SLIDE 20 Mode coupling through the EP term (1).
Hybrid reduced O(ε0
3) MHD equations (HMGC) (Briguglio et al., Phys. Plasmas 2, 3711 (1995);
Wang et al., Phys. Plasmas 18, 052504 (2011)).
equilibrium-distribution-function terms in the same equation. Linear evolution of the system is exactly obtained by linear- izing the source terms and retaining only the unperturbed phase-space orbits, instead of looking at the low-amplitude stage of the nonlinear evolution, like for standard particle
- codes. The code has been validated by comparing numerical
results with the analytical prediction for the linear stability of toroidal Alfvin
- modes. This has been done for the case
k1 pH4 1 (pN = ,eLH , pdH , pBH), for which original analytical results are derived and presented for the first time, showing that finite k,pH terms play an analogous role to core-plasma ion finite-Larmor-radius terms, as expected. In the general case, we demonstrate that, for typical equilibrium parameters, a nonperturbative regime occurs, with stable (or weakly unstable) TAE and unstable upper
- KTAE. The dependence of the KTAE growth rate on the
ratio between the typical hot-particle velocity and the Alfven velocity is also considered. At a fixed value of PH, defined as &= 8 mHTHlB2, with nH and TH the hot-particle den- sity and temperature, respectively, the growth rate is maxi- mum at values uH-uA, which is consistent with the reso- nant character of the energetic particle drive. There is a very good qualitative and quantitative agreement between these results and the findings of the analytical treatment. ’ The paper is organized as follows. In Sec. II the govern- ing set of MHD and gyrokinetic equations is presented. In
- Sec. III the essential features of the hybrid code are de-
scribed and it is shown that theoretical predictions for the energetic-particle response in the perturbative limit are well
- reproduced. Section IV is devoted to the nonperturbative
problem; linear-simulation results are discussed and com- pared with the analytical solution. Conclusions are reported in Sec. V.
We begin the derivation of the fluid model equations starting from the resistive MHD equations, in which a driv- ing term related to an energetic-particle population is in- cluded:
p &7P-v.l-r~++5,
dB x=-cVxE, E= vJ- ;vxB, (5) J= &VxB, (6) V.B=O.
- Phys. Plasmas, Vol. 2, No. 10, October 1995
(7) In the above equations, v is the fluid velocity, J the plasma current, E the electric field, p and P are, respectively, the mass density and the scalar pressure of the bulk plasma, & is the stress tensor of the hot particles, y the ratio of the specific heats, c the speed of light, and dIdPa/& +v-V. Since tokamak plasmas are characterized by values of the safety factor 4 ( r) = 0( 1) and inverse aspect ratio E much lower than unity, the MHD equations can be simplified by expanding in powers of E. This procedure has been widely used, since the first paper of Strauss,*” both for analytical and for numerical work. At the leading order in E, 0( E*), and considering the low-p approximation, p- 0 ( e2), the reduced-MHD equations describe the plasma in the cylindri- cal approximation. The toroidal corrections enter the equa- tions at the next order in the inverse aspect ratio. These equa- tions, without the term representing the coupling to the energetic particle population, have already been used,30 and their derivation is only briefly reported here. Following the low-p tokamak ordering, it is possible to write VI BL 6.v
VA B, VL
!k, v-v, VW,)
vA V‘4 la BP
& FQ $)
,.
where beBIB is the unit vector of the equilibrium magnetic
- field. A cylindrical-coordinate
system (R&q) has been used, and the subscript 1 denotes components perpendicular to cp. The magnetic field can be written as. where @ is the poloidal-magnetic-field stream function, T,=RoBo j B. .is the vacuum magnetic field at R=R,, and I= 0( ~~1~) is given, at the leading order, by equilibrium
- corrections. Substituting Eq. (8) and Ohm’s law, Eq. (5), in
Faraday’s law, Eq. (3), we obtain where U is proportional to the scalar potential. Taking the cross product by Vq, Eq. (9) can be solved with respect to v, : v,=(R21Ro]VUxV~+O(~3vA). (10) Equation (10) states that, at the lowest order, the perpendicu- lar velocity is given by the EXB drift. Then, taking the cp component of Eq. (9). the following equation for the evolu- tion of the magnetic stream function is obtained:
+ O(E~VAB~>,
with the Grad-Shafranov operator A* defined by a I a a= A*=RZRZ+dZ2.
Briguglio et al.
