src/equilibrium.f90: use the exact toroidal flux
To improve the analytical model correctness this changes the formula of the toroidal flux to lift the large aspect ratio approximation (a << R₀). In fact, Φ(r) = B₀πr² is technically inconsistent with the field varying as B₀R₀/R. The exact expression is: Φ(r) = B₀πr² 2/[1 + √(1 - r²/R₀²)], which is approximately equal to the former for r << R₀. Note that this change introduces a divergence in the poloidal field at r=R₀ (since ∂Φ/∂r → +∞), so the domain of the equilibrium has been restricted to r<R₀, as expected. This should not be a concern because the field outside the plasma boundary is never directly, particularly not by the integrator.
This commit is contained in:
Michele Guerini Rocco
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@ -766,7 +766,7 @@ contains
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implicit none
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real(wp_) :: dq
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real(wp_) :: dq, gamma
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btaxis = model%B0
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rmaxis = model%R0
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@ -782,9 +782,14 @@ contains
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zmxm = zbsup
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zmnm = zbinf
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! FIXME: it should be that Φ(r) = B₀πr² 1/√(1-ε²)
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! where ε=r/R₀ is the tokamak aspect ratio
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phitedge = model%B0 * pi * model%a**2
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! Toroidal flux at r=a:
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!
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! Φ(a) = B₀πa² 2γ/(γ + 1)
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!
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! where γ=1/√(1-ε²),
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! ε=a/R₀ is the tokamak aspect ratio
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gamma = 1/sqrt(1 - (model%a/model%R0)**2)
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phitedge = model%B0 * pi * model%a**2 * 2*gamma/(gamma + 1)
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! In cocos=3 the safety factor is
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!
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@ -828,7 +833,8 @@ contains
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! derivatives wrt (R, z) up to the second order.
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!
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! Note: all output arguments are optional.
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use gray_params, only : iequil
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use gray_params, only : iequil
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use const_and_precisions, only : one, pi
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implicit none
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@ -838,21 +844,75 @@ contains
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ddpsidrr, ddpsidzz, ddpsidrz
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! local variables
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real(wp_) :: rho_t, rho_p ! √Φ_n, √ψ_n
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real(wp_) :: r_g, rho_t, rho_p ! geometric radius, √Φ_n, √ψ_n
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real(wp_) :: gamma ! γ = 1/√(1 - r²/R₀²)
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real(wp_) :: dpsidphi ! (∂ψ_n/∂Φ_n)
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real(wp_) :: ddpsidphidr, ddpsidphidz ! ∇(∂ψ_n/∂Φ_n)
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real(wp_) :: dphidr, dphidz ! ∇Φ_n
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real(wp_) :: phi_n ! Φ_n
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real(wp_) :: dphidr, dphidz ! ∇Φ_n
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real(wp_) :: ddphidrdr, ddphidzdz ! ∇∇Φ_n
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real(wp_) :: ddphidrdz !
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real(wp_) :: q, dq ! q(ρ_p), Δq=(q₁-q₀)/(α/2 + 1)
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real(wp_) :: dqdr, dqdz ! ∇q
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real(wp_) :: dphidr2, ddphidr2dr2 ! dΦ_n/d(r²), d²Φ_n/d(r²)²
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if (iequil < 2) then
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! Analytical model
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!
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! The normalised poloidal flux ψ_n(R, z) is computed as follows:
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! 1. ψ_n = ρ_p²
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! 2. ρ_p = ρ_p(ρ_t), using `frhopol`, which in turns uses q(ψ)
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! 3. ρ_t = √Φ_n
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! 4. Φ_n = Φ(r)/Φ(a), where Φ(r) is the flux of B_φ=B₀R₀/R
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! through a circular surface
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! 5. r = √[(R-R₀)²+(z-z₀)²] is the geometric minor radius
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r_g = hypot(R - model%R0, z - model%z0)
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! ρ_t is mapped to the normalised geometrical radius,
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! so ρ_t = √[(R-R₀)² + (z-z₀)²]/a and ρ_p(ρ_t)
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rho_t = hypot(R - model%R0, z - model%z0)/model%a
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! The exact flux of the toroidal field B_φ = B₀R₀/R is:
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!
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! Φ(r) = B₀πr² 2γ/(γ + 1) where γ=1/√(1 - r²/R₀²).
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!
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! Notes:
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! 1. the function Φ(r) is defined for r≤R₀ only.
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! 2. as r → 0, γ → 1, so Φ ~ B₀πr².
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! 3. as r → 1⁻, Φ → 2B₀πr² but dΦ/dr → -∞.
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! 4. |B_R|, |B_z| → +-∞.
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!
