In this chapter we consider some non-rotating turbulent flows that are a bit more complicated than the flow of water down a slope considered in § 23.6. These include turbulent katabatic winds driven by thermal buoyancy (§ 24.1), avalanches driven by snow suspended in air (§ 24.2) and cumulonimbus clouds driven by the release of latent heat as water vapor condenses (§ 24.3).

Turbulent Katabatic Winds

In § 22.2.5, we investigate the katabatic wind down a slope in the case that the flow is laminar and, using typical parameter values, found that the flow is very likely turbulent rather than laminar. When flow is turbulent, the diffusivity coefficients for momentum and heat are not constant, but instead vary linearly with elevation. In this section, we will revisit the problem formulated in § 22.2.5, but with variable diffusivities, using Reynolds analogy to set the turbulent thermal diffusivity equal to the turbulent diffusivity of momentum.1

The governing equations now are

where z is elevation above the ground, u is the downslope speed, is the dimensionless perturbation temperature, is the reduced gravity, s the down-slope thermal gradient, is the temperature contrast, is the turbulent diffusivity, is a small dimensionless parameter is the velocity scale and is the roughness scale. As before, these equations are to be solved on the domain subject to the conditions u And as before, we can combine the two equations into a single complex equation, although the scalings are somewhat different. In the present case

while the complex equation for W= T ∗ −iu/U is

where is the dimensionless vertical distance and is the scaled boundary roughness, subject to conditions W(0) = 1 and W(∞) = 0.

The problem for the turbulent katabatic winds is a bit more challenging than for the laminar winds because our complex ordinary differential equation now has a variable coefficient. We can get this equation in “standard” form by introducing a new independent variable; let