Energetics of Slope Flows: Linear and Weakly Nonlinear Solutions of the Extended Prandtl Model

Abstract

The Prandtl model succinctly combines the 1D stationary boundary-layer dynamics and thermodynamics of simple anabatic and katabatic flows over uniformly inclined surfaces. It assumes a balance between the along-the-slope buoyancy component and adiabatic warming/cooling, and the turbulent mixing of momentum and heat. In this study, energetics of the Prandtl model is addressed in terms of the total energy (TE) concept. Furthermore, since the authors recently developed a weakly nonlinear version of the Prandtl model, the TE approach is also exercised on this extended model version, which includes an additional nonlinear term in the thermodynamic equation. Hence, interplay among diffusion, dissipation, and temperature-wind interaction of the mean slope flow is further explored. The TE of the nonlinear Prandtl model is assessed in an ensemble of solutions where the Prandtl number, the slope angle and the nonlinearity parameter are perturbed. It is shown that nonlinear effects have the lowest impact on variability in the ensemble of solutions of the weakly nonlinear Prandtl model when compared to the other two governing parameters. The general behavior of the nonlinear solution is similar to the linear solution, except that the maximum of the along-the-slope wind speed in the nonlinear solution reduces for larger slopes. Also, the dominance of PE near the sloped surface, and the elevated maximum of KE in the linear and nonlinear energetics of the extended Prandtl model are found in the PASTEX-94 measurements. The corresponding level where KE>PE most likely marks the bottom of the sublayer subject to shear-driven instabilities. Finally, possible limitations of the weakly nonlinear solutions of the extended Prandtl model are raised. In linear solutions, the local storage of TE term is zero, reflecting the stationarity of solutions by definition. However, in nonlinear solutions, the diffusion, dissipation and interaction terms (where the height of the maximum interaction is proportional to the height of the low-level jet by the factor ≈4/9) do not balance and the local storage of TE attains nonzero values. In order to examine the issue of non-stationarity, the inclusion of velocity-pressure covariance in the momentum equation is suggested for future development of the extended Prandtl model.

Thermally Driven Winds on Mars: A Review and a Slope Effect Numerical Study

“…The discrepancies between our results and those from other researchers and the Prandtl model can be due to many factors, among them: (a) the Prandtl model assumes laminar flow and motionless, stably stratified atmosphere, constant eddy diffusivity with height, etc., while this is not exactly the case in this research, for example, the Figure 14 shows the turbulent kinetic energy k profiles versus vertical distance over the ground along the Z-axis at 6 h M and 18 h M . For the major part, the profiles are qualitatively similar to those in Barcons et al (2019), Brun et al (2017), Giometto et al (2017), and Güttler et al (2016. Moreover, as found in our study, Giometto 2017) also report that, in general, k decreases with the slope angle.…”

On the turbulence structure of deep katabatic flows on a gentle mesoscale slope

“…Such unsteadiness is also found in the solutions of the non‐linear Prandtl model (cf. Güttler et al ., ). The largest difference in the budget between the deeper and shallower case is in the buoyancy forcing, which above 20 m is larger for the deeper case than for the shallower IOP1, which in its turn has a larger positive residual.…”

Anabatic Flow along a Uniformly Heated Slope Studied through Large-Eddy Simulation

“…One of the most accepted analytical models was proposed by Prandtl [19] to address the steady laminar flow over an infinite plate in a stratified atmosphere with Boussinesq approximation. Although rather simplified, the Prandtl model is used as a base for following the improvement of the theory, e.g., adding conditions of flow unsteadiness [20], heterogeneity [21], varying eddy viscosity [22], multiple dynamical balances [9] Coriolis force [23], periodicity [24] and weakly nonlinear effects [25]. On the other hand, pioneer field experiments have paved the way for key studies on the comprehension of the local processes driving the onset of the anabatic flows [26], the evolution of the anabatic layer [9] and its unsteadiness [27].…”