Thayer Watkins
Silicon Valley
& Tornado Alley

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,

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

This can be rearranged to

Du/Dt = ∂u/∂t
+ u(∂u/∂x) + u(∂u/∂x)
+ v(∂u/∂y) + u(∂v/∂y)
+ w(∂u/∂z) + u(∂w/∂z)
which is equivalent to

Du/Dt = ∂u/∂t
+ (∂u2/∂x) + (∂uv/∂y) + (∂uw/∂y)

The derivative of the expected value of u is then

Du/Dt = ∂u/∂t
+ ∂/∂x[uu + u'u']
+ ∂/∂y[uv + u'v']
+ ∂/∂z[uw + u'w']

Carrying out the differentiation and regrouping the terms gives

Du/Dt = ∂u/∂t + uu/∂x + vu/∂y + wu/∂z
+ u[∂u/∂x + ∂v/∂y + ∂w/∂z]
+ ∂u'u'/∂x + ∂u'v'/∂y + ∂u'w'/∂z

The term [∂u/∂x + ∂v/∂y + ∂w/∂z] is equal to zero under the Boussinesq assumption so the equation reduces to:

Du/Dt = ∂u/∂t + uu/∂x + vu/∂y + wu/∂z
+ ∂u'u'/∂x + ∂u'v'/∂y + ∂u'w'/∂z

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]


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 = -Kα(∂α/∂n).

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

u'α' = -Kα(∂α/∂x)
v'α' = -Kα(∂α/∂y)
w'α' = -Kα(∂α/∂z)

These forms are not derived from theory but instead are working hypotheses.

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