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# PS01 - Problem Set 1 Solutions H 4.3 An air parcel at 30o N...

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Problem Set 1 – Solutions H 4.3 An air parcel at 30 o N moves northward conserving absolute vorticity. If its initial relative vorticity is 5x10 -5 s -1 , what is its relative vorticity upon reaching 90 o N? 0 ) ( = + dt f d ζ , therefore 2 2 1 1 ζ ζ + = + f f ) 30 sin( 2 1 o f = , 1 5 1 10 5 - - = s x ζ , ) 90 sin( 2 2 o f = H 4.5 Find the average vorticity within a cylindrical annulus of inner radius 200km and outer radius 400km if the tangential velocity distribution is given by V = A/r , where A = 10 6 m 2 s -1 and r is in meters. - = - = dl V dl V C C C k k k k annulus 200 400 200 400 0 2 2 1 1 2 2 = Π - Π = r r A r r A C annulus What is the average vorticity within the inner circle of radius 200km? H 4.7 Compute the rate of change of circulation about a square in the (x,y) plane with corners at (0,0), (0,L), (L,L), and (L,0) if temperature increases eastward at a rate of 1 o C per 200 km and pressure increases northward at a rate of 1 hPa per 200 km. Let L = 1000 km and the pressure at point (0,0) be 1000 hPa. 1 5 2 1 1 2 10 29 . 2 - - - = - + = s x f f ζ ζ r 1 =200km r 2 =400km 1 5 2 1 200 ___ 200 10 5 2 - - = Π Π = = s x r A Area C k k ζ 0 = = Area C annulus annulus ζ

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H 4.12 A westerly zonal flow at 45 o N is forced to rise adiabatically over a north- south oriented mountain barrier. Before striking the mountain, the westerly wind increases linearly toward the south at a rate of 10 m/s per 1000 km. The crest of the mountain range is at 800 hPa and the tropopause, located at 300 hPa, remains undisturbed. What is the initial relative vorticity of the air? What is the relative vorticity when it reaches the crest if it is deflected 5 o latitude toward the south during the forced ascent? Leg 1 Leg 2 Leg 3 Leg 4 - = p RTd dt dC a ln - - - - = 4 3 2 1 ln ln ln ln p RTd p RTd p RTd p RTd Legs (1) and (3) are at constant pressure, 0 ln 3 , 1 = p RTd - - = 2 4 ln ln p RTd p RTd + - + + - = 5 ln 5 ln ) 5 ( P P RT P P T R 2 2 16 . 7 5 ln 5 - - = + - = s m P P R 1 5 6 1 10 10 10 - - - = - - = - = s m ms y u x v ζ
Relative vorticity affected by two things: deflection of air column toward the south (f decreases), and riding up over mountain top (h decreases): 2 2 2 1 1 1 h f h f ζ ζ + = + If the current assumes a uniform speed of 20 m/s, during its ascent to the crest, what is the radius of curvature of the streamlines at the crest? Additional Problems: 1) In the winter in the mid-latitudes temperature decreases towards the pole. a) Write an expression for the baroclinicity vector implied by the temperature decrease in terms of T and P . By the Ideal Gas Law: T RT P 1 ρα ρ = , therefore P T P × × 1 α ρ . Since temperature decreases toward pole, density must increase toward pole: The baroclinicity vector points to the WEST.

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PS01 - Problem Set 1 Solutions H 4.3 An air parcel at 30o N...

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