# Parabolic Equations for Atmospheric Waves

J.F.Lingevitch,1 M.D.Collins,1 D.K.Dacol,1 D.P.Drob,1 J.C.W.Rogers,2 and W.L.Siegmann3
1Acoustics Division
2Polytechnic University
3Rensselaer Polytechnic Institute

Introduction: The Earth's atmosphere is a complex dynamical system supporting a wide variety of wave phenomena. The study and understanding of such waves is aided by computer simulations in which controlled experiments under ideal conditions can be conducted to isolate physical effects and test hypotheses governing the real atmosphere. Even so, the numerical solution of the equations governing atmospheric waves is computationally expensive and requires simplification of the full hydrodynamic wave equations. A common approach in the study of atmospheric waves is to solve a linearized model about a representative atmospheric state using the methods of ray theory. Ray theory is an efficient technique for high frequencies but becomes less accurate when the length scales of atmospheric features are comparable to the acoustic wavelength. In this article we demonstrate another approach, the method of parabolic equations, for solving atmospheric wave equations. The parabolic equation method is an efficient and accurate solution technique and is not constrained by the asymptotic frequency restrictions of ray theory. We show that the parabolic equation method can be applied to atmospheric acousto-gravity (AG) waves and to acoustic waves in horizontal shear flow. This work improves previous implementations of parabolic equations, which were restricted to narrow propagation angles and low Mach number.

Parabolic Equation Method: The parabolic equation method was pioneered in the 1940s for the study of radio waves in the atmosphere. Since that time, the method has been extended to a wider class of wave phenomena, including ocean acoustics, geoacoustics, electromagnetics, and scattering problems. The method is based on factoring the wave equation into incoming and outgoing components. When one component of the wave dominates, as is often the case for a wave generated by a localized source, the factored equation can be solved orders of magnitude more efficiently than the full elliptic wave equation. This is important when the scale of the computational domain is many acoustic wavelengths. A parabolic equation is efficiently solved by advancing the field in range with a marching algorithm.

Applications to Atmospheric Problems: We apply the parabolic equation method to several problems involving atmospheric waves using a realistic model of the Earth's atmosphere. First, we consider the case of infrasonic AG waves in a stratified atmosphere with no horizontal background flow. A two-dimensional geometry is considered; the vertical coordinate is altitude above a rigid Earth, and the horizontal coordinate is range from an infrasonic acoustic source. The absorption in the upper atmosphere is inversely proportional to molecular density, so energy propagating at high angles is rapidly attenuated.

FIGURE 1
(a)Model atmospheric profiles for density, sound speed, and buoyancy frequency obtained by smoothing tabulated standard atmospheric profiles. (b)Intensity plots of 3-MHz (left) AG and (right) pure gravity waves. The AG wave contains a large amount of energy near the ground due to a contribution of the Lamb wave; dynamic range of the plots is 50 dB.

Figure 1(a) shows the density, sound speed, and buoyancy frequency profiles. A parcel of fluid displaced vertically from its equilibrium position will oscillate at the local buoyancy frequency. Figure 1(b) contains two intensity plots for the 3 MHz source located 10 km above the rigid Earth. The bottom left figure shows an AG wave, and the bottom right figure shows a pure gravity wave in which the compressibility effects are neglected. At infrasonic frequencies close to the buoyancy frequency of the atmosphere, the combined effects of gravity and medium compressibility lead to hybrid AG waves that are significantly different from pure gravity or acoustic waves. The main difference in this case is due to a surface wave that decays exponentially with altitude (a Lamb wave) in the AG wavenumber spectrum that is not present in the gravity wavenumber spectrum.

Parabolic equations are also applicable to problems involving range-varying atmospheric profiles or topography. Range-dependent propagation is handled by enforcing single-scattering continuity conditions on the pressure and displacement at ranges where the profiles are updated. This method accounts for first-order scattering caused by the range dependence. Multiple scattering effects are negligible if the range dependence is sufficiently weak.

FIGURE 2
Intensity plot for a 3 MHz Lamb wave propagating over variable topography.The range-dependence induces mode coupling as can be seen from the modal interference pattern;dynamic range of the plot is 25 dB.

Figure 2 shows an example demonstrating mode coupling resulting from range-dependent propagation over a simulated mountain range. The topographical variations occur at ranges between 100 and 340 km, with a maximum altitude of 3 km. A 3 MHz Lamb wave is incident on a simulated mountain range. Coupling induced by the mountains excites a mode that interferes with the original Lamb wave in the upper atmosphere.

Horizontal shear flow in the atmosphere can significantly affect infrasonic acoustic propagation. The top panel of Fig. 3 shows density, sound speed, and wind speed profiles obtained from measurements and models developed by the Upper Atmospheric Physics group at NRL (Code 7640). The noncommutativity of the shear and acoustic operators in the advected wave equation complicates the derivation of the parabolic equation in this case. An approximate factorization that is accurate to leading order in the commutator is derived from the spectral solution. The bottom panel of Fig. 3 shows the upwind/downwind comparison of the acoustic field and illustrates the dramatic influence of a wind profile on an infrasonic acoustic wave.

FIGURE 3
(a)Representative density, sound speed, and wind speed profiles obtained from measurement and models (top row). Intensity plots for an outward-propagating wave are due to a 0.5 Hz source. (b)The left shows the energy propagating in the upwind direction (to the left) and the right is the energy propagating in the downward direction (to the right). The dynamic range of the plot is 70 dB.

Summary: We have derived parabolic equations for problems involving acoustic and gravity waves in the atmosphere. The effects of horizontal shear have also been incorporated into a wide-angle high-Machnumber parabolic equation. Parabolic equation methods combine accuracy and efficiency for solving rangedependent wave propagation problems and are applicable at infrasonic frequencies where ray methods are inaccurate.

References
1J.F.Lingevitch,M.D.Collins,and W.L.Siegmann, "Parabolic Equations for Gravity and Acousto-gravity Waves," J.Acoust.Soc.Am.105 ,3049-3056 (1999).
2J.F.Lingevitch,M.D.Collins,D.K.Dacol,D.P.Drob, J.C.W.Rogers,and W.L.Siegmann, "High Mach Number Parabolic Equation," J.Acoust.Soc.Am. (in press).