One-dimensional dynamics of zonal jets on rapidly rotating spherical shells

Abstract

A one-dimensional dynamical model is derived which represents the evolution of mean zonal flow on a rapidly rotating spherical shell. This model is discussed and analyzed with reference to large-scale jet dynamics observed in geophysical and astrophysical systems. The reduced amplitude equation is obtained from a base model consisting of an equatorially forced Kolmogorov flow through application of an asymptotic expansion augmented by a closure hypothesis. This method of derivation allows one to generalize previous β-plane results obtained in the limit of weakly supercritical Reynold’s number by expanding the parameter range covered by the reduced model and by incorporating the effects of sphericity. Higher order spherical harmonic forcing functions with strong latitude–longitude anisotropy may generate emergent jet patterns which are confined in latitude on the slow scale. Both the spherical equation and β-plane analogues are numerically analyzed with a quadratic finite element discretization. No Lyapunov functional exists for the spherical case and chaotic dynamics are generated for some parameter values by processes associated with spatial variation in forcing amplitude. Oscillations can also occur at the boundaries in β-plane dynamics if one imposes a sharp transition between a homogeneously forced interior flow region and a damped exterior flow region. The nontrivial time dependence is explained in further analyses of a simplified damped Cahn–Hilliard-like equation in which spatial modulation is introduced in the form of a self-adjoint diffusion operator. Complex dynamics result because a damped Cahn–Hilliard jet formation process occurs coupled to a spatially inhomogeneous advection process.