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if (r_g > model%R0) then
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if (present(psi_n)) psi_n = -1
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if (present(dpsidr)) dpsidr = 0
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if (present(dpsidz)) dpsidz = 0
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if (present(ddpsidrr)) ddpsidrr = 0
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if (present(ddpsidzz)) ddpsidzz = 0
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if (present(ddpsidrz)) ddpsidrz = 0
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return
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end if
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gamma = 1 / sqrt(1 - (r_g/model%R0)**2)
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phi_n = model%B0 * pi*r_g**2 * 2*gamma/(gamma + 1) / phitedge
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rho_t = sqrt(phi_n)
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rho_p = frhopol(rho_t)
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! For ∇Φ_n and ∇∇Φ_n we also need:
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!
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! ∂Φ∂(r²) = B₀π γ(r)
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! ∂²Φ∂(r²)² = B₀π γ³(r) / (2 R₀²)
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!
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dphidr2 = model%B0 * pi * gamma / phitedge
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ddphidr2dr2 = model%B0 * pi * gamma**3/(2 * model%R0**2) / phitedge
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! ∇Φ_n = ∂Φ_n/∂(r²) ∇(r²)
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! where ∇(r²) = 2[(R-R₀), (z-z₀)]
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dphidr = dphidr2 * 2*(R - model%R0)
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dphidz = dphidr2 * 2*(z - model%z0)
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! ∇∇Φ_n = ∇[∂Φ_n/∂(r²)] ∇(r²) + ∂Φ_n/∂(r²) ∇∇(r²)
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! = ∂²Φ_n/∂(r²)² ∇(r²)∇(r²) + ∂Φ_n/∂(r²) ∇∇(r²)
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! where ∇∇(r²) = 2I
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ddphidrdr = ddphidr2dr2 * 4*(R - model%R0)*(R - model%R0) + dphidr2*2
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ddphidzdz = ddphidr2dr2 * 4*(z - model%z0)*(z - model%z0) + dphidr2*2
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ddphidrdz = ddphidr2dr2 * 4*(R - model%R0)*(z - model%z0)
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! ψ_n = ρ_p(ρ_t)²
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if (present(psi_n)) psi_n = rho_p**2
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@ -871,10 +931,6 @@ contains
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dq = (model%q1 - model%q0) / (model%alpha/2 + 1)
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dpsidphi = (model%q0 + dq) / q
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! Since Φ_n = ρ_t²(R,z): ∇Φ_n = 2/a² [R-R₀, z-z₀]
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dphidr = 2*(R - model%R0)/model%a**2
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dphidz = 2*(z - model%z0)/model%a**2
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! Using the above, ∇ψ_n = ∂ψ_n/∂Φ_n ∇Φ_n
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if (present(dpsidr)) dpsidr = dpsidphi * dphidr
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if (present(dpsidz)) dpsidz = dpsidphi * dphidz
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@ -890,18 +946,18 @@ contains
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! ∇q = α/2 (q-q₀) ∇ψ_n/ψ_n
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! = α/2 (q-q₀)/ψ_n (∂ψ_n/∂Φ_n) ∇Φ_n.
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!
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dqdr = model%alpha/2 * (q - model%q0)/rho_p**2 * dpsidphi * dphidr
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dqdz = model%alpha/2 * (q - model%q0)/rho_p**2 * dpsidphi * dphidz
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dqdr = model%alpha/2 * (model%q1 - model%q0)*rho_p**(model%alpha-2) * dpsidphi * dphidr
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dqdz = model%alpha/2 * (model%q1 - model%q0)*rho_p**(model%alpha-2) * dpsidphi * dphidz
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ddpsidphidr = - dpsidphi * dqdr/q
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ddpsidphidz = - dpsidphi * dqdz/q
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! Finally, ∇∇Φ_n = ∇∇ ρ_t²(R,z) = 2/a² I, so:
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! Combining all of the above:
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!
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! ∇∇ψ_n = ∇(∂ψ_n/∂Φ_n) ∇Φ_n + (∂ψ_n/∂Φ_n) 2/a² I
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! ∇∇ψ_n = ∇(∂ψ_n/∂Φ_n) ∇Φ_n + (∂ψ_n/∂Φ_n) ∇∇Φ_n
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!
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if (present(ddpsidrr)) ddpsidrr = ddpsidphidr * dphidr + dpsidphi * 2/model%a**2
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if (present(ddpsidzz)) ddpsidzz = ddpsidphidz * dphidz + dpsidphi * 2/model%a**2
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if (present(ddpsidrz)) ddpsidrz = ddpsidphidr * dphidz
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if (present(ddpsidrr)) ddpsidrr = ddpsidphidr * dphidr + dpsidphi * ddphidrdr
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if (present(ddpsidzz)) ddpsidzz = ddpsidphidz * dphidz + dpsidphi * ddphidzdz
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if (present(ddpsidrz)) ddpsidrz = ddpsidphidr * dphidz + dpsidphi * ddphidrdz
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else
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! Numerical data
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if (R <= rmxm .and. R >= rmnm .and. &
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