Grad-B Sources and Losses

This page analyzes grad-B drift particle flux in post-processing from Gkeyll gyrokinetic turbulence simulations, including source term implementations and energy loss calculations.

Overview

Grad-B source: Source term introduced on the right-hand-side of the gyrokinetic equation
Grad-B loss: Radial flux pointing outward at the lower x boundary where particles are absorbed by zero distribution function ghost cell conditions at f(x = -Δx) = 0

Particle Loss

Grad-B Drift Velocity

The grad-B particle drift is expressed as:

v_{∇B,s} = (μ B × ∇B)/(q_s B²) = (1/2) m_s v_⊥² (b × ∇B)/(q_s B²)

Particle Flux Density

To obtain the grad-B particle flux density, integrate over velocity space with the distribution function f_s:

Γ_{∇B,s}(x,t) = ∫ dv³ v_{∇B} f_s(x, v_∥, μ, t)
             = p_{⊥s}/(q_s) × (b × ∇B)/B²

Where the perpendicular pressure is defined as:

p_{⊥s}(x,t) = ∫ dv³ (1/2) m_s v_⊥² f_s(x, v_∥, μ, t)

Total Particle Flux Through Flux Surface

The total particle flux through a given flux surface is:

S_{∇B,s}(x,t) = ∫∫ dz dy Γ_{∇B,s}(x,t) · ∇x

The integrand can be decomposed as:

Γ_{∇B,s}(x,t) · ∇x = M_s(x,t) G_{∇B}(x)

Where:

• M_s(x,t) = p_{⊥s}/q_s: Moment part (time and species dependent)
• G_{∇B}(x) = (b × ∇B)_x/B²: Magnetic equilibrium part

Source Models

Adapted Source Rate

The adapted grad-B source automatically sets source rate to match particle loss rate at initial conditions:

s_{ada}(x) = -S_{∇B,loss,0} × s_{ana}(x,t)/S_{ana}

Where:

S_{∇B,loss,0} = (2 n_{s0} T_{s0})/(3 q_s) × (b × ∇B)_x/B²

Using the relation p_{⊥s} = (2/3) n_s T_s.

True Flux Shape Source Rate

A grad-B source model using the exact shape G_{∇B} instead of the η₂(z) envelope:

s_{tf}(x,t) = -M_s(x,0) G_{∇B}(x) max(-1/((2π v_{t0}²)^{3/2}) exp[-(v_∥-u_{∥0})²+μB/m)/(2v_{t0}²)])

Profile comparison of s_ada and s_tf along radial coordinate (left) and field line (right). Velocity space dependency removed by integration.

Source vs Loss Comparison

Comparison between losses from simulation without GB source (48×32×16×16×6 TCV PT) and different grad-B source models.

Grad-B Energy Loss

Heat Flux Definition

The grad-B heat flux for species s:

Q_{∇B,s} = n_s T_s u_{∇B,s} = T_s Γ_{∇B,s}

Units: [m/s × J/m³]

Total Energy Loss

Total energy loss due to grad-B loss by integrating heat flux crossing a flux surface:

dW_{∇B,s}/dt = ∫_S Q_{∇B,s} · dσ = ∫∫ dz dy Q_{∇B,s} · ∇x

Generalized Flux Through Any Surface

Heat Flux Components

Heat flux can be generally defined as:

Q_s = n_s · T_s · u_s

Where the total advection velocity includes all drift contributions:

u_s = u_{E×B} + u_{∇B,s} + u_{∥,s}

E×B Drift

u_{E×B} = (E × B)/B² = -(∇φ × B)/B²

Component-wise expression:

(u_{E×B})^i = -(1/JB)(∂_j φ b_k - ∂_k φ b_j)

Grad-B Drift

u_{∇B,s} = (T_{⊥,s})/(q_s B) × (b × ∇B)/B²

Component-wise expression:

(u_{∇B,s})^i = (T_{⊥,s})/(q_s JB) × (b_j ∂B/∂k - b_k ∂B/∂j)

Total Power Through Surface

Total power passing through surface S with normal surface element dσ = dσ ∇i:

P_{S,i} = Σ_s ∫∫_S Q_s · ∇i dσ = Σ_s ∫∫_S (Q_s)^i dσ

Total Domain Power Loss

Total power leaving the numerical domain:

P_{out} = P_{x=0,x} + P_{x=L_x,x} + P_{y=0,y} + P_{y=L_y,y} + P_{z=0,z} + P_{z=L_z,z}

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Part of the Gkeyll Theory and Analysis documentation