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Parabolic equation modeling of azimuthally advected gravity waves
Wave Motion 31 (2000) 131–138
Parabolic equation modeling of azimuthally advected gravity waves
夽
Michael D. Collins a,∗ , W.A. Kuperman b , B. Edward McDonaldc,1 , William L. Siegmann d b
a Naval Research Laboratory, Washington, DC 20375, USA Scripps Institution of Oceanography, La Jolla, CA 92093, USA c Saclant Undersea Research Centre, La Spezia, Italy d Rensselaer Polytechnic Institute, Troy, NY 12180, USA
Received 17 September 1998; received in revised form 27 June 1999
Abstract An adiabatic mode solution is derived for azimuthally advected gravity waves. Horizontal variations in the medium are assumed to be sufficiently gradual so that the coupling of energy between modes can be neglected. The wind speed is assumed to be small relative to the wave speed. The mode coefficients satisfy the same horizontal wave equation for both gravity and acoustic waves, which satisfy three-dimensional wave equations that are fundamentally different. The horizontal wave equation can be solved efficiently with the parabolic equation method. The adiabatic mode solution is used to model the propagation of gravity waves in the atmosphere of Jupiter. ©2000 Elsevier Science B.V. All rights reserved.
1. Introduction The parabolic equation method [1–3] and the adiabatic mode solution [4–7] are useful for solving nonseparable wave propagation problems when horizontal variations in the properties of the medium are gradual. The parabolic equation method is based on factoring an operator to obtain an outgoing wave equation. The adiabatic mode solution is based on a local separation of variables and the solution of horizontal wave equations for the mode coefficients. Some three-dimensional problems can be solved efficiently using a combination of these techniques in which the parabolic equation method is used to solve the horizontal wave equations [8,9]. For the acoustic problem, leading-order correction terms have been derived to account for horizontal advection [10] and mode coupling [11]. In this paper, we derive an advection term for gravity waves. In Section 2, we derive a wave equation for advected gravity waves. In Section 3, we derive the adiabatic mode solution. In Section 4, we use the adiabatic mode solution to model the propagation of gravity waves in the atmosphere of Jupiter. 夽
Sponsored by the Office of Naval Research. Corresponding author. E-mail address: [email protected] (M.D. Collins). 1 Work done while at the Naval Research Laboratory. ∗
0165-2125/00/$ – see front matter ©2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 2 1 2 5 ( 9 9 ) 0 0 0 4 0 - 2
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M.D. Collins et al. / Wave Motion 31 (2000) 131–138
2. The advected gravity wave equation In this section, we derive an equation for linear gravity waves in an incompressible medium with an ambient flow that can be regarded as a perturbation. We simplify the derivation by neglecting the Coriolis effect, which is small at sufficiently high frequencies. We work in a local Cartesian coordinate system, where x and y are the horizontal coordinates and z is the depth below a reference point. The velocity u = (u, v, w), pressure p, and density ρ satisfy the equations for Newton’s second law, incompressibility, and conservation of mass [12], Du + ∇p = ρg, Dt Dρ = 0, Dt ∇ · u = 0, ρ
