3.The Atmospheric Boundary Layer

3.1 Boundary Layer Equations

Let us write the Navier-Stokes equations as

\begin{displaymath} {{\rm D} {\bf u} \over {\rm D}t}=-{1\over \rho}\nabla p-g{\bf k}+\nabla .{{\leavevmode\hbox{\boldmit \char'34}}} \end{displaymath}

where ${{\leavevmode\hbox{\boldmit \char'34}}}$ is the stress tensor. In laminar flows, ${{\leavevmode\hbox{\boldmit \char'34}}}$ 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

\begin{displaymath} {\partial u\over \partial t}+{\partial\over \partial x}(u^2)... ...z}(uw)= -{1\over \rho}{\partial p\over \partial x},\eqno{(29)} \end{displaymath}


\begin{displaymath} {\partial v\over \partial t}+{\partial\over \partial x}(uv)+... ...z}(vw)= -{1\over \rho}{\partial p\over \partial y}.\eqno{(30)} \end{displaymath}

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

\begin{displaymath} \left .\eqalign{u&=\overline u+u',\cr v&=\overline v+v',\cr w&=\overline w+w'.\cr}\right\}\eqno{(31)} \end{displaymath}

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

\begin{displaymath} {\partial\overline u\over \partial t}+{\partial\over \partia... ...verline {u'v'})- {\partial\over \partial z}(\overline {u'w'}), \end{displaymath}


\begin{displaymath} {\partial\overline v\over \partial t}+{\partial\over \partia... ...verline {v'^2})- {\partial\over \partial z}(\overline {v'w'}). \end{displaymath}

Close to the surface, variations in the vertical dominate horizontal variations so that we can neglect $x$ and $y$ derivatives. Furthermore, the mean flow is almost horizontal so that $w=0$. Therefore

\begin{displaymath} {\partial\overline u\over \partial t}= -{1\over \rho}{\parti... ...ver \partial x}- {\partial\over \partial z}(\overline {u'w'}), \end{displaymath}


\begin{displaymath} {\partial\overline v\over \partial t}= -{1\over \rho}{\parti... ...ver \partial y}- {\partial\over \partial z}(\overline {v'w'}). \end{displaymath}

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 $\Omega =2\pi/(\hbox{1 day})$. In the rotating frame of reference the equations of motion become

\begin{displaymath} {\partial\overline u\over \partial t}-2\Omega\overline v\sin... ...l x}- {\partial\over \partial z}(\overline {u'w'}),\eqno{(32)} \end{displaymath}


\begin{displaymath} {\partial\overline v\over \partial t}+2\Omega\overline u\sin... ...l y}- {\partial\over \partial z}(\overline {v'w'}).\eqno{(33)} \end{displaymath}

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

As for the case of scalar transport, we assume that

\begin{displaymath} \left .\eqalign{\overline {u'w'}&=-K{\partial\overline u\ove... ...K{\partial\overline v\over \partial z}.\cr}\right\}\eqno{(34)} \end{displaymath}

$K$ is called the eddy viscosity. We often write

\begin{displaymath} {{\leavevmode\hbox{\boldmit \char'34}}}=-\rho (\overline {u'... ...l z},{\partial\overline v\over \partial z} \right )\eqno{(35)} \end{displaymath}

where ${{\leavevmode\hbox{\boldmit \char'34}}}$ is the turbulent stress.

3.2 Constant Stress Layer --- The Surface Layer

There is a thin region close to the surface where the stress ${{\leavevmode\hbox{\boldmit \char'34}}}$ 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

\begin{displaymath} \left .\eqalign{\tau_x&=-\rho K{{\rm d}\overline u\over {\rm... ...o K{{\rm d}\overline v\over {\rm d}z}.\cr}\right\} \eqno{(36)} \end{displaymath}

Let the wind speed $U$ be given by

\begin{displaymath} U^2=\overline u^2+\overline v^2 \end{displaymath}

and let

\begin{displaymath} \tau^2=\tau_x^2+\tau_y^2 \end{displaymath}

where $\tau$ 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

\begin{displaymath} \tau =-\rho K{{\rm d}U\over {\rm d}z}.\eqno{(37)} \end{displaymath}

