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

so that the value of Γ may conceivably have some relevance for f**in**ite amplitude disturbances. Apply**in**g the TF approximation for longitud**in**al salt sheets [i.e. sett**in**g k =0, ˆv =0**and** ∂/∂z =0, solv**in**g for û **and** ˆb T as was done **in** the previous subsection, **and** substitut**in**g **in**to (49)], one obta**in**s 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** comb**in**e to reproduce the f**in**ite-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 dom**in**ate 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 **in**stability. The latter class of modes, be**in**g transverse, is not described by (53), but typically has Γ ∼ 0.2 or less. 3 For example, the shear **in**stability shown **in** figure 2b,c has Γ=0.174. Results for the fastest-grow**in**g salt sheet **in**stability 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 with**in** 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, f**in**ite 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 m**in**imum 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 l**in**es **in**dicate 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 **in**crease **in** Γ at high Ri is greater **in** the observations. The observed dependence of Γ on R ρ is opposite to the theoretical prediction, show**in**g a decrease with **in**creas**in**g R ρ where all theories (50, 51, 52, 53) predict an **in**crease. In the l**in**ear case, shear effects become dom**in**ant when Ri b < Pr −1 = 1/7, whereas **in** the observations the change **in** Γ occurs near Ri =1. This discrepancy reflects the approximations **in**herent **in** l**in**ear theory, but it is also due **in** part to a difference **in** the def**in**ition of the Richardson number. Here, Ri b is the m**in**imum 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. Averag**in**g **in**evitably 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 l**in**ear regime is relevant at f**in**ite 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 f**in**gers 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 Longitud**in**al salt sheets have k =0**and** 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 f**in**ger modes could **in**teract with the mean shear **in** other ways, possibly generat**in**g a strong **momentum** flux, **and**/or rapid k**in**etic energy dissipation, despite their relatively low growth rates. In an environment with fluctuat**in**g shear, e.g. **in**ertial oscillations, salt sheets are always be at least slightly oblique, **and** hence have the potential to **in**teract 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. Increas**in**g k from zero causes a small decrease **in** growth rate, but differences **in** Sc s **and** Γ are **in**significant. In conclusion, obliquity does not appear to alter the conclusions drawn previously **in** this section. Figure 9: Test of the “tall f**in**ger” (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** Pr**and**tl 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 **in**teract with shear? The VC approximation is **in**valid **in** the **diffusive** convection regime because σ i is generally not ≪ Prl 2 . We will, however, extend the TF approximation to obta**in** a Pr**and**tl number for **diffusive** convection **in** a shear flow. [Because thermal buoyancy is the driv**in**g force beh**in**d **diffusive** convection, flux parameterizations are generally expressed **in** terms of a thermal diffusivity **and** a Pr**and**tl 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 sal**in**e buoyancy with thermal buoyancy. 13

- 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 and 10: than 1.25, and the resulting intens
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- Page 17 and 18: Radko, T., 2003: A mechanism for la