the mean flow: ν ss u w = 2 ŵ 2 U z σ + Prl 2 e2σrt. (38) 7

8 This may be combined with the saline diffusivity derived from (34) to form a Schmidt number: 0 4 tanh TF VC Sc ss u w /U z = σ + τl2 b S w /B Sz σ + Prl 2. (39) This Schmidt number is independent both of U and of time, and is bounded by Sc <Sc ss <. The limiting case σ =0, for which Sc ss = Sc, was described by Ruddick (985). Doing the same with the thermal buoyancy yields a Prandtl number that has similar form and is bounded by Pr <Pr sf <. The Schmidt number is of primary interest here, though, because saline buoyancy is the force that drives salt sheets. If, in addition, we now make the VC assumption (33), the Schmidt number becomes Sc ss = f(r ρ) Pr. (40) The Schmidt number is now independent not only of the mean flow but also of Gr. For a given Pr, the Schmidt number depends only on R ρ. Equivalent results for the localized gradients considered in this paper must be obtained numerically, but based on these approximations we may anticipate that, at sufficiently high Gr, the salt sheet Schmidt number will be, first, approximately independent of the mean flow and, second, bounded approximately within the interval (Sc, ). Moreover, when (33) holds, Sc ss will depend only on R ρ and Pr in accordance with (40). In figure 4, we show results for a case in which Gr = 0 8 is at the low end of the observed range. Nevertheless, this value exceeds 0 3 by five orders of magnitude, so we anticipate that the TF approximation will work well. The shear is relatively strong: the Reynolds number is 0 4 and the bulk Richardson number is 0.4. The TF approximation predicts both growth rates (figure 4a) and wavenumbers (figure 4b) of salt sheet instabilities accurately over the entire range of R ρ. Not surprisingly, the VC approximation is less successful. It gives accurate results for a limited range of R ρ in which (33) is valid (figure 4c). At lower R ρ, the wavenumber is underpredicted, while both growth rate and wavenumber are overpredicted at higher R ρ. Similar results are found for the flux ratio and the Schmidt number (figure 5). Here, we focus on the range σ max l max f(r ρ ) (a) (b) (c) R ρ Figure 4: Test of the tall finger (TF) and viscous control (VC) approximations for sheared salt sheets. Gr =0 8 ; Re = 0 4. Panels show FGM growth rate (a), spanwise wavenumber (b) and f(r ρ) (c) versus density ratio. Upper and lower dashed lines on (c) indicate f = Pr and f = τ. < R ρ 4, where SS growth rates are significant. The TF approximation is highly accurate, while the VC is much less so. There exists a range of small but oceanographically common values of R ρ in which the VC approximation (40) seriously overpredicts both the flux ratio and the Schmidt number. Unlike the other mode properties considered here, the Schmidt number is not guaranteed to be independent of the background shear by Squire s theorem. That independence is, however, a consequence of the TF approximation. A particular consequence of TF that is confirmed in figure 5a is the bound Sc <Sc ss <. (Note that Sc =.4 0 3, which is off the scale.) At lower Gr, we expect both the TF and VC approximations to be less accurate, and that is the case. In figure 6, we show γ sf (R ρ ) and Sc ss (R ρ ) for Gr =0 4. Re is set to, so the shear is very weak. (The bulk Richardson 8

9 γ sf 0.6 γ sf (a) (a) tanh TF VC fit Sc ss tanh TF VC fit Sc ss 0 (b) R ρ Figure 5: Test of the tall finger (TF) and viscous control (VC) approximations for sheared salt sheets. Gr =0 8 ; Re = 0 4. Panels show flux ratio (a) and Schmidt number (b) versus density ratio. γ sf Sc ss (a) tanh TF VC fit (b) R ρ Figure 6: Flux ratio (a) and Schmidt number (b) versus density ratio for weak salt sheets (Gr =0 4 ) and weak shear (Re = ). Numerical results are compared with the tall finger (TF) and viscous control (VC) approximations and with the empirical fit (40) for the Schmidt number. (b) R ρ Figure 7: Flux ratio (a) and Schmidt number (b) versus density ratio for weak salt sheets (Gr =0 4 ) and strong shear(re = 0 4 ). Numerical results are compared with the tall finger (TF) and viscous control (VC) approximations and with the empirical fit (40) for the Schmidt number. number is 400, so there is no shear instability.) The TF approximation exhibits a small but distinct departure from the numerical result, while the VC approximation is considerably less accurate. In contrast to the high-gr case, these results are sensitive to the background shear flow. Figure 7 shows results for Re = 0 4, and we see that the mismatch between the approximate and the numerical results is greater still. The flux ratio is overpredicted by about 50%, while Sc ss is overpredicted by a factor of two over much of the R ρ range. Because the simple formulas that result from the VC approximations are invalid in this parameter range, we propose an empirical fit for the Schmidt number: ( ) Rρ Sc ss = S(Gr, Re) ln. (4) R ρ The multiplicative factor S is determined via a best fit to computed values of Sc ss in the range <R ρ < 3. Asexemplified in figures 5b, 6b and 7b, this fit does reasonably well over the parameter range of interest, showing significant deviations from the numerical result only at very low values of R ρ. The dependence of S on Gr and Re provides a compact representation of the dependence of the Schmidt number 9

