Double diffusive convection

Numerical simulations results by Singh and Srinivasan^[1] show concentration fields at different Rayleigh numbers for fixed value of R_ρ = 6. The parameters are: (a) Ra_T = 7×10⁸ , t=1.12×10⁻², (b) Ra_T =3.5×10⁸, t=1.12×10⁻², (c) Ra_T =7×10⁶, t=1.31×10⁻², (d) Ra_T=7×10⁵, t=3.69×10⁻². It is seen from the figure that finger characteristics such as width, evolution pattern are a function of Rayleigh numbers.

Double diffusive convection is a fluid dynamics phenomenon that describes a form of convection driven by two different density gradients, which have different rates of diffusion.^[2]

Convection in fluids is driven by density variations within them. These density variations may be caused by gradients in the composition of the fluid, or by differences in temperature (through thermal expansion). Thermal and compositional gradients can often diffuse with time, reducing their ability to drive the convection, and requiring that gradients in other regions of the flow exist in order for convection to continue. A common example of double diffusive convection is in oceanography, where heat and salt concentrations exist with different gradients and diffuse at differing rates. An effect that affects both of these variables is the input of cold freshwater from an iceberg.

Double diffusive convection is important in understanding the evolution of a number of systems that have multiple causes for density variations. These include convection in the Earth's oceans (as mentioned above), in magma chambers,^[3] and in the sun (where heat and helium diffuse at differing rates).

Two quite different types of fluid motion exist—and therefore are classified accordingly—depending on whether the stable stratification is provided by the density-affecting component with the lowest or the highest molecular diffusivity. If the stratification is provided by the component with the lower molecular diffusivity (for example in case of a stable salt-stratified ocean perturbed by a thermal gradient due to an iceberg), the stratification is called to be of "diffusive" type (see external link below), otherwise it is of "finger" type, occurring frequently in oceanographic studies as salt-fingers.^[4] These long fingers of rising and sinking water occur when hot saline water lies over cold fresh water of a higher density. A perturbation to the surface of hot salty water results in an element of hot salty water surrounded by cold fresh water. This element loses its heat more rapidly than its salinity because the diffusion of heat is faster than of salt; this is analogous to the way in which just unstirred coffee goes cold before the sugar has diffused to the top. Because the water becomes cooler but remains salty, it becomes denser than the fluid layer beneath it. This makes the perturbation grow and causes the downward extension of a salt finger. As this finger grows, additional thermal diffusion accelerates this effect.

Role of salt fingers in oceans

Double diffusion convection plays a significant role in upwelling of nutrients and vertical transport of heat and salt in oceans. Salt fingering contributes to vertical mixing in the oceans. Such mixing helps regulate the gradual overturning circulation of the ocean, which control the climate of the earth. Apart from playing an important role in controlling the climate, fingers are responsible for upwelling of nutrients which supports flora and fauna. The most significant aspect of finger convection is that they transport the fluxes of heat and salt vertically, which has been studied extensively during the last five decades.^[5]

Governing Equations

The conservation equations for vertical momentum, heat and salinity equations (under Boussinesq’s approximation) have following form for double diffusive salt fingers:^[6]

${\displaystyle {\nabla }\cdot U=0}$

${\displaystyle {\frac {\partial U}{\partial t}}+U\cdot \nabla U=\nu {\nabla }^{2}U-g(\beta \Delta S-\alpha \Delta T)\mathbf {k} }$

${\displaystyle {\frac {\partial T}{\partial t}}+U\cdot \Delta T=k_{T}{\nabla }^{2}T}$

${\displaystyle {\frac {\partial S}{\partial t}}+U\cdot \Delta S=k_{s}{\nabla }^{2}s}$

Where, U and W are velocity components in horizontal (x axis) and vertical (z axis) direction; k is the unit vector in the Z-direction, k_T is molecular diffusivity of heat, k_s is molecular diffusivity of salt, α is coefficient of thermal expansion at constant pressure and salinity and β is the coefficient of saline expansion 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, velocity (U, W), salinity (S), temperature (T) and time (t) are non-dimensionalised as^[7]

$x=\frac{X}{H} , z=\frac{Z}{H}, u=\frac{U}{k_T /H}, w= \frac{W}{k_T /H}, S^*= \frac{S-S_B}{S_T-S_B} , T^*= \frac{T-T_B}{T_T-T_B}, t^* = \frac{t}{H^2 /k_T }.$

Where, (T_T, S_T) and (T_B, S_B) 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:

${\displaystyle {\nabla }\cdot u=0}$

${\displaystyle {\frac {\partial u}{\partial t^{*}}}+u\cdot \nabla u=Pr{\nabla }^{2}u-\left[PrRa_{T}({\frac {S^{*}}{R_{\rho }}}-T^{*})\right]\mathbf {k} }$

