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. € 2 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 = Vr + Vr [ ( )e ] f [ ( )e ]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 (4) ∫ x,y,z 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) ∫ 4 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) or 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) or 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.