10 S Gr= Gr=0 0 Gr= Gr=0 6 Gr= Re Figure 8: Parameter S(Gr, Re) used in the parameterization (4). on those parameters. Over a wide range of parameter values relevant to oceanic double diffusion, S remains close to S is insensitive to the mean flow when Re is less than a certain value, which increases with increasing Gr (figure 8). Above this Reynolds number, the Schmidt number appears to decrease dramatically, though this primarily signals the breakdown of the empirical relation (4). For Gr > 0 5, this maximum Reynolds number is approximated by 000Gr /2, which corresponds to a minimum Richardson number of 0 6 /P r. In the ocean, the restriction Gr > 0 5 is the most stringent. Thus (4) with S =0.08 provides an alternative to (40) that is valid over a much broader region of parameter space. σ max l max γ dc Pr dc (a) (b) (c) tanh real tanh imag TF real TF imag tanh TF Kelley(990) tanh TF.3R ρ 9 2 (d) R ρ Figure 9: Test of the tall finger (TF) approximation for weakly sheared diffusive convection cells. Gr =0 8 ; Re = 0 4. Panels show FGM growth rate (a), spanwise wavenumber (b), flux ratio (c) and Prandtl number (d) versus density ratio. Also shown in (c) is the flux ratio for diffusive convection in the laboratory (Kelley, 990). Also shown on (d) is an empirical fit Pr dc =.3Rρ 9. is d. How do diffusive convection cells interact with shear? The VC approximation is invalid in the diffusive convection regime because σ i is generally not Prl 2. We will, however, extend the TF approximation to obtain a Prandtl number for diffusive convection in a shear flow. (Because thermal buoyancy is the driving force behind diffusive convection, flux parameterizations are generally expressed in terms of a thermal diffusivity and a Prandtl number, (e.g. Walsh and Ruddick, 998).) We simply extend (37-39) to allow for complex growth rates, and replace saline buoyancy with thermal buoyancy. The result Pr dc u w /U z = [/(σ + Prl2 )] r b T w /B Tz [/(σ + l 2. (42) )] r The addition of oscillatory modes expands the range of possible Prantdl numbers from Pr <Pr ss < (found in the salt fingering case) to Pr <Pr dc <Pr, i.e. the TF approximation leads us to expect that the Prandtl number for diffusive convection may be greater than unity. The truth of this prediction is demonstrated in figure 9, which shows a case with relatively strong double diffusive instability (Gr =0 8 ) and shear (Re =0 4 ). The TF approximation predicts growth rates, oscillation frequencies and wavenumbers very accurately in this case (figures 9a,b). Also well predicted are the flux ratio and 0

