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- 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 ...

In the **in**terleav**in**g case, the assumption that shear varies vertically on the same scale as stratification is reasonable. If shear is imposed by an **in**ternal wave field, however, it is likely that the vertical scales will be very different. If thermohal**in**e properties vary on a much smaller scale than velocity, there exists the potential for other classes of shear **in**stability besides KH. If the shear field is aligned so that maximum shear co**in**cides with a stratified **in**terface, the result may be Holmboe **in**stability (e.g. Smyth **and** W**in**ters, 2003). On the other h**and**, if the shear maximum is located between adjacent stratified layers, Taylor **in**stability may result (e.g. Lee **and** Caulfield, 2001). These will be the subject of a separate publication. b. Nondimensionalization We nondimensionalize the problem us**in**g length scale h **and** time scale h 2 /κ T . This nondimensionalization **in**troduces four dimensionless parameters: the molecular Pr**and**tl 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 **in**verse 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 conta**in**s a contribution from a second scalar (sal**in**ity) **in** addition to the one whose diffusivity appears **in** the denom**in**ator (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-Galerk**in** 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) =s**in** . (31) The **in**ner product operator of any two functions a(z) **and** b(z) is def**in**ed by so that = 2 H ∫ H = δ mn ; 0 a(z)b(z)dz, (32) = δ mn (1 + δ m0 ). (33) The govern**in**g 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 Pr**and**tl number **and** the diffusivity ratio have the values Pr =7**and** τ =10 −2 , appropriate for seawater. We also def**in**e the molecular Schmidt number, Sc = Pr/τ = 700. Gr is assumed to be positive, **in**dicat**in**g stable stratification. Observations **in** a thermohal**in**e staircase east of Barbados, summarized by Kunze (2003), suggest values **in** the range [10 8 , 2 × 10 9 ] (depend**in**g on how one relates h to the thickness of a **double**-**diffusive** layer). Double-**diffusive** **in**stability requires (Pr + τ)/(Pr + 1)

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