66
STEPHEN GRIFFIES
Coordinate Definition
Range
- - -
z-sigma
a = ( ~ - r ] ) / ( H + r ] )
- 1 < u < O
Table 2.1. Table of vertical coordinates based on depth. These coordinates are naturally used for discretizing the Boussinesq equations.
6.2
Pressure based coordinates
The second class of vertical coordinates that we discuss is based on
pressure. Pressure based coordinates provide a straightforward way to
generalize Boussinesq depth based models to non-Boussinesq pressure
models (Huang et al., 2001, DeSzoeke and Samelson, 2002, Marshall
et al., 2003, Losch et al., 2004). The reason is that there is an isomorphism between the Boussinesq equations written in depth based coordinates and non-Boussinesq equations written in pressure based coordinates.
Pressure based vertical coordinates of interest include the following:
s = p
pressure
pressure-sigma
pb - Pa
s = P: (E)
pressurest ar.
In these equations, p is the hydrostatic pressure, pa is the pressure applied at the ocean surface from any media above the ocean, such as
the atmosphere and sea ice, pb is the hydrostatic pressure at the solidearth lower boundary, and p: is a time independent reference pressure,
usually taken to be the bottom pressure in a resting ocean.17 Since
p,, = -pg < 0 is single signed for the hydrostatic fluid, pressure provides a well defined vertical coordinate. Strengths and weaknesses of the
corresponding depth based coordinates also hold for the pressure based
coordinates, with the main difference being that pressure based models
are non-Boussinesq.
A technical reason that the pressure based coordinates considered
here are so useful for non-Boussinesq hydrostatic modelling is that p z,,
1 7 ~ o t e
that equation (11.64) of Griffies, 2004 used the time dependent pb rather than the
time independent reference pressure p i . The former vertical coordinate has not been used in
practice, and so we focus here on that coordinate defined with the reference pressure p i .
STEPHEN GRIFFIES
Coordinate Definition
Range
- - -
z-sigma
a = ( ~ - r ] ) / ( H + r ] )
- 1 < u < O
Table 2.1. Table of vertical coordinates based on depth. These coordinates are naturally used for discretizing the Boussinesq equations.
6.2
Pressure based coordinates
The second class of vertical coordinates that we discuss is based on
pressure. Pressure based coordinates provide a straightforward way to
generalize Boussinesq depth based models to non-Boussinesq pressure
models (Huang et al., 2001, DeSzoeke and Samelson, 2002, Marshall
et al., 2003, Losch et al., 2004). The reason is that there is an isomorphism between the Boussinesq equations written in depth based coordinates and non-Boussinesq equations written in pressure based coordinates.
Pressure based vertical coordinates of interest include the following:
s = p
pressure
pressure-sigma
pb - Pa
s = P: (E)
pressurest ar.
In these equations, p is the hydrostatic pressure, pa is the pressure applied at the ocean surface from any media above the ocean, such as
the atmosphere and sea ice, pb is the hydrostatic pressure at the solidearth lower boundary, and p: is a time independent reference pressure,
usually taken to be the bottom pressure in a resting ocean.17 Since
p,, = -pg < 0 is single signed for the hydrostatic fluid, pressure provides a well defined vertical coordinate. Strengths and weaknesses of the
corresponding depth based coordinates also hold for the pressure based
coordinates, with the main difference being that pressure based models
are non-Boussinesq.
A technical reason that the pressure based coordinates considered
here are so useful for non-Boussinesq hydrostatic modelling is that p z,,
1 7 ~ o t e
that equation (11.64) of Griffies, 2004 used the time dependent pb rather than the
time independent reference pressure p i . The former vertical coordinate has not been used in
practice, and so we focus here on that coordinate defined with the reference pressure p i .