(1) (2) (3)
where D/Dt ≡ ∂/∂t + u · ∇ is the material derivative and g ≡ (0, 0, g) is the acceleration due to gravity. We let (u1 , p1 , ρ1 ) be a perturbation to a steady ambient state (u0 , p0 , ρ0 ), which satisfies Eqs. (1)–(3). We assume that horizontal variations in the ambient state are gradual and neglect horizontal derivatives of u0 , p0 , and ρ0 . Approximations of this type are often used in the derivation of parabolic equation techniques. We also assume that w0 = 0 and that |u0 | is small relative to a representative wave speed. Substituting the ambient solution and the perturbation into Eqs. (1)–(3) and retaining linear terms in the perturbation, we obtain ρ0 Dt u1 + ρ0 D t ρ1 +
∂u0 w1 + ∇p1 = ρ1 g, ∂z
∂ρ0 w1 = 0, ∂z
∇ · u1 = 0,
(4) (5) (6)
where Dt ≡ ∂/∂t + u0 · ∇⊥ and ∇⊥ ≡ (∂/∂x, ∂/∂y, 0). Taking the inner product of ∇⊥ and Eq. (4) and applying Eq. (6), we obtain 2 p1 = ρ0 Dt ∇⊥
∂w1 ∂u0 − ρ0 · ∇⊥ w1 . ∂z ∂z
Using Eq. (5) to eliminate ρ1 from the z component of Eq. (4), we obtain ∂p1 = ρ0 Dt2 w1 + N 2 w1 , −Dt ∂z N2 ≡
g ∂ρ0 , ρ0 ∂z
(7)
(8) (9)
where N is the buoyancy frequency. Combining Eqs. (7) and (8), neglecting higher-order terms in the ambient flow, and dropping subscripts, we obtain the gravity wave equation: 2 ∂w ∂u 1 ∂ 1 ∂ ∂ ∂w ∂ 2 2 2 ∇ ρ + N ρ · ∇⊥ . (10) + 2u · ∇ w + ∇ w = ⊥ ⊥ ⊥ 2 ∂t ρ ∂z ∂z ρ ∂z ∂z ∂t ∂t The term on the right-hand side of Eq. (10) can be neglected when u depends weakly on depth. In this case, Eq. (10) is asymptotically equivalent to the equation: 1 ∂ ∂w 2 2 w+ w = 0, (11) ρ + N 2 ∇⊥ Dt2 ∇⊥ ρ ∂z ∂z
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which corresponds to the unperturbed gravity wave equation with Dt in place of ∂/∂t. Transforming Eq. (10) to the frequency domain, we obtain 1 ∂ ∂w iω ∂ ∂u 2 2 2 2 (12) ρ =− ρ · ∇⊥ w. N − ω − 2iωu · ∇⊥ ∇⊥ w − (ω + 2iωu · ∇⊥ ) ρ ∂z ∂z ρ ∂z ∂z It follows from the unperturbed gravity wave equation: ∂w ω2 1 ∂ 2 ρ = 0, w− 2 ∇⊥ ∂z N − ω2 ρ ∂z that Eq. (12) is asymptotically equivalent to the equation: ω2 ∂w 2iωN 2 u · ∇⊥ 1 ∂ ∂w iω ∂u 1 ∂ 1 ∂ 2 w− 2 ρ = ρ − ρ · ∇⊥ w. ∇⊥ 2 ∂z ∂z ∂z N − ω2 ρ ∂z N 2 − ω2 ρ ∂z N 2 − ω2 ρ ∂z
(13)
(14)
3. The adiabatic mode solution In this section, we derive the adiabatic mode solution, which is based on the assumption of gradual horizontal variations in the properties of the medium. The derivation involves the neglect of energy coupling between modes and terms that involve horizontal derivatives of the modes or properties of the medium. We first consider the unperturbed case. Expanding the solution of Eq. (13) in terms of the local modes and neglecting mode coupling, we obtain X wn (x, y)φn (z; x, y), (15) w= n
∂φn 1 ∂ ω2 ρ + kn2 (x, y)φn = 0, ∂z N 2 − ω2 ρ ∂z 2 wn + kn2 wn = 0, ∇⊥
where the kn2 are eigenvalues and the modes φn are orthonormal with respect to the inner product, Z ρ N 2 − ω2 φm φn dz = δmn . ω2
(16) (17)
(18)
The mode coefficients wn satisfy initial conditions at (x, y) = (0, 0) that depend on the nature of the source. It is practical to solve Eq. (17), the horizontal wave equation, for some global-scale problems [9,10]. It is relatively difficult to solve three-dimensional wave equations such as Eq. (13) even over relatively small regions. The modal phase speed is defined by cn = ω/kn . We obtain an expression for the modal group speed Cn using a perturbation approach that is based on the approximations: kn (ω + ␦ω) ∼ kn (ω) + ␦kn ,
(19)
φn (ω + ␦ω) ∼ φn (ω) + ␦φn .
