Fundamentals of Atmospheric Science

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 water vapor, and the vertical 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 water vapor (kinematic) flux is given by:

F v = w'q' ¯


This flux has units of m  s 1  kg( H 2 O) kg ( air ) 1  =m  s 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 that the latent heat flux should carry specific humidity from the surface, where it is greatest, to the free troposphere, where it is less.

However, we usually want to compare energy fluxes caused by different processes as in Lesson 7.3, so we multiply the flux of specific humidity 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 = ρ air l v w'q' ¯


with units k g air   m 3  J k g 1   K 1  m  s 1  k g v  k g air 1  = J  m 2   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 speed is the vector sum of the wind speeds in the x-direction and the y-direction.

V  ¯ = u ¯ 2 + v ¯ 2


The horizontal momentum flux is basically vertical turbulent eddies bringing horizontal wind eddies 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 near-surface air up and windier residual layer air down.

The equation for (kinematic) horizontal momentum flux is:

F m = u'w' ¯   or   F m = v'w' ¯


where the units are m2 s-2 and where u’ and v’ are wind speed perturbations in the x and y directions.

Note that the horizontal wind speed, V, is zero at Earth's surface because of molecular viscosity 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 momentum (i.e., velocity) to heights where the mean momentum is lower. That is, the horizontal momentum is moved from the upper troposphere to Earth's surface, where it is dissipated by molecular viscosity and friction.

Let's assume for simplicity that the wind is eastward in the upper troposphere, with a mean wind speed u ¯ . So just as the heat flux is equal to a constant times a gradient of the mean potential temperature (Equation [11.9]), the horizontal momentum flux can be written as:

u'w' ¯ =K u ¯ z

where K is the eddy diffusivity. K is proportional to the level of turbulence, which is proportional to the vertical gradient of the total mean wind and the vertical gradient of u ¯ is proportional to u ¯ . So:

K| V ¯ z | and  u ¯ z u ¯

Putting this all together and calling the constant of proportionality C d , the drag coefficient, the horizontal momentum flux above the surface can be written as the bulk aerodynamic formulae:

u'w' ¯ = C d | V | h u ¯ v'w' ¯ = C d | V | h v ¯

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