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f67abf1ee3 Jeff*0001 Ocean
0002 -----
0003
0004 In the ocean we interpret:
0005
0006 .. math::
0007 r=z\text{ is the height}
0008 :label: ocean-z
0009
0010 .. math::
0011 \dot{r}=\frac{Dz}{Dt}=w\text{ is the vertical velocity}
0012 :label: ocean-w
0013
0014 .. math::
0015 \phi=\frac{p}{\rho _{c}}\text{ is the pressure}
0016 :label: ocean-p
0017
0018 .. math::
0bad585a21 Navi*0019 b(\theta ,S,r)=\frac{g}{\rho _{c}} \left( \vphantom{\dot{W}} \rho (\theta,S,r) - \rho_{c}\right)
0020 \text{ is the buoyancy}
f67abf1ee3 Jeff*0021 :label: ocean-b
0022
0bad585a21 Navi*0023 where :math:`\rho_{c}` is a fixed reference density of water and
f67abf1ee3 Jeff*0024 :math:`g` is the acceleration due to gravity.
0025
0026 In the above:
0027
0bad585a21 Navi*0028 At the bottom of the ocean: :math:`R_{\rm fixed}(x,y)=-H(x,y)`.
f67abf1ee3 Jeff*0029
0bad585a21 Navi*0030 The surface of the ocean is given by: :math:`R_{\rm moving}=\eta`
f67abf1ee3 Jeff*0031
0032 The position of the resting free surface of the ocean is given by
0033 :math:`R_{o}=Z_{o}=0`.
0034
0035 Boundary conditions are:
0036
0037 .. math::
0bad585a21 Navi*0038 w=0~\text{at }r=R_{\rm fixed}\text{ (ocean bottom)}
f67abf1ee3 Jeff*0039 :label: fixed-bc-ocean
0040
0041 .. math::
0bad585a21 Navi*0042 w=\frac{D\eta }{Dt}\text{ at }r=R_{\rm moving}=\eta \text{ (ocean surface)}
f67abf1ee3 Jeff*0043 :label: moving-bc-ocean
0044
0045 where :math:`\eta` is the elevation of the free surface.
0046
0047 Then equations :eq:`horiz-mtm`- :eq:`humidity-salt` yield a
0048 consistent set of oceanic equations which, for convenience, are written
0049 out in :math:`z-`\coordinates in :numref:`ocean_appendix` - see eqs. :eq:`eq-ocean-mom`
0050 to :eq:`eq-ocean-salt`.
0051
