Monday's discussion on evapotranspiration - Part I

Last Monday at lunch, I and my students discussed about evapotranspiration. I already talk about it in various comment here. However, the starting point was the impression, coming from one of my student that the topic of transpiration is still in its infancy. I agree with him and I offered my synthesis.

1. In hydrology we use Dalton’s law (here, slide 21) or the derived equations named Penman-Monteith and Priestley Taylor (forgetting all the empirical formulas).
2. Dalton’s law  put together, diffusive vapor flux, vapor storage and turbulent transport.
3. We should have a water vapor budget equation instead written for some control volume where all the stuff is at its right place.

There is no difficult to recognize that control volume is limited in our case by the open atmosphere and the surfaces that are emitting vapor as consequence of  their thermodynamics. There is also no difficult to recognize that such limiting surfaces can be very complicate, as those of a canopy, for instance, which is also varying in shape and form with time. Out of this surfaces comes a vapor flux which is dependent on the thermodynamics and physiology existing below them, the holes in number and distribution through which the vapor is emitted, the water availability (and dynamics) in the storage below the surface and, last but not least, the vapor content in the control volume which, as in Dalton’s law commands the driving force.

It is not easy actually to account well for all of these factors. We can, maybe, for a single leaf. It is more complicate for the canopy of a single tree. It is even more difficult for a forest. Unless some goddess  acts to simplify the vapor budget, over the  billions of details and reduces all to some tretable statistics (we can call this statistics the Holy Grail of evapotranspiration - or the whole Hydorlogy itself) 

Assume we can deal with it, and  we have the fluxes right. Then the vapor budget seems cristalline simple to obtain, the variation of vapor in the control volume is given by the incoming vapor flux, minus the outcoming vapor flux, minus, in case, the vapor condensation. the good old mass conservation. Unfortunately, also the output flux is not that easy to estimate, because the transport agent is atmospheric turbulence, which is affected by its own complications, and by the interaction of the air fluid with the above mentioned complex terrain/vegetation surfaces.

 If we roughly see the action of turbulence as a diffusive one, we can say that it spreads out vapor much more effectively than molecular diffusion, and therefore we could think that the surface emitted vapor is efficiently spread by turbulence. Diffusion does not create vapor, nor it destroys it. Therefore it just dilutes vapor to adjacent control volumes. Any control volume, in fact, can receive or give vapor from/to the  next ones. I claim that hydrology should take care of the water vapor budget not just of a mimesis of fluxes with parameters to fix. 

Literature instead treats the problem as a flux, forget the real mechanics of fluxes, and simplify turbulence according to similarity theory, essentially due to Prandtl work at the beginning of the last century and, in case, complicated with the additions of Monin-Obukhov theory, The hypothesis of similarity theory are easily broken and the velocities distributions are rarely those expected from this theory in the case of rugged terrain covered by a complex vegetation (even forgetting itspeculiarity as vapor emitter). All of this makes also unreliable the transport theory, and this was what the work, for instance, of Raupach and coworkers shows.

So, even if we would know the inputs, the transport is complicate because turbulence interacting with complex surfaces is complicate. Numerically is a problem whose numerical solution is still open (a full branch of science, indeed), and we do not know how to model the rustling of leaves (“… and the icy cool of the far, far north, with rustling cedars and pines)

If inputs were easy to get, they would be a boundary condition for Navier-Stokes equations, and Navier-Stokes will describe the transport. This would be THE solution of our evaporation/transpiration problem. In fact, the models we use for what we usually call potential evapotranspiration, are an extreme simplification of this view. 

However, this interpretation forgets the role itself of air and soil temperature. We were thinking, in fact, only to the mass budget and the momentum budget (the latter is what NS equation is),  but there is no doubt that evaporation and traspiration are commanded also, and in many ways, by the energy budget. Turbulence itself is modified by temperature gradients, but also water vapor tension which concurs to establish the quantity of water vapor at vapor emitting surfaces. Finally energy is conserved as well, and this conservation includes the phase of the matter transported (the so-called latent heat). So necessarily, in relevant hydrological cases, we have to solve besides the mass and momentun conservation, the energy conservation itself.

Looking for simplified versions of mass, momentum and energy budget, would require a major rethinking of all the derivations and new impulse to proper measurements.