A Linear Theory of Local Thermal Circulations

Abstract

The dynamics of local thermal circulations (LTCs) are examined by constructing a linear theory based on Boussinesq equations in the planetary boundary layer (PBL). Linear theory arranges LTCs into a sixth-order partial differential equation of temperature, which can be solved by using the Fourier transform and inverse Fourier transform. The analytic solution suggests that the horizontal distribution of the temperature anomaly is basically determined by surface heating, while the vertical distribution of the temperature anomaly is a combination of exponential decay and an Ekman spiral. For shallow PBL cases where the Ekman elevation is much smaller than the vertical scale of motion, the higher-order partial differential terms that represent the Ekman spiral structure can be ignored so that the equations reduce to a second-order partial differential equation. Compared with the numerical results, this so-called low-order approximation does not induce dramatic errors in the temperature distribution. However, to avoid distortion in the forced atmospheric circulation, the eddy viscosity in the motion equations should be replaced with the Rayleigh form, which is the common practice in LTCs. For deep PBL cases where the Ekman elevation is comparable to the vertical scale of motion, both the exponential decay and Ekman spiral structure play roles in the forced atmospheric circulation. The most significant influence is that there exist three additional compensating forced circulation cells that surround the direct forced circulation cell.