### 11.9 What other fluxes are important?

We have focused on the sensible heat flux up to now, but turbulence creates other vertical fluxes. There are many vertical turbulent fluxes, but two important ones are the **latent heat flux**, which involves the vertical transport of water vapor, and the** horizontal momentum flux**, which involves the vertical transport of horizontal wind.

### Latent Heat Flux

For the purpose of this discussion, use the specific humidity, *q*. There is the mean value for *q* at different heights, and then there is the kinematic eddy flux. Using the same methods as before, we find that the kinematic water vapor flux (or specific humidity flux) is given by:

$${F}_{v}=\overline{w\text{'}q\text{'}}$$

This flux has SI units of ${\text{kg}}_{\text{water}}{\text{kg}}_{\text{air}}{}^{-1}{\text{ms}}^{-1}$. Usually the specific humidity is greatest near Earth's surface and decreases with height. Using the same logic as for the sensible heat flux, we expect a flux of water vapor from the surface, where the specific humidity is greater, to the free troposphere, where the specific humidity is less.

However, we usually want to compare energy fluxes caused by different processes as in Lesson 7.3, so we multiply the specific humidity flux by the terms necessary to convert it into an energy flux that would result from the condensation of that water vapor. We end up with a latent heat flux:

$${F}_{LH}={\rho}_{air}{l}_{v}\overline{w\text{'}q\text{'}}$$

with SI units of $({\text{kg}}_{\text{air}}{\text{m}}^{-3})({\text{Jkg}}_{\text{water}}{}^{-1})({\text{kg}}_{\text{water}}{\text{kg}}_{\text{air}}{}^{-1}{\text{ms}}^{-1})={\text{Jm}}^{-2}{\text{s}}^{-1}$.

Note that we have multiplied the specific humidity flux by the density of air and the latent heat of vaporization to put the specific humidity flux in terms of an energy flux, which we see is comparable to the sensible heat flux and is a significant fraction of the global energy balance at Earth’s surface. In fact, on a global scale, the latent heat flux is about five times larger than the sensible heat flux and is about half the total absorbed solar irradiance.

Latent heat flux is the primary way that water vapor gets into the atmosphere and is thus the primary source of water vapor for convection and clouds. Predicting convection and precipitation depends on knowing the latent heat flux.

### Horizontal Momentum Flux

The mean horizontal wind velocity is the vector sum of the wind components in the *x*-direction and the *y*-direction. The magnitude of the mean horizontal wind velocity is given by:

$$\overline{{V}_{}}=\sqrt{{\overline{u}}^{2}+{\overline{v}}^{2}}$$

The horizontal momentum flux is basically vertical turbulent eddies bringing high-wind-velocity air down from above. You all have experienced this phenomenon if you have ever been out early in the morning, just as the solar heating of the surface has begun to create convection and mix calm near-surface air up and windier residual layer air down.

The equations for the (kinematic) vertical fluxes of *x*-momentum and *y*-momentum air are, respectively:

$${F}_{mx}=\overline{{u}^{\prime}{w}^{\prime}}\text{and}{F}_{my}=\overline{{v}^{\prime}{w}^{\prime}}$$

where the SI units are m^{2} s^{–2} and where *u'* and *v'* are wind speed perturbations in the *x* and *y* directions, respectively.

Note that the horizontal wind speed, *V*, is zero at Earth's surface (because of molecular friction) and increases with height. Just as the turbulent heat flux moves air with a higher potential temperature to heights where the potential temperature is lower, the turbulent momentum flux moves air with higher horizontal momentum (i.e., horizontal velocity) to heights where the mean horizontal momentum is lower. That is, the horizontal momentum is moved downward through the boundary layer to the Earth's surface, where it is dissipated by molecular friction.

Just as the heat flux is equal to a constant times the vertical gradient of the mean potential temperature (Equation [11.9]), the *x*-momentum flux is equal to a constant times the vertical gradient of the mean *x*-wind:

where *K* is the eddy diffusivity.

Just as the change with time of the mean potential temperature is related to the negative of the vertical gradient of the kinematic heat flux (Equation [11.11]), so is the change with time of the mean velocity related to the negative of the vertical gradient of the kinematic momentum flux. Thus, the *x*-component momentum equation in the boundary layer becomes (ignoring other terms for now, such as the pressure gradient force and the Coriolis force):

$$\frac{\partial \overline{u}}{\partial t}=-\frac{\partial \left(\overline{{u}^{\prime}{w}^{\prime}}\right)}{\partial z}$$

Just as we assumed the mean potential temperature is constant with height in the boundary layer, we can assume that the mean *x*-momentum (i.e., zonal velocity, *u*) is constant with height in the boundary layer. We can then integrate the above equation from the surface (*z* = 0) to the top of the boundary layer (*z* = *h*) and make the same assumptions about the flux at the top being relatively small to get:

$$\frac{\partial \overline{u}}{\partial t}=-\frac{1}{h}{\left(\overline{{u}^{\prime}{w}^{\prime}}\right)}_{0}$$

where ${\left(\overline{{u}^{\prime}{w}^{\prime}}\right)}_{0}$ is the vertical flux of *x*-momentum at the surface. As Equation [11.16b] indicates, this flux depends on the local eddy diffusivity and the local vertical gradient of the mean *u* velocity. The eddy diffusivity near the surface will increase as the mean wind speed in the boundary layer increases because vertical shear, which is responsible for the mechanical generation of turbulence, will be greater as the mean wind speed increases. Also, the vertical gradient of the mean *u* velocity near the surface will increase as the mean *u* of the boundary layer increases. Thus, we expect

$${\left(\overline{{u}^{\prime}{w}^{\prime}}\right)}_{0}\propto \overline{V}\overline{u}$$

The coefficient of proportionality is called the drag coefficient, ${C}_{d}$, which depends on the roughness of the surface and the thermal stability at the surface. Hence, combining Equations [11.16d] and [11.16e], we have for both horizontal components:

These are the turbulent resistance terms that we introduced as friction in Lesson 10. They come from the downward transfer of horizontal momentum to the surface, where it is dissipated by molecular friction at Earth's surface.