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

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

We

We begin the discussion of salt sheets by establishing an analytical framework within which to interpret the numerical results, based on existing theories for salt fingers in an unsheared environment with uniform scalar gradients. We show that, even in a sheared environment with localized background gradients, these approximations can work quite well provided Gr is sufficiently large. We then investigate the diapycnal momentum flux due to salt sheets, as described by the Schmidt number, Sc s , the ratio of the effective viscosity of the salt sheets to their effective saline diffusivity. We find that Sc s is much less than unity in the linear regime, indicating that salt sheets may be relatively inefficient at fluxing momentum. Finally, we describe the dissipation ratio Γ, whose value is commonly used to distinguish double diffusive mixing from mechanically driven turbulence in complex ocean environments. The linear approximation to Γ turns out to bear significant resemblance both to its finite-amplitude, steady-state form and to observational estimates. (a) Comparison with salt sheets in uniform gradients 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 (1975). 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 h is the layer thickness and c is a dimensionless scaling factor. Assuming that salt sheets can be vertically uniform only inasmuch as the mean profiles (19) - (21) are, we expect that c will be at most O(1) and probably smaller. The condition for validity of the TF approximation is then, in nondimensional terms, l ≫ 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 equal to the gradients evaluated at z =0: B Tz0 = R ρ GrP r/(R ρ − 1), B Sz0 = −GrP r/(R ρ − 1) and U z0 = ReP r. With these assumptions, the eigenvalue problem (23-25) reduces to a cubic polynomial for the growth rate (Stern, 1975): (σ + τl 2 )(σ + Prl 2 )(σ + l 2 ) + GrP rσ +(R ρ τ − 1) GrP r R ρ − 1 l2 =0. (37) While the TF approximation simplifies the problem considerably, it still requires the solution of a cubic poynomial. 2 To make further progress, we 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 (Stern, 1975). 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: σ = Gr 1/2 f(R ρ ); f(R ρ )= and ( ) 1/2 Rρ − 1 (38) R ρ − 1 l = Gr 1/4 . (39) The assumption l ≫ 2π/c now becomes Gr ≫ (2π/c) 4 = 1559/c 4 . Given the uncertainty of c, all we can say is that Gr must be greater than O(10 3 ), possibly by several orders of magnitude, for these approximations to be valid; however, that condition is frequently satisfied in the ocean (section 2d). The opposite extreme, Gr ≪ 10 3 , corresponds to the thin interface limit, which is known to be a poor description of double-diffusive instability in the ocean (e.g. Kunze, 2003). The condition for the analytical solution (38,39) to be valid, τl 2 ≪ σ ≪ Prl 2 , becomes τ ≪ f(R ρ ) ≪ Pr. (40) Common oceanic values of R ρ at which salt fingering is observed range between 1.25, for which f =1.24, and 3, for which f =0.22. With τ =0.01 and Pr =7, these values are within the range specified in (40), so we may reasonably expect that the analytical solution (38,39) will be relevant. In limited regions, such as certain sublayers of a thermohaline interleaving complex, R ρ may be less 2 For complete analytic solution see Schmitt (1979) and Schmitt (1983). 8

than 1.25, and the resulting intense instability may not be well described by (38-40). The TF approximation allows explicit solution of the buoyancy equations (24,25) for longitudinal modes: ˆbT = −ŵB Tz0 σ + l 2 ; ˆbS = −ŵB Sz0 σ + τl 2 . (41) The flux ratio (34) thus becomes γ s = σ + τl2 σ + l 2 R ρ, (42) where once again the growth rate is assumed to be real. (These results will be generalized to include complex growth rates in section 6.) We cannot evaluate γ s explicitly without knowledge of σ and l, but we can deduce the bounds R ρ τ ≤ γ s

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