__Circulation and Vorticity__

1. Conservation of Absolute Angular
Momentum

The
tangential linear velocity of a parcel on a rotating body is related to angular
velocity of the body by the relation

(1)

If
equation (1) is applied to a point on the rotating earth, w is the angular velocity of the earth and
r is the radial distance to the axis of rotation, r = R cos ¿ where R is the
radius of the earth and ¿ is latitude.[1]

Angular momentum is defined as Vr and, in the absence of torques, absolute angular momentum (that is, angular momentum relative to a stationary observer in space) is conserved

(2)

where
V_{e} is the tangential velocity of the earth surface.

Equation
(2) states that the absolute angular momentum of a parcel of air is the sum of
the angular momentum imparted to the air parcel by the rotating surface of the
earth and angular momentum due to the motion of the air parcel relative to the
rotating surface of the earth (where the subscript ÒrÓ for Òrelative to the
earthÓ is dropped.

Put
(1) into (2)

(3)

Example Problem:

An air parcel at rest with respect to the
surface of the earth at the equator in the upper troposphere moves northward to
30N because of the Hadley Cell circulation. Assuming that absolute angular momentum
is conserved, what tangential velocity would the air parcel possess relative to
the earth upon reaching 30N?

(1)

Note that w is positive if rotation is counterclockwise
relative to North Pole. Thus, V is
positive if the zonal motion vector is oriented west to east.

(2)

Solve for V_{f}, the tangential velocity
relative to the earth at the final latitude.

_{ }(3)

r
= radial distance to axis of rotation = (4)

(5)

where
is the angular velocity of the earth, 7.292 X10^{-5} s^{-1}.

Substitute (5) into (3) and simplify by inserting initial
V_{i }= 0 and remembering that the average radius of the earth is 6378
km we get

V_{f} = 482.7 km h^{-1}

Clearly,
though such wind speeds are not observed at 30N in the upper troposphere, this
exercise proves that there should be a belt of fast moving winds in the upper
troposphere unrelated to baroclinic considerations (i.e., thermal wind) and
only related to conservation of absolute angular momentum. In the real atmosphere, such speeds are
not observed (the subtropical jet stream speeds are on the order of 200 km/hr)
because of viscosity/frictional effects.

2. Circulation: General

Circulation
is the macroscopic measure of ÒswirlÓ in a fluid. It is a precise measure of the average
flow of fluid along a given closed curve.
Mathematically, circulation is given by

(4)

where
is the position
vector. In natural coordinates, . For purely
horizontal flow, equation (4) reduces to

(5)

where,
for a closed curve,

(6)

For
an air column with circular cross-sectional area ¹r^{2} turning with a constant
angular velocity w, where V = w r, the distance Æs is given by the circumference 2¹r, the circulation VÆs is given by

(7a)

or

(7b)

Note
that the "omega" in equations (7a and b) represent the air parcel's
angular velocity relative to an axis perpendicular to the surface of the earth.

Equations
(3) and (7a) tell us that
circulation is directly proportional to angular momentum. The fundamental definition of vorticity
is (2w),
that is, twice the local angular velocity.
Thus, rearranging (7a) shows that circulation per unit area is the
vorticity, and is directly proportional to (but not the same as) angular
velocity of the fluid. Vorticity,
then, is the microscopic measure of swirl and is the vector measure of the
tendency of the fluid element to rotate around an axis through its center of
mass.

At
the North Pole, an air column with circular cross sectional area at rest with
respect to the surface of the earth would have a circulation relative to a
stationary observer in space due to the rotation of the earth around the local
vertical, Equation (7c).

(7c)

or

(7d)

Thus,
the circulation imparted to a an air column by the rotation of the earth is
just the Coriolis parameter times the area of the air column. Dividing both sides by the area shows
that the Coriolis parameter is just the "earth's vorticity."

An
observer in space would note that the total or absolute circulation experienced
by the air column is due to the circulation imparted to the column by the
rotating surface of the earth AND the circulation that the column possesses
relative to the earth.

** C _{a
}= C_{e }+ C **(8)

Thus,
dividing (8) by the area of the air column yields

(9)

which
states that absolute vorticity is the relative vorticity plus earthÕs vorticity
(Coriolis parameter).

2. Applications

Since
circulation is proportional to angular momentum, this means that both absolute
circulation and absolute vorticity are analagous to angular momentum. Since, in the absence of torques,
absolute angular momentum is conserved, then it can be stated that, in the
absence of torques

(10)

Of
course, although this may be true at the synoptic and macroscales, this
assumption fails, as we will see, in general. Yet it allows us to make some useful
observations of the way the atmosphere behaves.

