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

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

f(R ρ ) σ max l max γ

f(R ρ ) σ max l max γ s 10 0 10 −1 10 −2 10 2 10 1 10 0 10 −1 10 8 6 4 2 1 0.9 0.8 0.7 0.6 (a) (b) (c) (d) tanh TF VC 0.5 10 0 10 1 10 2 R ρ Figure 5: Same as figure 4, but for Gr =10 3 . bined with the saline diffusivity derived in the same manner from (41) to form the effective Schmidt number: Sc s ≡ −u′ w ′ /U z = σ + τl2 −b ′ S w′ /B Sz σ + Prl 2 . (46) This Schmidt number provides a useful route to estimation of the eddy viscosity of salt fingers via the more easily measured saline diffusivity. As a ratio of fluxes, it is independent of time, so that an estimate based on linear theory may be relevant at finite amplitude. (46) shows that, in the TF approximation, Sc s is also independent of the background shear and is bounded by Sc −1 ≤ Sc s < 1. Recall that Sc is the molecular Schmidt number, equal to 700 in this study. The limiting case σ = 0, for which Sc s takes its lower bound, Sc −1 , was described by Ruddick (1985). Note that the lower bound is positive, so that negative eddy viscosities are excluded. The upper bound Sc s < 1 shows that eddy viscosity cannot exceed saline diffusivity. Doing the same with the thermal buoyancy yields a Prandtl number that has similar form and is bounded by Pr −1

Sc s 10 0 10 −1 tanh TF VC fit of magnitude. The fit (48) shows significant deviations from the numerical result only at extremely low values of R ρ . The latter can be remedied by capping Sc s at 0.3 for R ρ < 1.02. Sc s 10 −2 10 0 10 −1 10 −2 10 0 (a) Gr=10 6 Re=1 Ri b =1.4× 10 5 (b) Gr=10 2 Re=1 Ri b =14 (c) The dissipation ratio The dissipation ratio Γ (Oakey, 1985) is frequently used as a surrogate for mixing efficiency in observational data analysis (e.g. Moum, 1996; Ruddick et al., 1997; Smyth et al., 2001), and also for distinguishing mixing due to salt fingering from that due to shear-driven turbulence (e.g. McDougall and Ruddick, 1992; St.Laurent and Schmitt, 1999). The dissipation ratio may be defined in terms of either temperature or salinity, but the former is used more commonly for ease of measurement: Sc s 10 −1 (c) Gr=10 2 Re=10 4 Ri b =1.4× 10 −7 10 −2 1 1.5 2 2.5 3 3.5 4 R ρ Figure 6: Schmidt number versus density ratio for sheared, longitudinal salt sheets. TF, VC, and empirical (48) approximations are identified in the legend to (a). TF is valid at high Gr (a). Even at low Gr, the Schmidt number is essentially independent of the mean shear (b,c). the TF approximation that makes mean shear irrelevant is not valid for these cases, the mean shear has no discernible effect. Because the VC approximation (47) is invalid at low R ρ , we propose an empirical fit for the Schmidt number: Sc s =0.08 ln ( Rρ R ρ − 1 ) . (48) This is shown by the dotted curves on figure 6. (48) is a good fit to the TF approximation to Sc s , and is therefore valid wherever the latter is. Our previous scaling considerations led to the criterion Gr ≫ 10 3 , while numerical results suggest that Gr > 10 4 is sufficient. This condition is commonly satisfied in the ocean; for example, the observed range in C-SALT was 10 8 ≤ Gr ≤ 2×10 9 (section 2b), exceeding the requirement by at least four orders Γ= χB z 2ɛBTz 2 . (49) Here, χ represents the dissipation rate of thermal buoyancy variance, χ = 〈2| ∇b ⃗ T | 2 〉 in nondimensional form, and ɛ = 〈2Pr e 2 ij 〉 is the dissipation rate of perturbation kinetic energy. The tensor e ij =(∂u i /∂x j +∂u j /∂x i )/2 quantifies the strain rate. Angle brackets indicate a spatial average, over the wavelength of the disturbance in the present context. At steady state, balances of kinetic energy and scalar variance imply Γ= R f R f − 1 R ρ − 1 R ρ γ s 1 − γ s , (50) where R f = −(b ′ T + b′ S )w′ /u ′ w ′ U z is the flux Richardson number (St.Laurent and Schmitt, 1999). For unsheared salt fingers, this takes the simpler form Γ= R ρ − 1 γ s (51) R ρ 1 − γ s (Hamilton et al., 1989; McDougall and Ruddick, 1992). Shear-driven turbulence typically gives values of Γ near 0.2 (Moum, 1996), whereas higher values are expected for salt fingers. St.Laurent and Schmitt (1999) found Γ ranging up to 0.6, and occasionally higher, in regions of strong thermal microstructure but weak shear, and concluded that the microstructure was due to salt fingering. In the linear regime of interest to us here, both χ and ɛ grow exponentially in time, but their ratio is constant, 11

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