Abstract
We provide a procedure for deriving Hamiltonian reduced fluid models for plasmas, starting from a Hamiltonian gyrokinetic system in the
δ
f
approximation. The procedure generalizes, to a ...considerable extent, previous results. In particular, the evolution of moments with respect to the magnetic moment coordinate is also taken into account, together with background density and magnetic inhomogeneities. In the limit of vanishing Finite Larmor Radius (FLR) effects, an infinite family of reduced electron drift-fluid equations is derived, evolving all the electron moments
g
i
j
e
, with
i
=
0
,
…
,
N
and
j
=
0
,
…
,
M
, where
N
and
M
are arbitrary non-negative integers counting the maximum order of the moments taken with respect to the parallel velocity and to the magnetic moment coordinates, respectively. An analogous result is found in the gyrofluid case and applied to the ion species. The gyrofluid result holds for
M
⩽
1
, finite FLR effects and a low ratio
β
e
between electron internal pressure and magnetic guide field pressure. In both the drift and gyrofluid case, the key for the identification of the Hamiltonian structure resides in changes of variables based on orthogonal matrices that diagonalize the Jacobi matrices associated with Hermite and Laguerre polynomials. In terms of the transformed variables, the drift and gyrofluid equations are cast in a simple form, which reduces to advection equations for Lagrangian invariants in the two-dimensional case with homogeneous background. Because the procedure requires to evolve, for a particle species
s
, all and only the moments
g
i
j
s
, with
i
=
0
,
…
,
N
and
j
=
0
,
…
,
M
, not every choice for the set of moments is admissible, for fixed
N
and
M
. This might also explain the scarcity of Hamiltonian reduced fluid models obtained so far which account for anisotropic temperature fluctuations.
We provide a general framework for deriving Hamiltonian electromagnetic gyrofluid models from a Hamiltonian system of gyrokinetic equations. The presented procedure permits to derive gyrofluid models ...for an arbitrary number of moments with respect to the velocity coordinate parallel to an equilibrium magnetic field. The resulting gyrofluid models account, in particular, for finite Larmor radius effects, equilibrium temperature anisotropies and fluctuations of the magnetic field in the direction parallel to the equilibrium magnetic field, thus generalizing Hamiltonian gyrofluid models previously presented in the literature. The Hamiltonian reduction procedure leading from the parent gyrokinetic model to the gyrofluid models is formulated in two stages. In the first step, after having shown that the parent gyrokinetic system indeed posseses a Hamiltonian structure, a Hamiltonian system is derived, by means of a Poisson sub-algebra argument, which describes the evolution of the perturbation of the gyrocenter distribution function, averaged with respect to the magnetic moment coordinate. The second stage brings from the latter model to the gyrofluid models by means of a closure relation, applicable at an arbitrary order in the moment hierarchy, which guarantees the preservation of a Hamiltonian structure. Casimir invariants of the noncanonical Poisson brackets of the gyrofluid models are provided. It is also shown how, in the two-dimensional limit, the gyrofluid model equations can be cast in the form of advection equations for Lagrangian invariants transported by generalized incompressible velocity fields, thus extending results obtained for previous Hamiltonian gyrofluid and drift-fluid models. The Hamiltonian reduction procedure is applied to derive a five-field gyrofluid model evolving the first two moments for the electron species and the first three moments for the ion species. The Casimir invariants and the Lagrangian advection formulation are provided explicitly for the five-field model. Remarks concerning possible variants of the procedure are discussed. As an example, it is shown how, by means of a variant of the procedure, it is possible to derive an isothermal two-field model for kinetic Alfvén waves including equilibrium electron temperature anisotropy effects.
