In plasma physics, the Hasegawa–Mima equation, named after Akira Hasegawa and Kunioki Mima, is an equation that describes a certain regime of plasma, where the time scales are very fast, and the distance scale in the direction of the magnetic field is long.
The equation was introduced in Hasegawa and Mima's paper submitted in 1977 to Physics of Fluids, where they compared it to the results of the ATC tokamak.
The form of the equation is: Although the quasi neutrality condition holds, the small differences in density between the electrons and the ions cause an electric potential.
In the limit where the wavelength of a perturbation of the electric potential is much smaller than the gyroradius based on the sound speed, the Hasegawa–Mima equations become the same as the two-dimensional incompressible fluid.
The entire equation without normalization is: Although the time derivative divided by the cyclotron frequency is much smaller than unity, and the normalized electric potential is much smaller than unity, as long as the gradient is on the order of one, both terms are comparable to the nonlinear term.
The unperturbed density gradient can also be just as small as the normalized electric potential and be comparable to the other terms.
These Poisson brackets are defined as: Using these Poisson bracket, the equation can be re-expressed as: Often the particle density is assumed to vary uniformly just in one direction, and the equation is written in a sightly different form.