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

z 1 0 (a) b′w′ (b) b′w′ (c) u′w′ (d) b′w′ (e) u′w′ (f) b′w′ (g) u′ w′ nance of the positive buoyancy flux due to sal**in**ity (figure 2d). For this example, γ s =0.55. In addition, convective motions with**in** the salt sheets tilt aga**in**st the shear, so that the velocity flux (figure 2e) draws energy from the mean flow to the perturbation. It is important to recognize that this energy transfer amplifies only motions **in** the x direction, **and** thus does not supply energy to the salt sheets per se. In fact, the background flow has no effect whatsoever on the salt sheet **in**stability, **in** accordance with Squire’s theorem. −1 −50 0 50 −4 0 4 −4 −2 0 −4 0 4 −2 0 −1 0 1 −1 0 Figure 2: Vertical fluxes of buoyancy **and** velocity for the three illustrative cases shown by bullets **in** figure 1, plus an example of sheared **diffusive** convection. (a) Pure salt f**in**ger**in**g: Re = 0, Gr = 10 6 , R ρ = 1.6, k = 22.36, l = 22.36 (bullet on figure 1a). The velocity flux is not shown as it is identically zero. (b,c) Pure KH: Re = 300,Ri b =0.1, R ρ = 100, k = 0.47, l = 0 (bullet on figure 1b). (d,e) Sheared salt f**in**gers: Re = 100, Ri b =0.1, Gr = 7000 R ρ =1.6, k =0, l =9.15 (bullet on figure 1c). (f,g) Sheared **diffusive** convection: Re = 100, Ri b =0.1, Gr = 7000 R ρ =0.96, k =0, l =9.15. (a,b,d,f) buoyancy fluxes: total (solid), thermal (dashed), sal**in**e (dotted). All buoyancy fluxes have been divided by 1000 for plott**in**g convenience. (c,e,g) velocity flux. has the value 0.59 for this example. This value agrees well with measurements of salt f**in**gers **in** lab **and** numerical experiments (Kunze, 2003) **and** **in** the ocean (Schmitt, 2003), despite hav**in**g been derived from l**in**ear theory. This flux ratio also obta**in**s **in** the vastly different parameter regime of sugar-salt f**in**gers (Schmitt, 1979). KH **in**stability grows because the velocity flux (figure 2c) acts to reduce the background shear **and** thus draws energy from the mean flow **in**to the perturbation. Because the stratification is almost purely thermal **in** this example, the same is true of the buoyancy flux (figure 2b). In fact, the flux ratio for this case is 93. This is similar to the value of the density ratio R ρ = 100, **in**dicat**in**g that the difference **in** the diffusivities of the two scalars plays no significant role **in** the dynamics, consistent with the fact that the mechanism of KH **in**stability is essentially **in**viscid. Salt sheets, like salt f**in**gers, grow because of the domi- The **momentum** **and** buoyancy fluxes may be expressed **in** terms of effective diffusivities via the usual fluxgradient formalism. The ratio of **momentum** to thermal diffusivity, here called the salt sheet Pr**and**tl number Pr s has the value 0.19, while the Schmidt number Sc s , the ratio of **momentum** to sal**in**e diffusivity, is 0.07. Diffusive convection is an oscillatory **in**stability driven by the dom**in**ance of the positive thermal buoyancy flux over the negative sal**in**e buoyancy flux (figure 2f). The f**in**ite amplitude expression of **diffusive** convection is a non-oscillatory convective layer**in**g, suggest**in**g a qualitative change **in** the physics between the small **and** large amplitude regimes, so that results from l**in**ear theory must be **in**terpreted with caution. The flux ratio is def**in**ed oppositely to that for salt f**in**gers: γ d = −b ′ S w′ /b ′ T w′ , (35) **and** has the value 0.21 for this example. Like salt sheets, sheared **diffusive** convection generates a vertical **transport** of **momentum** (figure 2f) via longitud**in**al convection cells. In contrast with the salt sheet case, **momentum** diffusivities are larger than scalar diffusivities. For the example shown **in** figure 2e,f, the Pr**and**tl number of the **diffusive** convection is Pr d =1.46 **and** the Schmidt number is Sc d =7.07. In subsequent sections, we will explore **in** detail the relationships between sheared, doubly **diffusive** flow profiles **and** the **in**stabilities they generate. 6

4 How are KH billows affected by **double**-**diffusive**ly unstable stratification? 2 1.5 3 3.5 3.5 3.25 For most oceanographic parameter values, **diffusive**ly unstable stratification has negligible effect on KH **in**stability. While this result is not obvious a priori, it is not surpris**in**g as the KH **in**stability mechanism is **in**viscid. An exception occurs where R ρ ∼ 1. There, **double**-**diffusive** **in**stability becomes so strong that, even when damped by shear, it overwhelms shear **in**stability. (In this case the longitud**in**al salt sheet **in**stability, be**in**g free of the damp**in**g effect of shear, is the dom**in**ant mode. Growth rates of salt sheets **and** KH billows are compared explicitly **in** section 7.) An example is given **in** figure 3, which shows growth rate as a function of R ρ **and** k for purely transverse modes with shear strong enough that the Richardson number is subcritical (Ri b = 0.14). The black contour signifies σ i =0; modes to the right of that contour are oscillatory. The white, dotted contour **in**dicates the fastest-grow**in**g mode (FGM) at each R ρ . For R ρ > 1.6, the FGM is stationary **and** has wavenumber near 0.45, as is characteristic of KH **in**stability (Hazel, 1972). At wavenumbers smaller than this fastest-grow**in**g KH mode, growth rates are **in**dependent of R ρ for all R ρ , as **in**dicated by vertical contours. In ord**in**ary stratification with Ri b =0.14, wavenumbers greater than 0.83 would be stable. Instead, a secondary maximum **in** σ r is found at much higher wave numbers (k ∼ 10 2 ), where salt f**in**ger**in**g **in**stabilities are damped **and** rendered oscillatory by the shear. The latter occurs when f**in**ger amplitudes are maximized not at z = 0, as **in** the example shown **in** figure 2a, but at some other depth where the background flow is nonzero. The background flow therefore advects the **in**stability, Dopplershift**in**g the oscillation frequency away from zero. For 1

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- Page 3 and 4: ackground flow by transporting mome
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- Page 13 and 14: 6000 σ r 5800 10 4 Sc s 5600 0.078
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