(20)
Substituting Eqs. (19) and (20) into Eq. (16) and neglecting higher-order terms, we obtain 2kn2 N 2 ␦ω ∂ ␦φn 1 ∂ ω2 2 φn − 2kn ␦kn φn . ␦ φ = ρ + k n n ∂z N 2 − ω2 ρ ∂z ω N 2 − ω2
(21)
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A bounded solution of Eq. (21) exists if the right-hand side is orthogonal to φn . To enforce this solvability condition [13], we apply Eq. (18) and obtain Z 1 ρN 2 2 ␦kn 1 = ≡ φ dz. (22) Cn ␦ω cn ω2 n For the perturbed case, the adiabatic mode solution is of the form: X wn (x, y)ψn (z; x, y), w=
(23)
n
2 wn ∼ − kn2 + Ln wn , ∇⊥
(24)
ψn ∼ φn + ␦φn ,
(25)
where |␦φn | |φn | and Ln is an operator such that |Ln wn | |kn2 wn |. Substituting Eq. (23) into Eq. (14) and applying Eqs. (16), (24) and (25), we obtain 1 ∂ ∂ ␦φn ω2 2 ρ + kn ␦φn wn ∂z N 2 − ω2 ρ ∂z iω ∂u 1 ∂ 2ikn2 N 2 φn u · ∇⊥ wn + 2 (26) ρ · ∇⊥ wn − φn Ln wn . = ∂z N − ω2 ρ ∂z ω N 2 − ω2 Applying a solvability condition to the right-hand side of Eq. (26), we obtain Ln wn = 2ikn Fn · ∇⊥ wn , Z ∂u ρN 2 u cn ∂ ρ φn2 dz, + Fn ≡ ∂z cn ω 2 2ω2 ∂z
(27) (28)
where Fn is the modal Froude number. From Eqs. (22) and (28), we conclude that Fn = u/Cn when the depth dependence of u can be neglected over the support of φn . It makes sense that the modal group speed appears in the expression for Fn in this case since advection translates energy. Substituting Eq. (27) into Eq. (24), we obtain the horizontal wave equation: 2 wn + 2ikn Fn · ∇⊥ wn + kn2 wn = 0. ∇⊥
(29)
The adiabatic mode solutions for gravity and acoustic waves are closely related. The horizontal wave equation for the acoustic problem is identical to Eq. (29), with cn and Cn corresponding to the acoustic modal phase and group speeds, the modal Mach number Mn in place of Fn , and Mn = u/Cn when the depth dependence of u can be neglected [10]. As described in [10], the parabolic equation method may be used to efficiently generate the solution of Eq. (29) on the surface of a sphere. This approach involves factoring the operator into a product of incoming and outgoing operators, where the outgoing direction is toward the antipode of the source position, and assuming that the outgoing energy dominates to obtain an outgoing wave equation. The outgoing wave equation is then solved by approximating the square root of an operator with a rational function and applying numerical techniques. The solution is obtained by marching from the source location to the antipode. Since energy eventually turns around and becomes incoming, the solution breaks down in a neighborhood of the antipode that can be relatively large when refraction is strong.
4. Application to the Jovian problem The impact of Comet Shoemaker-Levy 9 motivated the modeling of several types of wave propagation in the atmosphere of Jupiter, including large-scale oscillations [14,15], seismic waves [16–18], gravity waves [19], and
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Fig. 1. Wind speed as a function of latitude that was obtained by extrapolating Voyager data [23] to the poles as described in [10].
acoustic waves [10]. There is an evidence of both gravity waves [20,21] and acoustic waves [10,22] near the impact sites. The adiabatic mode solution can be used to determine promising locations to search the data for evidence of waves far from the impact sites. Appearing in Fig. 1 are the zonal winds [23], which give rise to horizontal caustics in which acoustic energy is enhanced by an order of magnitude [10]. In this section, we use the adiabatic mode solution to map out the caustics for gravity waves. Although compressibility is an important effect for some atmospheric modes, the idealized gravity wave model should provide representative results for many modes. We assume that the relevant modes are concentrated near the cloud decks, where the zonal winds are known. Such modes have been proposed as an explanation for the observed waves [21]. It was hoped that data from the Galileo probe [24,25] would confirm this hypothesis, but it turned out that the evidence was inconclusive since the probe descended in a relatively dry region of the atmosphere. We consider a problem involving a 0.002 Hz gravity wave source at the impact latitude of 44◦ S. We selected a modal group speed of 450 m/s, which was measured by Hammel et al. [8], and a modal phase speed of 500 m/s. Although the value of the modal phase speed is uncertain, an exact value is not required to determine qualitative behavior. We also consider the 0.02 Hz acoustic problem of [10], which involves a modal group speed of 908.5 m/s and a modal phase speed of 1009.0 m/s. Solutions of these problems appear in Fig. 2. Caustics exist along intense beams that follow nearly great circle paths for both waves. The beams are formed by wind shear in the cells between reversals in the zonal winds. Since the Froude number is about twice the Mach number, the gravity wave beams meander more than the acoustic beams as they cross the strong equatorial winds. One of the gravity wave beams is trapped in a wind cell and winds around the planet at the impact latitude. Similar trapping of gravity waves has been modeled in Neptune’s wind cells [26]. Since caustics are formed by geometrical effects, they occur for a wide range of frequencies, modal phase speeds, and modal group speeds. For modes trapped near the cloud decks, they should be the most prominent features in the gravity wave and acoustic fields far from the impact sites. If the beams are detected in some data set, they might provide information regarding the vertical structure of the zonal winds because the details of their appearance depends on this information. The most promising places to search for evidence of waves are the intersections of
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Fig. 2. The horizontal distribution of energy in a 0.002 Hz gravity wave mode (top) and a 0.02 Hz acoustic mode (bottom) in the atmosphere of Jupiter. The impact site is to the lower right (the impact longitude is 45◦ from the limb). The dynamic range is 20 dB, with black corresponding to high intensity and white corresponding to low intensity. The intense beams are formed by horizontal refraction due to wind shear. The energy in the beams is an order of magnitude greater than the energy in the neighboring areas. The gravity wave solution contains a caustic that is trapped in a wind cell near the impact latitude. A wave propagating far from an impact site would appear nearly circular but would have relatively large amplitude at the intersections with the beams.