For
example, suppose an air column is at rest with respect to the surface of the
earth at the north pole.
Conceptually, what relative circulation would develop (if any), if this
air column moved to the equator?

Inclass
Exercise 6

1. An air column
at rest with respect to the surface of the earth at the equator has a radius of
100 km. This air column moves to the North Pole. Determine (a) what
relative circulation, if any, the air column will develop, and (b) what the
tangential velocity (in km/h) would develop at the periphery of the air column
upon its arrival at the North Pole. Assume no real torques and that the
area of the air column does not change.

2. An air column
initially at rest with respect to the surface of the earth at 60N expands to
twice its original surface area because of horizontal divergence. What
tangential velocity relative to the earth (in km/h) will develop at the
periphery of the air column.

3. Real Torques

Remembering
that

(11)

and
that in natural coordinates the wind components are V and w the position vector
components are ds and dz, absolute circulation can be written

(12)

The
change in absolute circulation (assuming that ds and dz do not change) would be
given by

(13)

For
frictionless, non-curved flow, the equations of motion in natural coordinates
are

(14)

LetÕs
make the assumption that the pressure pattern is not changing (not a good
assumption for periods longer than an hour or so). LetÕs also remember surfaces
of g are parallel to z contours and evaluation of the line integral of gdz will
result in 0. Then substitution of
(14) into (10) and collection of terms yields

(15)

where
dp is the variation of pressure along the length of the circuit being
considered. The term to the right
of the equals sign is known as the solenoid term. A solenoid is the trapezoidal figure
created if isobars and isopycnics intersect. At a given pressure, density is
inversely proportional to temperature.
Hence, a solenoid is the trapezoidal figure created if isobars and
isotherms intersect.

Equation
(15) states that circulation will develop (increase or decrease) only when
isotherms are inclined with respect to isobars (known as a ÒbaroclinicÓ
state). When isotherms are parallel
to isobars (known as a ÒbarotropicÓ state), no circulation development can
occur. (Remember, we are assuming no frictional torques.)

4. BjerkenesÕ Circulation Theorem

Taking
the time derivative of (8), solving for the relative circulation after
substitution of equation (15) yields

(16)

which
is known as BjerkenesÕ Circulation Theorem. Equation (16) answers the important
question, how does circulation develop relative to the earthÕs surface? The solenoid term is very important near
fronts, sea-breeze interfaces, outflow boundaries, jet streaks etc., all
mesoscale or low-end synoptic scale features. For most synoptic and macroscale
features, the solenoid term can be neglected on an order of magnitude
basis. BjerknesÕ Circulation
Theorem still excludes circulation (and vorticity changes) due to tilting,
however.

5. Simplified Vorticity Equation

From
the discussions above absolute circulation can be stated as

(1)

where
z is
the absolute vorticity

Taking
the time derivative of both sides

(2)

Assuming
no torques

(3)

Applying
the fundamental definition of divergence

(4)

Equation
(4) is the simplified vorticity equation.
It states that the change in absolute vorticity (proportional to
absolute angular velocity) experienced by an air parcel is due to divergence or
convergence. This analgous to the
principle of conservation of absolute angular momentum applied at a microscopic
level. This is the so-called
Òballet dancerÓ effect applied to a fluid parcel. Please remember that (4) is
simplified. It applies only in
extremely restrictive circumstances.
Near fronts, sea-breeze boundaries, outflow boundaries etc., equation
(4) will not work, since it does not contain the solenoidal effects discussed
in class.

Equation
(4) can also be derived directly by obtaining the curl of the equation of
motion and doing synoptic-scaling (in which the tilting, stretching and
solenoid terms are dropped out on an order of magnitude basis) and
synoptic-scaling is performed.

Equation
(4) can be expanded using the definition of the Lagrangian/total derivative to
the Simplified Vorticity Equation.

(4a,b,c)

where 4(b) and 4(c) are the versions in rectangular
and natural coordinates, respectively.

Because
vertical velocities are small compared to horizontal velocities and the
vertical gradient of absolute vorticity is one to two orders of magnitude
smaller than the horizontal gradients of absolute vorticity, the last term on
the right of 4(b) and 4(c) can be neglected on an order of magnitude basis.

The
resulting simplified vorticity equation (often called the Barotropic Vorticity
Equation) in natural coordinates can be rewritten as follows:

(5)

Equation
(4a) states that air parcels experience changes in vorticity because of
divergence/convergence (at the synoptic scale). But equation (5) is a version of the
equation that allows us to relate vorticity advection patterns to divergence
and convergence patterns, if the synoptic scaling arguments made above are
valid.