Pressure in the MITgcm

PressureMITgcm.ipynb

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      "source": "# Pressure in the `MITgcm`\n\nI use the linear equation of state in the ocean.  Hopefully things are consistent for other equations of state.\n\nThe variable written in the files `PH.*.data` is `totPhiHyd`.  This is set in `diags_phi_hyd.F` as the sum of `phyHydC` at the local level and the surface pressure calculated from `EtaN` and the surface buoynacy anomaly.\n\nThe variable `phyHydC` is calculated in `calc_phi_hyd.F` and is calculated from \n```fortran\n        phiHydC(i,j) = phiHydF(i,j)\n     &             + dRlocM*gravity*alphaRho(i,j)*recip_rhoConst\n```\nSo the units are $m^2s^{-2}$.  `alphaRho(i,j)` is filled before this as: `alphaRho(i,j) = rhoInSitu(i,j,k,bi,bj)`, and `rhoInSitu` is set in `do_ocean_physics.F`.  There it is set by calling `calc_rho_2d`:\n```fortran\n          DO k=1,Nr\n            CALL FIND_RHO_2D(\n     I                iMin, iMax, jMin, jMax, k,\n     I                theta(1-OLx,1-OLy,k,bi,bj),\n     I                salt (1-OLx,1-OLy,k,bi,bj),\n     O                rhoInSitu(1-OLx,1-OLy,k,bi,bj),\n     I                k, bi, bj, myThid )\n          ENDDO\n```\nHere the first `k` is `kref` in `find_rho_2d` and the actual calculation for a linear EOS is simply:\n```fortran\n       refTemp=tRef(kRef)\n       refSalt=sRef(kRef)\n\n       dRho = rhoNil-rhoConst\n\n       DO j=jMin,jMax\n        DO i=iMin,iMax\n         rhoLoc(i,j)=rhoNil*(\n     &     sBeta*(sFld(i,j)-refSalt)\n     &   -tAlpha*(tFld(i,j)-refTemp) )\n     &        + dRho\n        ENDDO\n       ENDDO\n```\n\nSo what does this mean in math?  It means that `PH` is:\n\n$$ Ph = \\frac{1}{\\rho_0}\\int_z^{\\eta} (\\rho(x,y,z,t)-\\rho_R(z)) g\\ dz$$ \n\nIf `tRef` is a constant with depth, the $\\rho_R(z)$ is a constant, but if `tRef` is a profile (as I usually do) then it depends on $z$.\n\n\n\n\n\n"
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      "source": "## `PHrefC`\n\nThis variable, in my setup, is just $\\int_z^0 g\\ dz$  Why the model saves this, I don't understand.  It is written in `write_grid.F` from `phiRef`, which is a `2Nr` long variable with all the pressures.  This indeed does not include the reference density \"anomally\"\n\n\n\n\n"
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