Unformatted text preview: 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. The Potential Vorticity Equation
Recall that the continuity equation may be written: dh + hδ = 0 dt The vorticity equation may be written: d (ζ + f ) + ( ζ + f )δ = 0 . dt Taking logarithms, we may write these in the form d d log(ζ + f ) = −δ , log h = −δ . dt dt We eliminate δ between these equations to get: d d log(ζ + f ) − log h = 0 . dt dt This may also be put in the following form: Gravity-Inertia Waves. Free Rossby Waves. Forced Rossby Waves. Lee-side Trough. d ζ +f dt h
3 = 0.
4 This is the equation of conservation of potential vorticity. Gravity Waves: A First Look Elementary Applications of PV Conservation
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 pressure at the centre • Increasing h also implies increasing ζ • ζ > 0 implies Cyclonic ﬂow • Cyclonic ﬂow around high pressure is unbalanced • PGF and Coriolis force act outwards • ∴ Divergent ﬂow is induced. The restoring forces give rise to Inertia-gravity waves.
5 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. Constant Absolute Vorticity (CAV) Trajectories. d(ζ + f ) = 0. dt
Suppose a parcel, initially at latitude y − y0 with f = f0 and ζ = 0. Suppose that it moves North-eastward. • Increasing f means decreasing ζ • Negative ζ corresponds to anticyclonic ﬂow • Flow curves back towards y = y0 • Then f = f0 and ζ = 0 again • Parcel continues SE and opposite half-cycle occurs. Throughout the motion, ζ + f keeps the same value.
7 Fluid parcels follow trajectories on which ζ + f remains constant.
8 Na¨ argument concerning movement of Rossby waves ıve Formation of a Lee-side Trough This qualitative argument indicates westward propagation.
A mountain chain may produce a train of forced Rossby waves.
9 10 The Circulation Theorem
The momentum equation in vector form is: ∂V + V. V + f k × V + Φ = 0 ∂t We easily prove the following vector identity: (7) More Conservation Properties V· V= ( 1 V · V) + ζ k × V 2 Using this, the momentum equation may be written: ∂V + ( 1 V · V ) + ( f + ζ )k × V + Φ = 0 (8) 2 ∂t Assume ﬂuid system is contained in region D with boundary C , with no ﬂow across C . We integrate equation (8) around the contour C . The cross-product term vanishes, because k × V is perpendicular to V and thus to s. The gradient terms integrate to zero, because the contour is closed.
11 12 Thus, the integral of (8) gives ∂V d · ds = V · ds = 0 dt C C ∂t The integral of V around C is called the circulation. This result shows that the circulation around the boundary of the domain remains constant. Alternative view: the vorticity equation can be written ∂ζ + ·(ζ + f )V = 0 ∂t Integrating this over D: d ζ da = − ·(ζ + f )V da = (ζ + f )V · n ds = 0 dt D D C Thus, the integral of vorticity over the domain is a constant. Put another way, the average vorticity is conserved. Contour deﬁned by the ﬂow velocity V
13 14 Conservation of Mass.
We write the continuity equation in ﬂux form: ∂h + ·hV = 0 ∂t We multiply by ρ and integrate over the domain ∂ ρh + ·ρhV da = 0 ∂t D By the divergence theorem, the second term vanishes: ·ρhV da =
D C Conservation of Energy
The potential energy of a column of ﬂuid is:
h (9) P=
0 h ρgz dz = 1 ρgh2 = 2 ρ Φ2 . 2g The kinetic energy of the column is ρ ΦV · V . 2g 0 Multiply the continuity equation (9) by Φ to get g ∂P ∂ = ( 1 Φ2) = −Φ · ΦV . ρ ∂t ∂t 2 Next, multiply the momentum equation ∂V + (ζ + f )k × V + (Φ + 1 V · V) = 0 2 ∂t by ΦV to obtain (use V · k × V = 0) ∂V ΦV· + ΦV · Φ + ΦV · ( 1 V · V) = 0 2 ∂t K=
1 ρV · V dz = 1 ρhV · V = 2 2
16 ρhV · n ds = 0 Thus we get ∂ρh d da = ρh da = 0 dt D ∂t D But the mass of ﬂuid over an element of area da is dM = ρh da. Thus, the equation expresses conservation of total mass.
