Thayer Watkins
Silicon Valley
& Tornado Alley

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.

The order of magnitude of the horizontal pressure gradient is problematical. The horizontal pressure gradient is taken to be about 10 mb = 1000 Pa over the length scale of 1000 km = 106 m; i.e., 103/106 = 10-3 Pa/m. The pressure gradient component of the acceleration, which is (1/ρ)(∂p/∂x), is thus of the order of magnitude of 10-3. Dutton in The Ceaseless Wind uses 10 mb per 1000 km as the order of magnitude of the pressure gradient but uses cm instead of m as the unit of length which requires corresponding changes in pressure units in addition to the conversion of mb to Pa. Since these conversions are not made explicit it is not clear whether his results were derived from the scale analysis or the scale of the terms was set to be consistent with desired result; i.e., that the pressure gradient component of the acceleration being the same order of magnitude as the Coriolis term. Holton in his An Introduction to Dynamic Meteorology uses 10 mb \ 1000 Pa per 1000 km as the order of magnitude of the pressure gradient. This value makes the magnitude of the pressure gradient term in the horizontal momentum equation equal to the Coriolis term but 10 mb per 1000 km seems excessively small. The answer to the problem is that the momentum equations should be spatially averaged over a scale comparable to that of the length scale before a scale analysis is to be carried out. As they are given the momentum equations could include micro phenomena such as sound waves as well as meso-scale meteorological phenomena.

The order of magnitude of the vertical pressure gradient is given by the fact that the 500 mb is about 6 km above the 1000 mb surface pressure. Thus the order of magnitude would seem to about 50000 Pa/6000 = 8.33 Pa/m = 83.3 mb/km. This estimate of the vertical pressure gradient when divided by the density of 1 kg/m3 gives a vertical acceleration due to the vertical pressure gradient of 8.3 m/s2, comparable to the gravity acceleration term of 9.8 m/s2. Therefore in the scale analysis the order of magnitude of the vertical pressure gradient term is taken equal to that of the gravity term.

Below is given the order of magnitude analysis for the eastward component of the momentum equation. The order of magnitude for each term in the equation is given directly below within square brackets. Directly below those terms are the ratios of the orders of magnitude of the terms to the maximum order of magnitude term in the equation.

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