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 10 Rossby Wave Packets 2 Introduction
First, we consider wave interactions, and introduce the concept of group velocity. Then we deﬁne Rossby wave packets and study their behaviour. Finally, we illustrate the importance of group velocity for Rossby waves, using real atmospheric data. Interference of Two Waves
The simplest case to study is the superposition of two waves. We assume the two components have equal amplitudes and approximately the same wavenumbers and frequencies: ψ (x, t) = cos(k1x − ω1t) + cos(k2x − ω2t) . The components move with respective phase speeds c1 = ω1/k1 and c2 = ω2/k2 . By elementary trigonometry, ψ may be written k − k2 ω − ω2 k + k2 ω + ω2 ψ (x, t) = 2 cos 1 x− 1 t · cos 1 x− 1 t 2 2 2 2 We write the mean values and the diﬀerences ¯ ¯ k = (k1 + k2)/2 and ω = (ω1 + ω2)/2 ∆k = (k1 − k2)/2 and ∆ω = (ω1 − ω2)/2 . Then the wave combination is ¯ ¯ ψ (x, t) = 2 cos (∆k · x − ∆ω · t) · cos kx − ω t . . Again: ¯ ¯ ψ (x, t) = 2 cos (∆k · x − ∆ω · t) · cos kx − ω t ¯ ∆ω ω ¯ = 2 cos ∆k x − · t · cos k x − ¯ t . ∆k k The second term here represents a wave with wavenumber ¯ k moving with phase speed ¯ ¯¯ c = ω /k which is close to the phase speeds of the two components. The ﬁrst term is slowly varying in space: it has wavenumber ∆k and frequency ∆ω , and it moves with a speed cg , called the group velocity, cg = ∆ω . ∆k
Two wave components of approximately equal wavelength. The envelope amplitude of the sum is clear.
6 The group velocity may be radically diﬀerent from the phase ¯ velocity c, and of the opposite sign!
5 Group Velocity of Rossby Waves
The matlab program gv1.m shows the evolution of this waveform in time. We consider only a simple case here. A much more detailed discussion may be found in Pedlosky (2003). For nondivergent quasigeostrophic ﬂow on a beta plane of a wave which is independent of the y -coordinate, the Rossby phase speed is β ¯ c=u− 2. k ¯ Here u is the mean zonal ﬂow. Let us compute the group velocity: ∂ω d(kc) β ¯ cg = = =u+ 2. ∂k dk k We have the surprising result that the group velocity is directed towards the east (relative to the mean ﬂow) whereas the phase velocity is towards the west. The group velocity for a pair of waves was deﬁned ∆ω . cg = ∆k More generally, there is a dispersion relation ω = ω (k ) , and the group velocity is deﬁned by cg = ∂ω . ∂k More generally, a Rossby wave may be travelling in a direction other than westward. If we assume ψ = ψ0 exp[i(kx + y − ωt)] the dispersion relation is kβ ¯ ω = ku − 2 . k+2 and the phase speed for wavenumber k is thus β ¯ c (k ) = u − 2 . k+2 ¯ The mean ﬂow u simply transports wave patterns eastward ¯ (for u > 0) at a constant speed, so we will ignore this eﬀect ¯ by assuming u = 0. The components of group velocity in the x and y directions are: dω k2 − 2 β cgx = =+ 2 + 2 k2 + 2 dk k β dω 2k cgy = =+ 2 d k + 2 k2 + 2
9 dω k2 − 2 β cgx = =+ dk k2 + 2 k2 + 2 dω β 2k cgy = =+ 2 d k + 2 k2 + 2 The group velocity in the x-direction may be eastward or westward, depending on the sign of k 2 − 2: for waves which are large-scale in x (small k ) cgx is negative; for waves which are small-scale (large k ) it is positive. The group speed in the y -direction depends on the sign of k . However, the phase speed is cy = ω/ , so the ratio is cgy 22 =− 2 < 0. cy k+2 Thus the group velocity in the y -direction is in the opposite sense to the phase velocity. Exercise: Plot the phase and group speeds as functions of the wavenumbers k and .
10 Extraction of the Envelope
The envelope of a wave packet may be extracted using ideas based on the Hilbert transform. For full details, see Bracewell (1978, pp. 267–272). Several appications of this technique are presented in Zimin, et al., (2003). Let ψ (λ) be a function on a periodic domain 0 ≤ λ < 2π . We perform the following operations in sequence: ˆ • Compute the Fourier coeﬃcients: ψk = 1 2π e−ikλψ (λ) dλ.
2π 0 A simple example illustrates the technique. Suppose ψ (λ) = A cos nλ. There are just two nonvanishing terms in the fourier series: A cos nλ = 1 A[exp(nλ) + exp(−nλ)]. Elimination of the 2 negative frequency part leaves 1 A exp(nλ) and twice the ab2 solute value of this is A, as expected. The generalization for a function ψ (x) which is not periodic is straightforward: the Fourier series is replaced by the Fourier transform: ˆ • Compute the Fourier transform: ψ (ω ) = ∞ e−iωxψ (x) dx.