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the caustics with circles corresponding to the group speeds of gravity and acoustic waves. Although factors such as the visibility of waves outside the debris field are important, there is one factor that favors detecting waves far from the impact sites. Evidence of waves was observed in images that were obtained more than 2 h after impact. The caustics are well formed after about 10 h for acoustic waves [10] and 20 h for gravity waves. The cylindrical spreading of energy that occurs between 2 and 20 h is counterbalanced by the enhancement (by about an order of magnitude) of energy in the caustics.
5. Conclusion We have derived an adiabatic mode solution for azimuthally advected gravity waves. The same horizontal wave equation applies to both gravity and acoustic waves, which satisfy three-dimensional wave equations that are fundamentally different. The horizontal wave equations can be solved efficiently with the parabolic equation method. The adiabatic mode solution was used to model gravity wave propagation in the atmosphere of Jupiter. The zonal winds give rise to horizontal caustics, one of which gets trapped in a wind cell. The intersections of these features and circles corresponding to the group speeds of acoustic and gravity waves are the most promising locations to search for waves far from the impact sites. Possible applications of the adiabatic mode solution include analyzing gravity waves excited by storms in the atmosphere of Jupiter and other planets. Possible directions for extending this work include deriving a mode coupling correction and generalizing to acousto-gravity waves. References [1] M.A. Leontovich, V.A. Fock, Solution of the problem of propagation of electromagnetic waves along the earth’s surface by the method of parabolic equation, J. Exp. Theor. Phys. 16 (1946) 557–573. [2] F.D. Tappert, The parabolic approximation method, in: J.B. Keller, J.S. Papadakis (Eds.), Wave Propagation and Underwater Acoustics, Lecture Notes in Physics, vol. 70, Springer, New York, 1977, pp. 224–280. [3] F.B. Jensen, W.A. Kuperman, M.B. Porter, H. Schmidt, Computational Ocean Acoustics, American Institute of Physics, New York, 1994, pp. 343–412. [4] A.D. Pierce, Extension of the method of normal modes to sound propagation in an almost-stratified medium, J. Acoust. Soc. Amer. 37 (1965) 19–27. [5] D.M. Milder, Ray and wave invariants for SOFAR channel propagation, J. Acoust. Soc. Amer. 46 (1969) 1259–1263. [6] H. Weinberg, R. Burridge, Horizontal ray theory for ocean acoustics, J. Acoust. Soc. Amer. 55 (1974) 63–79. [7] W.A. Kuperman, M.B. Porter, J.S. Perkins, R.B. Evans, Rapid computation of acoustic fields in three-dimensional ocean environments, J. Acoust. Soc. Amer. 89 (1991) 125–133. [8] M.D. Collins, The adiabatic mode parabolic equation, J. Acoust. Soc. Amer. 94 (1993) 2269–2278. [9] M.D. Collins, B.E. McDonald, K.D. Heaney, W.A. Kuperman, Three-dimensional effects in global acoustics, J. Acoust. Soc. Amer. 97 (1995) 1567–1575. [10] M.D. Collins, B.E. McDonald, W.A. Kuperman, W.L. Siegmann, Jovian acoustics and Comet Shoemaker-Levy 9, J. Acoust. Soc. Amer. 97 (1995) 2147–2158. [11] A.T. Abawi, W.A. Kuperman, M.D. Collins, The coupled mode parabolic equation, J. Acoust. Soc. Amer. 102 (1997) 233–238. [12] A.E. Gill, Atmosphere–Ocean Dynamics, Academic Press, San Diego, 1982, pp. 128–130. [13] A.H. Nayfeh, Introduction