Energy Input from Maxwellian Particle Sources

This page describes the theoretical framework for Maxwellian particle sources in Gkeyll simulations, including energy input calculations and adaptive source implementations.

Maxwellian Source Function

The Maxwellian source rate function is given by:

S_M(x,v) = Γ(x) / √((2π v_{th0}²)³) × exp(-v²/(2 v_{th0}²))

Where:

• $Γ(x)$ is the particle source rate per unit volume [m⁻³ s⁻¹]
• $v_{th0} = √(T₀/m)$ is the thermal velocity
• $T₀$ is the temperature of the injected particles
• $m$ is the particle mass

Local Power Density

The power injected per unit volume is calculated by integrating over velocity space:

P(x) = ∫ (1/2) m v² S_M(x,v) J_v d³v

Where J_v is the velocity space Jacobian.

Since the integral of (1/2) m v² over a Maxwellian distribution is well-known, in 3D velocity space or (v_parallel, μ) gyrokinetic coordinates:

∫ (1/2) m v² f_M(v) J_v d³v = (3/2) T₀ n

This gives us:

Q(x) = (3/2) T₀ Γ(x)

Total Power Injected

The particle source follows a Gaussian spatial envelope:

Γ(x) = Γ₀ η(x)

Where:

η(x) = C exp(-(x-x₀)²/(2σ_x²) - (y-y₀)²/(2σ_y²) - (z-z₀)²/(2σ_z²))

With C being the normalization constant such that:

∫_Ω η(x) J_x d³x = 1

Where Ω is the configuration space domain and J_x is the configuration space Jacobian.

The total injection power is:

P = ∫ Q(x) d³x = (3/2) T₀ Γ₀

Therefore, given a particle injection rate Γ₀ and total power P, the source temperature is:

T₀ = 2P/(3Γ₀)

Adaptive Sources

In Gkeyll inner wall limited 3×2v gyrokinetic simulations, plasma turbulence evolution predicts kinetic profiles self-consistently, given a magnetic equilibrium and injection power.

Sourcing Criteria

The simulation setup must satisfy:

1. Heat injection that matches experimental power: P_sources = P_exp
2. No global particle injection or loss (high recycling regime)

Boundary Compensation

Absorbing boundary conditions at x = x_lo and x = x_up create undesired sinks. These boundaries absorb outward flux and prevent particle/heat injection due to the f = 0 constraint outside boundaries.

Core Source (S_core)

At the inner radial boundary, all heat and particle losses are compensated:

P_{core} = P_{x,lo}^{loss}\\
Γ_{core} = Γ_{x,lo}^{loss}

Source parameters:

(Γ_{core}, T_{core}) = (Γ_{x,lo}^{loss}, 2P_{x,lo}^{loss}/(3Γ_{x,lo}^{loss}))

Recycling Source (S_recy)

At vessel walls (x_up, z_lo, z_up), heat flows but particles recycle through ion-neutral interactions:

Γ_{recy} = Γ_{x,up}^{loss} + Γ_{z,lo}^{loss} + Γ_{z,up}^{loss} = Γ_{wall}^{loss}

Using the power criterion:

P_{core} + P_{recy} = P_{exp}

This yields:

T_{recy} = (2/3) × (P_{exp} - 2P_{x,lo}^{loss}/(3Γ_{x,lo}^{loss})) / Γ_{wall}^{loss}

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Part of the Gkeyll Theory and Analysis documentation