Circulation and Vorticity

1. Measures of Vortical Motion (Swirl) in a Fluid

Vortical motion is defined as that motion in which each individual particle rotates around its own axis. This is also called rotational motion or swirl.

The macroscopic measure of swirl in a fluid is called “Circulation” and the microscopic measure is called “Vorticity.” As you will see, each is directly related to the angular velocity and angular momentum of the particles. For the purposes of this handout, these particles can be called air parcels.

2. Conservation of Absolute Angular Momentum

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

V = ωr (1) where w is the angular velocity at the position of the air parcel and r is its radial distance to the local axis of rotation. If the local axis of rotation is the axis of the earth, then r = R cos ø where R is the radius of the earth (see Figure 1). and ø is latitude.1 €

Figure 1: Schematic diagram of the earth showing the relationship of the radial distance to the axis of rotation to the actual radius of the earth.

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

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. The absolute angular momentum of an air parcel moving, say, to the east relative to the earth is the sum of the angular momentum imparted to the air parcel by the underlying surface of the earth and the angular momentum the air parcel has relative to the earth, as given in Equation (2)

Vr Vr Vr constant ( )a = [ + ( )e ] = (2)

(Vr)a is the absolute angular momentum of a moving air parcel relative to a stationary observer in space, (Vr) is the angular momentum of the air parcel relative€ to the earth, and (Vre) is the angular momentum imparted to the air parcel by the underlying surface of the earth, 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)

ωr2 = constant ( )a (3)

Equations (3) and (2) are equivalent expressions.

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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? ωr2 = (Vr) = Vr + (Vr) = constant ( )a a [ e ] (1)

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

€ Vr + (Vr)e = Vr + (Vr)e [ ] f [ ]i (2)

Solve for Vf, the tangential velocity relative to the earth at the final latitude.

Vr + (Vr) − (Vr) € ([ e ]i [ e ] f ) V f = rf (3)

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

€ V = ΩRcosϕ e (5)

where Ω is the angular velocity of the earth, 7.292 X10-5 s-1 and R is the average radius, 6378 km. Substitute (5) into (3) and€ simplify by inserting initial Vi = 0 we get

-1 Vf = 482.7 km h

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. This is known as the SUBTROPICAL JET STREAM. . 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.

3 3. Circulation: General

There is a connection between angular momentum and a kinematic quantity known as “circulation.” 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. Essentially, it is the sum of the components of the velocity vector that act tangent to that closed path for every point along it, as shown in Fig. 2,

Figure 2: Schematic diagram illustrating counterclockwise flow along a curved path, estimated by Equation (4).

Mathematically, circulation is given by

  C = V • dl ∫ x,y,z (4) where is the position vector.

In natural€ coordinates, for purely horizontal flow, equation (4) reduces to

  C = V • ds ∫ (5)

Please note that Circulation is scalar.

For a closed curve,

  V • ds ≈ (VΔs) ∫ ∑ (6)

It is easy to visualize an air column that has a circular cross-sectional area. For such an air column, the cross-sectional area is πr2. Say the air column is turning with a constant angular velocity w, where V = w r (Equation 1), and the distance ∆s is given by the circumference 2πr, then then Equation (6) gives

2 C = 2πωr (7a)

C = 2ω = ζ πr2 (7b)

Note that r in equations (7a,b) is the radius of the air column and not the radial distance to the axis of rotation. Also, 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, since angular momentum is Vr or wr2. The circulation in Equation (7a) is the circulation around a vertical axis through the air column and the vorticity in (7b) is the relative vertical vorticity of one air parcel at the center of that column.

Rearranging (7a) into (7b) shows that circulation per unit area is the vorticity, and is directly proportional to (but not the same as) angular velocity of the fluid. This gives rise to the fundamental definition of vorticity as (2w), that is, twice the local angular velocity. 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.

5 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. This can be obtained from (7a) by substituting the angular velocity of the earth, Ω, and more generally for any latitude, Ω sin ∅. 2

2 2 2 Ce = 2πω er = 2πΩsinφr = fπr (7c)

C e = 2Ωsinφ = f = ζ πr2 e (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" since vorticity is Circulation per unit area.

An observer in space would note that the total or absolute horizontal 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

2 Please note that if we progress from treating an air parcel to conceptualizing the circulation of an air column, r is the radius of the air column, not the radius of the earth.

6 # C & # C & % 2 ( = f +% 2 ( $ πr ' a $ πr '

or (9)

ζa = f +ζ

which states that absolute vorticity is the relative vorticity plus earth’s vorticity €(Coriolis parameter), where � is the absolute vorticity.

4. Implications

Since circulation is proportional to angular momentum, this means that both absolute circulation and absolute vorticity are analogous to angular momentum. This is a kinematic statement. Recall from your physics courses that a “torque” is the rotational equivalent of a “force”. Since, in the absence of torques, absolute angular momentum is conserved, then it can be stated that

dCa dt = 0

or (10)

(Ce + C)i = (Ce + C)f

This is one of the important “conservation theorems” in meteorology. Of course, although this may be true in some circumstances at the synoptic and macroscales, the “absence of torques” aspect of the beginning assumptions makes it fail in many important “weather development” circumstances. Yet it allows us to make some useful observations of the way the atmosphere behaves.

