Numerical Solution of the Lorenz Equations with Spatial Inhomogeneity

Abstract

This chapter discusses a numerical scheme for the solution of the partial differential equations that generalize the classical Lorenz equations, and whether the numerical solution is a meaningful approximation to the corresponding analytic one. It examines numerical solutions to the “spatial” Lorenz equations for a range of parameter values, and discusses the relevance of the results to the behaviour of real systems. The interaction of modes in the spatially varying Lorenz equations, which leads to either stable or unstable solutions, can be investigated in greater detail by examining a simplified model in which only two Fourier modes are present. In conclusion, then it appears that the absence of damping mechanisms for high wavenumber modes, and indeed the fact that the degree of instability grows with wavenumber (leading to a migration of energy from lower to higher modes), means that the Lorenz equations including spatial variation are not a viable model for weakly nonlinear phenomena.