Weakly nonlinear surface waves in magnetohydrodynamics. I

2020 ◽  
pp. 1-35
Author(s):  
Olivier Pierre ◽  
Jean-François Coulombel
Keyword(s):  
Surface Waves ◽  
High Frequency ◽  
Rayleigh Waves ◽  
Three Dimensional ◽  
Type Equation ◽  
Approximate Solutions ◽  
Vortex Sheet ◽  
Solvability Conditions ◽  
Weakly Nonlinear ◽  
High Frequency Oscillations

This work is devoted to the construction of weakly nonlinear, highly oscillating, current vortex sheet solutions to the system of ideal incompressible magnetohydrodynamics. Current vortex sheets are piecewise smooth solutions that satisfy suitable jump conditions on the (free) discontinuity surface. In this article, we complete an earlier work by Alì and Hunter (Quart. Appl. Math. 61(3) (2003) 451–474) and construct approximate solutions at any arbitrarily large order of accuracy to the three-dimensional free boundary problem when the initial discontinuity displays high frequency oscillations. As evidenced in earlier works, high frequency oscillations of the current vortex sheet give rise to ‘surface waves’ on either side of the sheet. Such waves decay exponentially in the normal direction to the current vortex sheet and, in the weakly nonlinear regime which we consider here, their leading amplitude is governed by a nonlocal Hamilton–Jacobi type equation known as the ‘HIZ equation’ (standing for Hamilton–Il’insky–Zabolotskaya (J. Acoust. Soc. Amer. 97(2) (1995) 891–897)) in the context of Rayleigh waves in elastodynamics. The main achievement of our work is to develop a systematic approach for constructing arbitrarily many correctors to the leading amplitude. We exhibit necessary and sufficient solvability conditions for the corrector equations that need to be solved iteratively. The verification of these solvability conditions is based on mere algebra and arguments of combinatorial analysis, namely a Leibniz type formula which we have not been able to find in the literature. The construction of arbitrarily many correctors enables us to produce infinitely accurate approximate solutions to the current vortex sheet equations. Eventually, we show that the rectification phenomenon exhibited by Marcou in the context of Rayleigh waves (C. R. Math. Acad. Sci. Paris 349(23–24) (2011) 1239–1244) does not arise in the same way for the current vortex sheet problem.

How rapidly oscillating collapsible tubes extract energy from a viscous mean flow

2008 ◽  
Vol 601 ◽  
pp. 199-227 ◽  
Author(s):  
MATTHIAS HEIL ◽  
SARAH L. WATERS
Keyword(s):  
Reynolds Number ◽  
High Frequency ◽  
Computational Analysis ◽  
Functional Relationship ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Mean Flow ◽  
Collapsible Tubes ◽  
High Frequency Oscillations ◽  
Reynolds Number Flows

We present a combined theoretical and computational analysis of three-dimensional unsteady finite-Reynolds-number flows in collapsible tubes whose walls perform prescribed high-frequency oscillations which resemble those typically observed in experiments with a Starling resistor. Following an analysis of the flow fields, we investigate the system's overall energy budget and establish the critical Reynolds number, Recrit, at which the wall begins to extract energy from the flow. We conjecture that Recrit corresponds to the Reynolds number beyond which collapsible tubes are capable of performing sustained self-excited oscillations. Our computations suggest a simple functional relationship between Recrit and the system parameters, and we present a scaling argument to explain this observation. Finally, we demonstrate that, within the framework of the instability mechanism analysed here, self-excited oscillations of collapsible tubes are much more likely to develop from steady-state configurations in which the tube is buckled non-axisymmetrically, rather than from axisymmetric steady states, which is in agreement with experimental observations.

Short surface waves in the presence of a finite dock. II

1964 ◽  
Vol 60 (4) ◽  
pp. 985-1012 ◽  
Author(s):  
R. L. Holford
Keyword(s):  
Green’S Function ◽  
Surface Waves ◽  
Green's Function ◽  
High Frequency ◽  
Short Waves ◽  
High Frequency Oscillations

AbstractThe generalized wave-making problem for the forced high-frequency oscillations of a finite dock is solved through use of the Green's function obtained in the first part of this paper. The specific cases of heave and roll are considered with particular reference to the forces on the dock and the amplitude and phase of the radiated waves. These results are then utilized to solve the problem of transmission of short waves under a fixed dock.