Abstract
We review the progress made, during the last decade, on the analysis of formal stability for Hamiltonian fluid models for plasmas, carried out by means of the energy-Casimir (EC) method. The ...review begins with a tutorial section describing the essential concepts on the Hamiltonian formalism for fluid models and on the EC method, which will be frequently used in the article. Subsequently, a nonlinear stability analysis applied to reduced magnetohydrodynamics (MHD) is described, as paradigmatic example for the application of the EC method. The review of the recent results begins with the equilibrium and formal stability analysis of MHD with general helical symmetry, followed by the treatment of extended MHD. Applications of the EC method to a hybrid MHD-Vlasov model with pressure coupling and to a reduced fluid model accounting for electron temperature anisotropy are described next. The formal stability analysis of compressible reduced MHD is then presented and used to show the connection between the EC method and the classical
δW
method for MHD stability. The concept of negative energy mode (NEM) is also briefly reviewed and applied to a model for electron temperature gradient (ETG) instability. In the context of the search for equilibria by a variational procedure, which is part of the EC method, we discuss a recent interpretation of the classical tearing modes in terms of singular equilibria of MHD linearized about Beltrami equilibria. Finally, we mention some possible directions for future developments.
We present an infinite family of Hamiltonian electromagnetic fluid models for plasmas, derived from drift-kinetic equations. An infinite hierarchy of fluid equations is obtained from a Hamiltonian ...drift-kinetic system by taking moments of a generalized distribution function and using Hermite polynomials as weight functions of the velocity coordinate along the magnetic guide field. Each fluid model is then obtained by truncating the hierarchy to a finite number N+1 of equations by means of a closure relation. We show that, for any positive N, a linear closure relation between the moment of order N+1 and the moment of order N guarantees that the resulting fluid model possesses a Hamiltonian structure, thus respecting the Hamiltonian character of the parent drift-kinetic model. An orthogonal transformation is identified which maps the fluid moments to a new set of dynamical variables in terms of which the Poisson brackets of the fluid models become a direct sum and which unveils remarkable dynamical properties of the models in the two-dimensional (2D) limit. Indeed, when imposing translational symmetry with respect to the direction of the magnetic guide field, all models belonging to the infinite family can be reformulated as systems of advection equations for Lagrangian invariants transported by incompressible generalized velocities. These are reminiscent of the advection properties of the parent drift-kinetic model in the 2D limit and are related to the Casimirs of the Poisson brackets of the fluid models. The Hamiltonian structure of the generic fluid model belonging to the infinite family is illustrated treating a specific example of a fluid model retaining five moments in the electron dynamics and two in the ion dynamics. We also clarify the connection existing between the fluid models of this infinite family and some fluid models already present in the literature.
Full text
Available for:
GEOZS, IJS, IMTLJ, KILJ, KISLJ, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UL, UM, UPCLJ, UPUK
The linear stability of chains of magnetic vortices in a plasma is investigated analytically in two dimensions by means of a reduced fluid model assuming a strong guide field and accounting for ...equilibrium electron temperature anisotropy. The chain of magnetic vortices is modeled by means of the classical 'cat's eyes' solutions and the linear stability is studied by analysing the second variation of a conserved functional, according to the energy-Casimir method. The stability analysis is carried out on the domain bounded by the separatrices of the vortices. Two cases are considered, corresponding to a ratio between perpendicular equilibrium ion and electron temperature much greater or much less than unity, respectively. In the former case, equilibrium flows depend on an arbitrary function. Stability is attained if the equilibrium electron temperature anisotropy is bounded from above and from below, with the lower bound corresponding to the condition preventing the firehose instability. A further condition sets an upper limit to the amplitude of the vortices, for a given choice of the equilibrium flow. For cold ions, two sub-cases have to be considered. In the first one, equilibria correspond to those for which the velocity field is proportional to the local Alfvén velocity. Stability conditions imply: an upper limit on the amplitude of the flow, which automatically implies firehose stability, an upper bound on the electron temperature anisotropy and again an upper bound on the size of the vortices. The second sub-case refers to equilibrium electrostatic potentials which are not constant on magnetic flux surfaces and the resulting stability conditions correspond to those of the first sub-case in the absence of flow.
