- Text
- Shear,
- Instability,
- Flux,
- Ratio,
- Diffusive,
- Buoyancy,
- Convection,
- Linear,
- Approximation,
- Schmidt,
- Diapycnal,
- Momentum,
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Instability and diapycnal momentum transport in a double-diffusive ...

cluded **in**flectional profiles (i.e. flows **in** which shear is vertically localized, our ma**in** focus here). Kunze (1990, 1994) considered equilibration of f**in**ite-amplitude salt f**in**gers **in** shear, **in**clud**in**g the oceanically important case of **in**ertially rotat**in**g shear. Wells et al. (2001) conducted lab experiments on salt f**in**gers **in** a vertically periodic shear **and** showed how the strength of the salt f**in**gers adjusted to create an equilibrium state. Other **in**vestigators have looked **in**to the spontaneous generation of shear by divergent **double**-**diffusive** fluxes (Paparella **and** Spiegel, 1999; Stern, 1969; Holyer, 1981). The ability of **double****diffusive** motions to flux horizontal **momentum** **in** the vertical has been addressed experimentally by Ruddick (1985) **and** Ruddick et al. (1989), **and** observationally by Padman (1994). Here, we consider **in**stability **in** a vertically localized layer where **diffusive**ly unstable stratification **and** shear co**in**cide. The **in**flectional nature of the imposed shear yields the possibility of Kelv**in**-Helmholtz (KH) **in**stability **in**dependent of **double** diffusion. We del**in**eate regions of parameter space **in** which each class of **in**stability is likely to dom**in**ate, **and** also consider the effects of **double****diffusive** stratification on KH **in**stability. We exam**in**e the effect that the f**in**ite vertical extent of the stratified layer has on the characteristics of **double**-**diffusive** **in**stability relative to the previously studied case of uniform gradients. One of the most useful outcomes that any study of ocean mix**in**g can produce is a prediction of the fluxes due to the mix**in**g process **in** question. Predictions of **double****diffusive** fluxes are needed for the **in**terpretation of observational data (e.g. Gregg **and** Sanford, 1987; Padman **and** Dillon, 1987; Hebert, 1988; Rudels et al., 1999; Timmermans et al., 2003) **and** for models rang**in**g from global simulations (Zhang et al., 1999; Merryfield et al., 1999) to small scale processes studies (e.g. Walsh **and** Ruddick, 1998; Merryfield, 2000). In the l**in**ear (small amplitude) limit, fluxes grow exponentially. In reality, this exponential growth is arrested with the onset of nonl**in**ear effects, which generally lead the flow to a turbulent state. Fluxes occurr**in**g at f**in**ite amplitude cannot be predicted from l**in**ear theory. The situation becomes more encourag**in**g if one considers ratios of fluxes. As a l**in**ear eigenmode grows exponentially **in** time, any flux ratio will rema**in** constant. One therefore knows the value of the flux ratio at the onset of nonl**in**ear effects, **and** this provides a plausible estimate of the flux ratio that will perta**in** **in** fully nonl**in**ear flow. For example, the ratio of thermal to sal**in**e buoyancy fluxes **in** exponentially grow**in**g salt f**in**gers is a simple function of R ρ that agrees well with the flux ratio **in** the fully nonl**in**ear state (e.g. Schmitt, 1979; Kunze, 2003). Here, we attempt to parameterize **momentum** fluxes **in** terms of the better understood buoyancy fluxes, us**in**g flux ratios **in** the form of Pr**and**tl **and** Schmidt numbers. Results are **in** reasonable agreement with available laboratory **and** observational evidence, but provide a more comprehensive view of the parameter dependences of these flux ratios. The dissipation ratio Γ is frequently used to characterize mix**in**g processes (e.g. McDougall **and** Ruddick, 1992; Ruddick et al., 1997; St.Laurent **and** Schmitt, 1999). Like the Pr**and**tl **and** Schmidt numbers, Γ is constant **in** the l**in**ear regime. We compute Γ for small amplitude salt sheets **in** shear, **and** f**in**d that the results compare reasonably well with observations of f**in**ite amplitude salt f**in**ger**in**g **in** weakly sheared flow (St.Laurent **and** Schmitt, 1999). Our primary focus is on the salt f**in**ger**in**g regime, **in** which warm, salty fluid overlies cooler, fresher fluid. In the opposite scenario, l**in**ear theory predicts oscillatory **diffusive** convection. Diffusive convection as it occurs **in** the laboratory **and** the ocean is an **in**tr**in**sically f**in**iteamplitude phenomenon **and** is poorly modeled by its l**in**ear counterpart. Nevertheless, the l**in**ear dynamics are not without **in**terest **and** are **in**cluded here for completeness. Our basic tool is l**in**ear stability analysis, whose application is described **in** detail **in** section 2. The description of the results beg**in**s **in** section 3 with an overview of the ma**in** classes of unstable modes that may occur **in** a **diffusive**ly unstable, sheared flow: salt f**in**gers, salt sheets, KH **in**stabilities, **and** **diffusive** convection. We discuss the physics of each **and** def**in**e Schmidt **and** Pr**and**tl numbers with which to describe the **momentum** fluxes. In section 4, we look at the effects of **double**-**diffusive** stratification on KH **in**stabilities. Background shear does not affect **double**-**diffusive** **in**stabilities directly (except to select their orientation), but the **in**stabilities may affect the 2