to Perturbation Techniques, Wiley-Interscience, New York, 1981, pp. 412–414. [14] J. Harrington, R.P. LeBeau, K.A. Backes, T.E. Dowling, Dynamic response of Jupiter’s atmosphere to the impact of Comet Shoemaker-Levy 9, Nature 368 (1994) 525–527. [15] U. Lee, H.M. Van Horn, Global oscillation amplitudes excited by the Jupiter–Comet collision, Astrophys. J. Lett. 428 (1994) 41–44. [16] M.S. Marley, Seismological consequences of the collision of Shoemaker-Levy/9 with Jupiter, Astrophys. J. Lett. 427 (1994) 63–66. [17] D.M. Hunten, W.F. Hoffman, A.L. Sprague, Jovian seismic waves and their detection, Geophys. Res. Lett. 21 (1994) 1091–1094. [18] D. Deming, Prospects for Jovian seismological observations following the impact of comet Shoemaker-Levy 9, Geophys. Res. Lett. 21 (1994) 1095–1098. [19] A.P. Ingersoll, H. Kanamori, T.E. Dowling, Atmospheric gravity waves from the impact of Comet Shoemaker-Levy 9 with Jupiter, Geophys. Res. Lett. 21 (1994) 1091–1094. [20] H.B. Hammel, R.F. Beebe, A.P. Ingersoll, G.S. Orton, J.R. Mills, A.A. Simon, P. Chodas, J.T. Clarke, E. De Jong, T.E. Dowling, J. Harrington, L.F. Huber, E. Karkoschka, C.M. Santori, A. Toigo, D. Yeomans, R.A. West, HST imaging of atmospheric phenomena created by the impact of Comet Shoemaker-Levy 9, Science 267 (1995) 1288–1296.
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[21] A.P. Ingersoll, H. Kanamori, Waves from the collisions of comet Shoemaker-Levy 9 with Jupiter, Nature 374 (1995) 706–708. [22] M.D. Collins, B.E. McDonald, W.A. Kuperman, W.L. Siegmann, Jovian acoustic matched-field processing, J. Acoust. Soc. Amer. 102 (1997) 2487–2493. [23] S.S. Limaye, Jupiter: new estimates of the mean zonal flow at the cloud level, Icarus 65 (1986) 335–352. [24] A. Seiff, D.B. Kirk, C.D. Knight, R.E. Young, J.D. Mihalov, L.A. Young, F.S. Milos, G. Schubert, R.C. Blanchard, D. Atkinson, Thermal structure of Jupiter’s atmosphere near the edge of a 5 mm hot spot in the north equatorial belt, J. Geophys. Res. 103 (1998) 22857–22889. [25] D.H. Atkinson, J.B. Pollack, A. Seiff, The Galileo Probe Doppler wind experiment: measurement of the deep zonal winds on Jupiter, J. Geophys. Res. 103 (1998) 22911–22928. [26] T.E. Dowling, Personal communication.
Parabolic equation modeling of azimuthally advected gravity waves
夽
Michael D. Collins a,∗ , W.A. Kuperman b , B. Edward McDonaldc,1 , William L. Siegmann d b
a Naval Research Laboratory, Washington, DC 20375, USA Scripps Institution of Oceanography, La Jolla, CA 92093, USA c Saclant Undersea Research Centre, La Spezia, Italy d Rensselaer Polytechnic Institute, Troy, NY 12180, USA
Received 17 September 1998; received in revised form 27 June 1999
Abstract An adiabatic mode solution is derived for azimuthally advected gravity waves. Horizontal variations in the medium are assumed to be sufficiently gradual so that the coupling of energy between modes can be neglected. The wind speed is assumed to be small relative to the wave speed. The mode coefficients satisfy the same horizontal wave equation for both gravity and acoustic waves, which satisfy three-dimensional wave equations that are fundamentally different. The horizontal wave equation can be solved efficiently with the parabolic equation method. The adiabatic mode solution is used to model the propagation of gravity waves in the atmosphere of Jupiter. ©2000 Elsevier Science B.V. All rights reserved.
