src/equilibrium.f90: improve handling of the ψ normalisation
- Hide the implementation of the re-normalisation of the ψ(R,z) spline by adjusting the spline coefficients, instead of shifting and rescaling after each evaluation. - Correct the value of `psia` = ψ(X point) - ψ(O point) after the ψ(R,z) spline has been re-normalised. This fixes another instance of decoupling between the values of X and ∇X that introduce a systematic error in the numerical integration of the raytracing equations. Here the issue is caused by the different normalisations used, specifically X=X(ψ_n') and ∇X=∇X(ψ_n), where ψ_n' is the re-normalised spline of the normalised flux and ψ_n the spline of the normalised flux. See d0a5a9f for more details on problems resulting from this error. - Change `equian` and `equinum_psi` to return the normalised values for both the flux and its derivatives, to avoid confusions. Callers that needed the unnormalised derivatives now multiply explicitly by `psia`.
This commit is contained in:
Michele Guerini Rocco
parent
127d574be7
commit
98dda6d6fa
GPG Key ID:
BFBAF4C975F76450
@ -29,7 +29,7 @@ module equilibrium
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! Splines
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! Splines
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type(spline_1d), save :: fpol_spline ! Poloidal current function F(ψ)
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type(spline_1d), save :: fpol_spline ! Poloidal current function F(ψ)
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type(spline_2d), save :: psi_spline ! Poloidal flux ψ(R,z)
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type(spline_2d), save :: psi_spline ! Normalised poloidal flux ψ(R,z)
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type(spline_simple), save :: q_spline ! Safey factor q(ψ)
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type(spline_simple), save :: q_spline ! Safey factor q(ψ)
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type(linear_1d), save :: rhop_spline, rhot_spline ! Normalised radii ρ_p(ρ_t), ρ_t(ρ_p)
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type(linear_1d), save :: rhop_spline, rhot_spline ! Normalised radii ρ_p(ρ_t), ρ_t(ρ_p)
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@ -39,8 +39,6 @@ module equilibrium
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! More parameters
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! More parameters
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real(wp_), save :: fpolas ! Poloidal current at the edge (r=a), F_a
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real(wp_), save :: fpolas ! Poloidal current at the edge (r=a), F_a
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real(wp_), save :: psia ! Poloidal flux at the edge (r=a), ψ_a
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real(wp_), save :: psia ! Poloidal flux at the edge (r=a), ψ_a
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real(wp_), save :: psiant ! Normalised poloidal flux at the true edge
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real(wp_), save :: psinop ! Normalised poloidal flux at the true O point
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real(wp_), save :: phitedge ! Toroidal flux at the edge
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real(wp_), save :: phitedge ! Toroidal flux at the edge
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real(wp_), save :: btaxis ! B₀: B_φ = B₀R₀/R
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real(wp_), save :: btaxis ! B₀: B_φ = B₀R₀/R
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real(wp_), save :: rmaxis, zmaxis ! R,z of the magnetic axis (O point)
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real(wp_), save :: rmaxis, zmaxis ! R,z of the magnetic axis (O point)
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@ -65,8 +63,7 @@ module equilibrium
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public kspl, psi_spline, q_spline, points_tgo, model
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public kspl, psi_spline, q_spline, points_tgo, model
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! Members exposed to gray_core and more
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! Members exposed to gray_core and more
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public psia, psiant, psinop
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public psia, phitedge
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public phitedge
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public btaxis, rmaxis, zmaxis, sgnbphi
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public btaxis, rmaxis, zmaxis, sgnbphi
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public btrcen, rcen
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public btrcen, rcen
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public rmnm, rmxm, zmnm, zmxm
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public rmnm, rmxm, zmnm, zmxm
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@ -416,6 +413,7 @@ contains
