Discontinuous Galerkin Method for Linear Free-Surface Gravity Waves

Linear surface gravity waves on Maxwell viscoelastic fluids with finite depth are studied in this paper. A dispersion equation describing the spatial decay of the gravity wave in finite depth is derived. A dimensionless memory (time) number θ is introduced. The dispersion equation for the pure viscous fluid will be a specific case of the dispersion equation for the viscoelastic fluid as θ=0. The complex dispersion equation is numerically solved to investigate the dispersion relation. The influences of θ and water depth on the dispersion characteristics and wave decay are discussed. It is found that the role of elasticity for the Maxwell fluid is to make the surface gravity wave on the Maxwell fluid behave more like the surface gravity wave on the inviscid fluid.

Discontinuous Galerkin finite element method for solving population density functions of cortical pyramidal and thalamic neuronal populations

A discontinuous Galerkin finite element method for solving the Troesch’s problem

The kinetic theory of plasmas is governed by the Boltzmann equation, which describes the evolution of the distributions of electrons and ion species within six-dimensional phase space. When equipped with an appropriate model for the various binary collision events that can occur, this equation can be used to simulate a wide variety of plasma physics problems. Numerical solutions of the Boltzmann equation can be generated by either directly discretizing the six-dimensional phase space or by utilizing particle-based techniques, such as the Particle-in-Cell (PIC) method, to describe the statistical evolution of the distribution functions. However, both methods are inherently computationally expensive, and solutions in complex three-dimensional spatial domains remain intractable. For highly collisional plasmas in which the velocity distributions are nearly isotropic, a spherical harmonic expansion can be used to reduce the dimensionality of the phase space. Recent work at the Naval Research Laboratory 1 has re-examined this classical analytical approach in the context of low-temperature collisional plasmas. The authors derived a multi-term expansion that reduces the six-dimensional Boltzmann equation to a set as four-dimensional fluid-like equations. In this work, we describe the development of a general, high-order numerical solution method for this set of equations based on the discontinuous Galerkin (DG) finite element method. The electron velocity distribution function is discretized directly on the 4D phase-space mesh. Self-consistent electric and magnetic fields are simulated by solving Maxwell’s equations on a sub-dimensional manifold within the higher-dimensional phase-space mesh. A description of the numerical methodology will be presented along with a series of test problems that provide a demonstration of the algorithm and an initial examination of the benefits associated with using high-order phase-space discretizations.

Proof of unconditional stability for a single-field discontinuous Galerkin finite element formulation for linear elasto-dynamics

General solution for linear 2D anisotropic surface gravity water waves on uniform current

The authors have developed several three axis acoustic vorticity meters at the Woods Hole Oceanographic Institution for measuring shear and turbulence in ocean boundary layers. Such a measurement, in principle, filters out irrotational motion such as linear surface gravity waves which can swamp measurements of a flow's rotational motion. The authors have constructed one 15 cm acoustic path instrument, four 45 cm path instruments, and one 1.5 m path instrument and established their accuracy in a series of laboratory and ocean experiments. The 15 cm acoustic path instrument has a mean flow induced vorticity error of 0.034/s in a flow of 0.369 m/s and 20 dB rejection of waves. The 45 cm path instruments have an uncalibratable 10/sup -2//s mean flow induced vorticity error in a constant flow of 0.7 m/s consistent with the theoretical prediction that accuracy improves with increased path length. The 45 cm path instruments have been deployed in the ocean upper boundary layer to measure shear, in the thermocline to measure internal waves, and in the bottom boundary layer to measure turbulence.

Oscillatory flow produced near a seabed by linear surface gravity waves is analyzed. Using the boundary layer approximation, the problem is reduced to that of a boundary layer near a plate harmonically oscillating in a fluid at rest. Saffman's turbulence model is employed. Results on the ensemble averaged quantities are presented for Reynolds numbers ranging from laminar to fully developed turbulent regime, and for small, intermediate and large values of the roughness parameter. Based on the above model, a numerical fully nonlinear stability analysis of Stokes' solution is performed, and the unsteady transition from laminar to turbulent regime is investigated. The reliability of the analytical results is supported by a comparison with experimental data collected by others.

A space-time discontinuous Galerkin (DG) finite element method for nonlinear water waves in an inviscid and incompressible fluid is presented. The space-time DG method results in a conservative numerical discretization on time dependent deforming meshes which follow the free surface evolution. The dispersion and dissipation errors of the scheme are investigated and the algorithm is demonstrated with the simulation of waves generated by a wave maker.

