Omega Equation - Derivation

Derivation

The derivation of the equation is based on the vorticity equation and the thermodynamic equation. The vorticity equation for a frictionless atmosphere may be written as:

(2)

Here is the relative vorticity, the horizontal wind velocity vector, whose components in the and directions are and respectively, the absolute vorticity, the Coriolis parameter, the individual rate of change of pressure . is the unit vertical vector, is the isobaric Del (grad) operator, is the vertical advection of vorticity and represents the transformation of horizontal vorticity into vertical vorticity.

The thermodynamic equation may be written as:

(3)

where, in which is the supply of heat per unit-time and mass, the specific heat of dry air, the gas constant for dry air, is the potential temperature and is geopotential .

The equation (1) is then obtained from equation (2) and (3) by substituting values:

and

into (2), which gives:

(4)

Differentiating (4) with respect to gives:

(5)

Taking the Laplacian of (3) gives:

(6)

Adding (5) and (6), simplifying and substituting, gives:

(7)

Equation (7) is now a linear differential equation in, such that it can be split into two part, namely and, such that:

(8)

and

(9)

where is the vertical velocity due to the mean baroclinicity in the atmosphere and is the vertical velocity due to the non-adiabatic heating, which includes the latent heat of condensation, sensible heat radiation, etc. (Singh & Rathor, 1974).