Circulation and Vticity 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 tques, 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. 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 f 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 codinate system. 1
Example Problem: An air parcel at rest with respect to the surface of the earth at the equat in the upper troposphere moves nthward 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 Nth Pole. Thus, V is positive if the zonal motion vect is iented west to east. [ ] = Vr + Vr f [ ] i (2) Vr + ( Vr) e ( ) e Solve f 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 der 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 C = V x,y,z d l (4) where d l is the position vect. In natural codinates, d l = ds t + dzk. F purely hizontal flow, equation (4) reduces to C = V d s (5) where, f a closed curve, V d s VΔs ( ) (6) 3
F 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) C πr 2 = 2ω = ζ (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 proptional to angular momentum. The fundamental definition of vticity is (2ω), that is, twice the local angular velocity. Thus, rearranging (7a) shows that circulation per unit area is the vticity, and is directly proptional to (but not the same as) angular velocity of the fluid. Vticity, then, is the microscopic measure of swirl and is the vect measure of the tendency of the fluid element to rotate around an axis through its center of mass. At the Nth 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 (7c) 4
C e πr 2 = 2Ωsinφ = f = ζ e (7d) Thus, the circulation imparted to a an air column by the rotation of the earth is just the Ciolis parameter times the area of the air column. Dividing both sides by the area shows that the Ciolis parameter is just the "earth's vticity." An observer in space would note that the total 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 (8) Thus, dividing (8) by the area of the air column yields # C % ( $ πr 2 ' a # = f + C % ( $ πr 2 ' (9) ζ a = f +ζ which states that absolute vticity is the relative vticity plus earth s vticity (Ciolis parameter). 2. Applications Since circulation is proptional to angular momentum, this means that both absolute circulation and absolute vticity are analagous to angular 5
momentum. Since, in the absence of tques, absolute angular momentum is conserved, then it can be stated that, in the absence of tques dc a = 0 (10) (C e + C) i = (C e + C) f 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. F example, suppose an air column is at rest with respect to the surface of the earth at the nth pole. Conceptually, what relative circulation would develop (if any), if this air column moved to the equat? Inclass Exercise 6 1. An air column at rest with respect to the surface of the earth at the equat has a radius of 100 km. This air column moves to the Nth 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 Nth Pole. Assume no real tques 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 iginal surface area because of hizontal divergence. What tangential velocity relative to the earth (in km/h) will develop at the periphery of the air column. 3. Real Tques Remembering that 6
C = V d l (11) and that in natural codinates the wind components are V and w the position vect components are ds and dz, 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 dc a = dv ds + dw dz (13) F frictionless, non-curved flow, the equations of motion in natural codinates are dv = 1 ρ p s dw = 1 ρ p z g (14) Let s make the assumption that the pressure pattern is not changing (not a good assumption f periods longer than an hour 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 7
dc a = dp ρ (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 proptional to temperature. Hence, a solenoid is the trapezoidal figure created if isobars and isotherms intersect. Equation (15) states that circulation will develop (increase 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 tques.) 4. Bjerkenes Circulation Theem Taking the time derivative of (8), solving f the relative circulation after substitution of equation (15) yields dc = dp ρ d(2ωsinφa) (16) dc = ( ) dp ρ d fa which is known as Bjerkenes Circulation Theem. Equation (16) answers the imptant question, how does circulation develop relative to the earth s surface? The solenoid term is very imptant near fronts, sea-breeze interfaces, outflow boundaries, jet streaks etc., all mesoscale low-end synoptic scale features. F most synoptic and macroscale features, the solenoid term can be neglected on an der of magnitude basis. Bjerknes 8
Circulation Theem still excludes circulation (and vticity changes) due to tilting, however. 5. Simplified Vticity Equation From the discussions above absolute circulation can be stated as C a = ζ a A (1) where ζ is the absolute vticity Taking the time derivative of both sides dc a = d ( ζ A a ) # da # = ζ a % ( + A dζ a % $ ' $ ( (2) ' Assuming no tques 1 " da% A $ ' = # 1 " ζ $ a # dζ a % ' (3) Applying the fundamental definition of divergence DIV h = 1 $ ζ a % dζ a ' ) (4) ( 9
Equation (4) is the simplified vticity equation. It states that the change in absolute vticity (proptional to absolute angular velocity) experienced by an air parcel is due to divergence 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 wk, 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 der of magnitude basis) and synoptic-scaling is perfmed. Equation (4) can be expanded using the definition of the Lagrangian/total derivative to the Simplified Vticity Equation. DIV h = 1 " dζ a ζa $ # DIV h = 1 " ζ a ζa $ # t DIV h = 1 " ζ a ζa $ # t % ' = 1 " ζa $ # ζ a t + V h ζ a + w ζ a z + V ζ a s + w ζ a z + V % ζ a ' % ' % ' (4a,b,c) where 4(b) and 4(c) are the versions in rectangular and natural codinates, respectively. Because vertical velocities are small compared to hizontal velocities and the vertical gradient of absolute vticity is one to two ders of magnitude 10
smaller than the hizontal gradients of absolute vticity, the last term on the right of 4(b) and 4(c) can be neglected on an der of magnitude basis. The resulting simplified vticity equation (often called the Barotropic Vticity Equation) in natural codinates can be rewritten as follows: DIV h = 1 ζ a ζ a t V ζ a ζ a s (5) Equation (4a) states that air parcels experience changes in vticity because of divergence/convergence (at the synoptic scale). But equation (5) is a version of the equation that allows us to relate vticity advection patterns to divergence and convergence patterns, if the synoptic scaling arguments made above are valid. 11