This study uses ground-based recordings of low-frequency, inaudible acoustic waves (infrasound) to probe the wind and temperature fluctuations associated with internal gravity waves breaking in the middle atmosphere. Building on the approach introduced by Chunchuzov et al., the recorded waveforms are used to retrieve the effective sound speed fluctuations in an inhomogeneous atmospheric layer of infrasound backscattering. The retrieval procedure was applied to infrasound from controlled blasts related to the disposal of military explosives in Hukkakero, Finland and recorded at the IS37 station in Norway over a four-year period from 2014 to 2017. Our findings indicate that infrasound scattering occurs in the lower mesosphere between 50 and 75 km in altitude in a region where gravity waves interact due to strong nonlinear effects and form thin layers with strong wind shears. The retrieved effective sound speed fluctuations were then analysed in terms of the vertical wave number spectra. The analysis revealed that the spectra follow a kz–3 power law that corresponds to the "universal" saturated spectrum of atmospheric gravity waves within kz ∈ [2.1·10–3; 2.7·10–2] cycles/m. Based on this wavenumber range, we estimate the outer and inner vertical scale of atmospheric inhomogeneities that infrasound is sensitive to as Linner= 33 – 37 m, Louter = 382 – 625 m.   Furthermore, the spectra of the retrieved effective sound speed fluctuations were compared to theoretical linear and nonlinear gravity wave saturation theories as well as to independent wind measurements made by the Saura MF radar near the Andøya Space Center in Norway. The comparison showed a good agreement in terms of the amplitude and slopes of the vertical wavenumber spectra in both cases. The overall agreement allows us to suggest that the Saura radar and infrasound-based effective sound speed profiles represent the low- and high-wavenumber regimes of the same "universal" gravity wave spectrum. These results illustrate that the use of infrasound makes it possible to probe fine-scale motions that are not well captured by other techniques. The latter suggests that infrasound observations can be used as a complementary technique to probe internal gravity waves in the middle- and upper atmosphere. 

A conforming discontinuous Galerkin finite element method for Brinkman equations

Dispersion analysis of finite difference and discontinuous Galerkin schemes for Maxwell's equations in linear Lorentz media

Space–time discontinuous Galerkin method for nonlinear water waves

Stratified hydrostatic fluids have linear internal gravity waves with different phase speeds and vertical profiles. Here a simplified set of partial differential equations (PDE) is derived to represent the nonlinear dynamics of waves with different vertical profiles. The equations are derived by projecting the full nonlinear equations onto the vertical modes of two gravity waves, and the resulting equations are thus referred to here as the two-mode shallow water equations (2MSWE). A key aspect of the nonlinearities of the 2MSWE is that they allow for interactions between a background wind shear and propagating waves. This is important in the tropical atmosphere where horizontally propagating gravity waves interact together with wind shear and have source terms due to convection. It is shown here that the 2MSWE have nonlinear internal bore solutions, and the behavior of the nonlinear waves is investigated for different background wind shears. When a background shear is included, there is an asymmetry between the east- and westward propagating waves. This could be an important effect for the large-scale organization of tropical convection, since the convection is often not isotropic but organized on large scales by waves. An idealized illustration of this asymmetry is given for a background shear from the westerly wind burst phase of the Madden–Julian oscillation; the potential for organized convection is increased to the west of the existing convection by the propagating nonlinear gravity waves, which agrees qualitatively with actual observations. The ideas here should be useful for other physical applications as well. Moreover, the 2MSWE have several interesting mathematical properties: they are a system of nonconservative PDE with a conserved energy, they are conditionally hyperbolic, and they are neither genuinely nonlinear nor linearly degenerate over all of state space. Theory and numerics are developed to illustrate these features, and these features are important in designing the numerical scheme. A numerical method is designed with simplicity and minimal computational cost as the main design principles. Numerical tests demonstrate that no catastrophic effects are introduced when hyperbolicity is lost, and the scheme can represent propagating discontinuities without introducing spurious oscillations.

Bends in coastal mountain ranges may diffract propagating atmospheric Kelvin waves and trapped coastal currents. Analytic solutions exist for the diffraction of both linear Kelvin waves and linear nonrotating gravity waves. Within the context of the single-layer shallow-water equations, we examine the diffraction of nonlinear gravity waves and bores in a nonrotating reference frame and nonlinear Kelvin waves and coastally trapped bores in a rotating reference frame. The diffraction process can significantly decrease the amplitude of linear and nonlinear waves and bores in the nonrotating reference frame. Unlike for their linear counterpart, however, the diffraction-related amplitude decay for the nonrotating nonlinear waves takes place entirely within the region of the bend and does not produce a continuous decay after the bend. Moreover, theory predicts a critical bend angle at which bore amplitudes will be zero at the wall after propagation around the bend, but shallow-water model simulations do not confirm the existence of the critical angle. For Kelvin waves and trapped bores in the rotating reference frame, we find robust wave and bore propagation around coastal bends in all cases. No critical angles exist for the waves and bores in the rotating reference frame.