Abstract
A two-dimensional stochastic–diffusive energy balance model (EBM) formulated on a sphere by G. R. North et al. is explored and generalized. Instantaneous and frequency-dependent spatial autocorrelation functions and local temporal power spectral densities are computed for local sites and for spatially averaged surface temperature signals up to the global scale. On time scales up to the relaxation time scale given by the effective heat capacities of the ocean mixed layer and land surface, respectively, scaling features are obtained that are reminiscent of what can be derived from the observed temperature field. On longer time scales, however, the EBM predicts a transition to a white-noise scaling, which is not reflected in the observed records. A fractional generalization, which can be considered as a spatial generalization of the zero-dimensional, long-memory EBM of M. Rypdal and K. Rypdal, is proposed and explored. It is demonstrated that this generalized model describes qualitatively the main correlation characteristics of the temperature field reported in the literature and those derived herein from 500-yr-long control simulations of the NorESM Earth system model. A further generalization of the model, to include long-term persistence in the stochastic forcing, is also discussed.
Uniqueness and energy balance for isentropic Euler equation with stochastic forcing
