Role of Rayleigh numbers on characteristics of double diffusive salt fingers

Abstract

Double diffusion convection, driven by two constituents of the fluid with different molecular diffusivity, is widely applied in oceanography and large number of other fields like astrophysics, geology, chemistry and metallurgy. In case of ocean, heat (T) and salinity (S) are the two components with varying diffusivity, where heat diffuses hundred times faster than salt. Component (T) stabilizes the system whereas components (S) destabilizes the system with overall density remains stable and forms the rising and sinking fingers known as salt fingers. Recent observations suggest that salt finger characteristics such as growth rates, wavenumber, and fluxes are strongly depending on the Rayleigh numbers as major driving force. In this paper, we corroborate this observation with the help of experiments, numerical simulations and linear theory. An eigenvalue expression for growth rate is derived from the linearized governing equations with explicit dependence on Rayleigh numbers, density stability ratio, Prandtl number and diffusivity ratio. Expressions for fastest growing fingers are also derived as a function various non-dimensional parameter. The predicted results corroborate well with the data reported from the field measurements, experiments and numerical simulations.

Appendix: Power law for growth rate

We have derived a simple functional relationship for maximum growth rate and wavelength variation as a function of Ra_T and R_ρ and other system parameters. It is to be mentioned here that value of maximum growth rate obtained from exact solution (15) agrees within 1% of the value predicted by simple functional form in Eq. (6). In linear stability models the unstable mode with the largest finger growth rate is investigated. This approach is based on a sensible assumption: if a weak initial perturbation contains a full spectrum of normal modes, then the fastest growing mode is more likely to reach the level of nonlinear equilibration first and thereby determine the pattern of fully developed instability [3]. To find the maximum growth rate, we first note that there are three dimensionless timescales in the formulation (14). Since, $\frac{M^2}{Ra}>\frac{M^2}{Ra}>\frac{M^2}{Ra}$. Simplification result in M²v > M²k_T > M²k_S. To find positive growth rates, |σ| ≤ M²k_T implies that |σ| ≪ M²v. This implies steady state fingers [11]. Hence, we obtain the eigenvalue equation as

$$ {\sigma}^2+\left(\frac{M^2}{\Pr {Ra}_T}+\frac{M^2}{Sc\ {Ra}_T}+\frac{R_{\rho }-1}{M^2}\right)\sigma +\left(\frac{M^4}{PrSc{Ra_T}^2}\right)-\left(\frac{1-{\tau R}_{\rho }}{\mathit{\Pr}{Ra}_T}\right)=0 $$

(22)

The positive root for the above equation is

$$ \sigma =\frac{1}{2}\left(\frac{M^2}{\mathit{\Pr}{Ra}_T}+\frac{M^2}{Sc\ {Ra}_T}+\frac{R_{\rho }-1}{M^2}\right)\left(\sqrt{1-\frac{4\left(\frac{M^4}{\mathit{\Pr} Sc\ {Ra_T}^2}-\left(\frac{1-\tau {R}_{\rho }}{\mathit{\Pr}{Ra}_T}\right)\right)}{{\left(\frac{M^2}{\mathit{\Pr}{Ra}_T}+\frac{M^2}{Sc\ {Ra}_T}+\frac{R_{\rho }-1}{M^2}\right)}^2}}-1\right) $$

(23)

The value of M which maximizes σ in Eq. (9) is obtained by ∂σ/∂M=0, we get,

$$ \left[-\frac{M}{\mathit{\Pr}{Ra}_T}-\frac{M}{Sc{Ra}_T}+\frac{\left({R}_{\rho }-1\right)}{M^3}\right]+\left[\frac{\left(\frac{M^2}{2\mathit{\Pr}{Ra}_T}+\frac{M^2}{2 Sc{Ra}_T}+\frac{\left({R}_{\rho }-1\right)}{M^2}\right)\left(\frac{M}{\mathit{\Pr}{Ra}_T}+\frac{M}{Sc{Ra}_T}-\frac{\left({R}_{\rho }-1\right)}{M^3}\right)-\frac{8{M}^3}{PrSc{Ra_T}^2}}{\sqrt{{\left(\frac{M^2}{\mathit{\Pr}{Ra}_T}+\frac{M^2}{2 Sc{Ra}_T}+\frac{\left({R}_{\rho }-1\right)}{2{M}^2}\right)}^2-\frac{4{M}^4}{PrSc{Ra_T}^2}+4\left(\frac{1-\tau {R}_{\rho }}{\mathit{\Pr}{Ra}_T}\right)}}\right]=0 $$

(24)

It is observed that the second square bracketed term in equation (318) is always negative for various range of M, Sc, Pr, R_ρ and Ra_T, hence the first bracketed term has to be positive to satisfy the equality. Therefore, $\left[-\frac{M}{\mathit{\Pr}{Ra}_T}-\frac{M}{\mathit{\Pr}{Ra}_T}+\frac{M}{\mathit{\Pr}{Ra}_T}\right]>0$ or ${M}^4<\frac{PrSc{ Ra}_T\left({R}_{\rho }-1\right)}{\mathit{\Pr}+ Sc}$ which gives an expression for maximum wavenumber, as

$$ {M}_{Max}={C}_1{\left[\left({R}_{\rho }-1\right){Ra}_T\right]}^{1/4} $$

(25)

$$ {\delta}_{nf}={C}_2{\left[\left({R}_{\rho }-1\right){Ra}_T\right]}^{-1/4} $$

(26)

where, δ_nf is a dimensionless finger wavelength, C₁ = [PrSc(Pr + Sc)]^1/4 and C₂ = 2π[PrSc(Pr + Sc)]^−1/4 are constants. Note that an Eqs. (25) or (26) consists of all the relevant governing parameters that effect the finger wavenumber or finger width [13, 14]. These constants can be precisely determined for any double diffusive systems. The finger wavenumber increases or width decrease with increase in Rayleigh numbers and density stability ratio, which is consistent with the recent observations for various ranges of R_ρ and Ra_T [13, 14]. Though the expression (5) agrees in terms of power law with linear stability theory [17] which suggests that finger width is proportional to Ra_T^-1/4, it contains additional system constants that may be useful in predicting finger characteristics in other fields. In the next paragraph, we will validate the finger width prediction from Eq. (26) with the data reported in literature. Maximum growth rate σ_max can be obtained by substituting the value of M_max in (23), which is of the form

$$ {\upsigma}_{\mathrm{max}}\sim {C}_3{\left(\frac{R_{\rho }-1}{Ra_T}\right)}^{1/2} $$

(27)

where, C₃ = [( Pr² + Sc² + PrSc)(PrSc(Pr + Sc))]^1/2 . Notice that the similar functional form for maximum growth rate variation as σ_max~Ra_T^−1/2 was obtained by the order of magnitude analysis from Eq. (15).