1. Introduction The parabolic equation method [1–3] and the adiabatic mode solution [4–7] are useful for solving nonseparable wave propagation problems when horizontal variations in the properties of the medium are gradual. The parabolic equation method is based on factoring an operator to obtain an outgoing wave equation. The adiabatic mode solution is based on a local separation of variables and the solution of horizontal wave equations for the mode coefficients. Some three-dimensional problems can be solved efficiently using a combination of these techniques in which the parabolic equation method is used to solve the horizontal wave equations [8,9]. For the acoustic problem, leading-order correction terms have been derived to account for horizontal advection [10] and mode coupling [11]. In this paper, we derive an advection term for gravity waves. In Section 2, we derive a wave equation for advected gravity waves. In Section 3, we derive the adiabatic mode solution. In Section 4, we use the adiabatic mode solution to model the propagation of gravity waves in the atmosphere of Jupiter. 夽
Sponsored by the Office of Naval Research. Corresponding author. E-mail address: [email protected] (M.D. Collins). 1 Work done while at the Naval Research Laboratory. ∗
0165-2125/00/$ – see front matter ©2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 2 1 2 5 ( 9 9 ) 0 0 0 4 0 - 2
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2. The advected gravity wave equation In this section, we derive an equation for linear gravity waves in an incompressible medium with an ambient flow that can be regarded as a perturbation. We simplify the derivation by neglecting the Coriolis effect, which is small at sufficiently high frequencies. We work in a local Cartesian coordinate system, where x and y are the horizontal coordinates and z is the depth below a reference point. The velocity u = (u, v, w), pressure p, and density ρ satisfy the equations for Newton’s second law, incompressibility, and conservation of mass [12], Du + ∇p = ρg, Dt Dρ = 0, Dt ∇ · u = 0, ρ
(1) (2) (3)
where D/Dt ≡ ∂/∂t + u · ∇ is the material derivative and g ≡ (0, 0, g) is the acceleration due to gravity. We let (u1 , p1 , ρ1 ) be a perturbation to a steady ambient state (u0 , p0 , ρ0 ), which satisfies Eqs. (1)–(3). We assume that horizontal variations in the ambient state are gradual and neglect horizontal derivatives of u0 , p0 , and ρ0 . Approximations of this type are often used in the derivation of parabolic equation techniques. We also assume that w0 = 0 and that |u0 | is small relative to a representative wave speed. Substituting the ambient solution and the perturbation into Eqs. (1)–(3) and retaining linear terms in the perturbation, we obtain ρ0 Dt u1 + ρ0 D t ρ1 +
∂u0 w1 + ∇p1 = ρ1 g, ∂z
∂ρ0 w1 = 0, ∂z
∇ · u1 = 0,
(4) (5) (6)
where Dt ≡ ∂/∂t + u0 · ∇⊥ and ∇⊥ ≡ (∂/∂x, ∂/∂y, 0). Taking the inner product of ∇⊥ and Eq. (4) and applying Eq. (6), we obtain 2 p1 = ρ0 Dt ∇⊥
∂w1 ∂u0 − ρ0 · ∇⊥ w1 . ∂z ∂z
Using Eq. (5) to eliminate ρ1 from the z component of Eq. (4), we obtain ∂p1 = ρ0 Dt2 w1 + N 2 w1 , −Dt ∂z N2 ≡
g ∂ρ0 , ρ0 ∂z
(7)
(8) (9)
where N is the buoyancy frequency. Combining Eqs. (7) and (8), neglecting higher-order terms in the ambient flow, and dropping subscripts, we obtain the gravity wave equation: 2 ∂w ∂u 1 ∂ 1 ∂ ∂ ∂w ∂ 2 2 2 ∇ ρ + N ρ · ∇⊥ . (10) + 2u · ∇ w + ∇ w = ⊥ ⊥ ⊥ 2 ∂t ρ ∂z ∂z ρ ∂z ∂z ∂t ∂t The term on the right-hand side of Eq. (10) can be neglected when u depends weakly on depth. In this case, Eq. (10) is asymptotically equivalent to the equation: 1 ∂ ∂w 2 2 w+ w = 0, (11) ρ + N 2 ∇⊥ Dt2 ∇⊥ ρ ∂z ∂z
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which corresponds to the unperturbed gravity wave equation with Dt in place of ∂/∂t. Transforming Eq. (10) to the frequency domain, we obtain 1 ∂ ∂w iω ∂ ∂u 2 2 2 2 (12) ρ =− ρ · ∇⊥ w. N − ω − 2iωu · ∇⊥ ∇⊥ w − (ω + 2iωu · ∇⊥ ) ρ ∂z ∂z ρ ∂z ∂z It follows from the unperturbed gravity wave equation: ∂w ω2 1 ∂ 2 ρ = 0, w− 2 ∇⊥ ∂z N − ω2 ρ ∂z that Eq. (12) is asymptotically equivalent to the equation: ω2 ∂w 2iωN 2 u · ∇⊥ 1 ∂ ∂w iω ∂u 1 ∂ 1 ∂ 2 w− 2 ρ = ρ − ρ · ∇⊥ w. ∇⊥ 2 ∂z ∂z ∂z N − ω2 ρ ∂z N 2 − ω2 ρ ∂z N 2 − ω2 ρ ∂z