7 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? Equation (10) allows us to attempt an answer. You will do this in a class exercise.

It should be pointed out that, in the absence of torques, conservation of absolute circulation implies that absolute vorticity is a conservative quantity. This means that as an air parcel moves around it “conserves” its absolute vorticity.

� = � + � = ��

Or (12)

�� = 0 ��

Equation (12) is known as the Conservation of Absolute Vorticity. It implies that as air parcels change latitude, their relative vorticity increases or decreases creating troughs and ridges respectively.

5. Real Torques

Remembering that Circulation can be three dimensional, we rewrite (Equation 4) here

  C = V • dl ∫ (13)

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

C Vds wdz a = ∫ + ∫ (14)

8 where the first term on the right is the horizontal circulation and the far right hand term is vertical circulation (e.g., the Hadley Cell).

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

dCa dV dw dt = ∫ dtds + ∫ dtdz (15)

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

dV 1 ∂p dt = − ρ ∂s

dw 1 ∂p (16) dt = − ρ ∂z − g

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 (16) into (15) and collection of terms yields

dC dp a = − dt ∫ ρ (17) 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

9 temperature. Hence, a solenoid is the trapezoidal figure created if isobars and isotherms intersect.

Equation (17) 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.)

6. Bjerkenes’ Circulation Theorem

Taking the Lagrangian derivative of (8)

dCa /dt = dCe /dt + dC/dt solving for the Lagrangian derivative of the relative circulation

dC/dt = dCa /dt - dCe /dt

dCa dp and substitution of equations (17) ( = − ) and (7c) (Ce = fA) yields dt ∫ ρ

dC dp d(2ΩsinΦA) = − − dt ∫ ρ dt

or (18)

dC dp d( fA) = − − dt ∫ ρ dt which is known as Bjerkenes’ Circulation Theorem.

Equation (18) answers €the important question, “how does circulation develop relative to the earth’s surface?”

The solenoid term is very important in producing vertical circulations near fronts, sea-breeze interfaces, outflow boundaries, jet streaks etc., all

10 mesoscale or low-end synoptic scale features. It is also a factor in the development of horizontal circulation around pressure systems that have asymmetric density (temperature) distribution. However, 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.

Let’s stay away from areas in which the solenoid term can be important. In that case, equation (18) becomes

dC/dt = -f dA/dt - A df/dt (18a)

Dividing both sides of (18a) by the original area of the air column A, and inserting the fundamental definition of divergence

d (C/A)/dt = -f DIVh – df/dt (18b)

� (� + �) �� = −�� = (18�) ��

Equation (18c) states that if there is no divergence the absolute vorticity of air parcels is conserved, which is a restatement of equation (12). Equation (18c) also states that the basic reason for changes in the air parcel’s absolute vorticity at the synoptic scale is if that air parcel experiences horizontal divergence or convergence.

7. Simplified Vorticity Equation

Equation (18c) can be obtained directly from the definition of absolute circulation. From the discussions above ,absolute circulation can be stated as

Ca = ζa A (19)

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where za is the absolute vorticity

Taking the time derivative of both sides

dCa d(ζa A) # dA& # dζa & = = ζ % ( + A% ( dt dt a$ dt ' $ dt ' (20)

Assuming no torques

€ " % " % 1 dA 1 dζa $ ' = − $ ' (21) A# dt & ζa # dt &

Applying the fundamental definition of divergence €

$ ' 1 dζa DIVh = − & ) (22) ζa % dt (

Equation (22) 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 is analogous 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 Equation (22) 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.

12 Equation (22) 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 (22) can be expanded using the definition of the Lagrangian derivative to the Simplified Vorticity Equation in natural coordinates.

" % "  % 1 dζa 1 ∂ζa DIVh = − $ ' = − $ +V •∇ζa ' ζa # dt & ζa # ∂t & or "  % 1 ∂ζa ∂ζa DIVh = − $ +Vh •∇ζa + w ' ζa # ∂t ∂z & or (23a,b,c) "  % 1 ∂ζa ∂ζa ∂ζa DIVh = − $ +V + w ' ζa # ∂t ∂s ∂z &

where 23(b) and 23(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 23(b) and 23(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:

∂ζ V ∂ζ DIV = − 1 a − a h ζ (24) a ∂t ζa ∂s

13 €

Equation (24) states that air parcels experience changes in vorticity because of divergence/convergence (at the synoptic scale). But equation (24) 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.

At the synoptic scale, the changes in vorticity at a fixed location due to vorticity advection are almost exactly counter balanced by the horizontal divergence term, so that the Eulerian derivative term can be dropped on an order of magnitude basis. In that case, Equation (24) can be rewritten.

�� = − � (25)

Equation (25) states that if one knows the absolute vorticity at a location, the advection of absolute vorticity can be used to calculate the horizontal divergence at that location. This important result as this mathematical and conceptual interpretation: areas of positive vorticity advection diagnose3 areas of divergence and areas of negative vorticity advection diagnose convergence.

3 “diagnose” means to find and characterize and does not mean “to cause”.