Conservation equations for covariances - Turbulence in the Atmosphere - page 103

This is the analog of Eq. (5.1 5) fo r sca lar v ariance. It says that a positive eddy
diffusivity K , defined such that u i u j − δ ij u k u k / 3 =− KS ij , satsifies this constraint

locally by making the integrand positive definite (Problem 5.21) :

− u i u j S ij dV = KS ij S ij dV > 0 . (5.49)

volume volume

The major contributions to the rate of dissipation of TKE come from motions at
the smallest spatial scales. In Chapter 1 we identified these dissipative velocity and
length scales as the Kolmogorov scales υ and η :

ν υ 2

= ν 3 / 1 / 4 ,

(ν) 1 / 4 ,η

= = ∼

υ ν u i,j u i,j

η 2 .

(5.50)

In Chapter 1 we interpreted the dependence of υ and η on as meaning that a
turbulent flow adjusts the length and velocity scales of its smallest, viscous eddies
in order to dissipate TKE at the correct mean rate. We shall see in Chapter 6 that
this rate is set by the energy-containing turbulence, which establishes an “energy
cascade” to smaller eddies that terminates in viscous dissipation.

5.5.2 The mean-flow kinetic energy equation
Multiplying the mean-momentum equation (3.7) by U i and rearranging the terms
yields the equation for the mean-flow kinetic energy per unit mass, or the MKE
equation:

U i U i U j
2

∂
∂t

U i U i
2

∂
∂x j

∂
∂x j

U i U i
2

U i
ρ

∂P
∂x i

=− + U i u i u j − ν −

ν ∂U i
∂x j

∂U i
∂x j +

− u i u j S ij . (5.51)

The first term on the right is a divergence of (in order) the flux of MKE by the
mean flow, the turbulent flux of joint kinetic energy (including U i under the overbar
facilitates that interpretation), and the molecular flux of MKE. The next term, the
rate of production of MKE by mean flow down the mean pressure gradient, can also
be written as a divergence of PU i . Next is the rate of viscous dissipation of MKE,
and the final term is the negative of the rate of production of TKE. The viscous
dissipation term is of order νu 2 / 2 ; the last is of order u 3 / , a factor u/ν

=

R t
larger. Thus, we can neglect the viscous dissipation term in Eq. (5.51) except very
near solid boundaries.