Determinación de la cantidad total de correas necesarias

The gyrokinetic Vlasov equation must be combined with field equations to form a closed system of equations. In deriving Eq. (2.70), magnetic field fluctuations were neglected appropriate to the limit of low β, so only an equation for the elec- trostatic field is required to close the system. Whilst this equation is often called the ‘gyrokinetic Poisson equation’, the equation solved is actually the constraint of neutrality

∑ s

Zens(x)=0, (2.74)

where the sum is over all species. In reality, the charge density is not exactly zero, but it is assumed to be zero over the quasineutral length and timescales at which gyrokinetics is valid. By assuming zero charge density, the actual Poisson equation cannot be used to find the field and so is replaced by the quasineutrality closure. With the quasineutrality constraint, the Laplacian operator need not be inverted to find the field, which greatly simplifies the solution. The chief complication then in solving this equation is that the densities must be computed in physical parti- cle phase space(x,v), not in the gyrocentre phase space in which the gyrokinetic equation is formulated [86]. In gyrocentre phase space quasineutrality is represented as

¯

%(X)+%pol(X)=0, (2.75)

where ¯% is the gyrocentre charge density and %pol = −∇ ⋅Pgy is the gyrocentre po-

larisation density, with Pgy the gyrocentre polarisation vector. The gyrocentre po-

larisation density describes the difference between the actual charge density and the charge density of gyrocentres at a given point, which may be pictured as the Boltzmann density response of the gyrocentres due to the average variation in the potential over a gyro-orbit,φ(x)−⟨φ⟩(X). This can also be understood as the con- sequence of the polarisation drift Eq. (1.6) averaged over the gyro-orbit timescale.

To close the gyrokinetic Vlasov-Poisson system, a model for polarisation must be adopted which is consistent with the formulation of the gyrokinetic equation. If the field equations are obtained by variational principles from the same symplectic Lagrangian / Hamiltonian used to derive the equations of motion, then energy consistency can be guaranteed by including the next higher order energies in the field terms [54, 71, 72, 81]. The system Lagrangian is obtained by integrating the gyrocentre particle Lagrangian Γ Eq. (2.27) over all of phase space

L=∑

s ∫

d3v∫ d3xftotΓ. (2.76)

tional derivative [87] of this system Lagrangian with respect to δφ. Since Γ is in symplectic form, all dependence on the fields is contained in the Hamiltonian and the field equation may be written

∑ s

δ(ftotH)

δφ =0. (2.77)

For a consistent system, one order further must be kept inH than was used for the equations of motion [54,71,72, 81]. In the δf formulation, the field equation may be written as ∑ s δ[(FM+f)(H0+H1)] δφ =− ∑s δ[FMH2] δφ , (2.78)

which contains the simplest model of linearised polarisation. For the functional derivative, only the field dependent part of the Hamiltonian HE =H01E+H2E is

required, with H01E =Ze⟨φ⟩, H2E = Z2e2 2T (⟨φ⟩ 2_−φ2_), (2.79) where the first order part comes from Eq. (2.28), and the second order part is as used in other codes [72], and represents physically the kinetic energy in thevE drift with

a Taylor approximation for the gyroaverage operator. HE is therefore a function

of φ and its gyroaverage Gφ ≡ ⟨φ⟩ only, which are treated as separate variables for the purposes of the functional derivative. The exact form of the gyroaverage operator G will be demonstrated in Sec. 3.4, here it is sufficient to know that it is a spatially Hermetian operator. In this simple case then, all the functions of φ commute through the spatial integral in the system Lagrangian (and thus through theδ operator), and the functional derivative may be written

∑ s ∫ d3v[(FM +f) ∂H01E ∂Gφ δGφ+FM ∂H2E ∂φ δφ+FM ∂H2E ∂Gφ δGφ]=0, (2.80) since∂H01E/∂φ=0. Evaluating the nonzero derivatives and commuting the operator

Gleads to ∑ s ∫ d3v[Zse(GFM +Gf)+ Z_s2e2 Ts ( FMφ−GFMGφ)]δφ=0. (2.81)

TheFM in the first term is a background quantity on which the gyroaverageGhas

no effect. Performing the velocity space integral over this term yields the charge densityZens, which becomes zero when summed over the species. The field equation

can then be written as

∑ s ∫ d3v[ZseGf(X)+FM Z_s2e2 Ts ( 1−G2)φ(x)]=0. (2.82)

This equation is used to calculate the potential at a point x in real space. The first term represents the gyrocentre charge density ¯%integrated over all gyrocentres passing throughxwhilst the second represents the linearised gyrocentre polarisation density%pol for all the gyro-orbits passing through x(which is why the Goperator

acts twice on φ). These physical interpretations, may be seen more clearly in the alternative derivation which starts from quasineutrality in the real phase space; however, this requires full knowledge of the transformation to the gyrocentre phase space which is defined by the Lie transforms [54]. In Sec. 3.5, this equation is rewritten in a form suitable for an efficient numerical implementation, which allows the operatorGto be evaluated over the velocity space integral.