We usually write the constant stress $\tau$ as

\begin{displaymath} \tau =-\rho u_*^2\eqno{(38)} \end{displaymath}

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

\begin{displaymath} K\propto u_*z. \end{displaymath}

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

\begin{displaymath} K=ku_*z\eqno{(39)} \end{displaymath}

where $k$ is a constant called von Kármán's constant. It is usually taken as $k=0.4$. From Eq. (37), (38) and (39)

\begin{displaymath} {{\rm d}U\over {\rm d}z}={u_*\over kz}.\eqno{(40)} \end{displaymath}

Solving this for $u$ we get

\begin{displaymath} U={u_*\over k}\ln\left ({z\over z_0}\right )\eqno{(41)} \end{displaymath}

where $z_0$ is a constant. Note that as $z$ decreases towards zero, $U$ goes to zero at $z=z_0$. $z_0$ 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 $K=ku_*z$ no longer applies. Another way to think about the problem is that there is another length scale, in addition to $z$, introduced into the problem. This is

\begin{displaymath} L=-{\rho C_pu_*^3\overline T\over kgH}\eqno{(42)} \end{displaymath}

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

\begin{displaymath} K={ku_*z\over 1+\alpha{z\over L}} \end{displaymath}

so that

\begin{displaymath} {{\rm d}U\over {\rm d}z}={u_*\over kz}\left (1+\alpha{z\over L}\right ) \eqno{(43)} \end{displaymath}

where $\alpha$ is a constant. Typically, $\alpha\approx 5$. The solution of Eq. (43) is

\begin{displaymath} U={u_*\over k}\left [\ln\left ({z\over z_0}\right )+\alpha{z\over L}\right ]. \eqno{(44)} \end{displaymath}

3.4 The Ekman Layer

On a scale deeper than the surface layer, we can, to a crude approximation, take $K=\hbox{~constant}$. For neutral stratification, $K\approx 10$ m$^2$s$^{-1}$. If the flow is steady, Eq. (32)-(34) become

\begin{displaymath} -2\Omega\overline v\sin\phi= -{1\over \rho}{\partial\overline p\over \partial x}+ K{\partial^2\overline u\over \partial z^2}, \end{displaymath}


\begin{displaymath} +2\Omega\overline u\sin\phi= -{1\over \rho}{\partial\overline p\over \partial y}+ K{\partial^2\overline v\over \partial z^2}. \end{displaymath}

In the boundary layer we can assume that $\partial\overline p/\partial x$ and $\partial\overline p/\partial y$ are independent of height and are given by

\begin{displaymath} {\partial\overline p\over \partial x}=2\Omega\rho v_g\sin\phi , \end{displaymath}


\begin{displaymath} {\partial\overline p\over \partial y}=-2\Omega\rho u_g\sin\phi \end{displaymath}

where $u_g$ and $v_g$ are called the geostrophic wind components. For our purposes, we can take them to be the wind above the boundary layer. Let $f=2\Omega\sin\phi$ -- the Coriolis parameter. Then

\begin{displaymath} \left .\eqalign{K{\partial^2\overline u\over \partial z^2}=-... ...over \partial z^2}=f(\overline u-u_g).\cr} \right\}\eqno{(45)} \end{displaymath}

These are the Ekman equations. The solutions are

\begin{displaymath} \left .\eqalign{u &= u_{g}\Bigl [1-e^{-{z/\delta}}\cos\Bigl(... ...{z/\delta}}\sin\Bigl({z\over\delta}\Bigr)}\right\} \eqno{(46)} \end{displaymath}

where

\begin{displaymath} \delta =2\sqrt{{K\over f}}. \end{displaymath}

$\delta$ is a measure of the thickness of the atmospheric boundary layer. In mid-latitudes, $f\approx 10^{-4}$ s$^{-1}$ and we find that $\delta\approx$ 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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