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MODULE moments_eq_rhs
IMPLICIT NONE
PUBLIC :: compute_moments_eq_rhs
SUBROUTINE compute_moments_eq_rhs
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USE array
USE fields
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USE basic
USE prec_const
USE collision
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USE time_integration
USE geometry, ONLY: gradz_coeff, dlnBdz, Ckxky!, Gamma_phipar
USE calculus, ONLY : interp_z, grad_z, grad_z2
IMPLICIT NONE
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!compute ion moments_eq_rhs
CALL moments_eq_rhs(ips_i,ipe_i,ipgs_i,ipge_i,ijs_i,ije_i,ijgs_i,ijge_i,jarray_i,parray_i,&
xnipj, xnipp2j, xnipm2j, xnipjp1, xnipjm1, xnipp1j, xnipm1j,&
ynipp1j, ynipp1jm1, ynipm1j, ynipm1jm1, &
znipm1j, znipm1jp1, znipm1jm1, &
xphij_i, xphijp1_i, xphijm1_i, xpsij_i, xpsijp1_i, xpsijm1_i,&
kernel_i, nadiab_moments_i, ddz_nipj, interp_nipj, Sipj,&
moments_i(ipgs_i:ipge_i,ijgs_i:ijge_i,ikys:ikye,ikxs:ikxe,izgs:izge,updatetlevel),&
TColl_i, ddzND_nipj, diff_pi_coeff, diff_ji_coeff,&
moments_rhs_i(ips_i:ipe_i,ijs_i:ije_i,ikys:ikye,ikxs:ikxe,izs:ize,updatetlevel))
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!compute ion moments_eq_rhs
IF(KIN_E) &
CALL moments_eq_rhs(ips_e,ipe_e,ipgs_e,ipge_e,ijs_e,ije_e,ijgs_e,ijge_e,jarray_e,parray_e,&
xnepj, xnepp2j, xnepm2j, xnepjp1, xnepjm1, xnepp1j, xnepm1j,&
ynepp1j, ynepp1jm1, ynepm1j, ynepm1jm1, &
znepm1j, znepm1jp1, znepm1jm1, &
xphij_e, xphijp1_e, xphijm1_e, xpsij_e, xpsijp1_e, xpsijm1_e,&
kernel_e, nadiab_moments_e, ddz_nepj, interp_nepj, Sepj,&
moments_e(ipgs_e:ipge_e,ijgs_e:ijge_e,ikys:ikye,ikxs:ikxe,izgs:izge,updatetlevel),&
TColl_e, ddzND_nepj, diff_pe_coeff, diff_je_coeff,&
moments_rhs_e(ips_e:ipe_e,ijs_e:ije_e,ikys:ikye,ikxs:ikxe,izs:ize,updatetlevel))
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CONTAINS
!_____________________________________________________________________________!
!_____________________________________________________________________________!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!! moments_ RHS computation !!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
! This routine assemble the RHS of the moment hierarchy equations. It uses
! linear coefficients that are stored in arrays (xn*, yn* and zn*) computed in
! numerics_mod.F90. Otherwise it simply adds the collision term TColl_ that is
! computed in collision_mod.F90 and the nonlinear term Sapj_ computed in
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! nonlinear_mod.F90.
! All arguments of the subroutines are inputs only except the last one,
! moments_rhs_ that will contain the sum of every terms in the RHS.
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!_____________________________________________________________________________!