11 the Prandtl number. The flux ratio is significantly smaller than the laboratory-based empirical formulation of Kelley (990). We take this discrepancy as a caution that finite amplitude diffusive convection may not be well modeled by its linear counterpart. Pr dc increases from.2 in the limit R ρ to higher values at lower R ρ. This result is also consistent with Padman (994), who obtained <Pr dc < 3 from observations of a thermohaline staircase in the Arctic. To approximate the dependence of the diffusive convection Prandtl number on R ρ, we recommend the empirical fit Pr dc = {.3R 9 ρ, if 0.9 <R ρ < 3.4, if 0 <R ρ < 0.9. (43) This fit is generally good to within 0%, better at R ρ where diffusive convection is strong. It breaks down whenever Re is greater than about 00Gr /2,orPrRi b < 0 4. The exponential fit is terminated at R ρ =0.9 to prevent the overprediction of momentum fluxes at lower R ρ, where diffusive convection is inactive according to linear theory but not according to laboratory experiments (Kelley, 990). e. Which mode dominates? For oceanic parameter values, the KH growth rate may be approximated by σ KH =0.2ReP r( 4Ri b ). (44) Equating this with the salt sheet growth rate given by (3), we obtain 0.2Pr /2 ( 4Ri b )Ri /2 b = f(r ρ ), (45) which for any given Pr describes a curve on on the Ri b R ρ plane upon which the growth rates of the fastestgrowing KH and salt sheet modes are equal. For a given R ρ, (45) is easily solved for Ri c, the critical value of Ri b above which salt sheet instability dominates. Explicit calculation for the localized profiles (figure 0) confirms that Ri c based on (45) is accurate to within 0%. The growth rates of KH and double diffusive instabilities are comparable throughout the region where the individual instabilities are significant, and the double diffusive Ri b R ρ Figure 0: Ratio of double diffusive growth rate to KH growth rate for Gr =0 8. Dotted line is the analytical approximation (45). growth rate exceeds the KH growth rate over much of this region. We conclude that, even in the presence of inflectional shear with Ri b < /4, double diffusive instabilities are likely to play an important role. 4. Conclusions We have computed the linear stability characteristics of a stratified shear layer in which the stratification supports double diffusive instabilities, and compared the results with existing understanding of inflectional shear instabilities, salt sheets and diffusive convection. An important goal has been to parameterize momentum fluxes driven by double diffusive instabilities. Our main findings are as follows. The finite thickness of the stratified layer has a significant effect on the eigenmodes only when Gr < 0 5. For the larger values typical of oceanic double diffusion, stability characteristics are predicted accurately by the tall fingers approximation, which assumes that background gradients are uniform. Even in the presence of inflectional shear with Ri b < /4, double diffusive instabilities are strong enough

12 to play an important role in the dynamics. The surface on which the growth rates are equal is given approximately by (45). Properties of the linear KH instability are affected negligibly by diffusively unstable stratification unless R ρ is close to unity. Momentum flux by salt sheets is described by a Schmidt number Sc ss which, in accordance with the TF approximation, lies in the range Sc <Sc ss < and is by and large much smaller than unity. Sc ss is determined primarily by R ρ. For oceanically relevant parameter values, the parameterization (4) provides an accurate estimate of Sc ss. Both the Schmidt number and the flux ratio in salt sheets are decreased below their TF values when Gr is too small for the TF approximation to hold. This decrease is greater still when shear is applied. Momentum flux by diffusive convection cells is described by a Prandtl number Pr dc that, in accordance with the TF approximation, lies in the range Pr <Pr dc <Prand is generally greater than unity. Pr dc is determined primarily by R ρ. For oceanically relevant parameter values, the parameterization (43) provides a plausible estimate of Pr dc. These results are now being used in the modeling of thermohaline interleaving (Mueller et al., in preparation). The next stages in this line of research will involve () extension to more general background profiles, and (2) moving beyond linear perturbation theory and into the fully turbulent regime via direct numerical simulations. This will allow us to rigorously test the momentum flux parameterizations suggested by linear theory. Acknowledgements: This project has benefited from discussions with Rachael Mueller. Rita Brown provided editorial assistance. The work was supported by the National Science Foundation under grant OCE Appendix: Reduction of (4-6) to an algebraic eigenvalue problem Galerkin s method is used to discretize the z - dependence. Galerkin s method transforms our eigenvectors in the following manner. {ŵ(z), ˆb T (z), ˆb S (z)} = {û(z), ˆv(z), ˆπ(z)} = N {ŵ n, ˆb Tn, ˆb Sn }f n (z), (46) n= N {û n, ˆv n, ˆπ n }g n (z), (47) n= where ( nπz ) ( nπz ) f n (z) =sin ; g n (z) =cos. (48) H H The inner product operator of any two functions a(z) and b(z) is defined by H <a(z) b(z) >= 2 a(z) b(z)dz (49) H 0 where z =0and z = H are the lower and upper boundaries. Orthogonality of the inner product operation is expressed by <f m (z) f n (z) >= δ mn ; <g m (z) g n (z) >= δ mn ( + δ m0 ). (50) Applying Galerkin s method to the governing equations, we obtained: σ<f m 2 f n > ŵ n =( ik < f m U 2 f n > +ik < f m U zz f n > +ν <f m 4 f n >)ŵ n k 2 <f m f n > ˆb Tn k 2 <f m f n > ˆb Sn σ<f m f n > ˆb Tn = <f m B Tz f n > ŵ n ik < f m U f n > ˆb Tn + κ T <f m 2 f n > ˆb Tn σ<f m f n > ˆb Sn = <f m B Sz f n > ŵ n ik < f m U f n > ˆb Sn + κ S <f m 2 f n > ˆb Sn. This is a generalized matrix eigenvalue equation whose eigenvalue is σ and whose eigenvector is the concatenation of {ŵ,ˆb T, ˆb S }. 2

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