(avalanche)¶

In a spatially uniform fully ionized plasma with constant electric field, when the runaway generation is dominated by the avalanche mechanism—i.e. by multiplication through large-angle collisions—an analytical expression can be found for the resulting runaway electron distribution function. This distribution function, which was first derived in [1], takes the form

\[f(p, \xi) = \frac{A(p)}{2\pi m_e c\gamma_0 p^2} \frac{\exp\left[ -\gamma/\gamma_0 - A(p)(1+\xi) \right]}{1 - \exp\left[-2A(p)\right]}\]

with

\[\begin{split}A(p) &= \frac{\hat{E} + 1}{Z_{\rm eff} + 1} \gamma,\\ \gamma_0 &= \log\Lambda\sqrt{Z_{\rm eff}+5}\end{split}\]

where \(p\) is the electron momentum, \(\xi = \cos\theta_{\rm p}\) the electron pitch with respect to the magnetic field, \(\gamma = \sqrt{1 + p^2}\), \(\hat{E}\) the electric field strength normalized to the threshold electric field, \(Z_{\rm eff}\) the plasma effective charge, \(\log\Lambda\) the plasma Coloumb logarithm, \(m_e\) the electron rest mass and \(c\) the speed of light in vacuum.

[1]	Fülöp et al., 2006 “Destabilization of magnetosonic-whistler waves by a relativistic runaway beam”. Physics of Plasmas 13 (6), 062506 doi:10.1063/1.2208327.

Summary of options¶

The following parameters can be set on an avalanche distribution function.

Option	Description
`avalanche EHat`	Electric field strength, normalized to the threshold electric field.
`avalanche lnLambda`	Coloumb logarithm of the plasma.
`avalanche Zeff`	Plasma effective charge.
`avalanche radprof`	Name of configuration block for radial profile.

Example configuration¶

This example illustrates how an analytical avalanche distribution function can be defined and used in SOFT2 simulations:

# Global parameter specifying which distribution function to use
distribution_function = analytical_avalanche;

@DistributionFunction analytical_avalanche (avalanche) {
    EHat     = 10;            # Electric field strength (normalized to
                              # the Connor-Hastie critical electric field)
    lnLambda = 17;            # Coulomb logarithm
    Zeff      = 4;             # Effective plasma charge
}

All options¶

EHat¶

Default value:	None
Allowed values:	Any real number

Electric field strength, normalized to the Connor-Hastie critical electric field.

lnLambda¶

Default value:	None
Allowed values:	Any real numer

Value of the Coulomb logarithm.

radprof¶

Default value:	Uniform radial profile
Allowed values:	Name of any defined @RadialProfile

Specifies the radial profile object to use to generate a radial profile.

Zeff¶

Default value:	None
Allowed values:	Any real number

Effective plasma charge.