Abstract
The propagation of acoustic and gravity waves in planetary atmospheres is strongly dependent on both wind conditions and attenuation properties. This study presents a finite-difference modeling tool tailored for acoustic-gravity wave applications that takes into account the effect of background winds, attenuation phenomena (including relaxation effects specific to carbon dioxide atmospheres) and wave amplification by exponential density decrease with height. The simulation tool is implemented in 2D Cartesian coordinates and first validated by comparison with analytical solutions for benchmark problems. It is then applied to surface explosions simulating meteor impacts on Mars in various Martian atmospheric conditions inferred from global climate models. The acoustic wave travel times are validated by comparison with 2D ray tracing in a windy atmosphere. Our simulations predict that acoustic waves generated by impacts can refract back to the surface on wind ducts at high altitude. In addition, due to the strong nighttime near-surface temperature gradient on Mars, the acoustic waves are trapped in a waveguide close to the surface, which allows a night-side detection of impacts at large distances in Mars plains. Such theoretical predictions are directly applicable to future measurements by the INSIGHT NASA Discovery mission.
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Acknowledgements
We acknowledge Don Banfield and an anonymous reviewer for their detailed review of the manuscript. We thank the INSIGHT science team for fruitful discussions. We also thank the “Région Midi-Pyréenées” (France) and “Université fédérale de Toulouse” for funding the PhD grant of Quentin Brissaud. This study was also supported by CNES through space research scientific projects. Computer resources were provided by granted projects No. p1138 at CALMIP computing centre (Toulouse France), Nos. t2014046351 and t2015046351 at CEA centre (Bruyères, France). This is Insight Contribution Number 16.
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Appendices
Appendix 1: Analytical Solution for Acoustic Wave Propagation in a Windy Homogeneous Atmosphere with Vibrational Relaxation
In the framework of acoustic wave propagation in a windy homogeneous medium with vibrational relaxation, the state and momentum equations read
where, \(\forall t \in \mathbb{R}^{+}\)
with \(H\) the Heaviside function and \(\psi \) the relaxation function of the viscoelastic medium considered.
Taking the divergence of (22)-2 yields
where \(\Delta \) is the Laplacian operator. Thus, taking the total derivative \(D_{t}\) (where \(D_{t} = \partial_{t} + \mathbf{v}\nabla \)) of (22)-1 yields
By considering, in (25), plane wave solutions of the form \(p = \tilde{p}e^{-i\omega t}\) and \(\varPsi \), defined in (16), the relaxation function for a Zener mechanism (Carcione et al. 1988), we can perform the complex integration of the convolution term
then, from the plane wave form in the Fourier space \((\partial_{t} = -i \omega )\), and one can recast (25) as
The tilde denotes the time Fourier transform and \(\hat{K_{c}}\) is the complex bulk modulus, in the case of a single relaxation mechanism, which reads (Carcione et al. 1988)
where the complex velocity \(\hat{c}\) reads
The complex equation (27) can then be expanded as
with \(\hat{M}\) the Mach number, that is,
where \(\hat{c}\) is the complex velocity and \(k\) is the wavenumber \(\hat{k} = \frac{\omega }{\hat{c}}\). The hat over \(\hat{M}\) and \(\hat{k}\) is introduced to differentiate from the usual Mach and wave numbers in an inviscid medium.
Finally, we will consider
where \(\delta_{x}, \delta_{z}\) are the Kronecker delta symbols, and \(\tilde{A}\) is the Fourier transform of the amplitude of the time-dependent source function, that is,
We use the same method as in Ostashev et al. (2005) to obtain the pressure response in this framework, that is
where \((H^{1}_{i})_{i=1,2}\) are the Hankel functions of first kind, and \(\xi \) reads
Using Hankel function asymptotic (Abramowitz and Stegun 1964) by considering the approximation \(|\tilde{k}R| >> 1\), yields
Appendix 2: Numerical Implementation of ADE-PML for a Medium Subject to Gravity
This boundary condition corresponds to mesh stretching in the boundary layer which requires to recast the derivatives in the new stretched set of complex-valued coordinates (Komatitsch and Martin 2007). We do this using ADE-PML memory variables. In this new reference frame (19) read
with
Note that we wrote the pressure equation (37)-1 in terms of the ambient pressure \(p_{0}\) rather than in terms of its value coming from hydrostatic equilibrium (i.e. in the interior domain \(\nabla p_{0} = \rho \mathbf{g}\)). Indeed, as we stretch coordinates the hydrostatic equilibrium equation is also modified as
In this new reference frame derivatives will read, for any unknown \(\phi \)
where the memory variable is denoted \(Q_{x,z}^{\phi }\). Memory variables obey a first-order in time differential equation for \(x\)-derivatives and \(z\)-derivatives. We will only give details for the \(x-\)derivative since it is similar for the \(z\)-derivatives. We consider the four time sub-steps \((t^{n,i})_{i=1,4}\) as \(t^{n,i} = \Delta t (n+\nu_{i})\) for the four RK4 stages where \(\nu_{i} = 0, \nu_{2} = 0.5, \nu_{3} = 0.5\) and \(\nu_{4} = 1\). Then the memory variable \(Q_{x}^{\phi }\) obey the following differential equation
where
with
where the \(\theta \) parameter indicates if the computation is performed explicitly (\(\theta = 0\)), semi-implicitly (\(\theta = 0.5\)) or implicitly (\(\theta = 1\)). Here this parameter is taken as \(\theta = 0.5\). Since we are mainly interested in damping acoustic waves (most of the waves reaching the boundary will be acoustic waves) we take the same parameters as in Martin et al. (2010), meaning that
where \(f_{0}\) is the dominant frequency of the source, \(c_{max,x}\) the maximum effective velocity along \(x\) (sum of background flow velocity and adiabatic sound speed) and \(R_{c}\) the theoretical reflection coefficient taken as \(R_{c} = 0.0001\). The same formulation holds for derivatives along \(z\).
Appendix 3: “Image” Boundary Conditions
This boundary condition consists in duplicating the physical domain and sources at ghost points located below the bottom boundary in order to obtain a wave interference ensuring zero vertical velocity at the boundary (Morse and Ingard 1968, Sect. 7.4). One enforces that the pressure is equal on each side of the boundary
and the sign of gravity and wind is changed in the ghost domain as
while all other unknowns are duplicated from the real domain to the ghost one
As a consequence, the pressure being equal on both sides of the boundary, the pressure gradient in the momentum equation is zero at the boundary. Also, setting opposite gravity directions above and below the boundary leads to zero gravity at the surface, hence the gravity term in the momentum equation vanishes. Therefore, at the boundary, the momentum equation is verified since the zero velocity condition is imposed. Finally, we impose the boundary to be motionless and we choose
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Garcia, R.F., Brissaud, Q., Rolland, L. et al. Finite-Difference Modeling of Acoustic and Gravity Wave Propagation in Mars Atmosphere: Application to Infrasounds Emitted by Meteor Impacts. Space Sci Rev 211, 547–570 (2017). https://doi.org/10.1007/s11214-016-0324-6
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DOI: https://doi.org/10.1007/s11214-016-0324-6