Models Analytic of Atmospheric Fields
A common approach in dynamic meteorology is to assume that fields can be represented by analytic functions. When these functions are substituted into the governing equations, additional information can be learned about the atmosphere. These analytic functions should meet two primary requirements. First, they should generate a realistic structure of the atmosphere. Second, they should be easy to use - relatively simple, and easy to differentiate and integrate. The latter is particularly important, since the governing equations contain many complex mathematical operations. All too often, these requirements contradict each other. Thus, a compromise between the two is used. The geopotential height of an isobaric level (often called the geopotential), , is analytically modeled more than any other quantity. Perhaps this is because height contours on an isobaric surface clearly show the wind pattern and because many important quantities can be determined from the geopotential pattern. To achieve a high degree of reality, the model should include the mean thermodynamic structure, the jet stream, and waves. Mean Thermodynamic Structure The simplest representation of the mean thermodynamic structure, is given by its long-term, mean value at each isobaric level,
= (p) .
Such values may be determined by the U.S. Standard Atmosphere.
(1)
A more precise analytic relationship could be used, but the added complexity is usually not justified. For example, in a constant lapse rate atmosphere, the hypsometric equation can be integrated,
gT o = 1 -
Rd
P P o
g
.
Dr. W. A. Fingerhut's class notes - All rights reserved
Jet Stream The simplest representation of the jet stream is to use the geostrophic wind relationship, which relates the wind speed to the geopotential. If the wind direction is west to east, and the wind speed is constant,
foUo = -
y.
(2)
Not surprisingly, the geopotential must decrease at a constant rate towards the north. The subscript, o, refers to a reference latitude, where y = 0 . To make use of this information, multiply both sides by dy , and integrate. Then equation becomes,
(1) = ( p) - foUo y .
Since the jet stream was assumed to be constant, it does not vary with latitude. In some investigations, this does not affect the results, although this is clearly unrealistic. If realism is important, then the jet stream should decrease both north and south of the reference latitude. To accomplish this, U o should be multiplied be some function that has the desired north-south behavior. A function that does so and is easy to differentiate is the cosine. If L J is twice the width of the jet,
2 y foUo cos = - y LJ
would replace equation (2). Again, multiply both sides by dy , and integrate. Then equation (1) would become,
= ( p) - foUo
2 y LJ sin . 2 LJ
PJ . If the speed of the jet stream is zero
Another improvement is for the jet stream to decrease both upward and downward from some pressure level, at some level, PZ , equation (1) would become,
= ( p) - foUo
2 y 2 (P - PJ ) LJ cos sin . 2 LJ 4(PZ - PJ )
Dr. W. A. Fingerhut's class notes - All rights reserved
Waves Atmospheric waves generally vary with each of the four independent variables x, y, p, and t. Before attempting to model such a complex circulation, lets first consider a simple wave that varies only with x. If the amplitude is , and...