ASTR/ATOC-5410: Fluid Instabilities, Waves, and Turbulence September 11, 2016, Axel Brandenburg

Handout 6b: Double-diffusive instability

In the previous handout (Handout 6),^{1} we showed numerical solutions of σ in units of σ_{ν} = νk^{2}. This is useful for a first screening of the results and to find parameter values of where something interesting (such as oscillatory solutions) happens. Eventually one also wants to see the k dependence. This will be done here in the present Handout 6b. There will also be some general comments on how to do this.

1 Solving the dispersion relation numerically

In Handout 6, we derived the dispersion relation in the form

a_{3}σ^{3} + a_{2}σ^{2} + a_{1}σ + a_{0} = 0  (1)

with

a_{3} = 1,  (2)
a_{2} = (κ_{T} + κ_{C} + ν)k^{2},  (3)
a_{1} = [(κ_{T} κ_{C} + κ_{C}ν + νκ_{T})k^{4} + (N_{T}^{2} − N_{C}^{2})\frac{k_{⊥}^{2}}{k^{2}} ] ,  (4)
a_{0} = κ_{T} κ_{C}νk^{6} + (N_{T}^{2}κ_{C} − N_{C}^{2}κ_{T})k_{⊥}^{2} = 0,  (5)

where α_{T} β_{T} g = −N_{T}^{2} and α_{C}β_{C}g = −N_{C}^{2} correspond to the buoyancy frequencies if only one of the two gradients were different from zero. These quantities are also related to corresponding Rayleigh numbers, Ra_{T} = −N_{T}^{2}d^{4}/νκ_{T} (which is the standard Rayleigh number, and Ra_{C} = −N_{C}^{2}d^{4}/νκ_{T} (which is the Rayleigh number for salinity, as defined in the literature.^{2} During the lecture, the real and imaginary parts of σ/σ_{ν} for N_{T}^{2} = −3, N_{C}^{2} = −1, and k_{⊥}^{2}/k^{2} = 0.5 will be shown. We will see that Reσ > 0 (unstable) for κ_{C}/κ_{T} < ∼ 0.1. At the same time, Imσ ≠ 0, so the onset of instability occurred in an oscillatory fashion. This also implies that the principle of the exchange of stabilities is not valid in this case.

Figure 1: Real and imaginary parts of σ(k) for N_{T}^{2} = −3, N_{C}^{2} = −1, for k_{⊥}^{2}/k^{2} = 0.5. Note that Reσ > 0 (unstable) for k < 1, where Imσ ≠ 0 (corresponding to oscillatory solutions).

1Owing to a typo in the plotting routine, the y-title said σ_{ρ}, but it should have read σ_{ν}.

2A division by νκ_{C} instead would have made more sense, but for historical reasons the division by νκ_{T} has been kept.

To solve the cubic equation, one can just use a library routine, e.g. fz roots in idl. Here we call it for a certain number of k values, keeping however k_{⊥}^{2}/k^{2} = const, and sort the resulting values by increasing imaginary part and color the data points differently; see Figure 1.

As is common in Rayleigh-Bénard convection, instability occurs only for small values of k (here k < 1). Here the imaginary part is nonvanishing (red and orange data points). For k < 0.7 the imaginary part vanishes, but this does not necessarily imply anything about the solution in the nonlinear regime.

Figure 2 shows the corresponding case for N_{T}^{2} = −3, N_{C}^{2} = −1, which is the case where both T and C gradients are positive, so the temperature is stably stratified and salinity unstably. In this case, the onset of instability is non-oscillatory.

Figure 2: Real and imaginary parts of σ(k) for N_{T}^{2} = −3, N_{C}^{2} = −1, for k_{⊥}^{2}/k^{2} = 0.5. Note that Reσ > 0 (unstable) for k < 1, where Imσ ≠ 0 (corresponding to oscillatory solutions).

References

Stern, M. E., “The salt-fountain and thermohaline convection,” Tellus 12, 172-175 (1960).

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