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integer :: nr, nz, nrest, nzest, npsest, nrz, npsi, nbnd, ibinf, ibsup
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integer :: nr, nz, nrest, nzest, npsest, nrz, npsi, nbnd, ibinf, ibsup
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real(wp_) :: tension, rax0, zax0, psinoptmp, psinxptmp
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real(wp_) :: tension, rax0, zax0, psinoptmp, psinxptmp
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real(wp_) :: rbmin, rbmax, rbinf, rbsup, r1, z1
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real(wp_) :: rbmin, rbmax, rbinf, rbsup, r1, z1
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real(wp_) :: psiant, psinop
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real(wp_), dimension(size(data%psinr)) :: rhotn
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real(wp_), dimension(size(data%psinr)) :: rhotn
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real(wp_), dimension(:), allocatable :: rv1d, zv1d, fvpsi, wf
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real(wp_), dimension(:), allocatable :: rv1d, zv1d, fvpsi, wf
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integer :: ixploc, info, i, j, ij
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integer :: ixploc, info, i, j, ij
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@ -555,12 +553,13 @@ contains
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fpolas = fpol_spline%eval(data%psinr(npsi))
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fpolas = fpol_spline%eval(data%psinr(npsi))
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sgnbphi = sign(one ,fpolas)
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sgnbphi = sign(one ,fpolas)
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! Re-normalize ψ after spline computation
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! Re-normalize ψ_n after the spline computation
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! Note: this ensures 0 ≤ ψ_n'(R,z) < 1 inside the plasma
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! Start with un-corrected psi
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! Start with un-corrected ψ_n
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psia = data%psia
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psia = data%psia
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psinop = 0
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psinop = 0 ! ψ_n(O point)
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psiant = 1
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psiant = 1 ! ψ_n(X point) - ψ_n(O point)
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! Use provided boundary to set an initial guess
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! Use provided boundary to set an initial guess
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! for the search of O/X points
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! for the search of O/X points
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@ -643,6 +642,22 @@ contains
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call log_info(msg, mod='equilibrium', proc='set_equil_spline')
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call log_info(msg, mod='equilibrium', proc='set_equil_spline')
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end if
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end if
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! Adjust all the B-spline coefficients
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! Note: since ψ_n(R,z) = Σ_ij c_ij B_i(R)B_j(z), to correct ψ_n
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! it's enough to correct (shift and scale) the c_ij
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psi_spline%coeffs = (psi_spline%coeffs - psinop) / psiant
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! Same for the derivatives
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psi_spline%partial(1,0)%ptr = psi_spline%partial(1,0)%ptr / psiant
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psi_spline%partial(0,1)%ptr = psi_spline%partial(0,1)%ptr / psiant
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psi_spline%partial(2,0)%ptr = psi_spline%partial(2,0)%ptr / psiant
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psi_spline%partial(0,2)%ptr = psi_spline%partial(0,2)%ptr / psiant
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psi_spline%partial(1,1)%ptr = psi_spline%partial(1,1)%ptr / psiant
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! Do the opposite scaling to preserve un-normalised values
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! Note: this is only used for the poloidal magnetic field
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psia = psia * psiant
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! Save Bt value on axis (required in flux_average and used in Jcd def)
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! Save Bt value on axis (required in flux_average and used in Jcd def)
