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# Instability and diapycnal momentum transport in a double-diffusive ...

Instability and diapycnal momentum transport in a double-diffusive ...

## so that the value of Γ

so that the value of Γ may conceivably have some relevance for finite amplitude disturbances. Applying the TF approximation for longitudinal salt sheets [i.e. setting k =0, ˆv =0and ∂/∂z =0, solving for û and ˆb T as was done in the previous subsection, and substituting into (49)], one obtains Γ TF = Gr (σ + Prl 2 ) 2 (σ + l 2 ) 2 Re 2 Pr 2 +(σ + Prl 2 ) 2 . (52) In contrast to the Schmidt number, Γ is clearly a function of the mean shear (via Re) in the TF approximation. This is because perturbation velocities parallel to U (which are orthogonal to the salt sheet motions and do not affect their behavior in any way) make a significant contribution to ɛ. The TF approximation (52) can be shown to be bounded by 0 ≤ Γ TF ≤ Gr/(Re 2 + l 4 ). The upper bound corresponds to the steady state σ =0. As a consistency check, one may set σ = 0 in (37), (42), (46) and (52) and combine to reproduce the finite-amplitude, steady-state result (50). When the VC approximation is made, (52) becomes Γ VC = R ρ − 1 Gr R ρ Gr + Re 2 = R ρ − 1 PrRi b R ρ PrRi b +1 . (53) Note that the dependence on R ρ is the same as in the steady state case (50), but other details differ. As is clearest from the first equality in (53), the imposition of shear (Re > 0) acts to reduce Γ VC . The second equality shows that shear effects are negligible when Ri b ≫ Pr −1 , whereas shear effects dominate when Ri b ≪ Pr −1 and take Γ VC to zero in the limit Ri b → 0. In that case, the salt sheets are supplanted by shear instability. The latter class of modes, being transverse, is not described by (53), but typically has Γ ∼ 0.2 or less. 3 For example, the shear instability shown in figure 2b,c has Γ=0.174. Results for the fastest-growing salt sheet instability with Gr =10 8 are shown in figure 7. R ρ is set to the oceanographically common value 2. The TF approximation is accurate to within 1% of the numerical value for a wide range of Ri b , while the VC approximation is up to ∼10% low. For Ri b < 0.08, Γ is smaller than 0.2, the canonical value for shear-driven turbulence. 3 Note that significantly higher values are possible for preturbulent, finite amplitude KH billows, e.g. Smyth et al. (2001). Γ 0.5 0.4 0.3 0.2 0.1 0 10 −2 10 −1 10 0 10 1 Ri b tanh TF VC Figure 7: Dissipation ratio versus minimum Richardson number for salt sheets with R ρ =2, Gr =10 8 and k =0. Solid and dotted curves show the TF and VC approximations. The shaded region represents the range 0 < Γ ≤ 0.2 expected for shear-driven turbulence. Vertical lines indicate Ri b =0.25 and Ri b =1. At large Ri b (weak shear), Γ asymptotes to a value slightly higher than than the VC limit (R ρ −1)/R ρ =0.5. This is consistent with values observed at high Richardson number by St. Laurent and Schmitt (1999, their figure 9), although the increase in Γ at high Ri is greater in the observations. The observed dependence of Γ on R ρ is opposite to the theoretical prediction, showing a decrease with increasing R ρ where all theories (50, 51, 52, 53) predict an increase. In the linear case, shear effects become dominant when Ri b < Pr −1 = 1/7, whereas in the observations the change in Γ occurs near Ri =1. This discrepancy reflects the approximations inherent in linear theory, but it is also due in part to a difference in the definition of the Richardson number. Here, Ri b is the minimum value of the gradient Richardson number, evaluated precisely at the center of the shear layer. In contrast, the observational Ri is based on the shear of a velocity profile measured with ∼ 1m resolution. Averaging inevitably smooths away some fraction of the shear and thus leads to higher Ri. We conclude that the behavior of the salt sheet dissipation ratio in the linear regime is relevant at finite ampli- 12

6000 σ r 5800 10 4 Sc s 5600 0.078 0.076 0.074 0.072 0.07 σ max 10 3 10 2 80 60 (a) tanh real tanh imag TF real TF imag Γ 0.18 0.175 l max 40 20 1 (b) 0.17 0 0.05 0.1 0.15 0.2 0.25 0.3 k Figure 8: Real growth rate (a), Schmidt number (b) and dissipation ratio (c) versus wavenumber k, for salt fingers with Gr =10 8 , Ri b =0.1, Re =10 4 , l =99.8. tude. In particular, the known tendency of shear to reduce Γ is manifest in small amplitude salt sheets. γ dc Pr dc 0.8 0.6 0.4 0.2 0 6 4 (c) tanh TF Kelley(1990) tanh TF 1.3R ρ −9 2 (d) 0 0.88 0.9 0.92 0.94 0.96 0.98 1 R ρ (d) Obliquity effects Longitudinal salt sheets have k =0and are therefore not directly affected by the mean shear U(z). When k becomes nonzero, the mean flow acts to damp the growth rate, as shown on figure 1c. The possibility exists, however, that these oblique salt finger modes could interact with the mean shear in other ways, possibly generating a strong momentum flux, and/or rapid kinetic energy dissipation, despite their relatively low growth rates. In an environment with fluctuating shear, e.g. inertial oscillations, salt sheets are always be at least slightly oblique, and hence have the potential to interact with the mean shear in unexpected ways. Figure 8 shows the k-dependence of the growth rate, Schmidt number and dissipation ratio for a case with realistic Grashof number and strong shear (Ri b =0.1) The result is essentially negative. Increasing k from zero causes a small decrease in growth rate, but differences in Sc s and Γ are insignificant. In conclusion, obliquity does not appear to alter the conclusions drawn previously in this section. Figure 9: Test of the “tall finger” (TF) approximation for weakly sheared diffusive convection cells. Gr =10 8 ; Re = 10 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, 1990). Also shown on (d) is an empirical fit Pr d =1.3Rρ −9 . 6 How does diffusive convection 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, 1998). If needed, the Schmidt number can be computed as Sc d = Pr d R ρ /γ d .] We simply extend (44 - 46) to allow for complex growth rates, and replace saline buoyancy with thermal buoyancy. 13

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