Fundamentals of Atmospheric Science

11.7 Can we relate this turbulent flux to a molecular flux?

You have already encountered the idea of molecular fluxes in cloud physics, in the section on vapor deposition. Water vapor molecules are moving randomly all the time, but when water vapor molecules hit a liquid or ice surface and stick, the water vapor concentration near the drop is less than it is farther away. As a result, molecular diffusion tends to move higher water vapor concentration toward the drop where the lower water vapor concentration is. The greater the difference between the concentrations far away from the drop and the concentrations near the drop, the greater will be the flux of water molecules flowing to the drop.

F molecules = D v dn dr


where F molecules is the molecular flux  (molecules c m 2   s 1 ) , dn/dr is the change in concentration as a function of radial distance from the drop, and D v is the diffusion coefficient. When n increases with r , then the flux is negative, which means that the flux is toward the drop, in the negative r direction.

Molecular diffusion, by the way, is very slow at transferring molecules from one place to another in the troposphere. By solving the equations of motion for a simple case, we find that the characteristic time to travel from one location to another is:

τ= distance 2 D v MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes8a0jabg2da9maalaaabaGaaeizaiaabMgacaqGZbGaaeiDaiaabggacaqGUbGaae4yaiaabwgadaahaaWcbeqaaiaabkdaaaaakeaacaWGebaaaaaa@3EBD@


where distance is the linear distance the molecule travels, and D v is the molecular diffusion coefficient, where D v  ~ 2x 10 5   m 2   s 1 .

Check Your Understanding

By molecular diffusion, how long would it take water vapor molecules to move from Earth's surface to the top of the planetary boundary layer, 1 km away?

Click for answer.

Put 1000 m for the distance in Equation [11.8] along with the value for Dv. The resulting characteristic time is about 1600 years. So, it is clear that molecular diffusion cannot be responsible for transporting heat or anything else over distances on the scale of even the planetary boundary layer.

Molecular diffusion cannot transport heat or anything else fast enough for the atmosphere except on small scales of a centimeter or less. However, on the spatial and temporal scales of the planetary boundary layer, eddies are quite effective at moving heat and molecules. In the last section, we saw that heat flux tended to move heat from heights where it was greater to heights where it was less. Eddy "diffusion" shares this characteristic with molecular diffusion.

We can write the heat flux in the same way that we write the molecular flux:

F heat = ( w'Θ' ) ¯ =K Θ ¯ z MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAeadaWgaaWcbaGaamiAaiaadwgacaWGHbGaamiDaaqabaGccqGH9aqpdaqdaaqaamaabmaabaGaam4DaiaacEcacqqHyoqucaGGNaaacaGLOaGaayzkaaaaaiabg2da9iabgkHiTiaadUeadaWcaaqaaiabgkGi2kqbfI5arzaaraaabaGaeyOaIyRaamOEaaaaaaa@4617@


where K is the eddy diffusion coefficient. Since K is always positive, this equation makes it clear that the flux of any quantity goes from where there is more to where there is less of that quantity.