15 Add the continuity equation multiplied by 1 V · V: 2 ∂Φ 1 1V · V + V · V ·ΦV = 0 2 ∂t 2 to obtain the expression g ∂K + · [( 1 V · V)ΦV] + ·Φ2V − Φ ·ΦV = 0 . 2 ρ ∂t Finally, integrate the equations for P and K over the domain: d ρ ρ Φ2 da = − Φ ·ΦV da dt D 2g Dg ρ ρ d ΦV · V da = + Φ ·ΦV da . dt D 2g Dg Adding these gives the energy conservation equation: d ρ d ΦV · V + Φ2 da = [K + P ] = 0 . (10) dt dt D 2g This is the energy principle: the sum of the kinetic plus potential energy of the ﬂuid system remains constant.
17 Simpliﬁcation of the PV Equation
Conservation of potential vorticity implies that the quantity P = (ζ + f )/Φ, which we call the potential vorticity, is conserved following the motion. That is, the value of P for a particular parcel of ﬂuid remains constant as that parcel is carried along with the ﬂow. The conservation of potential vorticity is of great signiﬁcance. If the ﬂow is geostrophic, PV conservation provides a single equation for the dynamics. Let us assume geostrophic ﬂow: fk × V + ζ =k· ×V = Φ = 0. ·(1/f ) Φ The vorticity may then be written in terms of Φ: ·V×k= 18 The material time derivative takes the form ∂ ∂ ∂ ∂ 1 ∂Φ ∂ 1 ∂Φ ∂ d = + ug + v g = − + dt ∂t ∂x ∂y ∂t f ∂y ∂x f ∂x ∂y Then the potential vorticity equation (6) becomes an equation for a single dependent variable, Φ: To the ﬁrst order of approximation, we can move the factor 1/f inside the diﬀerential operator: Φ 1 k× Φ≈k× . f f Thus, the geopotential and stream function are related: Φ ≈ fψ . Now assume the deviation of geopotential from its mean value is small: ¯ ¯ Φ=Φ+Φ with Φ Φ. We can equate Φ with f ψ . Then we have ¯ 1 1 1 Φ 1 fψ =¯ ≈ ¯ 1− ¯ ≈ ¯ 1− ¯ ¯ Φ Φ(1 + Φ /Φ) Φ Φ Φ Φ Thus, the potential vorticity becomes P≡ ζ +f 1 fψ ≈ ¯ (ζ + f ) 1 − ¯ Φ Φ Φ f ζ f 2ψ ≈ ¯ + ¯ − ¯2 ΦΦ Φ
20 ∂ 1 ∂Φ ∂ 1 ∂Φ ∂ − + ∂t f ∂y ∂x f ∂x ∂y ·(1/f ) Φ + f = 0. Φ Although the above equation can be solved numerically, it is not convenient for analysis. We will derive a more amenable form now. Assume the ﬂow is quasi-geostrophic and quasi-nondivergent: 1 V ≈ k × Φ, V ≈ k× ψ. f
19 We can ignore the variation of f in the term containing ψ , so ¯ ΦP ≈ ζ + f − F ψ 2¯ where F ≡ f0 /Φ is a constant. Next, we use the nondivergent wind in the Lagrangian derivative: ∂α ∂α ∂α dα = +u +v dt ∂t ∂x ∂y ∂α ∂ ψ ∂α ∂ψ ∂α = + − ∂t ∂x ∂y ∂y ∂x ∂α = + J (ψ, α) . ∂t Using this together with the approximation for P derived above, the PV-equation may be written as The Barotropic QGPV Equation
The barotropic, quasi-geostrophic potential vorticity equation (the QGPV Equation) is ∂ ∂t 2 ψ − F ψ + J (ψ, 2 ψ) + β ∂ψ = 0. ∂x This is a single equation for a single variable, the stream function ψ . The simplifying assumptions have the eﬀect of eliminating high-frequency gravity wave solutions, so that only the slow Rossby wave solutions remain. We will study the wave-like solutions of this equation in the next lecture. We will also study numerical solutions of this equation using a program written in matlab and in C. ∂ ∂t 2 ψ − F ψ + J (ψ, 2 ψ) + β ∂ψ = 0. ∂x
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This note was uploaded on 01/31/2011 for the course MATH 21A taught by Professor Osserman during the Spring '07 term at UC Davis.
- Spring '07