Hamiltonian formulations of quasilinear theory for magnetized plasmas

Abstract

Hamiltonian formulations of quasilinear theory are presented for the cases of uniform and nonuniform magnetized plasmas. First, the standard quasilinear theory of Kennel and Engelmann (Kennel, Phys. Fluids, 1966, 9, 2377) is reviewed and reinterpreted in terms of a general Hamiltonian formulation. Within this Hamiltonian representation, we present the transition from two-dimensional quasilinear diffusion in a spatially uniform magnetized background plasma to three-dimensional quasilinear diffusion in a spatially nonuniform magnetized background plasma based on our previous work (Brizard and Chan, Phys. Plasmas, 2001, 8, 4762–4771; Brizard and Chan, Phys. Plasmas, 2004, 11, 4220–4229). The resulting quasilinear theory for nonuniform magnetized plasmas yields a 3 × 3 diffusion tensor that naturally incorporates quasilinear radial diffusion as well as its synergistic connections to diffusion in two-dimensional invariant velocity space (e.g., energy and pitch angle).

Electron Resonant Interaction With Coherent ULF Waves: Hamiltonian Approach

“…For systems with bounce motion and azimuthal drifts, there is a specific type of drift‐bounce resonance, $$\mathfrak{n}{\omega }_{b}+\mathfrak{m}\dot{\beta }=\omega $$. This resonance was considered for electron diffusive scattering by ULF kinetic Alfven waves (Chaston, Bonnell, Halford, et al., 2018; Chaston, Bonnell, Wygant, et al., 2018; Zhu et al., 2020), but was not discussed in context of nonlinear resonant interactions (see also Brizard & Chan, 2022, for discussion of general coupling of bounce and drift resonance in electron dynamics).…”

“…Drift theory equations (Northrop, 1963; Sivukhin, 1965) yield that the particle drift speed in inhomogeneous magnetic field is $$${\mathbf{v}}_{D}=\frac{\mu }{{m}_{e}\gamma }\frac{\left({\mathbf{B}}_{0}/{B}_{0}\right)\times \nabla {{\Omega }}_{0}}{{{\Omega }}_{0}}$$$ To construct Hamiltonian function from energy $$\mathcal{H}$$ we define a pair of conjugate variables. There is more than one approach to derive the gyro‐averaged Hamiltonian describing particle drifts, and the most developed one is the noncanonical guiding center theory (Brizard & Chan, 2022; Cary & Brizard, 2009). However, we prefer to develop a canonical guiding center theory (Gardner, 1959; Neishtadt & Artemyev, 2020) that may provide simpler tools for investigation of resonant wave‐particle interactions (e.g., Artemyev et al., 2022b; Degeling & Rankin, 2008).…”

Radial Transport in the Earth’s Radiation Belts: Linear, Quasi-linear, and Higher-order Processes

“…The theoretical framework to quantify and interpret the dynamical evolution of radiation belts on timescales of a few hours to several days relies exclusively on quasi-linear theories (Fälthammar 1965;Kennel & Engelmann 1966;Diamond et al 2010;Brizard & Chan 2022). The overwhelming reliance on quasi-linear models in radiation belt research is not fortuitous as it offers two benefits alternative computational and theoretical approaches lack:…”

Nonlinear equilibria and transport processes in burning plasmas

“…Thus, when looking at its nonlinear interaction with a generic fluctuation structure, including the n = 0 GAM/EGAM, we have where we have noted equations ( 14) and (15). Now recall equation (18) along with equations ( 37) and (38). Thus, when computing the contribution of the first term on the RHS above to the nonlinear interaction term in equation (A. where, for completeness, we have added long time scale dependences in the propagator together with the effect of ZFs on wave-particle decorrelation.…”

“…In this work, we first derive the PSZS evolution equation in conservative form using the equilibrium constants of motion as phase space coordinates. The orbit averaging approach, adopted here, has analogies with the methodologies that are used for neoclassical transport studies in stellarators in the weakly collisional regimes [16]; and in Hamiltonian formulations of quasi-linear transport [17,18]. However, our present approach takes into account that EP induced transport processes may be induced by a quasi-coherent fluctuation spectrum of non-perturbative nature and characterized by O(1) Kubo numbers [1][2][3], invalidating fundamental assumptions of quasi-linear theory.…”