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! and vacuum value B0 at ref. radius data%rvac (used in Jcd_astra def)
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! and vacuum value B0 at ref. radius data%rvac (used in Jcd_astra def)
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@ -804,12 +819,12 @@ contains
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if (R <= rmxm .and. R >= rmnm .and. &
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if (R <= rmxm .and. R >= rmnm .and. &
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z <= zmxm .and. z >= zmnm) then
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z <= zmxm .and. z >= zmnm) then
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if (present(psi)) psi = (psi_spline%eval(R, z) - psinop) / psiant
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if (present(psi)) psi = psi_spline%eval(R, z)
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if (present(dpsidr)) dpsidr = psi_spline%deriv(R, z, 1, 0) * psia
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if (present(dpsidr)) dpsidr = psi_spline%deriv(R, z, 1, 0)
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if (present(dpsidz)) dpsidz = psi_spline%deriv(R, z, 0, 1) * psia
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if (present(dpsidz)) dpsidz = psi_spline%deriv(R, z, 0, 1)
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if (present(ddpsidrr)) ddpsidrr = psi_spline%deriv(R, z, 2, 0) * psia
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if (present(ddpsidrr)) ddpsidrr = psi_spline%deriv(R, z, 2, 0)
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if (present(ddpsidzz)) ddpsidzz = psi_spline%deriv(R, z, 0, 2) * psia
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if (present(ddpsidzz)) ddpsidzz = psi_spline%deriv(R, z, 0, 2)
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if (present(ddpsidrz)) ddpsidrz = psi_spline%deriv(R, z, 1, 1) * psia
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if (present(ddpsidrz)) ddpsidrz = psi_spline%deriv(R, z, 1, 1)
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else
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else
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if (present(psi)) psi = -1
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if (present(psi)) psi = -1
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if (present(dpsidr)) dpsidr = 0
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if (present(dpsidr)) dpsidr = 0
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@ -863,13 +878,13 @@ contains
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! ψ_n = ρ_p²(R,z) = [(R-R₀)² + (z-z₀)²]/a²
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! ψ_n = ρ_p²(R,z) = [(R-R₀)² + (z-z₀)²]/a²
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if (present(psin)) psin = rho_p**2
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if (present(psin)) psin = rho_p**2
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! ∂ψ/∂R, ∂ψ/∂z, note that ψ = ψ_n⋅ψ_a
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! ∂ψ/∂R, ∂ψ/∂z
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if (present(dpsidr)) dpsidr = 2*(R - model%R0)/model%a**2 * psia
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if (present(dpsidr)) dpsidr = 2*(R - model%R0)/model%a**2
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if (present(dpsidz)) dpsidz = 2*(z - model%z0)/model%a**2 * psia
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if (present(dpsidz)) dpsidz = 2*(z - model%z0)/model%a**2
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! ∂²ψ/∂R², ∂²ψ/∂z², ∂²ψ/∂R∂z
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! ∂²ψ/∂R², ∂²ψ/∂z², ∂²ψ/∂R∂z
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if (present(ddpsidrr)) ddpsidrr = 2/model%a**2 * psia
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if (present(ddpsidrr)) ddpsidrr = 2/model%a**2
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if (present(ddpsidzz)) ddpsidzz = 2/model%a**2 * psia
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if (present(ddpsidzz)) ddpsidzz = 2/model%a**2
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if (present(ddpsidrz)) ddpsidrz = 0
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if (present(ddpsidrz)) ddpsidrz = 0
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! F(ψ) = B₀⋅R₀, a constant
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! F(ψ) = B₀⋅R₀, a constant
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@ -1120,8 +1135,7 @@ contains
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call log_error(msg, mod='equilibrium', proc='psi_raxis')
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call log_error(msg, mod='equilibrium', proc='psi_raxis')
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end if
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end if
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val=psin*psiant+psinop
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call sproota(psin, psi_spline%knots_x, psi_spline%nknots_x, &
