Scaling relationships for slope flows

Slope winds (flows), which from the standpoint of basic fluid dynamics are buoyantly driven boundary-layer-type flows along heated or cooled sloping surfaces in stratified fluid, are typical for vast areas of the Earth and often play an important role in climate of these areas. Commonly, meteorologists distinguish between anabatic winds, which are driven by surface heating, and katabatic winds, which result from surface cooling. From the viewpoint of formal description and modeling, slope flows represent challenging physical phenomena because, in the most general form, they conflate all four characteristic aspects of geophysical fluid dynamics: buoyant forcing, stratification, turbulence, and rotation. Although much progress has been made in the conceptual understanding and numerical simulation of the slope winds, there are still many open questions regarding the scale structure of these winds. Of particular interest for atmospheric applications is the scale structure of turbulent slope flows. Knowledge of basic scaling laws for such flows could be useful for designing parameterizations of slope-wind phenomena in atmospheric models.

We apply theoretical analysis in conjunction with direct numerical simulation (DNS) to investigate properties of the governing slope wind equations for cases of both laminar and turbulent flows in the absence of rotation, that is assuming flows of moderate temporal and spatial scales on which effects of the Coriolis force may be neglected. The simulated turbulent flows are shown to achieve quasi-steady periodic regimes at large times with low-frequency modulations of turbulent fluctuations by persistent oscillatory motions with frequency equal to the product of the ambient buoyancy frequency and the sine of the slope angle. These oscillatory wave-type motions result from interactions between turbulence and ambient stable stratification in the presence of a temporally constant surface buoyant forcing. Scaling analyses are applied to turbulent slope flows averaged in time and over horizontal planes parallel to the slope.

The invoked scaling hypotheses assume a scale separation for sufficiently high-Reynolds number flow. In the immediate vicinity of the slope, within the interior region, the flow is controlled by the surface energy production rate and molecular viscosity/diffusivity, whilst at larger distances from the wall, in the exterior region, molecular effects play no role and the flow structure is determined by the energy production and environmental stratification. In the limit of infinite Reynolds number, an intermediate distance range exists where flow structure is determined solely by the surface energy production. In this intermediate range, power-law asymptotic approximations are obtained for the profiles of mean-flow velocity and buoyancy, as well as for the corresponding profiles of turbulent fluxes, in the limit of a large Reynolds number. Scaled variances of the velocity components and buoyancy in the exterior region show a universal behavior as functions of dimensionless distance from the wall. The derived scaling relationships are verified using the DNS data.