(13)
(14)
3. The adiabatic mode solution In this section, we derive the adiabatic mode solution, which is based on the assumption of gradual horizontal variations in the properties of the medium. The derivation involves the neglect of energy coupling between modes and terms that involve horizontal derivatives of the modes or properties of the medium. We first consider the unperturbed case. Expanding the solution of Eq. (13) in terms of the local modes and neglecting mode coupling, we obtain X wn (x, y)φn (z; x, y), (15) w= n
∂φn 1 ∂ ω2 ρ + kn2 (x, y)φn = 0, ∂z N 2 − ω2 ρ ∂z 2 wn + kn2 wn = 0, ∇⊥
where the kn2 are eigenvalues and the modes φn are orthonormal with respect to the inner product, Z ρ N 2 − ω2 φm φn dz = δmn . ω2
(16) (17)
(18)
The mode coefficients wn satisfy initial conditions at (x, y) = (0, 0) that depend on the nature of the source. It is practical to solve Eq. (17), the horizontal wave equation, for some global-scale problems [9,10]. It is relatively difficult to solve three-dimensional wave equations such as Eq. (13) even over relatively small regions. The modal phase speed is defined by cn = ω/kn . We obtain an expression for the modal group speed Cn using a perturbation approach that is based on the approximations: kn (ω + ␦ω) ∼ kn (ω) + ␦kn ,
(19)
φn (ω + ␦ω) ∼ φn (ω) + ␦φn .
(20)
Substituting Eqs. (19) and (20) into Eq. (16) and neglecting higher-order terms, we obtain 2kn2 N 2 ␦ω ∂ ␦φn 1 ∂ ω2 2 φn − 2kn ␦kn φn . ␦ φ = ρ + k n n ∂z N 2 − ω2 ρ ∂z ω N 2 − ω2
(21)
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A bounded solution of Eq. (21) exists if the right-hand side is orthogonal to φn . To enforce this solvability condition [13], we apply Eq. (18) and obtain Z 1 ρN 2 2 ␦kn 1 = ≡ φ dz. (22) Cn ␦ω cn ω2 n For the perturbed case, the adiabatic mode solution is of the form: X wn (x, y)ψn (z; x, y), w=
(23)
n
2 wn ∼ − kn2 + Ln wn , ∇⊥
(24)
ψn ∼ φn + ␦φn ,
(25)
where |␦φn | |φn | and Ln is an operator such that |Ln wn | |kn2 wn |. Substituting Eq. (23) into Eq. (14) and applying Eqs. (16), (24) and (25), we obtain 1 ∂ ∂ ␦φn ω2 2 ρ + kn ␦φn wn ∂z N 2 − ω2 ρ ∂z iω ∂u 1 ∂ 2ikn2 N 2 φn u · ∇⊥ wn + 2 (26) ρ · ∇⊥ wn − φn Ln wn . = ∂z N − ω2 ρ ∂z ω N 2 − ω2 Applying a solvability condition to the right-hand side of Eq. (26), we obtain Ln wn = 2ikn Fn · ∇⊥ wn , Z ∂u ρN 2 u cn ∂ ρ φn2 dz, + Fn ≡ ∂z cn ω 2 2ω2 ∂z
(27) (28)
where Fn is the modal Froude number. From Eqs. (22) and (28), we conclude that Fn = u/Cn when the depth dependence of u can be neglected over the support of φn . It makes sense that the modal group speed appears in the expression for Fn in this case since advection translates energy. Substituting Eq. (27) into Eq. (24), we obtain the horizontal wave equation: 2 wn + 2ikn Fn · ∇⊥ wn + kn2 wn = 0. ∇⊥
(29)
The adiabatic mode solutions for gravity and acoustic waves are closely related. The horizontal wave equation for the acoustic problem is identical to Eq. (29), with cn and Cn corresponding to the acoustic modal phase and group speeds, the modal Mach number Mn in place of Fn , and Mn = u/Cn when the depth dependence of u can be neglected [10]. As described in [10], the parabolic equation method may be used to efficiently generate the solution of Eq. (29) on the surface of a sphere. This approach involves factoring the operator into a product of incoming and outgoing operators, where the outgoing direction is toward the antipode of the source position, and assuming that the outgoing energy dominates to obtain an outgoing wave equation. The outgoing wave equation is then solved by approximating the square root of an operator with a rational function and applying numerical techniques. The solution is obtained by marching from the source location to the antipode. Since energy eventually turns around and becomes incoming, the solution breaks down in a neighborhood of the antipode that can be relatively large when refraction is strong.
