Nonlinear dispersive regularization of inviscid gas dynamics

Ideal gas dynamics can develop shock-like singularities with discontinuous density. Viscosity typically regularizes such singularities and leads to a shock structure. On the other hand, in 1d, singularities in the Hopf equation can be non-dissipatively smoothed via KdV dispersion. Here, we develop a minimal conservative regularization of 3d ideal adiabatic flow of a gas with polytropic exponent $\gamma$. It is achieved by augmenting the Hamiltonian by a capillarity energy $\beta(\rho) (\nabla \rho)^2$. The simplest capillarity coefficient leading to local conservation laws for mass, momentum, energy and entropy using the standard Poisson brackets is $\beta(\rho) = \beta_*/\rho$ for constant $\beta_*$. This leads to a Korteweg-like stress and nonlinear terms in the momentum equation with third derivatives of $\rho$, which are related to the Bohm potential and Gross quantum pressure. Just like KdV, our equations admit sound waves with a leading cubic dispersion relation, solitary and periodic traveling waves. As with KdV, there are no steady continuous shock-like solutions satisfying the Rankine-Hugoniot conditions. Nevertheless, in 1d, for $\gamma = 2$, numerical solutions show that the gradient catastrophe is averted through the formation of pairs of solitary waves which can display approximate phase-shift scattering. Numerics also indicate recurrent behavior in periodic domains. These observations are related to an equivalence between our regularized equations (in the special case of constant specific entropy potential flow in any dimension) and the defocussing nonlinear Schrodinger equation (cubically nonlinear for $\gamma = 2$), with $\beta_*$ playing the role of $\hbar^2$. Thus, our regularization of gas dynamics may be viewed as a generalization of both the single field KdV & NLS equations to include the adiabatic dynamics of density, velocity, pressure & entropy in any dimension.Comment: 19 pages, 20 figure file

Whitham modulation theory for the Zakharov-Kuznetsov equation and transverse instability of its periodic traveling wave solutions

We derive the Whitham modulation equations for the Zakharov-Kuznetsov equation via a multiple scales expansion and averaging two conservation laws over one oscillation period of its periodic traveling wave solutions. We then use the Whitham modulation equations to study the transverse stability of the periodic traveling wave solutions. We find that all such solutions are linearly unstable, and we obtain an explicit expression for the growth rate of the most unstable wave numbers. We validate these predictions by linearizing the equation around its periodic solutions and solving the resulting eigenvalue problem numerically. Finally, we calculate the growth rate of the solitary waves analytically. The predictions of Whitham modulation theory are in excellent agreement with both of these approaches.Comment: 15 pages, 2 figure

Dispersive shock waves and modulation theory

There is growing physical and mathematical interest in the hydrodynamics of dissipationless/dispersive media. Since G. B. Whitham’s seminal publication fifty years ago that ushered in the mathematical study of dispersive hydrodynamics, there has been a significant body of work in this area. However, there has been no comprehensive survey of the field of dispersive hydrodynamics. Utilizing Whitham’s averaging theory as the primary mathematical tool, we review the rich mathematical developments over the past fifty years with an emphasis on physical applications. The fundamental, large scale, coherent excitation in dispersive hydrodynamic systems is an expanding, oscillatory dispersive shock wave or DSW. Both the macroscopic and microscopic properties of DSWs are analyzed in detail within the context of the universal, integrable, and foundational models for uni-directional (Korteweg–de Vries equation) and bi-directional (Nonlinear Schrödinger equation) dispersive hydrodynamics. A DSW fitting procedure that does not rely upon integrable structure yet reveals important macroscopic DSW properties is described. DSW theory is then applied to a number of physical applications: superfluids, nonlinear optics, geophysics, and fluid dynamics. Finally, we survey some of the more recent developments including non-classical DSWs, DSW interactions, DSWs in perturbed and inhomogeneous environments, and two-dimensional, oblique DSWs

