The *vertical* component of vorticity (which is the component
most meteorologists focus on) has its own governing equation: the
so-called *vorticity equation*.

This equation is derived by starting from the *u*- and
*v*-component momentum equations:

Notation here is conventional. Take the *x*-component partial
derivative of (b) and subtract from it the *y*-component partial
derivative of (a) - a process called cross-differentiation - which
gives:

The terms of this *vorticity equation* are:

- local time change of vorticity,
- horizontal vorticity advection,
- vertical vorticity advection,
- solenoid terms,
- tilting terms,
- divergence (or stretching) term,
- friction term.

Terms 1-3 are pretty obvious. Term 4 is the result of
cross-differentiating the pressure gradient terms of the momentum
equations, and represents the *solenoidal* contribution, one
true "source" term. Term 5 results from cross-differentiating the
vertical advection terms of the momentum equations, and represents
the *tilting* of horizontal vorticity into the vertical. Term 6,
the *divergence* (or "stretching") term, is the result of adding
the cross-differentiation of the horizontal advection terms to the
Coriolis terms in the momentum equations. Obviously, term 7 comes
from cross-differentiating the *friction* terms in the momentum
equations; it is another true "source" term.

What do I mean by a *source* term? If the flow starts out
without *any* vorticity in **three** dimensions, then this
equation shows that the only way to *create* vertical vorticity
is through the solenoid and friction terms. Hence, they are the only
true source terms. The advection terms simply *redistributes*
existing vertical vorticity, the divergence term increases or
decreases any existing vertical vorticity (including that due to
Coriolis) through the action of divergence or convergence, and the
tilting term changes the *orientation* of any existing
*horizontal* vorticity into the vertical. There are two more
such equations for the other two components of the vorticity vector,
of course.

Now any treatment of the horizontal flow must involve two
equations, since there are two independent momentum equations.
Combining them into one equation leaves only an incomplete
description of the flow. From the momentum equations, we can derive a
comparable equation to the vorticity equation by
cross-differentiating the other way: take the *x*-component
partial derivative of (a) and add it to the *y*-component
partial derivative of (b). This yields the *divergence* equation
(where *D* is the horizontal divergence; see
Mathematical Diversion - 1:

In this derivation, it is assumed that the *x*-coordinate is
aligned east-west, and the *y*-coordinate is aligned
north-south. The resulting equation is not very simple! *Some*
of these terms look analogous to terms in the vorticity equation but,
in general, the terms in the divergence equation are more difficult
to understand. See Hess (1957; §16.4 and §16.6). Under some
circumstances (notably, at synoptic-scale), most of these terms
(including all of the left-hand side) can be neglected, yielding a
so-called *balance* equation.

Finally, given the divergence and the vorticity, it can be shown
that any general two-dimensional wind field can be decomposed into a
*nondivergent* flow and an *irrotational* flow. This
finding is known as the **Helmholtz Theorem**. The nondivergent
flow is described by a so-called streamfunction (*y*) that satisfies:

and the irrotational flow is described by a so-called velocity
potential (*f*) that satisfies:

This partitioning is only to within a vector constant. It can be
shown (see Lynch 1989) that when this decomposition is done on a
finite domain, the solution can include a so-called harmonic part
that is *both* irrotational and nondivergent. It cn be described
by a either a streamfunction or a velocity potential (*c*_{h}) that satisfies the following:

The vector velocity, V, is given by:

where:

A major problem discussed in Lynch (1989) is the issue of boundary conditions when the domain is finite. I will not go into this, here.

Consider the 3- dimensional,vector equation of motion:

where: D/Dt denotes the Lagrangian time derivative (following the
air); **g*** is the "apparent gravity" that includes the effect of
centrifugal forces arising from the earth's rotation; **W** is the angular velocity vector of the
earth; and **F** is the "frictional" force. Now integrate this
equation around the closed boundary S of a
region G on some quasi-horizontal surface:

where *d***s** is the differential of distance around the
curve S (see
Fig. M4.1). The contribution from
"apparent gravity" vanishes for this horizontal part of the flow,
leaving only pressure gradient, Coriolis, and friction terms.

Fig. M4.1. Schematic showing the surface region G bounded by the curve S;kis a unit vector normal to the surface anddsis the increment of distance around S.

From this, it can be shown (see Bluestein 1992; pp. 253 ff.) that

Define the *absolute circulation* , *C _{a}* , to
be

where **r** is a position vector on the surface that includes
G, such that D**r**/Dt = **V**. The
*relative circulation* is *C* and the contribution to
circulation from Coriolis is *C _{c}* . The result of all
this is a

The relative circulation *C* is basically the integral of the
tangential component of the flow around the curve S bounding G. Observe
that on the basis of this theorem, there are only two ways to alter
the relative circulation: a term involving density and pressure
(which is related to the solenoid term in the standard meteorological
vorticity equation, above) and a term involving friction. Obviously,
the Coriolis contribution to the *absolute* circulation cannot
be changed, short of changing the rotation of the Earth! By applying
Stokes' Theorem:

where *d***A** is the differential area vector element of
the region G, such that *d***A** =
*dA* **k**, where **k** is the unit vector normal to G at any point. It can be seen that the relative
circulation *C* is simply an area integral of the vertical
component of the vorticity - put another way, relative vertical
vorticity is the relative *circulation per unit area* (see
Bluestein 1992, pp. 120 ff.):

Hess, S.L., 1957: *Introduction to Theoretical Meteorology*.
Holt, Rhinehart, and Winston, 362 pp.

Lynch, P., 1989: Partitioning the wind in a limited domain.
*Mon. Wea. Rev*., **117**, 1492-1500.