﻿ The Omega Equation for Estimating Vertical Flow
San José State University

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Thayer Watkins
Silicon Valley
USA

 The Omega Equation for Estimating Vertical Flow

## Background

In meteorology the symbol ω is used to represent the rate of change of pressure in a parcel over time, dp/dt. When pressure is used as the vertical coordinate dp/dt is analogous to the horizontal velocities, u=dx/dt and v=dy/dt. Thus dp/dt=ω is in the nature of a velocity. Under hydrostatic balance it is proportional to the vertical wind velocity w; i.e.,

#### ω = dp/dt = (dp/dz)(dz/dt) = (dp/dz)w

but under hydrostatic equilibrium

#### dp/dz = -ρg so ω = -ρgw

Both w and ω are difficult to measure directly so the field of meteorology has developed methods to estimate vertical wind flow. Three such methods are:

#### ω(p) = ω(ps) - ∫psp(∇· VH)dp

where VH is the horizontal wind velocity vector.

#### ω = (dT/dt)/Sp = [∂T/∂t + u∂T/∂x + v∂T/∂y]/Sp

where T is temperature and Sp is the static stability parameter (Γd-Γ)/ρg.

#### (∇2 + (f02/Sp)∂2/∂p2)ω = [∂/∂p(Vg·∇(∇2Φ+f0f) + ∇2(Vg·∇(-∂Φ/∂p)]/Sp

where Sp is again the static stability parameter and f and f0 are the Coriolis parameter (f as a function of latitude and f0 a constant value). Φ is geopotential height.

The problem with using the estimate of ω based upon the continuity equation is that the divergence term depends upon the imperfectly known ageostrophic wind in as much as the divergence of the geostrophic wind is necessarily zero. Furthermore the estimate of the ω field based upon the continuity equation might not be consistent with the thermodynamic equation. Likewise the problem with the estimate of the ω field based upon the thermodynamic equation is that it might not be consistent with the continuity equation. The ω equation which is derived from both the continuity equation and the thermodynamic equation is consistent with both. The problem in using the ω equation is that the equation gives a partial differential equation for ω which must be solved to obtain the values of ω. But this is a computational problem rather than a problem in consistency or measurement as in the case of the continuity and adiabatic methods.