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
V = ωr
(1)
If equation (1) is applied to a point on the rotating earth, ω is the angular
velocity of the earth and r is the radial distance to the axis of rotation, r = R
cos ø where R is the radius of the earth and ø 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
(Vr) a = [Vr + (Vr) e ] = constant
(2)
where Ve is the tangential velocity of the earth surface. This is the
quantitative basis for the “ballet dancer effect”.
€
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 “r” for
“relative to the earth” is dropped.
Put (1) into (2)
(ωr )
2
a
= constant
€
(3)
The symbol ω is also used to denote the vertical velocity in the x, y, p coordinate
system.
1
1
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?
(ωr ) = (Vr) = [Vr + (Vr) ] = constant
2
a
a
€
e
(1)
Note that ω is positive if rotation is counterclockwise
relative to North Pole. Thus, V is positive if the zonal
motion vector is oriented west to east.
[Vr + (Vr) ] = [Vr + (Vr) ]
e f
(2)
e i
Solve for Vf, the tangential velocity relative to the earth
at the final latitude.
€
Vf
[Vr + (Vr) ] − [(Vr) ] )
(
=
e i
e f
rf
r = radial distance to axis of rotation =
€
Ve = ΩR cos ϕ
(3)
Rcosϕ
(4)
(5)
where is the angular velocity
of the earth, 7.292 X10-5
€
s-1.
2
Substitute (5) into (3) and simplify by inserting initial
Vi = 0 and remembering that the average radius of the
earth is 6378 km we get
Vf = 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 “swirl” in a fluid. It is a precise
measure of the average flow of fluid along a given closed curve.
Mathematically, horizontal (around a vertical axis) circulation is given by


V
•
d
s ≈ ∑( VΔs)
∫
(6)
For an air column with circular cross-sectional area πr2 turning with a
constant angular velocity ω, where V = ω r, the distance ∆s is given by the
circumference 2πr, the circulation V∆s is given by
C = 2πωr
2
(7a)
or
3
C
2 = 2ω = ζ
πr
(7b)
Note that the "omega" in equations (7a and b) represents the air parcel's
angular velocity relative to an axis perpendicular to the surface of the earth.
€
Equations (3) and (5a) tell us that circulation is directly proportional to
angular momentum. The fundamental definition of vorticity is (2ω), 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).
2
2
Ce = 2πω e r = 2πΩsin φr = fπr
2
(8a)
or
Ce
2 = 2Ωsin φ = f = ζe
πr
€
€
(8b)
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."
4
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.
Ca = Ce + C
(9)
Where C is the circulation the air column has relative to the earth.
Thus, dividing (8) by the area of the air column yields
# C &
# C &
% 2 ( = f +% 2 (
$ πr ' a
$ πr '
or
(10a,b)
ζa = f + ζ
€ that absolute vorticity is the relative vorticity plus earth’s
which states
vorticity (Coriolis parameter).
2. Real Torques
In reality circulation can occur around the three coordinate axes (two for the
natural coordinate system). In natural coordinates the wind components are
V and w and absolute circulation can be written
Ca = ∫ Vds + ∫ wdz
(12)
The change in absolute circulation (assuming that ds and dz do not change)
would be given by
5
dC a
dt = ∫
dV
dt ds + ∫
dw
dt dz
(13)
Substitution of the horizontal and vertical equations of motion into (13)
dC a
dp
= −∫
dt
ρ
(14)
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 (14) states that circulation will develop (increase or decrease) only
when isotherms are inclined with respect to isobars (known as a “baroclinic”
state). When isotherms are parallel to isobars (known as a “barotropic”
state), no circulation development can occur. (Remember, we are assuming
no frictional torques.)
3. Simplified Vorticity Equation
From the discussions above absolute circulation can be stated as
Ca = ζa A
(1)
where ζα is the absolute vorticity
Taking the time derivative of both sides
€
6
# dA &
# dζa &
dCa d (ζa A)
=
= ζa % ( + A%
(
$ dt '
$ dt '
dt
dt
(2)
Assuming no torques so that absolute circulation is conserved dCa/dt = 0,
and
€
1 "$ dA %' = − 1 "$ dζa %'
A # dt &
ζa # dt &
(3)
Applying the fundamental definition of divergence
€
$ dζ '
DIVh = − 1ζ & a )
a % dt (
(4)
Equation (4) is the simplified vorticity equation. It states that the change in
€ vorticity (proportional to absolute angular velocity) experienced by
absolute
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 “ballet dancer” 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.
7