& Tornado Alley
Let u and v be the wind velocities in the horizontal west-to-east
and south-to-north directions, respectively. Suppose u and v are
stochastic with expected (mean) values of u and v. The deviations
from the means are denoted as u' and v'. The relationships are
u = u + u'
v = v + v'.
These relations imply that
However, u'v' is not necessarily zero. It could be a positive quantity or a negative one. It is called the covariance between u and v. Meteorologically it has a dual nature. It is simultaneously the net flux of eastward flow north and the net flux of northward flow east.
One of the non-obvious implications of the above relations is that
The Eulerian derivative of the wind velocity u is
If density is constant (the Boussinesq approximation) then
The left-hand side of this can be multiplied by u and the result added to the Eulerian derivative to yield,
This can be rearranged to
Du/Dt = ∂u/∂t
+ (∂u2/∂x) + (∂uv/∂y) + (∂uw/∂y)
The derivative of the expected value of u is then
Carrying out the differentiation and regrouping the terms gives
The term [∂u/∂x + ∂v/∂y + ∂w/∂z] is equal to zero under the Boussinesq assumption so the equation reduces to:
Let D/Dt be defined as ∂/∂t + u∂/∂x + v∂/∂y + w∂/∂z.
Then the previous equation becomes
A similar derivation in terms of v and w results in
Whereas D/Dt is defined in terms of u, v and w, D/Dt is defined in terms of u, v and w and hence is more appropriate for establishing equations for the time-averaged meteorological variables.
The general equation for any meteorological variable α is
In this formulation the fluxes serve as the forcing functions in the differential equations.
In diffusion processes the rate of transfer of a diffusible quantity in a particular direction is proportional to the gradient in that direction; i.e.,
where Kα is the coefficient of diffusion. In analogy with this form it is assumed that the rate of transfer of a quantity by eddy convection processes are of the form
These forms are not derived from theory but instead are working hypotheses.
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