plasma_plots.theory

Analytic theory to compare Struphy runs against.

Dispersion relations, plasma parameters, orbits, exact solutions and the properties of the numerical schemes.

Everything here is plain numpy (no scipy) and independent of Struphy. The modules:

• special: the Faddeeva and plasma dispersion functions, complete elliptic integrals.
• parameters: plasma frequencies, lengths, speeds and dimensionless numbers, in SI units.
• kinetic: kinetic dispersion relations: Langmuir waves and Landau damping, ion-acoustic waves, beam-plasma, bump-on-tail and two-stream instabilities, Weibel.
• waves: fluid, MHD and cold-plasma waves: light waves, MHD waves at any angle, dissipative and Hall-MHD Alfvén waves, cold-plasma (Stix) waves, cutoffs and resonances, Faraday rotation, cavity modes, drift waves and Hasegawa–Wakatani, Alfvén and slow continua and the TAE frequency.
• orbits: gyromotion, guiding-center drifts, trapped particles, bounce frequencies and banana widths in a large-aspect-ratio tokamak.
• exact: exact solutions to verify simulations: the Riemann problem of gas dynamics (with vacuum), dam break, diffusion, advection, pressureless flow and its caustics.
• numerics: accuracy of the numerics: amplification and phase errors of time integrators, and the numerical dispersion of spline finite elements.

Conventions

• Arrays: every function takes scalars or numpy arrays that broadcast against each other, and returns numpy arrays (a Python scalar for scalar input).
• Units: normalized units unless a function says otherwise. Kinetic results are in units of the plasma frequency ω_p (time), the Debye length λ_D (length) and the thermal speed v_th = √(T/m) (velocity); fluid and MHD results in whatever units the speeds and wavenumbers are given in. parameters works in SI units.
• Frequencies are complex for exp(i(k·x − ωt)): the real part is the frequency, a positive imaginary part a growth rate, a negative one a damping rate.
• Several branches come as a dict of branch names to complex frequencies, e.g. {"shear Alfvén": ..., "slow": ..., "fast": ...}. Such a function (or any callable that returns such a dict, including Struphy’s own struphy.dispersion_relations objects) can be passed to array.plasma.plot.dispersion(branches=...) as it is; for one branch, a lambda: branches={"fast": lambda k: mhd_waves(k, ...)["fast"]}.

Examples

The Landau damping of a Langmuir wave at k λ_D = 0.5, against a Vlasov–Ampère run’s measured frequency and damping rate:

>>> from plasma_plots.theory import kinetic
>>> omega = kinetic.langmuir(0.5)
>>> round(omega.real, 4), round(omega.imag, 4)
(1.4157, -0.1534)

Modules

• exactExact solutions to verify simulations: the Riemann problem of gas dynamics (with vacuum), the
• kineticKinetic dispersion relations of unmagnetized plasmas: Langmuir waves and Landau damping,
• numericsAccuracy of the numerics: amplification and phase errors of time integrators, and the numerical
• orbitsCharged-particle orbits: gyromotion, guiding-center drifts and trapped particles in a tokamak.
• parametersPlasma parameters in SI units: frequencies, lengths, speeds and Struphy's units.
• specialSpecial functions of plasma theory: the Faddeeva function, the plasma dispersion function and
• wavesFluid, MHD and cold-plasma waves: light waves, MHD waves at any angle, dissipative and Hall-MHD