Understanding Vorticity and Vortex Stretching in Fluid Dynamics
Explore the concept of vorticity in fluid dynamics, including rotational and shear components. Learn about vortex stretching and angular momentum conservation in vertical columns of incompressible fluid. Visualize vorticity through natural coordinates and understand its importance in synoptic-scale motion.
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EART30351 Lecture 9
Vorticity Vorticity is defined by: ? = ? ? ?? ?? ?? ?? ?? ?? ?? ?? ?? ? = ??, ??, ?? Vertical component of vorticity: ??=?? ?? ?? ??
Vorticity In two dimensions we can visualise using a small paddle wheel. If the flow is rotational: If the flow is sheared: >0 U U <0 T T T T T T So we have rotational and shear vorticity. For synoptic-scale motion we concentrate on z Streamlines of the flow R<0 <0 R>0 >0
Components of vorticity For synoptic-scale motion we concentrate on z as on this scale it is not coupled to the horizontal components. Magnitude of z ~ 10-4 s-1 similar to f Note that x, y are about 100 times larger! E.g. ??=?? ?? ?? ?? v increases from ~0 to ~50 ms-1 between ground and 10 km in a jet stream so v/ z~ 50 x 10-4 s-1 Dynamics of thunderstorms are profoundly dependent on tilting of horizontal vorticity to the vertical.
Natural coordinates This framework makes it easer to visualise z. n U s s and n are defined at each point of the flow pointing along and perpendicular to the flow ??=? ? ?? ?? Vorticity is the sum of the rotational and shear components
Vortex stretching Column of incompressible fluid stretched in the vertical. Angular momentum is conserved in this process 2 1 h1 h2 r1 r2 For solid-body rotation, U=r and U/ n = - U/ r = - . So ??=? ? ?? ??= 2 Note: here is the angular velocity of the cylinder, not the Earth!
Vortex stretching Column of incompressible fluid stretched in the vertical. Angular momentum is conserved in this process Angular momentum L about z axis ? = ? ?? ?(??????) ? = ? ?2 ????? = 2?? ?3?? = ?? ?4 2 1 h1 h2 r1 r2 For solid-body rotation, U=r and U/ n = - U/ r = - . So ??=? ? ?? ??= 2 Note: here is the angular velocity of the cylinder, not the Earth!
Vortex stretching Column of incompressible fluid stretched in the vertical. Angular momentum is conserved in this process Angular momentum L about z axis ? = ? ?? ?(??????) ? = ? ?2 ????? = 2?? ?3?? = ?? ?4 But volume also conserved, Vol= r2h ? =????2 2? This is vortex stretching stretching an air column increases and therefore z 2 1 h1 h2 r1 r2 For solid-body rotation, U=r and U/ n = - U/ r = - . So ??=? ? ?? ??= 2 Note: here is the angular velocity of the cylinder, not the Earth!
Barotropic vorticity equation From the basic vorticity equation: ( ) ( f dt + d f v u ) + + + = U . 0 H x p y p Away from fronts, the tilting terms are small so ( ) ( f dt + d f ) + + = U . 0 H Here f appears as the planetary vorticity, the vorticity existing because the Earth is spinning. +f, the absolute vorticity, is the key quantity
Potential Vorticity Apply the vorticity equation to: 2 1 pt p1 p2 pb ? ?? ? + ? = ? + ? ?.? ?? ?? = ? + ?
Potential Vorticity Let p = pt pb . Then ? ?? ? =??? = ?? ??=?? Apply the vorticity equation to: ?? ??? 2 ?? 1 ?? ? pt p1 p2 So ?? 1 ? ? ?? ??= pb ? Substitute in the vorticity equation ? ??? + ? ? + ? ? ? ?? ? = 0 ? ?? ? + ? = ? + ? ?.? ?? ?? By dividing through by p and integrating: = ? + ? ? ?? ? + ? ? = 0
Potential vorticity 2 The quantity ( +f)/ p is the Rossby form of the potential vorticity. It is exactly analogous to the angular momentum in the ice-skater model. A more exact form of the PV was derived by Ertel in 1942 directly from the momentum equations with no scaling: ? ?? =1 1 ?? + 2? .? ?? + 2? .??? ??+1 ???.? ?
