Fix density oscillations

[sajjadazimi]

Purpose

Fix density fluctuations by using downwind density info in updraft buoyancy eq

Here are $\rho^j - \rho$ plots after 1hr simulation of the single column trmm with prognostic EDMF when updating the updraft velocity by using density:

• upwind of the flow,
• interpolating the two sides,
• downwind of the flow,
  respectively:

[sajjadazimi]

Intuitive explanation of perturbation behavior (upwind vs downwind)

Consider an upward flow with a positive density perturbation at a cell center (ρ′ > 0 at i).

Upwind scheme

• Velocity at the face is updated using the upstream (upwind) density.
• A denser fluid leads to a reduction in velocity (u′ < 0).
• Reduced outflow from the perturbed cell causes the density perturbation to grow.
• Result: perturbations amplify → unstable.

Downwind scheme

• Velocity at the face is updated using the downstream (downwind) density.
• The negative velocity perturbation reduces inflow into the perturbed cell.
• The excess density gradually decays.
• Result: perturbations damp → stable.