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

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

In the

In the interleaving case, the assumption that shear varies vertically on the same scale as stratification is reasonable. If shear is imposed by an internal wave field, however, it is likely that the vertical scales will be very different. If thermohaline properties vary on a much smaller scale than velocity, there exists the potential for other classes of shear instability besides KH. If the shear field is aligned so that maximum shear coincides with a stratified interface, the result may be Holmboe instability (e.g. Smyth and Winters, 2003). On the other hand, if the shear maximum is located between adjacent stratified layers, Taylor instability may result (e.g. Lee and Caulfield, 2001). These will be the subject of a separate publication. b. Nondimensionalization We nondimensionalize the problem using length scale h and time scale h 2 /κ T . This nondimensionalization introduces four dimensionless parameters: the molecular Prandtl number Pr = ν/κ T , the diffusivity ratio τ = κ S /κ T , the Reynolds number Re = u 0 h/ν, and the Grashof number 1 Gr = B z0 h 4 /νκ T . The latter is just the additive inverse of the Rayleigh number, and as such is positive in statically stable stratification. It is also a scaled version of the buoyancy gradient B z0 , which is equal to the squared Brundt-Vaisala frequency. The equations needed for the analyses to follow become σû = −ikUû − ŵU z − ikˆπ + Pr∇ 2 û (22) σ∇ 2 ŵ = −ikU∇ 2 ŵ + ikŵU zz +Pr∇ 4 ŵ − ˜k 2ˆb. (23) σˆb T = −ŵB Tz − ikUˆb T + ∇ 2ˆbT (24) σˆb S = −ŵB Sz − ikUˆb S + τ∇ 2ˆbS , (25) with background profiles U = ReP r tanh z; (26) B Tz = R ρGrP r R ρ − 1 sech2 z; (27) 1 We use the term somewhat loosely, as the buoyancy gradient in Gr contains a contribution from a second scalar (salinity) in addition to the one whose diffusivity appears in the denominator (temperature). B Sz = −GrP r R ρ − 1 sech2 z. (28) In (22-28) and hereafter unless otherwise noted, all quantities are dimensionless. For the computations described here, boundaries are located at z = ±4. c. Numerical methods The Fourier-Galerkin method is used to discretize the z - dependence: {ŵ(z), ˆb T (z), ˆb S (z)} = where {û(z), ˆv(z), ˆπ(z)} = N∑ {ŵ n , ˆb Tn , ˆb Sn }f n (z), (29) n=1 N∑ {û n , ˆv n , ˆπ n }g n (z), (30) n=1 nπ(z − H/2) ; g n (z) = cos H nπ(z − H/2) H f n (z) =sin . (31) The inner product operator of any two functions a(z) and b(z) is defined by so that = 2 H ∫ H = δ mn ; 0 a(z)b(z)dz, (32) = δ mn (1 + δ m0 ). (33) The governing equations become: σ ŵ n =(−ik < f m ∗ U ∗ ¯∇ 2 f n > +ik < f m ∗ U zz ∗ f n > +Pr < f m ∗ ¯∇ 4 f n >)ŵ n −˜k 2 ˆb Tn − ˜k 2 ˆb Sn σ ˆb Tn = − ŵ n −ik < f m ∗ U ∗ f n > ˆb Tn + ˆb Tn σ ˆb Sn = − ŵ n −ik < f m ∗ U ∗ f n > ˆb Sn + τ ˆb Sn . This is a generalized algebraic eigenvalue problem whose eigenvalue is σ and whose eigenvector is the concatenation of {ŵ n , ˆb Tn , ˆb Sn }. Convergence requires up to 192 Fourier modes for boundaries located at z = ±4. 4

l d. Parameter values The molecular Prandtl number and the diffusivity ratio have the values Pr =7and τ =10 −2 , appropriate for seawater. We also define the molecular Schmidt number, Sc = Pr/τ = 700. Gr is assumed to be positive, indicating stable stratification. Observations in a thermohaline staircase east of Barbados, summarized by Kunze (2003), suggest values in the range [10 8 , 2 × 10 9 ] (depending on how one relates h to the thickness of a double-diffusive layer). Double-diffusive instability requires (Pr + τ)/(Pr + 1)

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