Higher Order Miller Geometry

This page describes higher order corrections to the Miller equilibrium geometry used in gyrokinetic simulations, providing enhanced geometric fidelity for tokamak modeling.

Overview

The higher order Miller geometry extends the standard Miller parametrization by including additional terms that account for more complex flux surface shapes, improving the accuracy of tokamak equilibrium representation.

Geometric Parametrization

Enhanced R and Z Coordinates

The higher order Miller geometry is defined as:

Major radius R(r,θ):

R(r,θ) = R₀ + r cos(θ) + 
         r(κ^{-1}-1)/(κ^{-1}+1) × [q(r)s(r)+6q_a]/[q_s s(a₀)+6q_a] cos(θ) + 
         δr²/(4a₀) × [q(r)s(r)+4q_a]/[q_s s(a₀)+4q_a] cos(2θ)

Vertical coordinate Z(r,θ):

Z(r,θ) = r sin(θ) - 
         r(κ^{-1}-1)/(κ^{-1}+1) × [q(r)s(r)+6q_a]/[q_s s(a₀)+6q_a] sin(θ) -
         δr²/(4a₀) × [q(r)s(r)+4q_a]/[q_s s(a₀)+4q_a] sin(2θ)

Parameter Definitions

• R₀: Major radius of magnetic axis
• r: Minor radius coordinate
• θ: Poloidal angle
• κ: Elongation parameter
• δ: Triangularity parameter
• q_a: Safety factor at magnetic axis
• q_s: Safety factor at separatrix
• a₀: Minor radius at outboard midplane
• q(r): Safety factor profile
• s(r): Magnetic shear profile

Safety Factor and Shear Profiles

Quadratic Safety Factor Profile

q(r) = q_a + (q_s - q_a)(r/a_{mid})²

This provides a realistic q-profile that increases quadratically from axis to edge.

Magnetic Shear Profile

s(r) = (r/q(r)) × dq/dr = 2(q_s - q_a)/q(r) × (r/a_{mid})²

The shear quantifies the rate of change of field line pitch with radius.

Geometric Derivatives

Radial Derivatives

∂R/∂r:

∂R/∂r = cos(θ) + 
        (κ^{-1}-1)/(κ^{-1}+1) × [q'(r)s(r)+q(r)s'(r)]/[q_s s(a₀)+6q_a] cos(θ) + 
        δr/(2a₀) × [q'(r)s(r)+q(r)s'(r)]/[q_s s(a₀)+4q_a] cos(2θ)

∂Z/∂r:

∂Z/∂r = sin(θ) - 
        (κ^{-1}-1)/(κ^{-1}+1) × [q'(r)s(r)+q(r)s'(r)]/[q_s s(a₀)+6q_a] sin(θ) - 
        δr/(2a₀) × [q'(r)s(r)+q(r)s'(r)]/[q_s s(a₀)+4q_a] sin(2θ)

Poloidal Derivatives

∂R/∂θ:

∂R/∂θ = -r sin(θ) - 
         r(κ^{-1}-1)/(κ^{-1}+1) × [q(r)s(r)+6q_a]/[q_s s(a₀)+6q_a] sin(θ) - 
         δr²[q(r)s(r)+4q_a]/[2a₀(q_s s(a₀)+4q_a)] sin(2θ)

∂Z/∂θ:

∂Z/∂θ = r cos(θ) - 
         r(κ^{-1}-1)/(κ^{-1}+1) × [q(r)s(r)+6q_a]/[q_s s(a₀)+6q_a] cos(θ) - 
         δr²[q(r)s(r)+4q_a]/[2a₀(q_s s(a₀)+4q_a)] cos(2θ)

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Part of the Gkeyll Theory and Analysis documentation