4. Application to the Jovian problem The impact of Comet Shoemaker-Levy 9 motivated the modeling of several types of wave propagation in the atmosphere of Jupiter, including large-scale oscillations [14,15], seismic waves [16–18], gravity waves [19], and
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Fig. 1. Wind speed as a function of latitude that was obtained by extrapolating Voyager data [23] to the poles as described in [10].
acoustic waves [10]. There is an evidence of both gravity waves [20,21] and acoustic waves [10,22] near the impact sites. The adiabatic mode solution can be used to determine promising locations to search the data for evidence of waves far from the impact sites. Appearing in Fig. 1 are the zonal winds [23], which give rise to horizontal caustics in which acoustic energy is enhanced by an order of magnitude [10]. In this section, we use the adiabatic mode solution to map out the caustics for gravity waves. Although compressibility is an important effect for some atmospheric modes, the idealized gravity wave model should provide representative results for many modes. We assume that the relevant modes are concentrated near the cloud decks, where the zonal winds are known. Such modes have been proposed as an explanation for the observed waves [21]. It was hoped that data from the Galileo probe [24,25] would confirm this hypothesis, but it turned out that the evidence was inconclusive since the probe descended in a relatively dry region of the atmosphere. We consider a problem involving a 0.002 Hz gravity wave source at the impact latitude of 44◦ S. We selected a modal group speed of 450 m/s, which was measured by Hammel et al. [8], and a modal phase speed of 500 m/s. Although the value of the modal phase speed is uncertain, an exact value is not required to determine qualitative behavior. We also consider the 0.02 Hz acoustic problem of [10], which involves a modal group speed of 908.5 m/s and a modal phase speed of 1009.0 m/s. Solutions of these problems appear in Fig. 2. Caustics exist along intense beams that follow nearly great circle paths for both waves. The beams are formed by wind shear in the cells between reversals in the zonal winds. Since the Froude number is about twice the Mach number, the gravity wave beams meander more than the acoustic beams as they cross the strong equatorial winds. One of the gravity wave beams is trapped in a wind cell and winds around the planet at the impact latitude. Similar trapping of gravity waves has been modeled in Neptune’s wind cells [26]. Since caustics are formed by geometrical effects, they occur for a wide range of frequencies, modal phase speeds, and modal group speeds. For modes trapped near the cloud decks, they should be the most prominent features in the gravity wave and acoustic fields far from the impact sites. If the beams are detected in some data set, they might provide information regarding the vertical structure of the zonal winds because the details of their appearance depends on this information. The most promising places to search for evidence of waves are the intersections of
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Fig. 2. The horizontal distribution of energy in a 0.002 Hz gravity wave mode (top) and a 0.02 Hz acoustic mode (bottom) in the atmosphere of Jupiter. The impact site is to the lower right (the impact longitude is 45◦ from the limb). The dynamic range is 20 dB, with black corresponding to high intensity and white corresponding to low intensity. The intense beams are formed by horizontal refraction due to wind shear. The energy in the beams is an order of magnitude greater than the energy in the neighboring areas. The gravity wave solution contains a caustic that is trapped in a wind cell near the impact latitude. A wave propagating far from an impact site would appear nearly circular but would have relatively large amplitude at the intersections with the beams.
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the caustics with circles corresponding to the group speeds of gravity and acoustic waves. Although factors such as the visibility of waves outside the debris field are important, there is one factor that favors detecting waves far from the impact sites. Evidence of waves was observed in images that were obtained more than 2 h after impact. The caustics are well formed after about 10 h for acoustic waves [10] and 20 h for gravity waves. The cylindrical spreading of energy that occurs between 2 and 20 h is counterbalanced by the enhancement (by about an order of magnitude) of energy in the caustics.
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