Energetics
Several possible forms of AQB energy equation may be derived from eqs.(39) to (41). The first step in these derivations is to take the scalar product of eq.(39) with giving:
This states that the kinetic energy per unit mass of fluid is changed by two processes : work done by the pressure gradient and buoyancy forces. The rate of working of the buoyancy force may be re-expressed as follows:
and on using eq.(41) and eq.(43) may be written as:
This may be expressed in flux form by multiplying by and using eq.(42) giving:
In a closed or periodic domain, the term of the above equation vanishes in a global integral (by Gauss’s theorem). Also, the term on the right-hand side vanishes when integrated across the domain on a
horizontal surface since there must be no net vertical mass flux -- consistent with eq.(40). Therefore, there exists the following expression of global energy conservation:
where represents an infinitesimal volume increment and on the integral sign denotes integration over the entire domain. A problem with the energy ()as defined in eq.(44) is that it is not necessarily positive. This is something of a disadvantage and it worth looking for other forms of energy equation which are positive definite.