Analytical Reduced Models for the Non-stationary Diabatic Atmospheric Boundary Layer

Abstract

Geophysical boundary-layer flows feature complex dynamics that often evolve with time; however, most current knowledge centres on the steady-state problem. In these atmospheric and oceanic boundary layers, the pressure gradient, buoyancy, Coriolis, and frictional forces interact to determine the statistical moments of the flow. The resulting equations for the non-stationary mean variables, even when succinctly closed, remain challenging to handle mathematically. Here, we derive a simpler physical model that reduces these governing unsteady Reynolds-averaged Navier–Stokes partial differential equations into a single first-order ordinary differential equation with non-constant coefficients. The reduced model is straightforward to solve under arbitrary forcing, even when the statistical moments are non-stationary and the viscosity varies in time and space. The model is successfully validated against large-eddy simulation for, (1) time-variable pressure gradients, and (2) linearly time-variable buoyancy. The new model is shown to have a superior performance compared to the classic Blackadar solutions (and later improvements on these solutions), and it covers a much wider range of conditions.

Appendix: Particular Solutions in Polar Coordinates

The particular solution of the reduced system in polar coordinates using Eq. 13 or 15 are

$$\begin{aligned}&\displaystyle M_{p}^{2}=\left( {u_{p}^{2}+v_{p}^{2}} \right) +2\hbox {e}^{-\alpha t}\left( {u_{p} \cos ft+v_{p} \sin ft} \right) , \end{aligned}$$

(19a)

$$\begin{aligned}&\displaystyle \tan \varphi _{p} =\frac{u_{p} \tan ft+v_{p} }{\hbox {e}^{-\alpha t}\cos ft+u_{p} }. \end{aligned}$$

(19b)

The transient parts in these equations appear due to the nonlinear relation between the forcing (pressure) and the velocity as in Eq. 8. For instance, the statistically-steady-state oscillations of Eqs.  13 and 15 are respectively

$$\begin{aligned} \lim _{t\rightarrow \infty } M(t)= & {} Qf\sqrt{\frac{\left( {\omega \sin (\omega t)+\alpha \cos (\omega t)} \right) ^{2}+f^{2}\cos ^{2}(\omega t)}{(\alpha ^{2}+f^{2}+\omega ^{2})^{2}-4f^{2}\omega ^{2}}}, \end{aligned}$$

(20a)

$$\begin{aligned} \lim _{t\rightarrow \infty } \varphi (t)= & {} \tan ^{-1}\left( {\frac{\sqrt{\alpha ^{2}+w^{2}} }{f}\frac{\cos (\omega t+\gamma _{2} )}{\cos (\omega t+\gamma _{1} )}} \right) , \end{aligned}$$

(20b)

$$\begin{aligned} \lim _{t\rightarrow \infty } M(t)= & {} \frac{Qf}{\sqrt{\alpha ^{2}+(f-\omega )^{2}} }, \end{aligned}$$

(21a)

$$\begin{aligned} \lim _{t\rightarrow \infty } \varphi (t)= & {} \tan ^{-1}\left( {\frac{\cos (\omega t+\gamma _{4} )}{\cos (\omega t+\gamma _{3} )}} \right) . \end{aligned}$$

(21b)