An analytic solution for periodic thermally driven flows over an infinite slope—Defant (1949) revisited

The flow generated as a consequence of a periodically varying surface temperature along an infinite slope is studied.

The set of equations derived by Defant (1949) - extending Prandtl's (1942) theory to include nonstationary surface temperature conditions - is adopted. In particular the flow properties are assumed to be invariant along the slope, and turbulent fluxes of momentum and heat are parameterized with a K-closure, with constant eddy viscosity K_m and eddy diffusivity K_h, and assuming a Prandtl number Pr = K_m/K_h = 1. Differently from Defant (1949), the initial problem of a flow starting from rest is solved. Solutions for both the transient and the subsequent steady periodic regime are calculated. Based on these, slope-normal profiles of the potential temperature anomaly, with respect to the basic state, and of the along-slope wind velocity can be evaluated at any time t.

The transient part can only be expressed in an integral form. Instead the solution for the periodic state can be explicitly expressed as a combination of exponential and sine/cosine functions of time and slope-normal coordinate n.

The resulting expressions are different from those found by Defant (1949). Let α be the slope angle, T the time period of the driving surface temperature cycle, ω=2π/T the related angular frequency and N_α=N/sinα, where N is the Brunt-Vaisala frequency of the stably stratified unperturbed atmosphere. Then it can be shown that three different regimes may occur, namely subcritical (N_α <ω), critical (N_α=ω), and supercritical (N_α >ω). The properties of the three regimes are shown and discussed.

Results from numerical simulations, based on the same equations and assuming either constant or variable eddy coefficients, compare well with the analytical solutions.