Compute reference density with derivative

[glwagner]

This PR computes the reference state density using a derivative, from the reference pressure, rather than evaluating the analytical formula. This ensures that

$$ \partial_z p_r = - \rho_r g $$

discretely exactly rather than just approximately.

Might make sense to polish this PR off with a test.

[kaiyuan-cheng]

For the test, it may make sense to have a still atmosphere as an initial condition to see if any perturbation is generated during the initialization.

[kaiyuan-cheng]

This code runs fine, and the difference in outcome is minimal. But it is good to have a consistent, discretized form of hydrostatic balance.

[navidcy]

are we sure it's not

... - ℑzᵃᵃᶠ(i, j, k-1, grid, ρᵣ) * g * Δzᶜᶜᶠ(i, j, k-1, grid)

?

[navidcy]

We want to interpolate the density on the face between cell center k-1 and k and add it to the integral, right? But that's face k-1, no?

[navidcy]

using Oceananigans
using CairoMakie
using Oceananigans.Operators: Δzᶜᶜᶠ, ℑzᵃᵃᶠ

grid = RectilinearGrid(size=20, z=(0, 5), topology=(Flat, Flat, Bounded))

ρ = CenterField(grid)

p₁ = CenterField(grid)
p₂ = CenterField(grid)
p_analytic = CenterField(grid)

p₀ = 123.2

set!(ρ, z -> exp(-2z))

# dp/dz = - ρ, p(0) = p₀
set!(p_analytic, z -> p₀ + 1/2 * (exp(-2z) - 1))

for f in (p₁, p₂, p_analytic)
    f[1] = p₀
end

i, j = 1, 1
for k = 2:grid.Nz
    @inbounds p₁[i, j, k] = p₁[i, j, k-1] - ℑzᵃᵃᶠ(i, j, k, grid, ρ) * Δzᶜᶜᶠ(i, j, k, grid)
    @inbounds p₂[i, j, k] = p₂[i, j, k-1] - ℑzᵃᵃᶠ(i, j, k-1, grid, ρ) * Δzᶜᶜᶠ(i, j, k-1, grid)
end

fig = Figure()
axρ = Axis(fig[1, 1], ylabel="ρ(z)")
axp = Axis(fig[1, 2], ylabel="p(z)")
axe = Axis(fig[1, 3], ylabel="error")

lines!(axρ, ρ)

scatterlines!(axp, p₁)
scatterlines!(axp, p₂)
scatterlines!(axp, p_analytic)

scatterlines!(axe, abs(p_analytic - p₁), label="|p_analytic - p₁|")
scatterlines!(axe, abs(p_analytic - p₂), label="|p_analytic - p₂|")
axislegend(axe, position=:rt)

fig

[navidcy]

Not sure which is better; given that it's only derivatives of pressure that matter the blue might be better since it's more "parallel" to the analytic? If abs value matters, seems like yellow is better as it doesn't have that offset error...

[glwagner]

oh you are right!

[glwagner]

pretty nice demo btw

[navidcy]

We can have a test to compare with analytic?

[glwagner]

We can have a test to compare with analytic?

well, there is a test that fails actually, which involves the potential temperature - energy equivalence.

I also realized that it will be simpler to compute the density from pressure (not the other way around) by computing the derivative. What do you think about that?

[navidcy]

I'm bit puzzled for the use of computing numerically since there are analytic expressions...
Is it so that there is consistency with the numerics?

[glwagner]

I'm bit puzzled for the use of computing numerically since there are analytic expressions... Is it so that there is consistency with the numerics?

that's the idea. but we don't know if it matters.

[navidcy]

Gotcha...

[glwagner]

Gotcha...

we'd have to read Pauluis 2008 more closely (we also should put in a total energy test)

[glwagner]

what do you think about my method using a derivative instead of integral?

[navidcy]

what do you think about my method using a derivative instead of integral?

Yes, I'm not sure.

Either methods use the analytical for p or ρ and compute the other numerically... But how do we pick which is better to be computed numerically vs analytically?

[glwagner]

I think the key property is that the vertical momentum equation is discretely satisfied (whereas it was not before). I believe this could enter into considerations about conservation of total discrete energy (although I am not sure, just speculating).

If that is what matters, then you can achieve it either way.

If there is another consideration, then I'm not sure. Using the analytical profiles might be fine too.