﻿ A Scale Analysis of the Primitive Momentum Equations of Air Flows

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# A Scale Analysis of the Primitive Momentum Equations of Air Flows

In this page the primitive momentum equations for planetary air flow are taken as given and a scale analysis based upon terrestrial conditions is carried out. The definitions of the variables of the primitive equations and their typical values are:

The Primitive Momentum Equations

## Scale Analysis

The scale of an acceleration such as du/dt is taken to be of the form u2/R. This form can be justified in two ways:

1. that all accelerations are of the order of the centripetal acceleration V2/R
2. dV/dt = ΔV/Δt and ΔV is on the order at maximum of V and Δt is R/V and thus ΔV/Δt is on the order of V/(R/V) = V2/R.

Thus du/dt and dv/dt are on the order of 102/106=10-4 m/s2. The order of magnitude of dw/dt is a different matter. The change in the value of w is on the order of a change in its sign over a twelve hour period; i.e., 2(0.05 m/s)/(12x3600) = 1.45x10-4.

The trigometric functions of latitute for the midlatitude range are on the order of unity.