Kinetic background

CanonicalMaxwellian2D

Canonical Maxwellian distribution function in

$(η_{1}, η_{2}, η_{3}, v_{∥}, μ)$ coordinates. Uses caching for evaluation of the canonical toroidal momentum in these coordinates.

The distribution is parameterized by the density and thermal speed as functions of the canonical toroidal momentum $ψ_{c}$ :

ψ_{c} = ψ + \frac{m _{s} F}{q _{s} B} v_{∥} - sign (v_{∥}) 2 (ϵ - μ B) \frac{m _{s} F}{q _{s} B} H (ϵ - μ B),

Energy

ϵ = \frac{1}{2} m_{s} v_{∥}^{2} + μ B,

Magnetic moment

μ = \frac{m _{s} v _{⊥}^{2}}{2 B},

where $ψ$ is the poloidal magnetic flux function, $F = F (ψ)$ is the poloidal current function and $H$ is the Heaviside function.

With the three constants of motion, a canonical Maxwellian distribution function is defined as

F (ψ_{c}, ϵ, μ) = \frac{n ( ψ _{c} )}{( 2 π ) ^{3/2} v _{th}^{3} ( ψ _{c} )} exp [- \frac{ϵ}{v _{th}^{2} ( ψ _{c} )}] .

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Phase space

Velocity dimension: 2
Velocity coordinates: vpara_mu
Axes: v∥ · μ
Volume form: yes — includes the velocity Jacobian

Velocity-space distribution

f₀ over (v∥, μ), at logical position η = (0.5, 0.5, 0.5).

f₀(v∥, μ)v∥–μ · 1.67e-10 to 7.93e-1

v∥: -5 to 5
μ: 0 to 5

Reduced profiles

The distribution along each velocity axis, with every other phase-space axis integrated out.

v∥-5 to 5 · peak 3.98e-1

μ0 to 5 · peak 2.00e+0

Parameters

Constructor defaults. Moments are given as a background value (a float or a callable) paired with an optional perturbation. This class cannot be built from defaults alone; the catalogue constructs it with equil.

Parameter	Default	Perturbation	Description
`n`	`1`	`None`	Moments of the canonical Maxwellian as tuples. The first entry defines the background (float for constant background or callable), the second entry defines a Perturbation (can be None).
`u_para`	`0`	`None`	—
`u_perp`	`0`	`None`	—
`vth_para`	`1`	`None`	—
`vth_perp`	`1`	`None`	—
`volume_form`	`true`	—	If ``True``, represent the distribution as a volume form and include the appropriate velocity-space Jacobian when evaluating it.
`B0`	`absB0`	—	—
`uniform_on_disc`	`false`	—	—
`vth`	`1`	`None`	Thermal-speed background and optional perturbation.
`equil`	`AdhocTorus`	—	Fluid equilibrium used to evaluate background profiles in the magnetic geometry.
`epsilon`	`1`	—	—
`cache_size`	`null`	—	Number of rows in the cache buffer for :math:`\psi_c` evaluation. If ``None``, no caching is used. Must be able to accomodate all markers on the current process.