SUBROUTINE moments_eq_rhs(ips_,ipe_,ipgs_,ipge_,ijs_,ije_,ijgs_,ijge_,jarray_,parray_,&
xnapj_, xnapp2j_, xnapm2j_, xnapjp1_, xnapjm1_, xnapp1j_, xnapm1j_,&
ynapp1j_, ynapp1jm1_, ynapm1j_, ynapm1jm1_, &
znapm1j_, znapm1jp1_, znapm1jm1_, &
xphij_, xphijp1_, xphijm1_, xpsij_, xpsijp1_, xpsijm1_,&
kernel_, nadiab_moments_, ddz_napj_, interp_napj_, Sapj_,&
moments_, TColl_, ddzND_napj_, diff_p_coeff_, diff_j_coeff_, moments_rhs_)
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IMPLICIT NONE
!! INPUTS
INTEGER, INTENT(IN) :: ips_, ipe_, ipgs_, ipge_, ijs_, ije_, ijgs_, ijge_
INTEGER, DIMENSION(ips_:ipe_), INTENT(IN) :: parray_
INTEGER, DIMENSION(ijs_:ije_), INTENT(IN) :: jarray_
REAL(dp), DIMENSION(ips_:ipe_,ijs_:ije_), INTENT(IN) :: xnapj_
REAL(dp), DIMENSION(ips_:ipe_), INTENT(IN) :: xnapp2j_, xnapm2j_
REAL(dp), DIMENSION(ijs_:ije_), INTENT(IN) :: xnapjp1_, xnapjm1_
REAL(dp), DIMENSION(ips_:ipe_), INTENT(IN) :: xnapp1j_, xnapm1j_
REAL(dp), DIMENSION(ips_:ipe_,ijs_:ije_), INTENT(IN) :: ynapp1j_, ynapp1jm1_, ynapm1j_, ynapm1jm1_
REAL(dp), DIMENSION(ips_:ipe_,ijs_:ije_), INTENT(IN) :: znapm1j_, znapm1jp1_, znapm1jm1_
REAL(dp), DIMENSION(ips_:ipe_,ijs_:ije_), INTENT(IN) :: xphij_, xphijp1_, xphijm1_
REAL(dp), DIMENSION(ips_:ipe_,ijs_:ije_), INTENT(IN) :: xpsij_, xpsijp1_, xpsijm1_
REAL(dp), DIMENSION(ijgs_:ijge_,ikys:ikye,ikxs:ikxe,izgs:izge,0:1),INTENT(IN) :: kernel_
COMPLEX(dp), DIMENSION(ipgs_:ipge_,ijgs_:ijge_,ikys:ikye,ikxs:ikxe,izgs:izge),INTENT(IN) :: nadiab_moments_
COMPLEX(dp), DIMENSION(ipgs_:ipge_,ijgs_:ijge_,ikys:ikye,ikxs:ikxe,izgs:izge),INTENT(IN) :: ddz_napj_
COMPLEX(dp), DIMENSION(ipgs_:ipge_,ijgs_:ijge_,ikys:ikye,ikxs:ikxe,izgs:izge),INTENT(IN) :: interp_napj_
COMPLEX(dp), DIMENSION( ips_:ipe_, ijs_:ije_, ikys:ikye,ikxs:ikxe, izs:ize), INTENT(IN) :: Sapj_
COMPLEX(dp), DIMENSION(ipgs_:ipge_,ijgs_:ijge_,ikys:ikye,ikxs:ikxe,izgs:izge),INTENT(IN) :: moments_
COMPLEX(dp), DIMENSION( ips_:ipe_, ijs_:ije_, ikys:ikye,ikxs:ikxe, izs:ize), INTENT(IN) :: TColl_
COMPLEX(dp), DIMENSION(ipgs_:ipge_,ijgs_:ijge_,ikys:ikye,ikxs:ikxe,izgs:izge),INTENT(IN) :: ddzND_napj_
REAL(dp), INTENT(IN) :: diff_p_coeff_, diff_j_coeff_
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!! OUTPUT
COMPLEX(dp), DIMENSION( ips_:ipe_, ijs_:ije_, ikys:ikye,ikxs:ikxe, izs:ize),INTENT(OUT) :: moments_rhs_
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INTEGER :: p_int, j_int ! loops indices and polynom. degrees
REAL(dp) :: kx, ky, kperp2
COMPLEX(dp) :: Tnapj, Tnapp2j, Tnapm2j, Tnapjp1, Tnapjm1 ! Terms from b x gradB and drives
COMPLEX(dp) :: Tnapp1j, Tnapm1j, Tnapp1jm1, Tnapm1jm1 ! Terms from mirror force with non adiab moments_
COMPLEX(dp) :: Tpar, Tmir, Tphi, Tpsi
COMPLEX(dp) :: Mperp, Mpara, Dphi, Dpsi
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COMPLEX(dp) :: Unapm1j, Unapm1jp1, Unapm1jm1 ! Terms from mirror force with adiab moments_