We derive the conditions under which the fluid models obtained from the first two moments of Hamiltonian drift-kinetic systems of interest to plasma physics, preserve a Hamiltonian structure. The ...adopted procedure consists of determining closure relations that allow to truncate the Poisson bracket of the drift-kinetic system, expressed in terms of the moments, in such a way that the resulting operation is a Poisson bracket for functionals of the first two fluid moments. The analysis is carried out for a class of full drift-kinetic equations and also for drift-kinetic systems in which a splitting between an equilibrium distribution function and a perturbation is performed. In the former case we obtain that the only closure that leads to a Poisson bracket, without involving operators or an explicit dependence on the spatial coordinates, corresponds to that of an ideal adiabatic gas made of molecules possessing one degree of freedom. In the latter case, Hamiltonian closures turn out to be those in which the second moment is a linear combination of the first two moments, which can be seen as a linearization of the Hamiltonian closure of the full drift-kinetic case. A number of weakly-3D Hamiltonian-reduced fluid models of interest, for instance for tokamak plasmas, can be derived in this way and, viceversa given a fluid model with a Hamiltonian structure of a certain type, a parent Hamiltonian drift-kinetic model can then be identified. We make use of this correspondence to identify the drift-kinetic models from which Hamiltonian fluid models for magnetic reconnection and compressible plasma dynamics in the presence of a static but inhomogeneous magnetic field can be derived. The Casimir invariants of the Poisson brackets of the derived fluid models are also discussed. It is also shown that the Poisson structure for the fluid model derived from the full drift-kinetic system coincides with that of a reduced fluid model, when using the fluid velocity instead of the momentum as a dynamical variable.
The linear and nonlinear evolutions of the tearing instability in a collisionless plasma with a strong guide field are analysed on the basis of a two-field Hamiltonian gyrofluid model. The model is ...valid for a low ion temperature and a finite $\beta _e$. The finite $\beta _e$ effect implies a magnetic perturbation along the guide field direction, and electron finite Larmor radius effects. A Hamiltonian derivation of the model is presented. A new dispersion relation of the tearing instability is derived for the case $\beta _e=0$ and tested against numerical simulations. For $\beta _e \ll 1$ the equilibrium electron temperature is seen to enhance the linear growth rate, whereas we observe a stabilizing role when electron finite Larmor radius effects become more relevant. In the nonlinear phase, stall phases and faster than exponential phases are observed, similarly to what occurs in the presence of ion finite Larmor radius effects. Energy transfers are analysed and the conservation laws associated with the Casimir invariants of the model are also discussed. Numerical simulations seem to indicate that finite $\beta _e$ effects do not produce qualitative modifications in the structures of the Lagrangian invariants associated with Casimirs of the model.
Reduced fluid models for collisionless plasmas including electron inertia and finite Larmor radius corrections are derived for scales ranging from the ion to the electron gyroradii. Based either on ...pressure balance or on the incompressibility of the electron fluid, they respectively capture kinetic Alfvén waves (KAWs) or whistler waves (WWs), and can provide suitable tools for reconnection and turbulence studies. Both isothermal regimes and Landau fluid closures permitting anisotropic pressure fluctuations are considered. For small values of the electron beta parameter
$\unicodeSTIX{x1D6FD}_{e}$
, a perturbative computation of the gyroviscous force valid at scales comparable to the electron inertial length is performed at order
$O(\unicodeSTIX{x1D6FD}_{e})$
, which requires second-order contributions in a scale expansion. Comparisons with kinetic theory are performed in the linear regime. The spectrum of transverse magnetic fluctuations for strong and weak turbulence energy cascades is also phenomenologically predicted for both types of waves. In the case of moderate ion to electron temperature ratio, a new regime of KAW turbulence at scales smaller than the electron inertial length is obtained, where the magnetic energy spectrum decays like
$k_{\bot }^{-13/3}$
, thus faster than the
$k_{\bot }^{-11/3}$
spectrum of WW turbulence.
PDF
