A reference state close to the initial condition

[kaiyuan-cheng]

The current design uses an isentropic atmospheric profile as the reference state, which may cause problems when simulating a non-isentropic atmosphere. (An isentropic profile is highly idealized and not representative of real atmospheric conditions.) I suspect this may be the root cause of why we aren’t getting the correct internal gravity wave, and possibly the same issue affecting the supercell case.

Could we instead use a reference state that is closer to the initial condition? For example, the model could take a user-defined atmospheric profile, such as a sounding or the horizontal average of a 3D realistic atmosphere, adjust it to be hydrostatically balanced, and use that as the reference state. I believe this is how CM1 handles the reference state in its anelastic solver.

[glwagner]

I would prefer not to automatically compute the reference state. However, we can build some tools to compute reference states that are close to some other state.

One thing that is missing is the ability to have a moist reference state. This would be a simple change.

However other than that there are no additional parameters to adjust. The hydrostatic constraint means that the solution is completely specified by the potential temperature and pressure at z=0, plus the moisture profile. Or am I missing something?

I am curious if there is any evidence that "closeness to the reference state" matters for fidelity?

[kaiyuan-cheng]

I am not complete sure if I understand hydrostatic constraint you were referring to. My understanding is that the hydrostatic constraint is applied locally and depends on the temperature profile (which varies with height) and surface pressure.

My concerns with the current reference state are twofold:

1. The violation of the anelastic approximation (i.e., small perturbation relative to the reference state). An isentropic profile represents a well-mixed profile and can be very different from a realistic atmospheric profile. When we initialize the model with a realisitc atmospheric profile, it is likely to generate large perturbations (relative to the isentropic reference state).

The figure below shows the ratio of the initial perturbation (initial condition - reference) to the isentropic reference for density.

(Disclaimer: this figure was calculated and produced by ChatGPT. The exact values might be off, but the behavior matches my experience.)

Whether this is acceptable depends on how we define “small” perturbations. If we take <10% as the criterion, the anelstic approximation is good below 17.5 km. However, these are the perturbations for a calm atmosphere. When there are motions, the perturbations can be larger. In any case, I would expect that the smaller the perturbation, the better.

2. The solution of the dynamics field strongly depends on the reference state. Correct me if I am wrong, but I expect that a different reference state will give a different dynamics field. If that’s the case, then starting from a highly idealized isentropic state could bias or distort the simulated dynamics.

For those reasons, I think it would be better to use a reference state that is close to the initial condition.

[glwagner]

I am not complete sure if I understand hydrostatic constraint you were referring to. My understanding is that the hydrostatic constraint is applied locally and depends on the temperature profile (which varies with height) and surface pressure

Doesn't the reference state need to be a solution to the equations of motion / thermodynamic equation? The anelastic equations are developed by perturbing around this basic state, but the basic state must be a solution or we will end up with more terms in the anelastic equations.

For those reasons, I think it would be better to use a reference state that is close to the initial condition.

I buy this!

I propose a utility that looks like this:

reference_state = optimal_reference_state(grid, constants; vertical_profile_information...)

This will be an alternative to using

reference_state = ReferenceState(grid, constants; parameters...)

whereby users input the base pressure, potential temperature, etc. This leaves both options open, with no difference in the amount of code required between the two options.

If we take <10% as the criterion, the anelstic approximation is good below 17.5 km.

My question is: even if the asymptotic conditions of the anelastic approximation are violated --- what is the impact on the simulated dynamics? It is sometimes the case that asymptotic approximations remain valid outside of their strict range of validity.

[kaiyuan-cheng]

I am not complete sure if I understand hydrostatic constraint you were referring to. My understanding is that the hydrostatic constraint is applied locally and depends on the temperature profile (which varies with height) and surface pressure

Doesn't the reference state need to be a solution to the equations of motion / thermodynamic equation? The anelastic equations are developed by perturbing around this basic state, but the basic state must be a solution or we will end up with more terms in the anelastic equations.

Yes, the reference state is part of the equations. That's why I think the choice of the reference state matters.

For those reasons, I think it would be better to use a reference state that is close to the initial condition.

I buy this!

I propose a utility that looks like this:

reference_state = optimal_reference_state(grid, constants; vertical_profile_information...)
This will be an alternative to using

reference_state = ReferenceState(grid, constants; parameters...)
whereby users input the base pressure, potential temperature, etc. This leaves both options open, with no difference in the amount of code required between the two options.

Looks great to me!

If we take <10% as the criterion, the anelstic approximation is good below 17.5 km.

My question is: even if the asymptotic conditions of the anelastic approximation are violated --- what is the impact on the simulated dynamics? It is sometimes the case that asymptotic approximations remain valid outside of their strict range of validity.

I did some research a while back, and it appears that the common practice is just to ignore the violation of the anelastic approximation during the simulation. I think such a limitation highlights the need for fully compressible equations.

[glwagner]

Yes, the reference state is part of the equations. That's why I think the choice of the reference state matters.

Ok so continuing on this, a horizontally-uniform solution must be in hydrostatic balance. If the reference state is dry, hydrostatic balance + ideal gas law reduce the adjustable state to just the surface pressure and surface temperature.

If we allow moisture, then a much wider range of solutions are admitted, since the specific humidity can vary arbitrarily in the vertical.

[kaiyuan-cheng]

Ok so continuing on this, a horizontally-uniform solution must be in hydrostatic balance. If the reference state is dry, hydrostatic balance + ideal gas law reduce the adjustable state to just the surface pressure and surface temperature.

If we allow moisture, then a much wider range of solutions are admitted, since the specific humidity can vary arbitrarily in the vertical.

I think the dry, hydrostatically balanced vertical profile derived from the initial condition should be sufficient.

[glwagner]

Ah ok well that is super easy then!