Lect-09-P4 - M.Sc in Computational Science Fundamentals of...

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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 9 Mixed Rossby & Gravity Waves 2 Introduction We continue our investigation of the linear solutions of the shallow water equations. First, we compare the properties of the simple wave solutions already found. Then we consider the case where gravity waves and Rossby waves occur simultaneously as solutions. Finally,, we introduce the concept of filtering the equations. 3 Comparison of Wave Speeds We can estimate the relative sizes of the phase speeds of the two types of pure wave solutions for parameter values typical of the atmosphere. In the absence of a mean flow, the phase speeds are √ βL2 β ¯ cR = − 2 = − 2 . cG = Φ = g H ; k 4π Taking approximate values g = 10 m s2 , H = 104 m , β = 10−11 m−1 s−1 , L = 106 m the phase speeds are cG ≈ 300 m/s ; cR ≈ 0.5m/s . Thus the gravity waves travel much faster than the Rossby or planetary waves: |cR| |c G | 4 Even Rossby waves with wavelength L = 107 m , i.e., comparable in size to the earth’s radius, have a phase speed of about 50 m/s, still much slower than the gravity waves). The other crucial distinction between the two types of waves: Mixed Gravity & Rossby Waves We consider now a more complicated case in which both gravity waves and Rossby waves are present. To keep things as simple as possible, we ignore variations with latitude (i.e., in the y -direction). Then the divergence, vorticity and continuity equations for the linearized perturbation variables are: ∂δ ∂δ ¯ α +u − f ζ + βu + Φxx = 0 ∂t ∂x ∂ζ ∂ζ ¯ +u + f δ + βv = 0 ∂t ∂x ¯ ∂Φ ∂Φ ∂ Φ ¯ ∂u ¯ +u +v +Φ = 0. ∂t ∂x ∂y ∂x ¯ ¯ Here δ = ux, ζ = vx and the basic state satisfies f u = −Φy . For gravity waves, the divergence is vital for the dynamics. The vorticity vanishes. For Rossby waves, the vorticity is the important quantity. The divergence vanishes. 5 Exercise: The coefficient α normally has the value 1. It is a tracer, which is carried through the analysis as a parameter, and allows us to examine the effect of omitting the term dδ/dt in the divergence equation by giving it the value zero in the final result. (This is just a handy trick to avoid repetition). We are interested in wave-like solutions of the form: u u0 v = v0 exp[ik (x − ct)] Φ Φ0 so that u = δ/ik , v = ζ/ik and the differential operators are: ∂ ∂ ∂2 ∼ −k 2 ; ∼ ik ; ∼ −ikc . 2 ∂x ∂t ∂x The form of the basic state allows us to write ¯ ζ fu ¯ ¯ v Φy = (−f u) = − ζ. ik ik Check that these equations are correct. 6 We can now substitute the exponential expression for the dependent variables into the equations, which may then be written in matrix form: β ζ 2 −f α[ik (¯ − c)] + u −k ik δ = 0 . [ik (u − c) + β ] +f 0 ik ¯ fu ¯ Φ ik (¯ − c) u − Φ ik Since this is a homogeneous system, there is a solution iff the determinant of the coeficient matrix vanishes. Expanding out this gives us the dispersion equation: β f2 ¯ f2 β2 ¯ ¯¯ α(¯ − c) − 2 (¯ − c) − 2 + Φ (u − c) + 2 u + Φ 2 = 0 u u k k k k This is a cubic equation for the phase speed c. We can solve the cubic analytically or numerically. However, a simpler way is to note the relative sizes of the terms and estimate the roots approximately. 7 8 High Frequency Roots First, suppose the magnitude of the phase speed is large. Specifically, let us suppose ¯ |u − c| ¯ |u | and ¯ |u − c| |β/k 2| . ¯ Then we may neglect f 2u/k 2 compared to (f 2/k 2)(¯ − c), and u also the terms involving β . The cubic then reduces to α(¯ − c)2 − u f2 k2 ¯ +Φ ¯ (u − c ) = 0 . In the case of no rotation, the inertia-gravity wave phase speed ¯f Φ+ 2 k reduces to that for pure gravity wave solutions with √ ¯ ¯ c = u ± Φ. ¯ c=u± In this case the phase speed is independent of the wavelength (or wavenumber k ). More generally, the phase speed is modified by the effect of rotation and depends on k , making the waves dispersive. Taking typical parameter values, we find that the phase speed of these solutions is of the order c = 300 m/s. Thus, our assumptions are justified a posteriori. 2 ¯ The solution c = u cannot be admitted, as we have assumed that c is large. Thus, the quadratic term must vanish, giving the two roots: ¯ c=u± ¯f Φ+ 2. k 9 2 Exercise: Calculate the group velocity of the inertia-gravity waves. Group velocity is defined two frames below 10 These are the inertia-gravity wave solutions. Low Frequency Roots Next, suppose the phase speed is small, specifically that it is much smaller than the pure gravity wave speed: ¯ ¯ ¯ c2 Φ and also |u − c|2 Φ. We also recall that the pure Rossby speed (β/k 2) is very much smaller that the gravity wave speed. Thus, the cubic term in the dispersion equation can be neglected compared to the linear term and the equation becomes − f2 ¯ f2 β ¯ ¯¯ + Φ (u − c ) + 2 u + Φ 2 = 0 . 2 k k k ¯¯ β + f 2u/Φ ¯ k 2 + f 2/Φ This solution is, of course, the Rossby wave solution, which always travels westward relative to the mean flow: ¯ c=u− ¯¯ β + f 2u/Φ ¯ k 2 + f 2/Φ . ¯ For large values of Φ, it is approximated by the previously obtained simple formula β ¯ c=u− 2. k Exercise: Calculate the group velocity of the Rossby waves. This has the following solution ¯ c=u− . Definition: The group velocity cg for a wave in the xdirection is defined by ∂ω ∂kc = . cg = ∂k ∂k ¯ For typical values of Φ, say 105 m2s−2, we find that c is quite small, thereby justifying our assumptions. 11 12 Filtering In the above analysis, we used a tracer α to mark the term dδ ∂δ ∂δ ¯ = +u dt ∂t ∂x in the divergence equation. If we now set α = 0, the dispersion equation becomes linear and only the Rossby wave solution remains. To rephrase, omission of the term corresponding to changes in the divergence is sufficient (in the present case) to eliminate solutions corresponding to gravity waves. This process of modifying the governing equations in such a way as to exclude certain solutions is called filtering. The Quasi-geostrophic Equations The question arises: How can we generalise this idea? How can the general nonlinear equations be modified so that the high frequency gravity wave solutions are eliminated, while the low frequency Rossby waves, which are the important solutions, are preserved? This question will not be addressed in this lecture series, but an affirmative answer can be obtained. The equations resulting from the filtering procedure are called the Quasigeostrophic Equations. The quasi-geostrophic equations provide a powerful basis for the elucidation of the dynamics of the atmosphere in middle latitudes. 13 14 15 16 ...
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