Turbulent fluxes: surface level vs atmospheric level properties

[olivierbonte]

When working on #112, I started thinking a bit more about the formulation of turbulent fluxes. The classic bulk formula, here for heat transfer, is

$$H = \rho_a c_p \frac{T_s - T_a}{r_a}$$

with $T_s$ the surface temperature, $T_a$ the temperature at the atmospheric reference level ($z_{atm}$), $r_a$ aerodynamic resistance. In most textbooks I come across, and also the CLM documentation on this although here formulated with potential temperature, both the air density $\rho_a$ and the (isobaric) specific heat capacity of air are assumed at $z_{atm}$. For CLM, $c_p$ is considered a constant assuming dry air (see here).

When looking at the ClimaLand paper or Breeze.jl, they don't use $\rho_a$ but instead use $\rho_s$, the surface air density. ClimaLand uses an adiabatic extrapolation for this (see here in the SurfaceFluxes.jl repo) and also Breeze.jl has several methods to estimate this variable (see e.g. this one).

Within Terrarium.jl, such an approach could also be adopted. At this point, it is not entirely obvious to me however why using $\rho_s$ over $\rho_a$ would be more correct, as classical references on this don't go in detail on this. Maybe @glwagner or @kmdeck can help me out here? I suppose that $c_p$ can still be be evaluated at the specific humidity $q$ from $z_{atm}$ (e.g. using cp_m), or would it also for this variable make more sense to evaluate it with $q_s$?

Many thanks in advance

[kmdeck]

I think you need rho_sfc regardless (to compute q_sat at the surface - or P_sfc). Then it is a question of how you parameterize it since it is not explicitly modelled in most/any land models. One is rho_s = rho_atmos, another is adiabatic extrapolation, etc.

With respect to cp_m, we use a gradient in dry static energy as driving the flux at the boundary, so we look at (cp_m T)_atmos - (cp_m T)_sfc. As for the rho out front, I'm not sure which is most correct. If the boundary condition (at the surface) is expressed proportional to rho d(cT) = rho(cT_atmos - cT_sfc), it seems like a modelling choice whether rho is evaluated at the face or center or a mix of the two, though you're right that in CL we use rho_sfc. What we do is described here: https://essopenarchive.org/doi/full/10.22541/essoar.173940262.23304403/v1

[glwagner]

My two cents is that this corresponds to a model for the energy or entropy flux (depending on formulation) at the bottom of the atmosphere. Therefore $\rho$ should be assessed at the surface (but on the air side).

One thought experiment to clarify the situation is to consider what happens when you change the measurement height for $T_a$. In that case, if the atmospheric variables indeed has a similarity profile, then the flux that you compute will be exactly the same. For this to be true, it requires that the variables which are not explicitly used to construct the similarity profile are evaluated at the surface.

When you use the density at the measurement height, you are invoking a model that the density is well-mixed.

I would not consult Breeze for documentation on surface fluxes. Those are only intended for idealized simulations, and are not constructed carefully as would be needed for coupled simulations. The NumericalEarth documentation is more comprehensive and documents more sophisticated models: https://numericalearth.github.io/NumericalEarthDocumentation/stable/interface_fluxes/

[olivierbonte]

Thanks to both of you for the replies!
Your thought experiment defnitely clarifies this choice for $\rho_s$ @glwagner, never thought of it this way before.
I also understand better now why ClimaLand uses the dry static energy as to be able to consistenly evaluate the specific heat capcity at both levels.