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call sproota(val, psi_spline%knots_x, psi_spline%nknots_x, &
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czc, zeroc, mest, m, ier)
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czc, zeroc, mest, m, ier)
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r1=zeroc(1)
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r1=zeroc(1)
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r2=zeroc(2)
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r2=zeroc(2)
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@ -1186,15 +1200,15 @@ contains
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select case(iflag)
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select case(iflag)
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case(1)
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case(1)
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call equinum_psi(x(1),x(2),dpsidr=dpsidr,dpsidz=dpsidz)
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call equinum_psi(x(1),x(2),dpsidr=dpsidr,dpsidz=dpsidz)
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fvec(1) = dpsidr/psia
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fvec(1) = dpsidr
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fvec(2) = dpsidz/psia
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fvec(2) = dpsidz
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case(2)
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case(2)
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call equinum_psi(x(1),x(2),ddpsidrr=ddpsidrr,ddpsidzz=ddpsidzz, &
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call equinum_psi(x(1),x(2),ddpsidrr=ddpsidrr,ddpsidzz=ddpsidzz, &
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ddpsidrz=ddpsidrz)
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ddpsidrz=ddpsidrz)
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fjac(1,1) = ddpsidrr/psia
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fjac(1,1) = ddpsidrr
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fjac(1,2) = ddpsidrz/psia
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fjac(1,2) = ddpsidrz
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fjac(2,1) = ddpsidrz/psia
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fjac(2,1) = ddpsidrz
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fjac(2,2) = ddpsidzz/psia
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fjac(2,2) = ddpsidzz
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case default
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case default
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write (msg, '("invalid iflag: ",g0)')
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write (msg, '("invalid iflag: ",g0)')
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call log_error(msg, mod='equilibrium', proc='fcnox')
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call log_error(msg, mod='equilibrium', proc='fcnox')
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@ -1260,14 +1274,14 @@ contains
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case(1)
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case(1)
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call equinum_psi(x(1),x(2),psinv,dpsidr)
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call equinum_psi(x(1),x(2),psinv,dpsidr)
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fvec(1) = psinv-f0(1)
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fvec(1) = psinv-f0(1)
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fvec(2) = dpsidr/psia-f0(2)
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fvec(2) = dpsidr-f0(2)
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case(2)
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case(2)
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call equinum_psi(x(1),x(2),dpsidr=dpsidr,dpsidz=dpsidz, &
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call equinum_psi(x(1),x(2),dpsidr=dpsidr,dpsidz=dpsidz, &
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ddpsidrr=ddpsidrr,ddpsidrz=ddpsidrz)
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ddpsidrr=ddpsidrr,ddpsidrz=ddpsidrz)
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fjac(1,1) = dpsidr/psia
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fjac(1,1) = dpsidr
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fjac(1,2) = dpsidz/psia
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fjac(1,2) = dpsidz
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fjac(2,1) = ddpsidrr/psia
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fjac(2,1) = ddpsidrr
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fjac(2,2) = ddpsidrz/psia
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fjac(2,2) = ddpsidrz
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case default
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case default
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write (msg, '("invalid iflag: ",g0)')
