Several Thermodynamics updates.

[tapios]

A few updates to Thermodynamics and the documentation:

• [ ] Update the thermodynamics documentation following the tracked changes in the design doc.
• [x] Complete and merge #112 (including the updated derivative ∂q_vap_sat_∂T)
• [x] Use the updated function for ∂q_vap_sat_∂T from #112 in saturation adjustment l. 1402. (This will change the gradient by about 5%; thus, it may lead to a minor improvement in convergence of saturation adjustment, but the changes should not be large.)
• [x] Add a docstring after l. 843 for the saturation_vapor_pressure function over a mix of liquid and ice in l. 914
• [x] Change the saturation adjustment convergence check generally (in all version of SA) to a solution (i.e., temperature) tolerance, requiring temperature changes ΔT from iteration to iteration to be small in relative terms, i.e., ΔT/T < 1e-4.
• [x] Enable passing an initial state for saturation adjustment
• [ ] Check average number of iterations needed with previous time step (or stage) as initial state and ΔT/T < 1e-4. Compare that with the current default initial state. (Expectation is that fewer iterations are needed with previous time step as initial state.)
• [ ] If previous test yields expected results (faster convergence with previous timestep as initial state), make that the default in ClimaAtmos.

Some other minor fixes:

• [x] Use liquid_frac and λ consistently. E.g., replace liquid_frac in ll. 914 ff. and ll. 1314 ff. by λ
• [x] Replace specific entropy in l. 2846 and l. 2865 by partial pressure

[charleskawczynski]

@tapios, I think we spoke about this, the current implementation uses ΔT < tol, not ΔT/T < 1e-4, is this okay? If not, we can add this capability into RootSolvers.jl and update Thermodynamics-- it's not a difficult change, but I just wanted to clarify this point.

Also, can you confirm if we've updated the particular part of the docs in the first bullet point? I believe we've addressed them all in #112

[tapios]

@tapios, I think we spoke about this, the current implementation uses ΔT < tol, not ΔT/T < 1e-4, is this okay? If not, we can add this capability into RootSolvers.jl and update Thermodynamics-- it's not a difficult change, but I just wanted to clarify this point.

ΔT < tol is ok if relative tolerance takes much work. But I'd prefer a relative tolerance (e.g., so that we have a relative tolerance on buoyancy, and can easily ran this in other planetary configurations.

Also, can you confirm if we've updated the particular part of the docs in the first bullet point? I believe we've addressed them all in #112

The docs (https://clima.github.io/Thermodynamics.jl/dev/Formulation/) do not seem to be updated yet. There were a number of changes in the LaTeX design doc (tracked) that need to be mirrored in the docs, e.g., formulation of saturation vapor pressure and derivatives of saturation specific humidity. (Downside of having two versions. We should abandon the LaTeX soon.)

[charleskawczynski]

Ah, we've already added RelativeSolutionTolerance to RootSolvers, so we just need to update it here.