Quantities of Interest

This page describes the formulas used in the Personal Gkeyll scripts Python library to compute multiple quantities of interest for gyrokinetic simulations.

Moment Diagnostics

Moment-Temperature Equivalency

The equivalency between Mi moments (i=0,1,2) and the Bimaxwellian distribution:

M₀ = n                              (density)
M₁ = n u_∥                         (parallel momentum density)
M_{2∥} = n T_∥/m + n u_∥²            (parallel energy density)
M_{2⊥} = 2n T_⊥/m                    (perpendicular energy density)
M₂ = n T/m + n u_∥²                (total energy density)

Where:

• m: particle mass
• n: density
• u_∥: fluid velocity parallel to magnetic field
• T_∥: parallel temperature
• T_⊥: perpendicular temperature
• T: isotropic temperature

Temperature Relations

From the moment definitions:

M₂ = M_{2∥} + (1/2) M_{2⊥}

And the temperature relation:

T = T_∥ + (1/2) T_⊥

Energy Diagnostics

Total Energy Decomposition

The total energy of species s is decomposed into three components:

W_s(x,t) = W_{k,s}(x,t) + W_{f,s}(x,t) + W_{p,s}(x,t)

Particle Kinetic Energy

The random thermal energy:

W_{k,s}(x,t) = ∫ dv³ (1/2) m_s v² f_s
           = n_s(x,t) × (1/3) × [T_{∥,s}(x,t) + 2T_{⊥,s}(x,t)]

This represents the thermal energy content of the plasma.

Fluid Kinetic Energy

The organized flow energy:

W_{f,s}(x,t) = n_s(x,t) × (1/2) × m_s u_{∥,s}(x,t)²

This represents the kinetic energy associated with bulk plasma flow.

Potential Energy

The electrostatic potential energy:

W_{p,s}(x,t) = n_s(x,t) × q_s × φ(x,t)

This represents the energy stored in the electrostatic field.

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Part of the Gkeyll Theory and Analysis documentation