An Analytical Model of an Urban Heat Island Circulation in Calm Conditions

We obtain an analytical model solution to one of the simplest problems of an urban heat island circulation (UHIC): the steady shallow convective flow of a viscous stably stratified fluid over a differentially heated lower boundary without system rotation (no Coriolis force) or background wind. The coupled linearized equations of motion and thermal energy are solved for flows forced by a surface buoyancy that varies laterally as a smooth function of a horizontal coordinate. Solutions are obtained for two-dimensional Cartesian (slab-symmetric) and axisymmetric geometries with surface buoyancies considered as Gaussian and parabolic arch functions. When suitably scaled, the governing equations and boundary conditions are free of governing parameters, and the solutions are universal. The scalings can be used to explain why the daytime UHIC can be stronger than the nighttime UHIC despite the relative weakness of urban-rural temperature contrasts during the daytime. The solutions show that the main updraft of the axisymmetric circulation is stronger, while the compensating downdraft and the inflow and outflow branches of the slab-symmetric circulation are stronger. These results are explained in terms of the response of the perturbation pressure to the thermal forcing.