- Text
- Shear,
- Instability,
- Flux,
- Ratio,
- Diffusive,
- Buoyancy,
- Convection,
- Linear,
- Approximation,
- Schmidt,
- Diapycnal,
- Momentum,
- Oce.orst.edu

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

We beg**in** the discussion of salt sheets by establish**in**g an analytical framework with**in** which to **in**terpret the numerical results, based on exist**in**g theories for salt f**in**gers **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 **in**vestigate 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 sal**in**e diffusivity. We f**in**d that Sc s is much less than unity **in** the l**in**ear regime, **in**dicat**in**g that salt sheets may be relatively **in**efficient at flux**in**g **momentum**. F**in**ally, we describe the dissipation ratio Γ, whose value is commonly used to dist**in**guish **double** **diffusive** mix**in**g from mechanically driven turbulence **in** complex ocean environments. The l**in**ear approximation to Γ turns out to bear significant resemblance both to its f**in**ite-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 f**in**gers” (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 scal**in**g factor. Assum**in**g that salt sheets can be vertically uniform only **in**asmuch 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 sal**in**ity 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 f**in**d the growth rate **and** wavenumber of the fastest-grow**in**g **in**stability: 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 uncerta**in**ty 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 th**in** **in**terface limit, which is known to be a poor description of **double**-**diffusive** **in**stability **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 f**in**ger**in**g 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 with**in** the range specified **in** (40), so we may reasonably expect that the analytical solution (38,39) will be relevant. In limited regions, such as certa**in** sublayers of a thermohal**in**e **in**terleav**in**g complex, R ρ may be less 2 For complete analytic solution see Schmitt (1979) **and** Schmitt (1983). 8

than 1.25, **and** the result**in**g **in**tense **in**stability may not be well described by (38-40). The TF approximation allows explicit solution of the buoyancy equations (24,25) for longitud**in**al 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 aga**in** the growth rate is assumed to be real. (These results will be generalized to **in**clude 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

- 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: 4 How are KH billows affected by do
- Page 11 and 12: Sc s 10 0 10 −1 tanh TF VC fit of
- 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