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Lect-07-P4 - M.Sc in Computational Science Fundamentals of...

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M.Sc. in Computational Science Fundamentals of Atmospheric Modelling Peter Lynch, Met ´ Eireann Mathematical Computation Laboratory (Opp. Room 30) Dept. of Maths. Physics, UCD, Belfield. January–April, 2004. Lecture 7 Potential Vorticity Conservation 2 Applications of PV Conservation We consider a few simple applications of Potential Vorticity conservation. The treatment is purely qualitative . A quantitative treatment will be undertaken in subsequent lectures. Gravity-Inertia Waves. Free Rossby Waves. Forced Rossby Waves. Lee-side Trough. 3 The Potential Vorticity Equation Recall that the continuity equation may be written: dh dt + = 0 The vorticity equation may be written: d dt ( ζ + f ) + ( ζ + f ) δ = 0 . Taking logarithms, we may write these in the form d dt log( ζ + f ) = - δ , d dt log h = - δ . We eliminate δ between these equations to get: d dt log( ζ + f ) - d dt log h = 0 . This may also be put in the following form: d dt ζ + f h = 0 . This is the equation of conservation of potential vorticity . 4
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Elementary Applications of PV Conservation 5 Gravity Waves: A First Look Suppose initially the flow is irrotational and is converging towards a point. Assume that f is constant. Consider a column of fluid. Convergence induces stretching Stretching implies increased pres- sure at the centre Increasing h also implies increas- ing ζ ζ > 0 implies Cyclonic flow Cyclonic flow around high pres- sure is unbalanced PGF and Coriolis force act out- wards Divergent flow is induced. The restoring forces give rise to Inertia-gravity waves . 6 Rossby Waves: A First Look Suppose initially the flow is nondivergent, so there is no vertical velocity and h is constant for a fluid parcel. Then absolute vorticity is conserved. d ( ζ + f ) dt = 0 . Suppose a parcel, initially at latitude y - y 0 with f = f 0 and ζ = 0 . Suppose that it moves North-eastward.
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