SIGNAL AND NOISE IN DEEP WELLS

Geophysics ◽  
1964 ◽  
Vol 29 (5) ◽  
pp. 721-732 ◽  
Author(s):  
Eduard J. Douze
Keyword(s):  
Surface Waves ◽  
High Frequency ◽  
Rayleigh Waves ◽  
Frequency Noise ◽  
Low Velocity ◽  
P Waves ◽  
High Frequency Noise ◽  
Deep Well ◽  
Velocity Wave ◽  
Deep Wells

Deep‐well measurements at Grapevine, Texas; Hobart, Oklahoma; and Orlando, Florida, show that the noise is composed of surface waves that decrease in amplitude with depth. At Hobart, a low‐velocity wave guide contains wave‐guided noise. Fundamental and higher mode Rayleigh waves appear to be present in the noise at each site. The amplitudes of incident P waves depend on the depth at which the deep‐well seismometer is operated. High‐frequency P waves from quarry blasts are clearly visible in recordings from the deep‐well seismometer because the high‐frequency noise is suppressed at depth.

scholarly journals Comparison of Hybrid-4DEnVar and Hybrid-4DVar Data Assimilation Methods for Global NWP

2015 ◽  
Vol 143 (1) ◽  
pp. 212-229 ◽  
Author(s):  
Andrew C. Lorenc ◽  
Neill E. Bowler ◽  
Adam M. Clayton ◽  
Stephen R. Pring ◽  
David Fairbairn
Keyword(s):  
Data Assimilation ◽  
Time Evolution ◽  
High Frequency ◽  
Three Dimensional ◽  
Variational Data Assimilation ◽  
Background Error ◽  
Hybrid Combination ◽  
High Frequency Oscillations ◽  
Climatological Model ◽  
Better Than

Abstract The Met Office has developed an ensemble-variational data assimilation method (hybrid-4DEnVar) as a potential replacement for the hybrid four-dimensional variational data assimilation (hybrid-4DVar), which is the current operational method for global NWP. Both are four-dimensional variational methods, using a hybrid combination of a fixed climatological model of background error covariances with localized covariances from an ensemble of current forecasts designed to describe the structure of “errors of the day.” The fundamental difference between the methods is their modeling of the time evolution of errors within each data assimilation window: 4DVar uses a linear model and its adjoint and 4DEnVar uses a localized linear combination of nonlinear forecasts. Both hybrid-4DVar and hybrid-4DEnVar beat their three-dimensional versions, which are equivalent, in NWP trials. With settings based on the current operational system, hybrid-4DVar performs better than hybrid-4DEnVar. Idealized experiments designed to compare the time evolution of covariances in the methods are described: the basic 4DEnVar represents the evolution of ensemble errors as well as 4DVar. However, 4DVar also represents the evolution of errors from the climatological covariances, whereas 4DEnVar does not. This difference is the main cause of the superiority of hybrid-4DVar. Another difference is that the authors’ 4DVar explicitly penalizes rapid variations in the analysis increment trajectory, while the authors’ 4DEnVar contains no dynamical constaints on imbalance. The authors describe a four-dimensional incremental analysis update (4DIAU) method that filters out the high-frequency oscillations introduced by the poorly balanced 4DEnVar increments. Possible methods for improving hybrid-4DEnVar are discussed.

scholarly journals Identification of subsonic P‐waves

Geophysics ◽  
2002 ◽  
Vol 67 (3) ◽  
pp. 909-920 ◽  
Author(s):  
Paul Michaels
Keyword(s):  
Surface Waves ◽  
High Frequency ◽  
Rayleigh Waves ◽  
Field Trial ◽  
Water Saturation ◽  
Near Field ◽  
Plasticity Index ◽  
P Waves ◽  
Sandy Silt

A field trial was conducted to test the existence of subsonic (Vp < 331 m/s) P‐waves previously reported in the literature. A 1‐m‐long reverse profile was acquired with three‐component (3C) geophones on a sandy silt (unified classification ML). The silt had a porosity of 54%, a degree of water saturation of 63%, and a plasticity index of 10. No subsonic P‐waves were observed, although high‐frequency (up to 1200 Hz) Rayleigh waves were identified by hodogram analysis. These surface waves were observed with horizontal velocities that varied from 40 to 200 m/s. Hodogram observations and theory suggest that a portion of the data were also in the near‐field.

scholarly journals Diffusion approximations for randomly arriving expert opinions in a financial market with Gaussian drift

2021 ◽  
Vol 58 (1) ◽  
pp. 197-216 ◽  
Author(s):  
Jörn Sass ◽  
Dorothee Westphal ◽  
Ralf Wunderlich
Keyword(s):  
Stock Returns ◽  
Financial Market ◽  
Discrete Time ◽  
High Frequency ◽  
Utility Maximization ◽  
Limit Theorems ◽  
Approximate Solutions ◽  
Diffusion Approximations ◽  
Expert Opinions ◽  
Arrival Intensity

AbstractThis paper investigates a financial market where stock returns depend on an unobservable Gaussian mean reverting drift process. Information on the drift is obtained from returns and randomly arriving discrete-time expert opinions. Drift estimates are based on Kalman filter techniques. We study the asymptotic behavior of the filter for high-frequency experts with variances that grow linearly with the arrival intensity. The derived limit theorems state that the information provided by discrete-time expert opinions is asymptotically the same as that from observing a certain diffusion process. These diffusion approximations are extremely helpful for deriving simplified approximate solutions of utility maximization problems.

Sign in / Sign up

Export Citation Format

Share Document