Energy Envelope and Attenuation Characteristics of High-Frequency (HF) and Very-High-Frequency (VF) Martian Events

ABSTRACT Since its deployment at the surface of Mars, the Seismic Experiment for Interior Structure (SEIS) instrument of the InSight mission has detected hundreds of small-magnitude seismic events. In this work, we highlight some features of two specific families: high-frequency (HF) and very-high-frequency (VF) events. We characterize the shape of the energy envelopes of HF and VF events with two parameters: (1) the delay time td between the onset and the peak of the dominant arrival; and (2) the quality factor Qc, which quantifies the energy decay rate in the coda. We observe that the envelope of HF and VF events is frequency independent. As a consequence, a single delay time suffices to characterize envelope broadening in the 2.5–7.5 Hz band. The typical coda decay time is also frequency independent, as attested by the close to linear increase of Qc with frequency. Finally, we use elastic radiative transfer theory to perform a series of inversion of seismogram envelopes for the attenuation properties of the Martian lithosphere. The good fit between synthetic and observed envelopes confirms that multiple scattering of elastic waves released by internal sources is a plausible explanation of the events characteristics. We quantify scattering and attenuation properties of Mars and highlight the differences and similarities with the Earth and the Moon. The albedo, that is, the contribution of scattering to the total attenuation, derived from VF events is very high, which we interpret as a signature of a mostly dry medium. Our results also suggest a stratification of the scattering and attenuation properties.

A Note on Stokes Approximations to Leray Solutions of the Incompressible Navier–Stokes Equations in ℝn

In the early 1980s it was well established that Leray solutions of the unforced Navier–Stokes equations in Rn decay in energy norm for large t. With the works of T. Miyakawa, M. Schonbek and others it is now known that the energy decay rate cannot in general be any faster than t−(n+2)/4 and is typically much slower. In contrast, we show in this note that, given an arbitrary Leray solution u(·,t), the difference of any two Stokes approximations to the Navier–Stokes flow u(·,t) will always decay at least as fast as t−(n+2)/4, no matter how slow the decay of ∥u(·,t)∥L2(Rn) might be.

A Note on Stokes Approximations to Leray Solutions of the Incompressible Navier-Stokes Equations in Rn

In the early 1980s it was well established that Leray solutions of the unforced Navier-Stokes equations in Rn decay in energy norm for large time. With the works of T. Miyakawa, M. Schonbek and others it is now known that the energy decay rate cannot in general be any faster than t^-(n+2)/4 and is typically much slower. In contrast, we show in this note that, given an arbitrary Leray solution u(.,t), the difference of any two Stokes approximations to the Navier-Stokes flow u(.,t) will always decay at least as fast as t^-(n+2)/4, no matter how slow the decay of || u(.,t) ||_L2 might happen to be.

Energy decay of some boundary coupled systems involving wave\ Euler-Bernoulli beam with one locally singular fractional Kelvin-Voigt damping

<p style='text-indent:20px;'>In this paper, we investigate the energy decay of hyperbolic systems of wave-wave, wave-Euler-Bernoulli beam and beam-beam types. The two equations are coupled through boundary connection with only one localized non-smooth fractional Kelvin-Voigt damping. First, we reformulate each system into an augmented model and using a general criteria of Arendt-Batty, we prove that our models are strongly stable. Next, by using frequency domain approach, combined with multiplier technique and some interpolation inequalities, we establish different types of polynomial energy decay rate which depends on the order of the fractional derivative and the type of the damped equation in the system.</p>

Decay rate of a weakly dissipative viscoelastic plate equation with infinite memory

In this paper, a weakly dissipative viscoelastic plate equation with an infinite memory is considered. We show a general energy decay rate for a wide class of relaxation functions. To support our theoretical findings, some numerical illustrations are presented at the end. The numerical solution is computed using the popular finite element method in space, combined with time-stepping finite differences.

Long frontal waves and dynamic scaling in freely evolving equivalent barotropic flow

We present a scaling theory that links the frequency of long frontal waves to the kinetic energy decay rate and inverse transfer of potential energy in freely evolving equivalent barotropic turbulence. The flow energy is predominantly potential, and the streamfunction makes the dominant contribution to potential vorticity (PV) over most of the domain, except near PV fronts of width $O(L_{D})$, where $L_{D}$ is the Rossby deformation length. These fronts bound large vortices within which PV is well-mixed and arranged into a staircase structure. The jets collocated with the fronts support long-wave undulations, which facilitate collisions and mergers between the mixed regions, implicating the frontal dynamics in the growth of potential-energy-containing flow features. Assuming the mixed regions grow self-similarly in time and using the dispersion relation for long frontal waves (Nycander et al., Phys. Fluids A, vol. 5, 1993, pp. 1089–1091) we predict that the total frontal length and kinetic energy decay like $t^{-1/3}$, while the length scale of the staircase vortices grows like $t^{1/3}$. High-resolution simulations confirm our predictions.