COMPLEX(dp) :: i_kx,i_ky,phikykxz, psikykxz
! Measuring execution time
CALL cpu_time(t0_rhs)
! Spatial loops
zloop : DO iz = izs,ize
kxloop : DO ikx = ikxs,ikxe
kx = kxarray(ikx) ! radial wavevector
i_kx = imagu * kx ! radial derivative
kyloop : DO iky = ikys,ikye
ky = kyarray(iky) ! binormal wavevector
i_ky = imagu * ky ! binormal derivative
psikykxz = psi(iky,ikx,iz)! tmp psi value
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! Kinetic loops
jloop : DO ij = ijs_, ije_ ! This loop is from 1 to jmaxi+1
j_int = jarray_(ij)
ploop : DO ip = ips_, ipe_ ! Hermite loop
p_int = parray_(ip) ! Hermite degree
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eo = MODULO(p_int,2) ! Indicates if we are on odd or even z grid
kperp2= kparray(iky,ikx,iz,eo)**2
IF((CLOS .NE. 1) .OR. (p_int+2*j_int .LE. dmaxe)) THEN ! for the closure scheme
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!! Compute moments_ mixing terms
! Perpendicular dynamic
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! term propto n^{p,j}
Tnapj = xnapj_(ip,ij)* nadiab_moments_(ip,ij,iky,ikx,iz)
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! term propto n^{p+2,j}
Tnapp2j = xnapp2j_(ip) * nadiab_moments_(ip+pp2,ij,iky,ikx,iz)
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! term propto n^{p-2,j}
Tnapm2j = xnapm2j_(ip) * nadiab_moments_(ip-pp2,ij,iky,ikx,iz)
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! term propto n^{p,j+1}
Tnapjp1 = xnapjp1_(ij) * nadiab_moments_(ip,ij+1,iky,ikx,iz)
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! term propto n^{p,j-1}
Tnapjm1 = xnapjm1_(ij) * nadiab_moments_(ip,ij-1,iky,ikx,iz)
! Perpendicular magnetic term (curvature and gradient drifts)
Mperp = imagu*Ckxky(iky,ikx,iz,eo)*(Tnapj + Tnapp2j + Tnapm2j + Tnapjp1 + Tnapjm1)
! Parallel dynamic
! ddz derivative for Landau damping term
Tpar = xnapp1j_(ip) * ddz_napj_(ip+1,ij,iky,ikx,iz) &
+ xnapm1j_(ip) * ddz_napj_(ip-1,ij,iky,ikx,iz)
Tnapp1j = ynapp1j_ (ip,ij) * interp_napj_(ip+1,ij ,iky,ikx,iz)
Tnapp1jm1 = ynapp1jm1_(ip,ij) * interp_napj_(ip+1,ij-1,iky,ikx,iz)
Tnapm1j = ynapm1j_ (ip,ij) * interp_napj_(ip-1,ij ,iky,ikx,iz)
Tnapm1jm1 = ynapm1jm1_(ip,ij) * interp_napj_(ip-1,ij-1,iky,ikx,iz)
Unapm1j = znapm1j_ (ip,ij) * interp_napj_(ip-1,ij ,iky,ikx,iz)
Unapm1jp1 = znapm1jp1_(ip,ij) * interp_napj_(ip-1,ij+1,iky,ikx,iz)
Unapm1jm1 = znapm1jm1_(ip,ij) * interp_napj_(ip-1,ij-1,iky,ikx,iz)
Tmir = dlnBdz(iz,eo)*(Tnapp1j + Tnapp1jm1 + Tnapm1j + Tnapm1jm1 +&
Unapm1j + Unapm1jp1 + Unapm1jm1)
! Parallel magnetic term (Landau damping and the mirror force)
Mpara = gradz_coeff(iz,eo)*(Tpar + Tmir)
!! Electrical potential term
IF ( p_int .LE. 2 ) THEN ! kronecker p0 p1 p2
Dphi =i_ky*( xphij_ (ip,ij)*kernel_(ij ,iky,ikx,iz,eo) &
+xphijp1_(ip,ij)*kernel_(ij+1,iky,ikx,iz,eo) &
+xphijm1_(ip,ij)*kernel_(ij-1,iky,ikx,iz,eo) )*phi(iky,ikx,iz)
ELSE
