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Flux and Transfer in Meteorology

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

u' = 0
v' = 0
and that
vu' = 0
uv' = 0

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

uv = uv + u'v'

The Eulerian derivative of the wind velocity u is

Du/Dt = ∂u/∂t + u(∂u/∂x) + v(∂u/∂y) + w(∂u/∂z)

If density is constant (the Boussinesq approximation) then

(∂u/∂x) + (∂v/∂y) + (∂w/∂z) = 0

The left-hand side of this can be multiplied by u and the result added to
the Eulerian derivative to yield,

Let D/Dt be defined as
∂/∂t + u∂/∂x
+ v∂/∂y
+ w∂/∂z.

Then the previous equation becomes

Du/Dt = Du/Dt
+ ∂u'u'/∂x
+ ∂u'v'/∂y
+ ∂u'w'/∂z

Or, equivalently,

Du/Dt = Du/Dt -
[ ∂u'u'/∂x
+ ∂u'v'/∂y
+ ∂u'w'/∂z]

A similar derivation in terms of v and w results in

Dv/Dt = Dv/Dt -
[ ∂v'u'/∂x
+ ∂v'v'/∂y
+ ∂v'w'/∂z]

and

Dw/Dt = Dw/Dt
-[ ∂w'u'/∂x
+ ∂w'v'/∂y
+ ∂w'w'/∂z]

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

Dα/Dt = Dα/Dt -
[ ∂α'u'/∂x
+ ∂α'v'/∂y
+ ∂α'w'/∂z]

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.,

flux in α in direction n = -K_{α}(∂α/∂n).

where K_{α} 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