The flux form of the PV equation has been discussed at length in Haynes and McIntyre (1987), Marshall and Nurser (1992), Rhines (1993), Schar (1993) and Bretherton and Schar (1993). Here we very briefly outline the approach, drawing out issues that are particularly pertinent to the ocean.
First let's clarify some issues related to the flux-form of the PV equation. Following Bretherton and Schar (1993) we can write, making use of the definition of PV and that :
(1) |
This shows that one can always write a conservation law in flux form from the definition of PV. This is true for any scalar field and non-divergent vector . If one adds any non-divergent vector to J, the flux-form equation will still be satisfied. The problem is then to set the gauge and this choice has to be made on physical grounds.
Expanding the partial derivative in J, one can write:
(2) |
(3) |
X=0 is not a satisfying choice because that would imply, as noted by BC, that J=0 in the steady state. Instead, we require that J reduces to the advective flux in the absence of diabatic and frictional forcings. This choice will uniquely determine X and hence J. Using the previous expression for J, X can be written:
Now we use the definitions of PV and the thermodynamic and momentum equations:
(6) |
(7) |
In (8), the density of the Boussinesq fluid is given by where is a constant reference density. is the geopotential and is the Bernoulli function written in the Boussinesq approximation:
In (9), p is the deviation of the pressure from that of a resting, hydrostatically balanced ocean. Replacing in (5), we have:
(10) |
(omitting the term ). The first three terms on the right hand side cancel leaving:
X sets the gauge for J. Replacing in (4), we find:
Note that Eq.(12):
To obtain an expression for
J that includes explicitely the frictional and buoyancy sources, we follow the same procedure as above, replacing in (12) the rates of change of
and
u from the thermodynamic and momentum equations but retaining the terms
and
F, respectively. We find:
In the remaining of this paper, unless noted otherwise, we will make use of the form (12) for J.