Journal of Physics: Conference Series OPEN ACCESS A review on Lamb's atmosheric oscillations using initial value roblem aroach To cite this article: Ángel De Andrea González 04 J. Phys.: Conf. Ser. 56 004 View the article online for udates and enhancements. This content was downloaded from IP address 48.5.3.83 on 09/0/08 at 9:53
4th International Worksho & Summer School on Plasma Physics 00 Journal of Physics: Conference Series 56 (04 004 doi:0.088/74-6596/56//004 A review on Lamb s atmosheric oscillations using initial value roblem aroach Ángel De Andrea González Deartamento de Física. Escuela Politécnica Suerior. Universidad Carlos III de Madrid. Av. Universidad, 30. 89 Leganés. Sain. E-mail: aandrea@fis.uc3m.es Abstract. Waves at a surface of discontinuity in the atmoshere were analysed in 90 by Lamb, who derived, using normal mode aroach, an analytical disersion relation for a disete mode (surface mode. Lamb examined the case of waves roagated along a horizontal lane where the equilibrium temerature is discontinuous. For simlicity, the uer and the lower regions are considered incomressible. The oscillations are treated in the ideal (dissiationless limit and the uniform gravitational acceleration is taken to be co-aligned with the revailing temerature gradient. In this work, in order to show how the modes aear in the resonse of a surface discontinuity to an initial erturbation, we consider the initial value roblem (IPV. The main difference from the standard analysis is that solutions to the linearized equations of motion which satisfy general conditions are obtained in terms of Fourier-Lalace transform of the hydrodynamics variables. These transforms can be inverted exlicitly to exress the fluid variables as integrals of Green s functions multilied by initial data. In addition to disete mode (surface mode, sets of continuum modes due to branch cuts in the comlex lane, not treated exlicitly in the literature, aears.. Introduction Several authors have urosed solutions of the initial value roblem through the use of a Lalace Transform in time [-4]. The rincial finding of the initial value roblem aroach is that, in addition to the disete eigenvalues linked to the normal modes, there exists a continuous sectrum of eigenvalues. Thus, the modal aroach cannot rovide a comlete solution.. Linear stability analysis We consider two infinitely incomressible lasmas in a steady state which are searated by a lane interface at y0 and subjected to a gravitational field g r directed erendicular to the interface, where q and q are both the inverse density-gradient scale length at each region. By following the aroach develoed by Lamb (90 [], it is ossible to reduce the linearized equations of continuity, momentum and energy for a olytroe to a set of artial differential equations. These equations can be easily combined into a stream-function equation: Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s and the title of the work, journal citation and DOI. Published under licence by Ltd
4th International Worksho & Summer School on Plasma Physics 00 Journal of Physics: Conference Series 56 (04 004 doi:0.088/74-6596/56//004 3 ψ ψ ψ ψ 0 y k y ( Where the dimensionless normalized arameters are: t kg, k k q, t t kg. We define the Lalace transform ψ ( y, ω ω in the comlex ω -lane: i σ i ωt ψ ( y, ω ω ψ ( y, t e dω π ( i σ Then, we consider the initial value roblem for equation (, and introduce the Lalace transform. So, we obtain the following equation: ψ ψ ψ ω Γ y y k y k ω ω ω ( (3 being: i ω F F i ω F F Γ ( y iω F F ω y k y (4 is: where the Lalace transform function ψ ( y, ( y, ψ L ω ψ s i ωf F With the following initial conditions: ψ F ψ ( y, ; F t 0 y ( y, 0 (5 (6 With Lalace contour σ chosen that the conditions at ψ s y at y 0. has no singularities in Re( ω ψ ω σ. Equation is subjected to y that ψ ω 0, the continuity of ψ ω at y 0 and the jum conditions on
4th International Worksho & Summer School on Plasma Physics 00 Journal of Physics: Conference Series 56 (04 004 doi:0.088/74-6596/56//004 We consider the Green s function associated with equation (3. So, the solution of equation ( is: (, (,, Γ( ψ y ω G y ω y y dy (7 s o o 0.. Results and discussion The Green s function for y > 0 : (, ω, y G y o y β y o α k kr ω e r ω α r ω β kr k (8 G ±,, y o 0, the square roots are defined so that Re(α > 0 y Re( β > 0 for Re (ω > σ. It is evident that Green s function has a ole (disete sectra. Exanding Green s function about α 0, and β 0, this function does not turn out to be an In accord with the boundary conditions ( ω even function of α and β. So, the Green s function has branch oints at continuous sectra neglected by Lamb: (associated with ω γ b 4k γ 4k b y en 4kr γ 4k r b (i.e., α 0 y β 0 (9 It is interesting to follow the migration of the ole as k is varied. For k > k, γ > γ b. If k is diminished, then the γ ole moves toward γ b. For k k, then γ γ b. For k < k the ole dros on to the lower sheet of β and the eigenmode disaears. The disaearance of the γ ole was neglected by Lamb too. Thus, there will be some itical wave number k such for k > k k < k eigenmodes exist (do not exist. Lamb obtained the following disersion relation []: ( ( ω ω - - 0 k ( r r r- (0 Being, r q.we show that Lamb s disersion relation, the itical wave number q k is: 3 r k r for r ( 3
4th International Worksho & Summer School on Plasma Physics 00 Journal of Physics: Conference Series 56 (04 004 doi:0.088/74-6596/56//004 w kg.0 0.8 0.6 0.4 0. k 0.0 0 4 6 8 0 q Figure. Dimensionless normalized growth rate versus dimensionless normalized wave number for r4. Solid line corresonds to the Lamb s results; dots corresond to our numerical results. There is an excellent agreement with Lamb s results for k > k. Dashed line and dot dashed line corresond to branch oints. 3. Conclusion Waves at a surface of discontinuity in the atmoshere are analysed using initial value roblem (IPV. So that, sets of continuum modes due to branch cuts in the comlex lane, not treated exlicitly in the literature, exlain the disaearance of the disersion relation. Thus, there will be some itical wave number k such that for k > k ( k < k eigenmodes exist (do not exist. 4. References [] Lamb H 90 Proc. Lond. Math. Soc. A 84 55-7 [] Ott E and Russell D A 978 Physical Review Letters 4 048-5 [3] Russell D A and Ott E 979 Journal of Geohysical Research 84 6573-79. [4] Bogdan T J and Cally P S 997 Proc. R. Soc. Lond. A, Math. Phys. Eng. Sci., 453 943-6 4