Collisionless Magnetic Reconnection in Space Plasmas

Magnetic reconnection requires the violation of the frozen-in condition which ties gyrating charged particles to the magnetic field inhibiting diffusion. Ongoing reconnection has been identified in near-Earth space as being responsible for the excitation of substorms, magnetic storms, generation of field aligned currents and their consequences, the wealth of auroral phenomena. Its theoretical understanding is now on the verge of being completed. Reconnection takes place in thin current sheets. Analytical concepts proceeded gradually down to the microscopic scale, the scale of the electron skin depth or inertial length, recognizing that current layers that thin do preferentially undergo spontaneous reconnection. Thick current layers start reconnecting when being forced by plasma inflow to thin. For almost half a century the physical mechanism of reconnection has remained a mystery. Spacecraft in situ observations in combination with sophisticated numerical simulations in two and three dimensions recently clarified the mist, finding that reconnection produces a specific structure of the current layer inside the electron inertial (also called electron diffusion) region around the reconnection site, the X line. Onset of reconnection is attributed to pseudo-viscous contributions of the electron pressure tensor aided by electron inertia and drag, creating a complicated structured electron current sheet, electric fields, and an electron exhaust extended along the current layer. We review the general background theory and recent developments in numerical simulation on collisionless reconnection. It is impossible to cover the entire field of reconnection in a short space-limited review. The presentation necessarily remains cursory, determined by our taste, preferences, and knowledge. Only a small amount of observations is included in order to support the few selected numerical simulations.Comment: Review pape

Internal waves in fluid flows. Possible coexistence with turbulence

Waves in fluid flows represents the underlying theme of this research work. Wave interactions in fluid flows are part of multidisciplinary physics. It is known that many ideas and phenomena recur in such apparently diverse fields, as solar physics, meteorology, oceanography, aeronautical and hydraulic engineering, optics, and population dynamics. In extreme synthesis, waves in fluids include, on the one hand, surface and internal waves, their evolution, interaction and associated wave-driven mean flows; on the other hand, phenomena related to nonlinear hydrodynamic stability and, in particular, those leading to the onset of turbulence. Close similarities and key differences exist between these two classes of phenomena. In the hope to get hints on aspects of a potential overall vision, this study considers two different systems located at the opposite limits of the range of existing physical fluid flow situations: first, sheared parallel continuum flows - perfect incompressibility and charge neutrality - second, the solar wind - extreme rarefaction and electrical conductivity. Therefore, the activity carried out during the doctoral period consists of two parts. The first is focused on the propagation properties of small internal waves in parallel flows. This work was partly carried out in the framework of a MISTI-Seeds MITOR project proposed by Prof. D. Tordella (PoliTo) and Prof. G. Staffilani (MIT) on the long term interaction in fluid flows. The second part regards the analysis of solar-wind fluctuations from in situ measurements by the Voyagers spacecrafts at the edge of the heliosphere. This work was supported by a second MISTI-Seeds MITOR project, proposed by D. Tordella (PoliTo), J. D. Richardson (MIT, Kavli Institute), with the collaboration of M. Opher (BU)

Plasma heating and kinetic instabilities in the terrestrial foreshock

The terrestrial foreshock, the area upstream of, and magnetically connected to, the bow shock is a complex system in which the turbulent, supersonic and superalfvénic solar wind encounters the Earth’s magnetosphere. As a result, particle populations stream sunwards against the solar wind flow, creating a kinetic two-stream instability that leads to a variety of linear and nonlinear plasma processes. Foreshock plasma is collisionless, and the instability supports a variety of ultra-low frequency (ULF) modes. A statistical technique, based on categorizing wavenumber-frequency pairs by their associated power, is used to determine the dispersion relations for ULF modes in a number of case studies using magnetic field data from two-point measurements of the Cluster mission. Sunward-propagating fast magnetosonic and beam resonant modes are identified, as well as Alfvén modes propagating both sunwards and anti-sunwards. The fast magnetosonic modes are advected towards the Earth by the solar wind, and due to a cubic nonlinearity, steepen into sharply peaked waves. Three examples of these nonlinear wavetrains are compared to solutions of the derivative nonlinear Schrödinger equation, and are found to be in good agreement. The impact of the waves on the form of the pseudopotential, a quantity related to core plasma parameters, is also discussed. Wave-wave interactions are investigated for a case study of Cluster data, with a focus on energy transfer between ULF modes and a band of frequencies centred at 1Hz. Evidence for three-wave processes, formed by quadratic nonlinearities that interact between triads of frequencies that satisfy the frequency (f1 + f2 + f3) and wavenumber (k1 + k2 + k3) resonance conditions, is presented. Evidence for four wave processes in the same interval is also discussed