Tphi = 0._dp
ENDIF
!! Vector potential term
IF ( (p_int .LE. 3) .AND. (p_int .GE. 1) ) THEN ! Kronecker p1 or p3
Dpsi =-i_ky*( xpsij_ (ip,ij)*kernel_(ij ,iky,ikx,iz,eo) &
+xpsijp1_(ip,ij)*kernel_(ij+1,iky,ikx,iz,eo) &
+xpsijm1_(ip,ij)*kernel_(ij-1,iky,ikx,iz,eo))*psi(iky,ikx,iz)
ENDIF
!! Sum of all RHS terms
moments_rhs_(ip,ij,iky,ikx,iz) = &
! Nonlinear term Sapj_ = {phi,f}
- Sapj_(ip,ij,iky,ikx,iz) &
! Perpendicular magnetic term
- Mperp &
! Parallel magnetic term
- Mpara &
! Drives (density + temperature gradients)
- (Dphi + Dpsi) &
! Collision term
+ TColl_(ip,ij,iky,ikx,iz) &
! Perpendicular pressure effects (electromagnetic term) (TO CHECK)
- i_ky*beta*dpdx * (Tnapj + Tnapp2j + Tnapm2j + Tnapjp1 + Tnapjm1)&
! Parallel drive term (should be negligible, to test)
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! -Gamma_phipar(iz,eo)*Tphi*ddz_phi(iky,ikx,iz) &
! Numerical perpendicular hyperdiffusion
-mu_x*diff_kx_coeff*kx**N_HD*moments_(ip,ij,iky,ikx,iz) &
-mu_y*diff_ky_coeff*ky**N_HD*moments_(ip,ij,iky,ikx,iz) &
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! Numerical parallel hyperdiffusion "mu_z*ddz**4" see Pueschel 2010 (eq 25)
-mu_z*diff_dz_coeff*ddzND_napj_(ip,ij,iky,ikx,iz)
! GX like Hermite hypercollisions see Mandell et al. 2023 (eq 3.23), unadvised to use it
IF (p_int .GT. 2) &
moments_rhs_(ip,ij,iky,ikx,iz) = &
moments_rhs_(ip,ij,iky,ikx,iz) - mu_p*diff_pe_coeff*p_int**6*moments_(ip,ij,iky,ikx,iz)
IF (j_int .GT. 1) &
moments_rhs_(ip,ij,iky,ikx,iz) = &
moments_rhs_(ip,ij,iky,ikx,iz) - mu_j*diff_je_coeff*j_int**6*moments_(ip,ij,iky,ikx,iz)
! fourth order numerical diffusion in vpar
! IF( (ip-4 .GT. 0) .AND. (num_procs_p .EQ. 1) ) &
! ! Numerical parallel velocity hyperdiffusion "+ dvpar4 g_a" see Pueschel 2010 (eq 33)
! ! (not used often so not parallelized)
! moments_rhs_(ip,ij,iky,ikx,iz) = &
! moments_rhs_(ip,ij,iky,ikx,iz) &
! + mu_p * moments_(ip-4,ij,iky,ikx,iz)
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ELSE
moments_rhs_(ip,ij,iky,ikx,iz) = 0._dp
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ENDIF
END DO ploop
END DO jloop
END DO kyloop
END DO kxloop
END DO zloop
! Execution time end
CALL cpu_time(t1_rhs)
tc_rhs = tc_rhs + (t1_rhs-t0_rhs)
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END SUBROUTINE moments_eq_rhs
!_____________________________________________________________________________!
!_____________________________________________________________________________!
END SUBROUTINE compute_moments_eq_rhs
SUBROUTINE add_Maxwellian_background_terms
! This routine is meant to add the terms rising from the magnetic operator,
! i.e. (B x k_gB) Grad, applied on the background Maxwellian distribution
! (x_a + spar^2)(b x k_gB) GradFaM
! It gives birth to kx=ky=0 sources terms (averages) that hit moments_ 00, 20,
! 40, 01,02, 21 with background gradient dependences.
