The conservation equations for vertical momentum, heat, and salinity (under Boussinesq's approximation) have the following form for double diffusive salt fingers: {\nabla}\cdot U =0 \frac{\partial U}{\partial t} + U\cdot\nabla U = \nu {\nabla}^2 U - g(\beta \Delta S-\alpha \Delta T)\mathbf{k} \frac{\partial T}{\partial t} + U\cdot\nabla T = k_T {\nabla}^2 T \frac{\partial S}{\partial t} + U\cdot\nabla S = k_S {\nabla}^2 S, where
U and
W are velocity components in horizontal (
x axis) and vertical (
z axis) directions,
k is the unit vector in the
z-direction,
kT is the molecular diffusivity of heat,
kS is the molecular diffusivity of salt, α is the coefficient of thermal expansion at constant pressure and salinity, and β is the
haline contraction coefficient at constant pressure and temperature. The above set of conservation equations governing the two-dimensional finger-convection system is non-dimensionalised using the following scaling: the depth of the total layer height
H is chosen as the
characteristic length, and velocity (
U,
W), salinity (
S), temperature (
T), and time (
t) are non-dimensionalised as x=\frac{X}{H} , z=\frac{Z}{H}, u=\frac{U}{k_T /H}, w= \frac{W}{k_T /H}, S^*= \frac{S-S_B}{S_T-S_B} , T^*= \frac{T-T_B}{T_T-T_B}, t^* = \frac{t}{H^2 /k_T }, where (
TT,
ST) and (
TB,
SB) are the temperature and concentration of the top and bottom layers respectively. On introducing the above non-dimensional variables, the above governing equations reduce to the following form: {\nabla}\cdot u =0 \frac{\partial u}{\partial t^*} + u\cdot\nabla u = Pr {\nabla}^2 u - \left[Pr Ra_T (\frac{S^*}{R_\rho}-T^*)\right] \mathbf{k} \frac{\partial T^*}{\partial t^*} + u\cdot\Delta T^* = {\nabla}^2 T^* \frac{\partial S^*}{\partial t^*} + u\cdot\Delta S^* = \frac{1}{Le} {\nabla}^2 S^*,where
Rρ is the density stability ratio,
RaT is the thermal
Rayleigh number,
Pr is the
Prandtl number, and
Le is the
Lewis number. These are defined as R_\rho = \frac{\alpha \Delta T }{\beta \Delta S}, Ra_T = \frac{g\alpha \Delta T H^3}{\nu k_T}, Pr=\frac{\nu}{k_T}, Le=\frac{k_T}{k_S} . Figure 1 shows the evolution of salt fingers in a heat-salt system for different Rayleigh numbers at a fixed
Rρ. It can be noticed that thin and thick fingers form at different
RaT. The fingers' flux ratio, growth rate, kinetic energy, evolution pattern, width, etc. are found to be functions of Rayleigh numbers and
Rρ. The flux ratio is another important non-dimensional parameter. It is the ratio of heat and salinity fluxes, defined as R_f=\frac{\alpha T'}{\beta S'}.Time-dependent analytic
self-similar solutions can be derived which contain
Gaussian and
error functions. ==Applications==