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Perturbation methods leading to the Gyrokinetic Maxwell-Vlasov equations
There exists two principal groups of methods that permits us to get reduced dynamical equations implemented inside those codes. The first one, referred to also as the standard method, consists in dealing with explicit gyroaveraging of the Vlasov equation expressed in lowest order reduced (guiding-center) coordinates. This is followed by separation of equilibrium and perturbed parts of the guiding-center distribution function. One of the serious disadvantages of such a method is its failure to provide a clear iterative algorithm. Another group of methods do not deal with Vlasov equation directly, but start with consideration of a single particle Lagrangian. They use Lie-transform techniques which provide near-identity coordinate transformations that decouple the gyration from the slower dynamics of interest. Such a method was formally introduced in [69] and applied for stationary electrostatic turbulence case. Later its application was expanded on the problem of a single particle motion in an external non-uniform magnetic [36] and electromagnetic [37] fields as well as to study of mechanics of magnetic field line flow [39]. Their first advantage with respect to the first group of reduction methods is that such a transformation is reversible, so the information about the fast dynamics is not lost and can be recovered when it is needed. The second strong point of such approaches is existence of a well defined iterative procedure that permits us at each order to derive gyroangle-independent dynamics. The more general among those methods, is the action-variational Lie perturbation method. This method deals with the phase-space Lagrangian (Poincar´e-Cartan fundamental one-form), which couples the symplectic structure and the Hamiltonian2: Γ ≡ L dt = p · dq − Hdt (1.27).
Physical motivations and the E × B model
Fusion plasma are sophisticated systems that combine the intrinsic complexity of neutral fluid turbulence and the self-consistent response of charged species, both electrons and ions, to magnetic fields. Regarding magnetic confinement in a tokamak, a large external magnetic field and a first order induced magnetic field are organised to generate the so-called magnetic equilibrium of nested toroidal magnetic surfaces [53]. On the latter, the plasma can be sustained close to a local thermodynamical equilibrium. In order to analyse turbulent transport we consider plasma perturbations of this class of solutions with no evolution of the magnetic equilibrium, thus excluding MHD instabilities. Such perturbations self-consistently generate electromagnetic perturbations that feedback on the plasma evolution. Following present experimental evidence, we shall assume here that magnetic fluctuations have a negligible impact on turbulent transport [54]. We will thus concentrate on electrostatic perturbations that correspond to the vanishing β limit, where β = p/(B2/2µ0) is the ratio of the plasma pressure p to the magnetic pressure. The appropriate framework for this turbulence is the Vlasov equation in the gyrokinetic approximation associated with the Maxwell-Gauss equation that relates the electric field to the charge density. When considering the Ion Temperature Gradient instability [55] that appears to dominate the ion heat transport, one can further assume the electron response to be adiabatic so that the plasma response is governed by the gyrokinetic Vlasov equation for the ion species. Let us now consider the linear response of such a distribution function f , to a given electrostatic perturbation, typically of the form Te φ e−iωt+ikr, (where f and φ are Fourier amplitudes of distribution function and electric potential). To leading orders one then finds that the plasma response exhibits a resonance:
f = ω+ω∗ − 1 φfeq (2.1) ω − k|| v||.
Here feq is the reference distribution function, locally Maxwellian with respect to v|| and ω ∗ is the diamagnetic frequency that contains the density and temperature gradient that drive the ITG instability [55]. Te is the electronic temperature. This simplified plasma response to the electrostatic perturbation allows one to illustrate the turbulent control that is considered to trigger off transport barriers in present tokamak experiments.
Let us examine the resonance ω − k|| v|| = 0 where k|| = (n − m/q)/R with R being the major radius, q the safety factor that characterises the specific magnetic equilibrium and m and n the wave numbers of the perturbation that yield the wave vectors of the perturbation in the two periodic directions of the tokamak equilibrium. When the turbulent frequency ω is small with respect to vth/(qR), (where vth = kB T /m is the thermal velocity), the resonance occurs for vanishing values of k||, and as a consequence at given radial location due to the radial dependence of the safety factor. The resonant effect is sketched on figure 2.1. In a quasilinear approach, empedding large scale turbulent transport and broad resonances favouring strong turbulent transport. the response to the perturbations will lead to large scale turbulent transport when the width of the resonance δm is comparable to the distance between the resonances ∆m,m+1 leading to an overlap criterion that is comparable to the well known Chirikov criterion for chaotic transport σm = (δm + δm+1)/∆m,m+1 with σ > 1 leading to turbulent transport across the magnetic surfaces and σ < 1 localising the turbulent transport to narrow radial regions in the vicinity of the resonant magnetic surfaces.
The present control schemes are two-fold. First, one can consider a large scale radial electric field that governs a Doppler shift of the mode frequency ω. As such the Doppler shift ω − ωE has no effect. However a shear of the Doppler frequency ωE , ωE = ω¯E + δrωE will induce a shearing effect of the turbulent eddies and thus control the radial extent of the mode δm, so that one can locally achieve σ < 1 in order to drive a transport barrier. Second, one can modify the magnetic equilibrium so that the distance between the resonant surfaces is strongly increased in particular in a magnetic configuration with weak magnetic shear (dq/dr ≈ 0) so that ∆m,m+1 is strongly increased, ∆m,m+1 δm, also leading to σ < 1. Both control schemes for the generation of ITBs can be interpreted using the situation sketched on figure 2.1. The initial situation with large scale radial transport across the magnetic surfaces (so called L-mode) is indicated by the dashed lines and is governed by significant overlap between the resonances. The ITB control scheme aims at either reducing the width of the islands or increasing the distance between the resonances yielding a situation sketeched by the plain line in figure 2.1 where the overlap is too small and a region with vanishing turbulent transport, the ITB, develops between the resonances.
