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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 6 Vorticity and Divergence 2 Introduction
In this Lecture we will continue our investigation of the properties of the Shallow Water Equations (SWE). We introduce vorticity and divergence and derive equations for them. We show that an arbitrary velocity ﬁeld may be partitioned into curl-free and divergence-free components. Also, we show that the velocity may be reconstructed from knowledge of vorticity and divergence. The most important result we derive is the Conservation of Potential Vorticity. Recall the form of the SWE: ∂u ∂u ∂Φ ∂u +u +v − fv + =0 (1) ∂t ∂x ∂y ∂x ∂v ∂v ∂v ∂Φ +u +v + fu + =0 (2) ∂t ∂x ∂y ∂y ∂Φ ∂Φ ∂Φ ∂ u ∂v +u +v +Φ + =0 (3) ∂t ∂x ∂y ∂x ∂y The geopotential is Φ = gh and the Coriolis parameter is f = 2Ω sin φ. Recall that we have neglected all eﬀects of spherical geometry except in the Coriolis term. We deﬁne the beta parameter: df 2Ω cos φ β= = . dy a For latitudes φ not too far from a central value φ0, we may assume that 2Ω cos φ 2Ω cos φ0 f = 2Ω sin φ ≈ 2Ω sin φ0 and β= ≈ a a are both constant, unless diﬀerentiated w.r.t. y .
4 3 “Spin” and “Spread”
The extent to which the ﬂuid is rotating may be measured by calculating the circulation around a small circle C and taking the limit as the area A goes to zero: 1 ζ = lim V · s ds . A→0 A C We may call this the Spin or, more usually, the Vorticity. The extent to which the ﬂuid is spreading may be measured by calculating the outward ﬂux from a small circle C and taking the limit as the area A goes to zero: 1 δ = lim V · n ds . A→0 A C We may call this the Spread or, more usually, the Divergence. Using Stokes’ and Gauss’s Theorems, we will obtain diﬀerential forms of the vorticity and divergence.
5 6 Tangent and normal unit vectors s and n. First, consider Stokes’ Theorem: V · s ds =
C A k· × V da . Assuming the area A of the circle is small, we get 1 V · s ds ≈ k · × V . AC Taking the limit A −→ 0, we deﬁne the vorticity as We deﬁne the vorticity and divergence as follows: ∂ v ∂u − ζ =k· ×V = ∂x ∂y ∂ u ∂v + ∂x ∂y Note that ζ is the vertical component of the vorticity and δ is the horizontal divergence. However, we use the words divergence and vorticity to mean δ and ζ . δ= ·V = We will derive equations for the vorticity and divergence by diﬀerentiating and combining the momentum equations. ζ =k·
Now recall Gauss’s Theorem V · n ds =
C ×V · V da .
Show that the ratio of the vertical to horizontal component of the (3-D) vorticity is of the order w/V so that, with the assumptions we have made, the vertical component dominates. If we relax the assumption ∂ V/∂z = 0, how does this aﬀect the conclusion?
7 8 Assuming the area A of the circle is small, we get 1 V · n ds ≈ · V . AC Taking the limit A −→ 0, we deﬁne the divergence as δ= ·V Exercise: Geostrophic Divergence
Suppose the wind is geostrophic. Derive expressions for vorticity and divergence in terms of geopotential. Assume that the scale of motion is synoptic, L = 106 m. Since a = 6371 km ≈ 107 m, we may assume L ∼ Ro 1. a Let us assume that the ﬂow is approximately geostrophic. Then V VL ζ∼ δ ∼ ∼ ζ. L a a Thus, for typical synoptic motions, the divergence is an order of magnitude smaller than the vorticity δ ∼ Ro ζ The geostrophic velocity is given by 1 ∂Φ 1 ∂Φ v=+ , u=− f ∂y f ∂x Calculating vorticity directly, we get ζ= ∂ v ∂u − ∂x ∂y = ∂ 1 ∂Φ + ∂x f ∂x − ∂ 1 ∂Φ − ∂y f ∂y = 1 f
2 β Φ + u. f For constant f the stream function is ψ = Φ/f . Calculating divergence directly, we get δ= ∂ u ∂v + ∂x ∂y = ∂ 1 ∂Φ − ∂x f ∂y + ∂ 1 ∂Φ + ∂y f ∂x β = − v. f The smallness of the divergence is due to approximate cancellation between inﬂux and outﬂow. The terms ∂u/∂x and ∂v/∂y are roughly equal in magnitude but opposite in sign. This makes accurate calculation of divergence very diﬃcult. For f constant, it follows immediately that δ = 0. Therefore, the geostrophic divergence depends on the variation of f , the beta-eﬀect.