${\displaystyle {\frac {\partial T^{*}}{\partial t^{*}}}+u\cdot \Delta T^{*}={\nabla }^{2}T^{*}}$

${\displaystyle {\frac {\partial S^{*}}{\partial t^{*}}}+u\cdot \Delta S^{*}={\frac {Pr}{Sc}}{\nabla }^{2}S^{*}}$

Where, R_ρ is the density stability ratio, Ra_T is the thermal Rayleigh number, Pr is the Prandtl number, Sc is the Schmidt number which are defined as

$R_\rho = \frac{\alpha \Delta T }{\beta \Delta S}, Ra_T = \frac{g\beta \Delta S H^3}{\nu k_T}, Pr=\frac{\nu}{\alpha}, Sc=\frac{\nu}{\beta} .$

Figure 1(a-d) shows the evolution of salt fingers in heat-salt system for different Rayleigh numbers at a fixed R_ρ. It can be noticed that thin and thick fingers form at different Ra_T. Fingers flux ratio, growth rate, kinetic energy, evolution pattern, finger width etc. are found to be the function of Rayleigh numbers and R_ρ.Where, flux ratio is another important non-dimensional parameter. It is the ratio of heat and salinity fluxes, defined as,

$R_f=\frac{\alpha T'}{\beta S'}.$

Applications

Double diffusion convection holds importance in natural processes and engineering applications. The effect of double diffusive convection is not limited to oceanography but also in various fields like geology,^[8] astrophysics, metallurgy.^[9]

See also

• Rayleigh number
• Prandtl number

References

1. Singh, O.P; Srinivasan, J. (2014). "Effect of Rayleigh numbers on the evolution of double-diffusive salt fingers". Physics of Fluids. 26 (062104): 1–18. Bibcode:2014PhFl...26f2104S. doi:10.1063/1.4882264.
2. Mojtabi, A.; Charrier-Mojtabi, M.-C. (2000). "13. Double-Diffusive Convection in Porous Media". In Kambiz Vafai. Handbook of porous media. New York: Dekker. ISBN 978-0-8247-8886-5.
3. Huppert, H E; Sparks, R S J (1984). "Double-Diffusive Convection Due to Crystallization in Magmas". Annual Review of Earth and Planetary Sciences. 12 (1): 11–37. Bibcode:1984AREPS..12...11H. doi:10.1146/annurev.ea.12.050184.000303.
4. Stern, Melvin E. (1969). "Collective instability of salt fingers". Journal of Fluid Mechanics. 35 (02): 209. Bibcode:1969JFM....35..209S. doi:10.1017/S0022112069001066.
5. Oschilies, A.; Dietze, H.; Kahlerr, P. (2003). "Salt-finger driven enhancement of upper ocean nutrient supply". Geophys. Res. Lett. 30: 2204–08. Bibcode:2003GeoRL..30.2204O. doi:10.1029/2003GL018552.
6. Schmitt, R.W. (1979). "The growth rate of supercritical salt fingers". Deep-Sea Research. 26A: 23–40. Bibcode:1979DSRA...26...23S. doi:10.1016/0198-0149(79)90083-9.
7. Sreenivas, K.R.; Singh, O.P.; Srinivasan, J. (2009). "On the relationship between finger width, velocity, and fluxes in thermohaline convection". Physics of Fluids. 21 (026601): 1–15. Bibcode:2009PhFl...21b6601S. doi:10.1063/1.3070527.
8. Singh, O.P; Ranjan, D.; Srinivasan, J. (September 2011). "A Study of Basalt Fingers Using Experiments and Numerical Simulations in Double-diffusive Systems". Journal of Geography and Geology. 3 (1). doi:10.5539/jgg.v3n1p42.
9. Schmitt, R.W. (1983). "The characteristics of salt fingers in a variety of fluid systems, including stellar interiors, liquid metals, oceans and [magmas". Physics of fluid. 26 (9): 2373–2377. Bibcode:1983PhFl...26.2373S. doi:10.1063/1.864419.

• Huppert, Herbert E.; Turner, J. Stewart (2006). "Double-diffusive convection". Journal of Fluid Mechanics. 106: 299. Bibcode:1981JFM...106..299H. doi:10.1017/S0022112081001614.

External links

• Oceanic Double-diffusion: Introduction
• Double Diffusion in Oceanography
• Diffusive-mode Double Diffusive Convection, Stability and Density-Driven Flows
• Stockman, H.W; Li, C.; Cooper, C.; 1997. Practical application of lattice-gas and lattice Boltzmann methods to dispersion problems. InterJournal of Complex Systems, manuscript no. 90.