Coupled diabatic and frictional forcing of single-layer vortical flow

Diabatic and frictional forcing of a primary balanced vortical flow involves both a direct vorticity tendency and the advection of vorticity by an induced secondary circulation. In turn, the forcing itself may be influenced by the primary vortical flow in various ways, depending on the physical system. The coupling is considered here in a simple single-layer model, beginning with the axisymmetric shallow-water equations with primary azimuthal vortical flow. Gradient wind is imposed as a balance constraint, filtering gravity waves and allowing the time evolution to be written in terms of a single prognostic equation for the potential vorticity. The secondary, radial circulation is determined from the diabatic heating and friction through a modified Sawyer–Eliassen equation, following the standard approach of axisymmetric balanced models, where the linearity of the equation allows a partition into the two components of the forcing. The model formulation is then extended, under suitable approximations, to the case of non-axisymmetric vortical flow, providing a simple, dynamically consistent model of diabatic and frictionally forced flow. To illustrate the coupling between forcing and vortical flow, a simple representation of a tropical cyclone is considered in which diabatic heating is linked instantaneously to the cyclone intensity through a crude proxy for boundary-layer convergence and moist static stability. Despite neglecting many aspects of tropical cyclone dynamics, the characteristic annular structure of a mature cyclone, with a central minimum of potential vorticity surrounded by a sharp maximum at finite radius, develops spontaneously from a broad initial vorticity distribution. The barotropic instability of the ring is also illustrated.