9 10 The Helmholtz Theorem (2D-Form)
A fundamental theorem due to Stokes (1849) states that a velocity ﬁeld can be decomposed into the sum of an irrotational (curl-free) and a non-divergent part. V = VD + VR = χ+ × A, where χ and A are called the scalar and vector potentials. It is possible to impose an additional constraint, · A = 0. For two-dimensional ﬂow, the decomposition is as follows: V = Vχ + Vψ = χ+k× ψ. Here ψ and χ are the stream function and velocity potential. We recall the vector identities (curl grad χ = 0 and div curl A = 0): × χ=0 · ( × A) = 0 . In 2-D, the latter implies · k × ψ = 0.
11 Thus, δ= ·V =
2χ ζ =k· ×V = ×V = 2ψ . Note the useful vector identity: k · · (V × k). Given the vorticity and divergence, we can recover the velocity ﬁeld provided appropriate boundary conditions are speciﬁed. Procedure: (1) Solve the two Poisson equations
2χ = δ , 2ψ = ζ , for the stream function and velocity potential. (2) Calculate the wind from V= χ+k× ψ. ∂χ ∂ψ + . ∂y ∂x
12 or, in component form ∂χ ∂ψ u= − , ∂x ∂y v= The Vorticity Equation
Recall the momentum equations ∂u ∂u ∂u ∂Φ +u +v − fv + =0 (1) ∂t ∂x ∂y ∂x ∂v ∂v ∂v ∂Φ +u +v + fu + =0 (2) ∂t ∂x ∂y ∂y Taking the x-derivative of (2) and subtracting from it the y -derivative of (1), we get an equation for ζ : ∂ζ ∂ζ ∂ζ + u + v + (ζ + f )δ + βv = 0 . ∂t ∂x ∂y Note that, since f is independent of time, df ∂f =v = βv . dt ∂y Thus the vorticity equation may also be written: The absolute vorticity η is deﬁned as the sum of relative vorticity and planetary vorticity: η
Absolute Vorticity = ζ
Relative Vorticity + f
Planetary Vorticity . The vorticity equation may now be written 1 dη + δ = 0. η dt The relative rate-of-change of absolute vorticity is equal to (minus) the divergence. We note the formal similarity to the continuity equation: 1 dh + δ = 0. h dt The relative rate-of-change of depth is equal to (minus) the divergence. We may illustrate this by considering a column of ﬂuid. d (ζ + f ) + ( ζ + f )δ = 0 . dt
13 14 The Continuity Equation may be interpreted pictorially. The Vorticity Equation may be interpreted pictorially. Convergence is associated with stretching of the column. Divergence is associated with shrinking of the column.
15 Convergence is associated with spin-up of the ﬂuid column. Divergence is associated with spin-down of the column.
16 The Divergence Equation
Recall again the momentum equations ∂u ∂u ∂u ∂Φ +u +v − fv + =0 ∂t ∂x ∂y ∂x (1) ∂v ∂v ∂v ∂Φ +u +v + fu + =0 (2) ∂t ∂x ∂y ∂y Taking the x-derivative of (1) and adding it to the y -derivative of (2), we get an equation for δ : Note: The derivation of the divergence equation in the above form is elementary, but it requires a page or two of algebraic manipulation. Since we will not make explicit use of the full divergence equation, we need not consider it further. Observation: for large-scale atmospheric ﬂow in middle latitudes, the divergence is much smaller than the vorticity: |δ | |ζ | . This allows us to make approximations to the equations.
2 ∂δ ∂δ ∂δ + u + v − ζf + δ 2 − 2J (u, v ) + βu + ∂t ∂x ∂y
The Jacobian term is deﬁned as ∂ u ∂v ∂v ∂u − J (u, v ) = ∂x ∂y ∂x ∂y . Φ = 0. New Deﬁnitions:
The ﬂow is cyclonic if f ζ > 0. The ﬂow is anticyclonic if f ζ < 0. 17 18 The Potential Vorticity Equation
The continuity equation may be written: dh + hδ = 0 dt The vorticity equation may be written: d (ζ + f ) + ( ζ + f )δ = 0 . dt Wo eliminate δ between the vorticity and continuity equations to get: 1 d(ζ + f ) 1 dh = . ζ +f dt h dt This may also be put in the following form (take logs): Exercise: Bottom Orography
We have assumed the bottom surface is ﬂat. Now we will relax this. Assume the height of the bottom boundary is hB(x, y ). Show that the Conservation of Potential Vorticity takes the form: d ζ +f dt h − hB = 0. This states that the following ratio is conserved: d Absolute Vorticity = 0. dt Fluid Depth This is an important exercise. The solution will not be given. The proof is straightforward, requiring only a minor adjustment of the derivation in the case hB(x, y ) ≡ 0.
20 d ζ +f dt h = 0. This is the equation of conservation of potential vorticity.
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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