Boundary Layer Meteorology Lecture 4 Turbulent Fluxes Energy Cascades Turbulence closures TKE Budgets

Turbulent Fluxes Terminology: –Stationary Turbulence is constant is independent of translation along the time axis. –Homogeneous Turbulence is independent of translation along any spatial axis. –Isotropic Turbulence is statistically independent of rotation, reflection or translation

Energy Cascades Kolmogorov Theory: –Existence of equilibrium range of scales of motion in which the average properties are determined uniquely by the fluid’s viscosity and the total dissipation.  k =      v    –Existence of an inertial subrange within the equilibrium range, but removed from the scales where viscous forces are important, where only inertial transfer of energy is important. 1/l s <<  << 1/  k

Turbulence Closures Equation 2.42 has unknowns like. How do we solve for it? Note that the problem arises from Reynolds Averaging. Why do we need Reynolds averaging? Because we don’t have enough computing power to span the whole inertial subrange, bridging the scales where viscosity matters to the scales we care about (10s of meters to a few kilometers in the vertical). As Garratt points out, we’d need a trillion grid points to do this explicitly. Can we fudge it?

First order closures: Can be as simple as assuming u’s’ = K ds/dx, with constant K Or as complicated as prescribing a vertical and time dependence of K (which will also vary depending on what’s being transported), where K might depend on the local shear and on the Richardson number.

One-and-a-half oder closure In this case we acknowedge that an eddy diffusivity (K) ought to depend on the kinetic energy of the eddy part of the flow; so we should maybe predict the TKE for each location. Further simplifications are introduced to make this possible (see chapter 8).

Second-order closure Maybe “if a crude assumption for nth order moments predicts (n-1)th moments adequately, perhaps a similar assumption for (n+1)th moments will predict nth moments just as well” (Lumley & Khajeh-Nouri, 1974, quoted by Garratt). u i u j u k terms usually parameterized by assuming down-gradient diffusion of u i u j terms. Dissipation usually parameterized in terms of TKE and a large-eddy length scale (probably at least a little arbitrary).

TKE budget equation De/Dt = S + B + T+ D S = shear production; B = bouyancy flux; T = transport + pressure work; D = dissipation See Bretherton Notes 3.