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write (msg, '("invalid iflag: ",g0)')
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call log_error(msg, mod='equilibrium', proc='fcntgo')
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call log_error(msg, mod='equilibrium', proc='fcntgo')
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@ -1258,7 +1258,7 @@ contains
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subroutine plas_deriv(xv, bres, xgcn, dens, btot, bv, derbv, &
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subroutine plas_deriv(xv, bres, xgcn, dens, btot, bv, derbv, &
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xg, yg, derxg, deryg, psinv_opt)
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xg, yg, derxg, deryg, psinv_opt)
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use const_and_precisions, only : zero
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use const_and_precisions, only : zero, cm
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use gray_params, only : iequil
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use gray_params, only : iequil
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use equilibrium, only : psia, equinum_fpol, equinum_psi, equian, sgnbphi
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use equilibrium, only : psia, equinum_fpol, equinum_psi, equian, sgnbphi
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use coreprofiles, only : density
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use coreprofiles, only : density
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@ -1281,7 +1281,7 @@ contains
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real(wp_), dimension(3) :: dbtot,bvc
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real(wp_), dimension(3) :: dbtot,bvc
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real(wp_), dimension(3,3) :: dbvcdc,dbvdc,dbv
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real(wp_), dimension(3,3) :: dbvcdc,dbvdc,dbv
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real(wp_) :: brr,bphi,bzz,dxgdpsi
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real(wp_) :: brr,bphi,bzz,dxgdpsi
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real(wp_) :: dpsidr,dpsidz,ddpsidrr,ddpsidzz,ddpsidrz,fpolv,dfpv,ddenspsin
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real(wp_) :: dpsidr,dpsidz,ddpsidrr,ddpsidzz,ddpsidrz,fpolv,dfpv,ddensdpsi
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xg = zero
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xg = zero
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yg = 99._wp_
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yg = 99._wp_
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@ -1342,14 +1342,14 @@ contains
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end if
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end if
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! compute xg and derivative
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! compute xg and derivative
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call density(psinv,dens,ddenspsin)
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call density(psinv, dens, ddensdpsi)
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xg = xgcn*dens
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xg = xgcn*dens
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dxgdpsi = xgcn*ddenspsin/psia
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dxgdpsi = xgcn*ddensdpsi
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! B = f(psi)/R e_phi+ grad psi x e_phi/R
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! B = f(psi)/R e_phi+ grad psi x e_phi/R
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bphi = fpolv/rrm
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bphi = fpolv/rrm
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brr =-dpsidz/rrm
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brr = -dpsidz*psia/rrm
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bzz = dpsidr/rrm
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bzz = +dpsidr*psia/rrm
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! bvc(i) = B_i in cylindrical coordinates
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! bvc(i) = B_i in cylindrical coordinates
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bvc = [brr, bphi, bzz]
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bvc = [brr, bphi, bzz]
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@ -1360,12 +1360,12 @@ contains
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bv(3)=bvc(3)
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bv(3)=bvc(3)
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! dbvcdc(iv,jv) = d Bcil(iv) / dxvcil(jv)
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! dbvcdc(iv,jv) = d Bcil(iv) / dxvcil(jv)
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dbvcdc(1,1) = -ddpsidrz/rrm - brr/rrm
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dbvcdc(1,1) = -ddpsidrz*psia/rrm - brr/rrm