(11)
3713 Upon applying the operator Vp.VXR2* * * to the mo- mentum equation, Eq. (2), the following equation for the evolution of the scalar potential is obtained: = -&B.VA*~+ ~v.[~~~v~+v~rr~~xVpl (12) where R2 D
6=7 P* K= &
RO
a+vyv,
Note that, in both Eq. (11) and Eq. (12), vP and f enter only at the fourth order in E. In Eq. (12) the dependence on the density gradient has been retained explicitly. With the par- ticular choice of the mass density pR* = 6Ri= cons& and us- ing the definition of v, given in Eq. (lo), the continuity equation, Eq. (I), is satisfied up to the third order. In the following, we will consider the pressure of the bulk plasma to be zero and the normalized mass density I; to be constant in space and time. Thus, only Eqs. (11) and (12) need to be
- evolved. As a boundary condition we take a rigid conducting
wail at the plasma edge. In order to close the set of reduced MHD equations (11) and (12), the hot-particle stress-tensor components can be evaluated by directly calculating the ap- propriate velocity momentum of the distribution function for the particle population moving in the perturbed fields + and u. In the view of numerical particle pushing, and in order to avoid too severe limitations
- n the time-step size, it is
worth3’532 following particle evolution in the gyrocenter- coordinate system g=( R,ti,p,&), where k is the gyro- center position, fi is the exactly conserved magnetic mo- mentum, j5 corresponds to the canonical parallel momentum, and 8 is the gyrophase. This corresponds to averaging the single-particle equations of motion over the fast Larmor pre- cession and allows one to retain the relevant finite Larmor radius effects without resolving the details of the gyromo- tion. The equations of motion in the gyrocenter coordinates can be derived by a straightforward extension of previous treatments.33-35 They take the form di! x=0, (13) di dt= m,L,
ii+$iixVlnB (
~)Vq]- g6.V In B. Here, 6?H, mH and fiH=eHB/mHc are, respectively, the energetic-particle charge, mass and Larmor frequency. Note that Eqs. (13) do not contain any dependence on the gy- rophase &. The fluctuating potentials (p and all are related to the stream functions U and $ by the relationships (14) (15) Note that j has been used, instead of the gyrocenter parallel velocity U=~-qlmH, in order to avoid the appearance of the time derivatives of the fluctuating vector potential in Eqs. (13). In terms of the gyrocenter coordinates, the hot-particle stress tensor can be written as i-I,(t,x) = -$
2
F,(t,ii,lii,j+ij(x-ii),
(161 where I is the identity tensor, Iij” Sij, FH(t,ii,ti,i) is the hot-particIe distribution function, and d62 includes the Jaco- bian of the transformation from canonical to gyrocenter co-
The distribution function F, satisfies the Vlasov equa- tion 07) with dkdt and d@dt given by Eqs. (13). It is convenient to define the perturbed distribution func- tion #H by the reIationship FH(t,fi,n;l,E7)=FHO(t,li,M,p)+S~,(t,fi,n;f,ls), (18) where RHO is an appropriate “‘Iowest-order” distribution
to be Maxwellian,
n&t? + $mH[i- (u$mH)]’
TH
, 09) where PZH( R) and TH are, respectively, the energetic-particle equilibrium density and (uniform) temperature, from Eqs. (13) and ( 17) the following equation for SF, is obtained: dp d&+,
dt a@ (20)
3714
- Phys. Plasmas, Vol. 2, No. 10, October
1995
with
Briguglio et al.
Psðt; xÞ ¼ 1 m2
s
ð d ZDZc!
Z
Fsðt; R; M; VÞ Xs M ms I þ bb
M ms
R Þ;
dR dt ¼ Vb þ es msXs b r/
msXs b rak þ
ms þ
Xs
ms
d M dt ¼ 0; d V dt ¼ 1 ms b es Xs
ms
ms rak
þ es msXs rak r/
M ms b r ln B: (15)
a||=(es/c)(R0/R)ψ; U=-cϕ/B0; ψ is the magnetic stream function; ϕ is the e.s. potential; “s” stay for EP species, thermal ions, … Zi=(R,M,V) are the gyrocenter coordinates, dZi/dt the phase-space velocities, (dZi/dt)pert the perturbed ones; Fs;eq the equilibrium distribution function of the “s” EP species. Vlasov eq. for gyrocenter distribution function Fs: Or, in term of δFs: ✓ ∂ ∂t + dZi dt ∂ ∂Zi ◆ ¯ Fs = 0 ¯ Fs = ¯ Fs;eq + δ ¯ Fs ✓ ∂ ∂t + dZi dt ∂ ∂Zi ◆ δ ¯ Fs = − ✓dZi dt ◆
pert
∂ ∂Zi ¯ Fs;eq
20
- G. Vlad - EUROfusion Science meeting - 16 May 2018
