## 3.The Atmospheric Boundary Layer

### 3.1 Boundary Layer Equations

Let us write the Navier-Stokes equations as

where is the stress tensor. In laminar flows, may be expressed in terms of the viscosity. In turbulent flows, molecular viscosity is not usually important because the Reynolds number is high. Instead, transport by turbulent eddies is more effective. Neglecting viscosity, we have for an incompressible flow

As for the case of turbulent transport of a tracer, we put

Substituting Eq. (31) into (29)-(30) and averaging

Close to the surface, variations in the vertical dominate horizontal variations so that we can neglect and derivatives. Furthermore, the mean flow is almost horizontal so that . Therefore

It turns out that on the scale of the whole boundary layer, we cannot neglect the effect of the rotation of the Earth. This effect is included by working in a frame of reference fixed relative to the surface of the Earth in the region of interest (this is of course a rotating, or non-inertial frame of reference). The frame of reference is rotating with angular speed . In the rotating frame of reference the equations of motion become

The extra terms which have been introduced are called the Coriolis force. is the latitude.

As for the case of scalar transport, we assume that

is called the eddy viscosity. We often write

where is the turbulent stress.

### 3.2 Constant Stress Layer --- The Surface Layer

There is a thin region close to the surface where the stress does not vary much from its surface value and so it can be assumed to be constant. This is called the surface layer. In the surface layer

Let the wind speed be given by

and let

where is the magnitude of the surface stress. Because the direction of the stress does not vary with height in the surface layer it follows from Eq. (36) that

We usually write the constant stress as

where is a quantity called the friction velocity. We can use arguments from dimensional analysis to evaluate the eddy viscosity . It turns out that the only scales available to us are and the height itself. In terms of these we have

Physically this indicates that at greater heights, larger turbulent eddies are acting to transport momentum. We put

where is a constant called von Kármán's constant. It is usually taken as . From Eq. (37), (38) and (39)

Solving this for we get

where is a constant. Note that as decreases towards zero, goes to zero at . is called the roughness length and it is a measure of the size of the so-called roughness elements making up the surface. These roughness elements could be grains of sand, blades of grass, bushes, trees, etc.

Eq. (41) usually applies up to a few tens of metres from the ground.

### 3.3 Effect of Stability

Under some circumstances (particularly at night when there is no cloud) the surface cools by radiation into space and the layer of air close to the surface becomes cold and dense.

Fig. 5. Temperature and density profiles close to the surface on a clear night. The air close to the surface is stably stratified.

This stable stratification inhibits vertical eddy motion. This can be thought of as reducing the size of the turbulent eddies so that no longer applies. Another way to think about the problem is that there is another length scale, in addition to , introduced into the problem. This is

where is the specific heat capacity at constant pressure, is an average temperature (absolute) and is the surface heat flux. For surface cooling, is negative and so is positive. is called the Monin-Obukhov length. It is found experimentally that for stable stratification a reasonable approximation is

so that

where is a constant. Typically, . The solution of Eq. (43) is

### 3.4 The Ekman Layer

On a scale deeper than the surface layer, we can, to a crude approximation, take . For neutral stratification, ms. If the flow is steady, Eq. (32)-(34) become

In the boundary layer we can assume that and are independent of height and are given by

where and are called the geostrophic wind components. For our purposes, we can take them to be the wind above the boundary layer. Let -- the Coriolis parameter. Then

These are the Ekman equations. The solutions are

where

is a measure of the thickness of the atmospheric boundary layer. In mid-latitudes, s and we find that 500 m.

The Ekman solution (46) shows that the wind direction varies within the boundary layer and that the wind vectors form a spiral with increasing height (see Fig. 7).

Fig. 7. Wind vectors in the Ekman boundary layer. The heights of the wind vectors are shown in metres.

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