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dbvcdc(2,1) = dfpv*dpsidr/rrm - bphi/rrm
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dbvcdc(2,1) = dfpv*dpsidr*psia/rrm - bphi/rrm
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dbvcdc(3,1) = ddpsidrr/rrm - bzz/rrm
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dbvcdc(3,1) = ddpsidrr*psia/rrm - bzz/rrm
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dbvcdc(1,3) = -ddpsidzz/rrm
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dbvcdc(1,3) = -ddpsidzz*psia/rrm
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dbvcdc(2,3) = dfpv*dpsidz/rrm
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dbvcdc(2,3) = dfpv*dpsidz*psia/rrm
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dbvcdc(3,3) = ddpsidrz/rrm
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dbvcdc(3,3) = ddpsidrz*psia/rrm
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! dbvdc(iv,jv) = d Bcart(iv) / dxvcil(jv)
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! dbvdc(iv,jv) = d Bcart(iv) / dxvcil(jv)
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dbvdc(1,1) = dbvcdc(1,1)*csphi - dbvcdc(2,1)*snphi
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dbvdc(1,1) = dbvcdc(1,1)*csphi - dbvcdc(2,1)*snphi
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@ -1403,12 +1403,12 @@ contains
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deryg = 1.0e-2_wp_*dbtot/Bres
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deryg = 1.0e-2_wp_*dbtot/Bres
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bv = bv/btot
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bv = bv/btot
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do jv=1,3
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do jv=1,3
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derbv(:,jv) = 1.0e-2_wp_*(dbv(:,jv) - bv(:)*dbtot(jv))/btot
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derbv(:,jv) = cm * (dbv(:,jv) - bv(:)*dbtot(jv))/btot
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end do
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end do
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derxg(1) = 1.0e-2_wp_*drrdx*dpsidr*dxgdpsi
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derxg(1) = cm * drrdx*dpsidr*dxgdpsi
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derxg(2) = 1.0e-2_wp_*drrdy*dpsidr*dxgdpsi
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derxg(2) = cm * drrdy*dpsidr*dxgdpsi
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derxg(3) = 1.0e-2_wp_*dpsidz *dxgdpsi
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derxg(3) = cm * dpsidz *dxgdpsi
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end subroutine plas_deriv
|
end subroutine plas_deriv
|
||||||
|
|
||||||
|
|
||||||
|
|||||||
@ -103,7 +103,7 @@ contains
|
|||||||
use const_and_precisions, only : wp_,zero,one,pi,ccj=>mu0inv
|
use const_and_precisions, only : wp_,zero,one,pi,ccj=>mu0inv
|
||||||
use gray_params, only : iequil
|
use gray_params, only : iequil
|
||||||
use equilibrium, only : btrcen,btaxis,rmaxis,zmaxis,phitedge,zbsup,zbinf, &
|
use equilibrium, only : btrcen,btaxis,rmaxis,zmaxis,phitedge,zbsup,zbinf, &
|
||||||
equian,equinum_psi,bfield,frhotor,fq,tor_curr
|
equian,equinum_psi,bfield,frhotor,fq,tor_curr,psia
|
||||||
use dierckx, only : regrid,coeff_parder
|
use dierckx, only : regrid,coeff_parder
|
||||||
implicit none
|
implicit none
|
||||||
|
|
||||||
@ -175,8 +175,8 @@ contains
|
|||||||
else
|
else
|
||||||
call equinum_psi(rmaxis,zmaxis,ddpsidrr=ddpsidrr,ddpsidzz=ddpsidzz)
|
call equinum_psi(rmaxis,zmaxis,ddpsidrr=ddpsidrr,ddpsidzz=ddpsidzz)
|
||||||
end if
|
end if
|
||||||
qq=abs(btaxis)/sqrt(ddpsidrr*ddpsidzz)
|
qq=abs(btaxis)/sqrt(ddpsidrr*ddpsidzz)/psia
|
||||||
ajphiav=-ccj*(ddpsidrr+ddpsidzz)/rmaxis
|
ajphiav=-ccj*(ddpsidrr+ddpsidzz)*psia/rmaxis
|
||||||
|
|
||||||
psicon(1)=0.0_wp_
|
psicon(1)=0.0_wp_
|
||||||
rcon(:,1)=rmaxis
|
rcon(:,1)=rmaxis
|
||||||
@ -448,7 +448,7 @@ contains
|
|||||||
use logger, only : log_warning
|
use logger, only : log_warning
|
||||||
use dierckx, only : profil, sproota
|
use dierckx, only : profil, sproota
|
||||||
use equilibrium, only : model, frhotor, psi_spline, &
|
use equilibrium, only : model, frhotor, psi_spline, &
|
||||||
kspl, psiant, psinop, points_tgo
|
kspl, points_tgo
|
||||||
use limiter, only : rwallm
|
use limiter, only : rwallm
|
||||||
|
|
||||||
implicit none
|
implicit none
|
||||||
@ -510,8 +510,7 @@ contains
|
|||||||
call log_warning(msg, mod='magsurf_data', proc='contours_psi')
|
call log_warning(msg, mod='magsurf_data', proc='contours_psi')
|
||||||
end block
|
end block
|
||||||
end if
|
end if
|
||||||
val=h*psiant+psinop
|
call sproota(h, psi_spline%knots_x, psi_spline%nknots_x, &
|
||||||
call sproota(val, psi_spline%knots_x, psi_spline%nknots_x, &
|
|
||||||
czc, zeroc, mest, m, ier)
|
czc, zeroc, mest, m, ier)
|
||||||
if (zeroc(1).gt.rwallm) then
|
if (zeroc(1).gt.rwallm) then
|
||||||
rcn(ic)=zeroc(1)
|
rcn(ic)=zeroc(1)
|
||||||
|
|||||||
Loading…
Reference in New Issue
Block a user