Experimental strategies in advanced scenarios comprising Internal Transport Barriers are based on means to enforce these two control schemes. In both cases they aim at modifying macroscopically the discharge conditions to fulfill locally the σ < 1 criterion. It thus appears interesting to devise a control scheme based on a less intrusive action that would allow one to modify the chaotic transport locally by the choice of an appropriate electrostatic perturbation hence leading to a local transport barrier.
The E × B model
For fusion plasmas, the magnetic field B is slowly variable with respect to the inverse of the Larmor radius ρL i.e: ρL|∇ ln B| 1. This fact allows the separation of the motion of a charged test particle into a slow motion (parallel to the lines of the magnetic field) and a fast motion (Larmor rotation). This fast motion is named gyromotion, around some gyrocenter. In first approximation the averaging of the gyromotion over the gyroangle gives the approximate trajectory of the charged particle. This averaging is the guiding-center approximation. In this approximation, the equations of motion of a charged test particle in the presence of a strong uniform magnetic field B = Bˆz, (where ˆz is the unit vector in the z direction) and of an external time-dependent electric field E = − ∇V1 are:
Table of contents :
1 Introduction
1.1 Particle dynamics: guiding center approach
1.2 Kinetic approach
1.3 Perturbation methods leading to the Gyrokinetic Maxwell-Vlasov equations
1.4 Continuous systems Hamiltonian formalism
1.4.1 Korteweg–de Vries
1.4.2 Maxwell-Vlasov
1.5 Hamiltonian perturbation theory
2 Barriers for the reduction of transport due to the E × B drift in magnetized plasmas
2.1 Introduction
2.2 Physical motivations and the E × B model
2.2.1 Physical motivations
2.2.2 The E × B model
2.3 Localized control theory of hamiltonian systems
2.3.1 The control term
2.3.2 Properties of the control term
2.4 Numerical investigations for the control term
2.4.1 Phase portrait for the exact control term
2.4.2 Robustness of the barrier
2.4.3 Energetical cost
2.5 Discussion and Conclusion
2.5.1 Main results
2.5.2 Discussion, open questions
3 Maxwell-Vlasov conservation law
3.1 Introduction and physical motivations
3.2 Maxwell-Vlasov equations and variational principles
3.3 Variational principle for perturbed Maxwell-Vlasov
3.3.1 Eulerian variations
3.3.2 Perturbed Maxwell-Vlasov equations
3.4 Momentum conservation law
3.4.1 Constrained variations
3.4.2 Noether method
3.4.3 Proof of Momentum conservation
3.4.4 Particle canonical momentum
3.4.5 Momentum conservation law in background separated form
3.5 Gyrokinetic variational principle
3.5.1 Eulerian variations
3.5.2 Gyrokinetic Maxwell-Vlasov equations
3.6 Gyrokinetic momentum conservation law
3.6.1 Noether Method
3.6.2 Proof of Gyrokinetic Momentum conservation
3.6.3 Gyrokinetic particle canonical momentum
3.7 Applications of the gyrokinetic momentum conservation law
3.7.1 Gyrokinetic momentum conservation law in background separated form .
3.7.2 Parallel momentum conservation law
3.7.3 Toroidal gyrokinetic momentum conservation law
3.7.4 Intrinsic plasma rotation mechanisms identification
3.7.5 Toroidal momentum evolution equation
3.8 Summary
4 Intrinsic guiding center theory
4.1 Noncanonical Hamiltonian structure
4.2 Dynamical reduction
4.2.1 Rescaled Hamiltonian dynamics
4.2.2 Gyrogauge transformation
4.2.3 Constant of motion and Hamiltonian normal form
4.3 Local dynamical reduction
4.3.1 Fixed and dynamical basis
4.3.2 Local Poisson bracket
4.3.3 Local equations of motion
4.3.4 Iterative construction of the constant of motion
4.4 Investigation of trapped particles trajectories
4.4.1 Dynamics in axisymmetric magnetic field
4.4.2 Trajectories
4.5 Intrinsic dynamical reduction
4.5.1 Hamiltonian normal form
4.5.2 Intrinsic basis
4.5.3 Intrinsic gyroaveraging
4.6 Intrinsic Hamiltonian normal form equation
4.6.1 Solution
4.6.2 Final result for second order solution
4.6.3 Discussion
4.7 Summary
5 Conclusions and discussion
6 Eulerian variations for Maxwell-Vlasov action
6.1 Eulerian variation for Maxwell part of action
6.2 Eulerian variation for Vlasov part of action
6.2.1 Noether’s term for Vlasov part
6.2.2 Vlasov equation on a 6 dimensional phase space
7 Proof of Momentum conservation
8 Particle canonical equation of motion
9 Gyrocenter magnetization
9.1 Functional dependence on B0
9.2 Hgc
9.3 φ1gc
10 Gyrokinetic momentum conservation application
10.1 Curvilinear coordinates
10.1.1 Covariant and contravariant representation
10.1.2 Metric tensor
10.1.3 Dyadic identity tensor and gradient
10.2 Momentum conservation law projection
10.2.1 ∂x ∂φ · ∇ · |E1|2I
10.2.2 ∂x ∂φ · ∇ · E1E1
10.2.3 Vlasov term
10.2.4 Final result: general axisymmetric geometry
10.2.5 Final result:cylindrical geometry
11 Local Poisson bracket
11.1 Calculation of the brackets {zi, zj}old
12 Hamiltonian Normal Form Series
12.1 Second order
13 Equations of motion
13.1 General axisymmetric geometry
13.2 Bi-cylindrical coordinates