−∞ ˜ ˆ • Set the coeﬃcients to zero for negative index: ψk = Hk ψk where Hk is the Heaviside sequence. ˜ • Compute the inverse transform: Ψ(λ) = k=∞ ψk eikλ.
k =−∞ ˜ ˆ • Set the transform to zero for negative ω : ψ (ω ) = H (ω )ψ (ω ) where H (ω ) is the Heaviside function. ˜ • Compute the inverse transform: Ψ(x) = 1 ∞ eiωxψ (ω ) dω .
2π −∞ • Double and take the absolute value: A(λ) = 2|Ψ(λ)|. In words, we calculate the Fourier series, throw away the negative frequencies, invert, double and take the absolute value. • Double and take the absolute value: A(x) = 2|Ψ(x)|. The theoretical explanation of the envelope extraction method is given in Bracewell (loc. cit.). The envelope extraction may be combined with low-pass or band-pass ﬁltering by replacing the Heaviside function by a suitable masking function, for example M (ω ) = 1, 0, ωL ≤ ω ≤ ωH otherwise Gaussian Wave-packet which eliminates all components except in the frequency band [ωL, ωH ]. 13 14 Suppose that we may express the streamfunction at the initial time t = 0 as
2 ψ (x, 0) = A exp[− 1 x2/σ0 ] exp(ik0x) , 2 The group velocity is d(kc) β ¯ =u+ 2. (2) dk k We suppose that the governing equation for ψ (x, t) is linear. Then each Fourier component will evolve independently of the others. So the solution may be written ∞ 1 ˆ ψ (k, 0) exp[i(kx − ωt)] dk . (3) ψ (x, t) = 2π −∞ cg = ˆ For large σ , the transform ψ is concentrated near k = k0 and we can approximate the frequency ω using the Taylor series 1 d2ω dω (k − k 0 ) + (k − k 0 )2 . dk k0 2 dk 2 k0 or, more brieﬂy, with obvious notation, ω ( k ) ≈ ω (k 0 ) + ω ≈ ω0 + ω0(k − k0) + 1 ω0 (k − k0)2 . 2 that is, as a rapidly varying wave function whose amplitude envelope varies slowly with x. A straightforward application of Fourier’s Theorem allows us to write this as ∞ 1 ˆ ψ (x, 0) = exp(ikx)ψ (k, 0) dk 2π −∞ where the spectral transform is given by ∞ √ 2 ˆ ψ (k, 0) = exp(−ikx)ψ (k, 0) dk = 2πσ0A exp[− 1 σ0 (k − k0)2] . 2
−∞ Assume that the mode with wavenumber k has frequency ω (k ), given by the Rossby wave dispersion relation. Then the phase velocity is β ¯ c=u− 2. (1) k Substituting this into (??) and evaluating the integral (about a page of calculus) we get ψ (x, t) = (x − ω0t)2 Aσ0 exp[i(k0x − ω0t)] . exp − 2 2(σ0 + iω0 t) 2 σ0 + iω0 t The ﬁrst component is a Gaussian envelope, centered at x = ω0t, whose width is given by
2 σ 2 = σ0 (1 + τ 2) (4) This solution has several points of interest. The ﬁrst term shows that, for large time, the amplitude decreases as t−1/2. The second term is the envelope, which we examine presently. The last term represents an oscillation with wavenumber k0 and frequency ω0, which travels with phase speed c0 = ω0/k0. The middle term on the right of (??) may be written (x − ω t)2 (x − ω0t)2 exp − 2 0 = exp − 2 exp i 2(σ0 + iω0 t) 2σ0 (1 + τ 2)
2 where τ = (ω0 /σ0 )t is re-scaled time. so it moves with the group velocity cg = ω0 and spreads as time increases. The second term is a chirp-function: its local wavenumber is zero at x = ω0t and increases linearly with distance from this point. The factor τ /(1 + τ 2) vanishes at t = 0 and for large time, reaching its maximum at τ = 1. The full solution is now written as a product of four components: ψ (x, t) = ξ2 exp − 2 exp i 2σ 2 σ0 + iω0 t Aσ0
Amplitude Gaussian τ (x − ω0t)2 2 2σ0 (1 + τ 2) τ ξ2 2σ 2 exp[ik0(x − c0t)] ,
W ave , C hirp where ξ = x − ω0t.
17 18 The properties of the solution ψ (x, t) = ξ2 exp − 2 exp i 2σ 2 σ0 + iω0 t Aσ0
Amplitude Gaussian References
τ ξ2 2σ 2 exp[ik0(x − c0t)] ,
W ave Bracewell, R, 1978: The Fourier Transform and its Applications. Second Edn., McGraw-Hill, New York. 444pp. Zimin, Aleksey V., Szunyogh, Istvan, Patil, D. J., Hunt, Brian R., Ott, Edward. 2003: Extracting Envelopes of Rossby Wave Packets. Monthly Weather Review, Vol. 131, 1011–1017. C hirp may be summarised as follows • Individual wave crests move with the phase velocity c0. • The overall amplitude decays as O(t−1/2). • The envelope moves with the group velocity cg = ω0. 2 • The spread of the envelope grows as σ 2 = σ0 (1 + τ 2). • What to say about the chirp part? ...
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