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

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 . b**in**ed with the sal**in**e 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 f**in**gers via the more easily measured sal**in**e diffusivity. As a ratio of fluxes, it is **in**dependent of time, so that an estimate based on l**in**ear theory may be relevant at f**in**ite amplitude. (46) shows that, **in** the TF approximation, Sc s is also **in**dependent 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 limit**in**g 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 sal**in**e diffusivity. Do**in**g the same with the thermal buoyancy yields a Pr**and**tl 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 capp**in**g 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 mix**in**g efficiency **in** observational data analysis (e.g. Moum, 1996; Ruddick et al., 1997; Smyth et al., 2001), **and** also for dist**in**guish**in**g mix**in**g due to salt f**in**ger**in**g from that due to shear-driven turbulence (e.g. McDougall **and** Ruddick, 1992; St.Laurent **and** Schmitt, 1999). The dissipation ratio may be def**in**ed **in** terms of either temperature or sal**in**ity, 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, longitud**in**al 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 **in**dependent 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 **in**valid 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 scal**in**g 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), exceed**in**g 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 k**in**etic energy. The tensor e ij =(∂u i /∂x j +∂u j /∂x i )/2 quantifies the stra**in** rate. Angle brackets **in**dicate a spatial average, over the wavelength of the disturbance **in** the present context. At steady state, balances of k**in**etic 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 f**in**gers, 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 f**in**gers. St.Laurent **and** Schmitt (1999) found Γ rang**in**g 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 f**in**ger**in**g. In the l**in**ear regime of **in**terest to us here, both χ **and** ɛ grow exponentially **in** time, but their ratio is constant, 11

- Page 1 and 2: Instability and diapycnal momentum
- Page 3 and 4: ackground flow by transporting mome
- Page 5 and 6: l d. Parameter values The molecular
- Page 7 and 8: 4 How are KH billows affected by do
- Page 9: than 1.25, and the resulting intens
- Page 13 and 14: 6000 σ r 5800 10 4 Sc s 5600 0.078
- Page 15 and 16: 8 Conclusions We have computed the
- Page 17 and 18: Radko, T., 2003: A mechanism for la