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 [ ] = constant (2) ( Vr) a = Vr + Vr ( ) e where V e 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) 1 The symbol ω is also used to denote the vertical velocity in the x, y, p coordinate system. 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 2 ) = Vr a [ ] = constant (1) ( ) a = Vr + ( Vr) e 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 f [ ] i (2) Vr + ( Vr) e ( ) e Solve for V f, the tangential velocity relative to the earth at the final latitude. ([ V f = Vr + ( Vr ) e ] i [( Vr) e ] ) f r f (3) r = radial distance to axis of rotation = Rcosϕ (4) V e = ΩRcosϕ (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 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 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 πr 2 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 πr 2 = 2ω = ζ (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). C e = 2πω e r 2 = 2πΩsinφr 2 = fπr 2 (8a) or C e πr 2 = 2Ωsinφ = f = ζ e (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. C a = C e + 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 & % ( $ πr 2 ' a # = f + C & % ( $ πr 2 ' or (10a,b) ζ a = f +ζ which states that absolute vorticity is the relative vorticity plus earth s 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 C a = Vds + wdz (12) The change in absolute circulation (assuming that ds and dz do not change) would be given by 5
dc a = dv ds + dw dz (13) Substitution of the horizontal and vertical equations of motion into (13) dc a = dp 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 C a = ζ a A (1) where ζ α is the absolute vorticity Taking the time derivative of both sides 6
dc a = d ( ζ a A) # da& # = ζ a % ( + A dζ a % $ ' $ & ( (2) ' Assuming no torques so that absolute circulation is conserved dc a / = 0, and 1 " da% A $ ' = # & 1 " ζ $ a # dζ a % ' (3) & Applying the fundamental definition of divergence DIV h = 1 $ ζ & a % dζ a ' ) (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. 7