USE prec_const
USE time_integration, ONLY : updatetlevel
USE model, ONLY: taue_qe, taui_qi, k_Ni, k_Ne, k_Ti, k_Te, KIN_E
USE array, ONLY: moments_rhs_e, moments_rhs_i
USE grid, ONLY: contains_kx0, contains_ky0, ikx_0, iky_0,&
ips_e,ipe_e,ijs_e,ije_e,ips_i,ipe_i,ijs_i,ije_i,&
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zarray, izs,ize,&
ip,ij
IMPLICIT NONE
real(dp), DIMENSION(izs:ize) :: sinz
sinz(izs:ize) = SIN(zarray(izs:ize,0))
IF(contains_kx0 .AND. contains_ky0) THEN
IF(KIN_E) THEN
DO ip = ips_e,ipe_e
DO ij = ijs_e,ije_e
SELECT CASE(ij-1)
CASE(0) ! j = 0
SELECT CASE (ip-1)
CASE(0) ! Na00 term
moments_rhs_e(ip,ij,iky_0,ikx_0,izs:ize,updatetlevel) = moments_rhs_e(ip,ij,iky_0,ikx_0,izs:ize,updatetlevel)&
+taue_qe * sinz(izs:ize) * (1.5_dp*k_Ne - 1.125_dp*k_Te)
CASE(2) ! Na20 term
moments_rhs_e(ip,ij,iky_0,ikx_0,izs:ize,updatetlevel) = moments_rhs_e(ip,ij,iky_0,ikx_0,izs:ize,updatetlevel)&
+taue_qe * sinz(izs:ize) * (SQRT2*0.5_dp*k_Ne - 2.75_dp*k_Te)
CASE(4) ! Na40 term
moments_rhs_e(ip,ij,iky_0,ikx_0,izs:ize,updatetlevel) = moments_rhs_e(ip,ij,iky_0,ikx_0,izs:ize,updatetlevel)&
+taue_qe * sinz(izs:ize) * SQRT6*0.75_dp*k_Te
END SELECT
CASE(1) ! j = 1
SELECT CASE (ip-1)
CASE(0) ! Na01 term
moments_rhs_e(ip,ij,iky_0,ikx_0,izs:ize,updatetlevel) = moments_rhs_e(ip,ij,iky_0,ikx_0,izs:ize,updatetlevel)&
-taue_qe * sinz(izs:ize) * (k_Ne + 3.5_dp*k_Te)
CASE(2) ! Na21 term
moments_rhs_e(ip,ij,iky_0,ikx_0,izs:ize,updatetlevel) = moments_rhs_e(ip,ij,iky_0,ikx_0,izs:ize,updatetlevel)&
-taue_qe * sinz(izs:ize) * SQRT2*k_Te
END SELECT
CASE(2) ! j = 2
SELECT CASE (ip-1)
CASE(0) ! Na02 term
moments_rhs_e(ip,ij,iky_0,ikx_0,izs:ize,updatetlevel) = moments_rhs_e(ip,ij,iky_0,ikx_0,izs:ize,updatetlevel)&
+taue_qe * sinz(izs:ize) * 2._dp*k_Te
END SELECT
END SELECT
ENDDO
ENDDO
ENDIF
DO ip = ips_i,ipe_i
DO ij = ijs_i,ije_i
SELECT CASE(ij-1)
CASE(0) ! j = 0
SELECT CASE (ip-1)
CASE(0) ! Na00 term
moments_rhs_i(ip,ij,iky_0,ikx_0,izs:ize,updatetlevel) = moments_rhs_i(ip,ij,iky_0,ikx_0,izs:ize,updatetlevel)&
+taui_qi * sinz(izs:ize) * (1.5_dp*k_Ni - 1.125_dp*k_Ti)
CASE(2) ! Na20 term
moments_rhs_i(ip,ij,iky_0,ikx_0,izs:ize,updatetlevel) = moments_rhs_i(ip,ij,iky_0,ikx_0,izs:ize,updatetlevel)&
+taui_qi * sinz(izs:ize) * (SQRT2*0.5_dp*k_Ni - 2.75_dp*k_Ti)
CASE(4) ! Na40 term
moments_rhs_i(ip,ij,iky_0,ikx_0,izs:ize,updatetlevel) = moments_rhs_i(ip,ij,iky_0,ikx_0,izs:ize,updatetlevel)&
+taui_qi * sinz(izs:ize) * SQRT6*0.75_dp*k_Ti
END SELECT
CASE(1) ! j = 1
SELECT CASE (ip-1)
CASE(0) ! Na01 term
moments_rhs_i(ip,ij,iky_0,ikx_0,izs:ize,updatetlevel) = moments_rhs_i(ip,ij,iky_0,ikx_0,izs:ize,updatetlevel)&
-taui_qi * sinz(izs:ize) * (k_Ni + 3.5_dp*k_Ti)
CASE(2) ! Na21 term
moments_rhs_i(ip,ij,iky_0,ikx_0,izs:ize,updatetlevel) = moments_rhs_i(ip,ij,iky_0,ikx_0,izs:ize,updatetlevel)&
-taui_qi * sinz(izs:ize) * SQRT2*k_Ti
END SELECT
CASE(2) ! j = 2
SELECT CASE (ip-1)
CASE(0) ! Na02 term
moments_rhs_i(ip,ij,iky_0,ikx_0,izs:ize,updatetlevel) = moments_rhs_i(ip,ij,iky_0,ikx_0,izs:ize,updatetlevel)&
+taui_qi * sinz(izs:ize) * 2._dp*k_Ti
END SELECT
END SELECT
ENDDO
ENDDO
ENDIF
END SUBROUTINE
END MODULE moments_eq_rhs