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 Ņ where R is the radius of the earth and Ņ 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 Ve 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 “r” for “relative to the earth” 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 Vf, 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 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, 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  Ļr2 turning with a constant angular velocity w, where V = w r, the distance ∆s is given by the circumference 2Ļr,  the circulation V∆s 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.

 

 

                           Ca = Ce + C                             (8)

 

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

 

                                                     (9)                      

 

 

which states that absolute vorticity is the relative vorticity plus earth’s 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)

 

Let’s make the assumption that the pressure pattern is not changing (not a good assumption for periods longer than an hour or so). Let’s 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 “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.)

 

4.  Bjerkenes’ Circulation Theorem

 

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

 

 

                                          (16)

 

which is known as Bjerkenes’ Circulation Theorem.  Equation (16) answers the important question, how does circulation develop relative to the earth’s 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.  Bjerknes’ 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 “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.

 

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.



[1] The symbol w  is also used to denote the vertical velocity in the x, y, p coordinate system.