[MITgcm-support] pressure gradient error

[Martin Losch]

Jean-Michel,

you warned me about the pressure gradient errors for the z*-coordinate. 
The question is now, how much error do I have to expect. I played 
around with a setup that is prone to such errors: a 2D section through 
an idealized Antarctic shelf with an abyss (3000m) as shelf break 
(1000m) and an shelf-ice cavern extending down to 1350m. I specify the 
shelf ice by an initial eta and a constant (in-time) pressure (pload), 
that I compute by integrating dp=-g*rho*dz "exactly" (by an iteration, 
as we did with the pressure coodinate model), so that the 3rd and 2nd 
last terms in eq.(16) in Adcroft+Campin(2004) should balance (grad of 
eta+grad_z* of p). They probably don't do that exactly.

If I have an initial stratification (theta and S varying linearily with 
depth), that is horizontally homogeneous, I have to do some 
interplation to get theta ans S onto the correct levels, that's a 
source of errors. Also, after computing the pressure at level tracer 
points, I have to interpolate it to get the pressure load at z=eta 
(regardless of stratification). Another source of errors, probably more 
severe, because I am dealing with pressure now.

When I let the system go, it should not move (theoretically), if I have 
done everything correctly. Of course everything moves, because
1. pressure gradient errors
2. interpolation errors described above that lead to an initial 
imbalance
3. other things that I have not thought about (including the typical 
mlosch-mistakes).

So I am not surprised about the system is starting to move. What I am 
surprised about is the magnitude of the motion. U and V can reach 10m/s 
along the sloping z*-coordinates. So my question is, what's the 
expected order of magnitude for the pressure gradient errors?

I have attached my gendata.m and data/data.pkg files, if you want to 
have a look at them. Also I attached a plot of the in-plane velocity 
after 40 days for a case with moderate slopes of z*. This is already 
close to a steady state in equilibrium. When I choose a configuration 
where the ice-load depresses the surface to almost the bottom at the 
end of the cavern, the velocity is much higher and reaches far down 
into the cavern, which then leads to a blow-up because of CFL. Am I 
making a serious mistake or should I just accept that with 
sigma/z*-coordinates, things like these happen? Is there a simple way 
to estimate the pressure gradient error?

Martin

PS. I also noticed something weird about the diagnostic variable PH. At 
the zeroth time step (PH.0000000000.data), it seems to have the surface 
pressure included, and after that, it doesn't. I could not figure out 
why that is so, but that probably has not effect on the solution.

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