A linear stability theory of double-diffusive horizontal intrusions in a temperature–salinity front

Abstract

The stability of a temperature–salinity front of finite width in which the density is exactly compensated for in the horizontal direction is studied by means of a linear theory. It is assumed that salt fingers are responsible for vertical transports of salinity and heat. The front is found to be always unstable even if viscosity is present. Horizontal intrusions are generated as a result of the instability, and cold/fresh water sinks while hot/salty water rises.

The stability of the front is described by three dimensionless parameters: a frontal stability parameter G = [g(1 − γ)βΔS]⁶/κ_e²a²N¹⁰, a stratification parameter $\mu = g(1-\gamma)\beta\overline{S}_z/N^2$ and a Schmidt number ε = ν/κ_e, where g is the acceleration due to gravity, β the salinity contraction coefficient, ΔS half of the salinity difference across the front, γ the density-flux ratio of temperature to salt due to salt fingers, N the Brunt–Väisälä frequency corresponding to the basic density stratification, a the half-width of the front, $\overline{S}_z$ the vertical gradient of the basic salinity, κ_e the eddy diffusivity of salt due to salt fingers, and ν is either molecular or eddy kinematic viscosity. When ε is not zero it is found that a modified frontal stability parameter R defined by R = G/ε plays an important role in determining the stability for a fixed value of μ. In particular, when ε is larger than 10, the stability is almost completely determined by R if μ is specified.

The dependence of the vertical scale h of the fastest-growing mode on external parameters varies according to the value of R for a given value of μ. When R is less than 40(1 + μ)^5.4 (for ε = 10⁻³-10³) h becomes independent of R and ε and is given by h = 2.2d/(1 + μ), where d = g(1 − γ)βΔS/N² is the scale suggested by Ruddick & Turner (1979) who performed an experiment on horizontal intrusions across a narrow front. When R > 2 × 105⁵ (1 + μ)^4.9, on the other hand, h becomes proportional to dR^−1/4, the scale suggested by Toole & Georgi (1978), who considered the stability of a temperature–salinity front of infinite width. The constant of the proportionality is a weak function of ε and varies from 2π[(2 + μ)/4(1 + μ)²]^−¼ to 2π[2(1 + μ + (1 + μ)^½)]^½ as ε goes from zero to infinity. Thus a front can be said to be narrow if R < 40(1 + μ)^5.4 and wide if R > 2 × 10⁵(1 + μ)^4.9.

The results of the theory explain those of Ruddick & Turner's experiment (1979) reasonably well.