The stable boundary layer - Turbulence in the Atmosphere - page 275

We require that z s not exceed the height z cw to which the cooling wave has diffused:

z s (t) ≤ z cw (t). (12.9)

We'll use neutral surface-layer scaling to approximate Eq. (12.8) as

a u 3

u 3 g

θ 0 Q 0 =−

a −

∗ ∗

=

k θ 0 Q 0 ∼

kz s , s (t)

aL(t), (12.10)

with L(t) the evolving M-O length. With the constraint (12.9) this gives

aL(t) ≤ z s (t) ≤ z cw (t). (12.11)

The shaded region z s z cw sketched in Figure 12.4 is influenced by the

growing stable stratification induced by surface cooling. Within an hour or so there
can be a shallow (no more than a few hundreds of meters deep) nocturnal SBL in
place.

≤ z ≤

12.2.2 The inertial oscillation aloft
As we discussed in Chapter 11 , in the quasi-steady, horizontally homogeneous
mixed layer the mean horizontal momentum equations are

∂U
∂t =−

∂uw
∂z +

f(V − V g ) 0 ,

(12.12)

∂V
∂t =−

∂vw
∂z +

f(U g − U) 0 .

In the late-afternoon transition the turbulence aloft, which is supported largely by
buoyant production, begins to decay, causing the stress-divergence terms in these
equations to decay. Within a few large-eddy turnover times the mean momentum
equations have lost important terms and so the mean wind components begin to
evolve in time:

∂U
∂t =

∂V
∂t =

f(V − V g ), f(U g − U). (12.13)

Solution of the coupled equations (12.13) is facilitated by defining a complex
mean horizontal velocity W(z,t) iV (z, t). † If the corresponding com-

plex geostrophic wind depends only on z , i.e., W g (z)

= U(z,t) +

= U g (z) + iV g (z) , the equation

for W ≡ W − W g is

∂W
∂t

=− if W, (12.14)

†

This is the conventional notation. The reader should not confuse W with the mean vertical velocity, which
vanishes here.