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An axisymmetric cyclonic vortex is in cyclostrophic balance (ƒ = 0) with a tan- gential velocity profile given by V = Vo(r/ro)", where Vo is the tangential velocity component at a distance ro from the vortex center. (a) Compute the circulation and vorticity about a streamline at radius r. What is special about the cases n = 1 and n = −1? Sketch V(r) for both cases. (b) Using cyclostrophic balance and assuming that density p is constant, show that the pressure at radius r is given by P -(V² – V₂), P-Po= 2n where po is the pressure at radius ro-

An axisymmetric cyclonic vortex is in cyclostrophic balance (ƒ = 0) with a tan- gential velocity profile given by V = Vo(r/ro)", where Vo is the tangential velocity component at a distance ro from the vortex center. (a) Compute the circulation and vorticity about a streamline at radius r. What is special about the cases n = 1 and n = −1? Sketch V(r) for both cases. (b) Using cyclostrophic balance and assuming that density p is constant, show that the pressure at radius r is given by P -(V² – V₂), P-Po= 2n where po is the pressure at radius ro-

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4. An axisymmetric cyclonic vortex is in cyclostrophic balance (f = 0) with a tan- gential velocity profile given by V = Vo(r/ro)", where Vo is the tangential velocity component at a distance ro from the vortex center. (a) Compute the circulation and vorticity about a streamline at radius r. What is special about the cases n = 1 and n=-1? Sketch V(r) for both cases. (b) Using cyclostrophic balance and assuming that density p is constant, show that the pressure at radius r is given by P-Po=(V²_V²), 2n where po is the pressure at radius ro. HINTS: The radial pressure gradient force is p-¹0p/ar, and the vertical vorticity in cylindrical polar coordinates is given by $ = V/Or + V/r when there is no radial velocity component. [~ Holton 4.11]
Transcribed Image Text:4. An axisymmetric cyclonic vortex is in cyclostrophic balance (f = 0) with a tan- gential velocity profile given by V = Vo(r/ro)", where Vo is the tangential velocity component at a distance ro from the vortex center. (a) Compute the circulation and vorticity about a streamline at radius r. What is special about the cases n = 1 and n=-1? Sketch V(r) for both cases. (b) Using cyclostrophic balance and assuming that density p is constant, show that the pressure at radius r is given by P-Po=(V²_V²), 2n where po is the pressure at radius ro. HINTS: The radial pressure gradient force is p-¹0p/ar, and the vertical vorticity in cylindrical polar coordinates is given by $ = V/Or + V/r when there is no radial velocity component. [~ Holton 4.11]
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