ackground flow by **transport****in**g **momentum** **in** the vertical. This process, **and** its parameterization, are the subjects of section 5. We then give an equivalent (though less detailed) discussion for **diffusive** convection **in** section 6. In 7, we del**in**eate the regions of parameter space **in** which each of these classes of **in**stabilities is expected to dom**in**ate. Conclusions are given **in** section 8. 2 Methodology a. The eigenvalue problem for l**in**ear normal modes We assume that density is a l**in**ear function of temperature **and** sal**in**ity **and** neglect **in**ertial effects of density variations **in** accordance with the Bouss**in**esq approximation. The velocity field ⃗u(x, y, z, t) ={u, v, w} is measured **in** a nonrotat**in**g, Cartesian coord**in**ate system {x, y, z}. The result**in**g field equations are: D⃗u Dt = −∇π ⃗ + bˆk + ν∇ 2 ⃗u (1) ∇·⃗u = 0 (2) Db T Dt = κ T ∇ 2 b T (3) Db S Dt = κ S ∇ 2 b S (4) b = b T + b S (5) The variable π represents the pressure scaled by the uniform characteristic density ρ 0 , **and** ˆk is the vertical unit vector. Buoyancy is def**in**ed as b = −g(ρ−ρ 0 )/ρ 0 , where g is the acceleration due to gravity. Buoyancy is the sum of a thermal component b T **and** a sal**in**e component b S . K**in**ematic viscosity **and** thermal **and** sal**in**e diffusivities are represented by ν, κ T **and** κ S , respectively. Boundary conditions are periodic **in** the horizontal, flux free at z =0 **and** z = H. The velocity, buoyancy **and** pressure terms are separated **in**to two parts, a background profile **and** a perturbation: ⃗u(x, y, z, t) = U(z)î + ε⃗u ′ (x, y, z, t) (6) b T (x, y, z, t) = B T (z)+εb ′ T (x, y, z, t) (7) b S (x, y, z, t) = B S (z)+εb ′ S(x, y, z, t) (8) π(x, y, z, t) = Π(x, y, z, t)+επ ′ (x, y, z, t) (9) We substitute (6-9) **in**to (1-4) then collect the O(ɛ) terms. The perturbations are assumed to take the normal mode form: w ′ (x, y, z, t) =ŵ(z)e σt+i(kx+ly) + c.c. (10) (for example), where k **and** l are wavenumbers **in** streamwise **and** spanwise directions, respectively **and** σ is the growth rate of the perturbation. Substitut**in**g these perturbations **in**to the O(ɛ) equations gives: σû = −ikUû − ŵU z − ikˆπ + ν ¯∇ 2 û (11) σˆv = −ikU ˆv − ilˆπ + ν ¯∇ 2ˆv (12) σŵ = −ikUŵ − ˆπ z + ˆb + ν ¯∇ 2 ŵ (13) i(kû + lˆv)+ŵ z =0 (14) σˆb T = −ikUˆb T − ŵB Tz + κ T ¯∇2ˆbT (15) σˆb S = −ikUˆb S − ŵB Sz + κ s ¯∇2ˆbS , (16) ˆb = ˆbT + ˆb S , (17) where the subscript z **in**dicates the derivative, ¯∇2 = d 2 /dz 2 − ˜k 2 **and** ˜k 2 = k 2 + l 2 . Elim**in**at**in**g ˆπ from (11- 14), we obta**in** σ ¯∇ 2 ŵ = −ikU ¯∇ 2 ŵ + ikŵU zz + ν ¯∇ 4 ŵ − ˜k 2ˆb. (18) Equations (15-18) form a closed, generalized, differential eigenvalue problem whose eigenvalue is σ **and** whose eigenvector is the concatenation of {ŵ,ˆb T , ˆb S }. The horizontal velocities **and** the pressure may then be obta**in**ed separately if needed. The background profiles appear**in**g **in** (11-18) are chosen to describe a st**and**ard stratified shear layer: U = u 0 tanh z h ; (19) B Tz = R ρB z0 R ρ − 1 sech2 z h ; (20) B Sz = −B z0 z R ρ − 1 sech2 h . (21) Here, u 0 is the half velocity change across a layer of half depth h, B z0 is the maximum background buoyancy gradient (or squared buoyancy frequency), **and** R ρ = −B Tz /B Sz is the density ratio. 3

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- Page 5 and 6: l d. Parameter values The molecular
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- Page 9 and 10: than 1.25, and the resulting intens
- 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
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- Page 17 and 18: Radko, T., 2003: A mechanism for la