The Force–Restore Model for Surface Temperatures and Its Generalizations

Atmospheric general circulation models require efficient approximation procedures for including the vertical diffusion or heat into an underlying soil. The most effective procedure for diurnal heating—the “force-restore” approach—uses a prognostic equation for temperature that reproduces exactly the response to periodic heating and uniform thermal properties. Errors result from the neglect of higher harmonies in time in the forcing and from inhomogeneous thermal properties of the underlying soil. Neglect of higher harmonics is tolerable, in part because the response decreases with an increased forcing frequency. Neglect of vertical variation of thermal conductivity and specific heat can introduce major errors in the force-restore treatment, a possibility illustrated by the response to diurnal periodic surface forcing with a snow layer overlying a soil layer. A generalization of the force-restore procedure is developed for inhomogeneous thermal properties and illustrated for a snow layer overlying the soil. A reasonably accurate formulation for coupling the diurnal and seasonal force-restore formalisms is presented.