Convection equations

Consider a fluid flow in a layer of uniform depth where the temperature difference, , between the upper and lower bounding planes is kept constant. Such a system has a steady-state solution in which there is no fluid motion and the temperature varies linearly. If this solution is unstable, convection should develop. When all motion is parallel to the -plane, the governing equations are [a1]:

where is the height of the layer (in the -direction),

stands for the Jacobian determinant, is a stream function for the two-dimensional fluid motion, and is the deviation of the temperature from the case where no convection occurs. The coefficients , , , are, respectively, kinematic viscosity, gravity acceleration, thermal expansion, and thermal conductivity. The part of the first equation that does not depend upon is the third component of the vorticity equation

where is the velocity vector and is the vorticity.

See [a4], [a5], and also Curl and Vector product.

By expanding and in double Fourier series with coefficients depending on and truncating to three terms, the Lorenz equations result [a2].

References

[a1]	B. Saltzman, "Finite amplitude free convection as an initial value problem. I" J. Atmos. Sci. , 19 (1962) pp. 329–341
[a2]	E.N. Lorenz, "Deterministic non-periodic flow" J. Atmos. Sci. , 20 (1963) pp. 130–141
[a3]	G.K. Batchelor, "An introduction to fluid dynamics" , Cambridge Univ. Press (1967)
[a4]	B.K. Shivamoggi, "Theoretical fluid dynamics" , Martinus Nijhoff (1985) pp. 13–14
[a5]	"Modern developments in fluid dynamics" S. Goldstein (ed.) , Dover, reprint (1965) pp. 114