C.7 Stability Analysis of the Nonconvective State

Versão em Português

The parameter a is determined by examining the conditions on the stability of the nonconvective state. The nonconvective state has ψ = 0 and τ = 0 and hence corresponds to X, Y, Z = 0. If we let x, y, and z represent the values of X, Y, and Z near this fixed point, and drop all nonlinear terms from the Lorenz equations, the dynamics near the fixed point is modeled by the following linear differential equations:

Note that z(t) is exponentially damped since the parameter b is positive. Thus, we need consider only the x and y equations. Using our now familiar results from Section 3.11, we see that the nonconvective fixed point becomes unstable when r > 1. Returning to the original Rayleigh number, we see that the condition is

We choose the parameter a to be the value that gives the lowest Rayleigh number for the beginning of convection. In a sense, the system selects the wavelength 2π/a by setting up a convection pattern with the wavelength 2π/a at the lowest possible Rayleigh number. This condition yields . Hence, the Rayleigh number at which convection begins is R = 27π⁴/4. The parameter b is then equal to 8/3, the value used in most analyses of the Lorenz model.