Electrostatic trapping as a key to the dynamics of plasmas, fluids and other collective systems

This review article focusses on the phenomenon of collective particle trapping in dilute plasmas and related fluid-like systems. A coherent electrostatic wave or fluctuation, being excited by some mechanism in a plasma, is able to trap collectively charged particles in its potential trough(s) with the ultimate feedback of stabilizing and manipulating the original cause of growth. henomenon is well-known from particle simulations of a current-driven two-stream instability and its subsequent quenching by particle trapping. But also the nonlinear Landau damping process resulting in a BGK-like (Bernstein, Green, Kruskal) trapped particle mode sets an example. However, as shown in this report, already a slightly driven plasma has many possibilities of generating trapped particle modes—the mentioned cases representing only two examples—through which it generally becomes nonlinearly unstable. A direct consequence of this feedback of particle trapping is that the macroscopic (dielectric) properties of such a structured plasma may have changed fundamentally such that the relationship to what is known from linear wave theory is lost. nce, have to deal with a nonlinear kinetic description which, in case of a collisionless, electrostatic plasma, is the Vlasov–Poisson description. esent report is devoted to a large extent to a 1D Vlasov–Poisson system but also consequences for other physical systems will be derived and mentioned. and other findings will be developed in some detail culminating in a new paradigm for plasma stability which says:(1) ent-carrying plasma is nonlinearly unstable in a much wider region of parameter space than predicted by linear wave theory with the consequence that the associated turbulence and anomalous transport are triggered much easier than suggested by standard linear wave analysis. sible for this new scenario are localized trapped particle modes-more specifically electron and ion holes of zero or negative energy—which are found to be excited well below the threshold of linear instability. er words, a current-driven plasma shows a much larger sensibility to fluctuations than thought before and described in textbooks. The analysis presented reveals that a plasma, becoming structured by the generation of such modes, resides in a lower free energy state than the one without structures, being therefore in a preferred state that acts as an attractor in the system. having this property will be briefly called negative energy holes (NEHs). For example, zero or negative energy ion holes are found to exist for any drift velocity between electrons and ions and for any temperature ratio. dependent codes, a Vlasov-code and a PIC-(particle in cell)code, are used to approve this new scenario of instability. Moreover, by adding a Fokker–Planck collision term to the Vlasov-code, holes are shown to resist weak collisions, turn out to be robust and not only found in purely collisionless plasmas and cause an increase of resistivity. ral outcome of this scenario, therefore, is that whenever free (kinetic) energy is available, holes (and double layers) are necessarily excited, penetrating intermittently the plasma. Satellite measurements, yielding holes and double layers as the most omnipresent structures found in space, provide a typical example. investigated classical plasmas this way, we show that many of these innovations can be transferred to other systems, as well. we perform a quantum-correction to electron holes by using the Wigner–Moyal description of quantum mechanics in phase-space. As a result we get a weakening of the hole for which tunneling of particles across the separatrix of the unperturbed, deterministic classical hole equilibrium is responsible. rmalism is then used to find a link between hole structures in classical plasmas and envelope solitons in nonlinear optical media. This gives rise to a new approximation method for wave envelope solutions of the nonlinear Schrödinger equation, which utilizes quasi-particle trapping and may be valuable in cases of nonlinearties for which a direct solution is missing. r important application are particle beams in circular accelerators and storage rings. We prove analytically the existence of localized and periodic structures in coasting beams, as have been found experimentally for instance at Fermilab and at CERN, which are quite analogous to holes in classical plasmas. We also present an improved criterion for focusing. For bunched beams we describe and apply an iterative numerical procedure to find solitary hump and hole structures superimposed on the particle bunch, the former of which having been found recently in the Relativistic Hadron-Ion Collider (RHIC) at Brookhaven. y, we stress the mathematical equivalence between the 1D Vlasov–Poisson system and the equations describing a 2D incompressible, ideal fluid or the perpendicular dynamics of a strongly magnetized plasma in fluid or MHD approximation and other more complex fluids, such as rotating fluids, inhomogeneous plasmas, etc. This implies that tiny fluid elements trapped in coherent patches of shear flow motion, such as in secondary (tertiary) states that govern the transition to turbulence in ordinary hydrodynamics, do play a similar role than trapped particles in electrostatic waves, violating any linear wave ansatz. Or, said in different words, whenever a continues spectrum arises in a linearized fluid-like system associated with singular perturbations and a resonance between (quasi-)particles and the field, one has to consider this as a hint that the neglect of nonlinearity is not justified and that nonlinear wave solutions have to be taken into account in describing the evolution of the system correctly. This statement holds true already at an infinitesimal energy level of the coherent perturbations. Nonlinearity, and with it trapping structures, turns out to be a necessary requisite in all stages of the dynamical evolution not only at finite wave amplitudes, as commonly believed. clusion, in this report we emphasize the importance of collective trapping in (nearly) ideal plasmas and related systems bringing in at any level of wave activity a fundamental nonlinearity which is missed in standard linear wave theories as described in textbooks. The associated trapped particle modes challenge standard flow theories playing a key role in the interpretation of turbulence and anomalous transport.