Fundamentals of Atmospheric Science

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


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 water vapor from regions of higher water vapor concentration far from the drop towards regions of lower water vapor concentration near the drop. The greater the difference between the concentration far away from the drop and the concentration near the drop, the greater will be the flux of water molecules to the drop.

F molecules = D v n r MathType@MTEF@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=ribbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqqaaaaaaaaGySf2yRbWdbiaadAeapaWaaSbaaSqaa8qacaWGTbGaam4BaiaadYgacaWGLbGaam4yaiaadwhacaWGSbGaamyzaiaadohaa8aabeaak8qacqGH9aqpcqGHsislcaWGebWdamaaBaaaleaapeGaamODaaWdaeqaaOWdbmaalaaapaqaa8qacqGHciITcaWGUbaapaqaa8qacqGHciITcaWGYbaaaaaa@4683@

where F molecules MathType@MTEF@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=ribbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqqaaaaaaaaGySf2yRbWdbiaadAeapaWaaSbaaSqaa8qacaWGTbGaam4BaiaadYgacaWGLbGaam4yaiaadwhacaWGSbGaamyzaiaadohaa8aabeaaaaa@3D3A@ is the molecular flux (SI units of molecules m–2 s–1), n/r MathType@MTEF@5@5@+=faaahmart1ev3aqaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=ribbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqqaaaaaaaaGySf2yRbWdbiabgkGi2kaad6gacaGGVaGaeyOaIyRaamOCaaaa@3909@ is the change in concentration n (SI units of molecules m–3) as a function of radial distance r from the drop (SI units of m), and D v is the molecular diffusion coefficient (SI units of m2 s–1). 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 a distance L by molecular diffusion is:

τ= L 2 D v MathType@MTEF@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=ribbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqqaaaaaaaaGySf2yRbWdbiabes8a0jabg2da9maalaaapaqaa8qacaWGmbWdamaaCaaaleqabaWdbiaaikdaaaaak8aabaWdbiaadseapaWaaSbaaSqaa8qacaWG2baapaqabaaaaaaa@3AB9@


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? A typical value for Dv is 2 x 10–5 m2 s–1.

Click for answer.

Put 1000 m for the L 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 the transport of water vapor (and, as it turns out, anything else, such as ozone, heat, and momentum) over distances on the scale of even the planetary boundary layer.

Molecular diffusion cannot transport anything 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, molecules, and momentum. In the last section, we saw that turbulence tends to move heat from heights where the air is warmer to heights where the air is cooler. 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@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=ribbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqqaaaaaaaaGySf2yRbWdbiaadAeapaWaaSbaaSqaa8qacaWGObGaamyzaiaadggacaWG0baapaqabaGcpeGaeyypa0ZdamaanaaabaWdbiqadEhapaGbauaapeGafqiUde3dayaafaaaa8qacqGH9aqpcqGHsislcaWGlbWaaSaaa8aabaWdbiabgkGi2kqbeI7aX9aagaqeaaqaa8qacqGHciITcaWG6baaaaaa@457C@


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 of that quantity to where there is less of that quantity.