Reduced Drift-MHD Model

Equations (4.69)–(4.74) constitute our reduced drift-MHD resonant plasma response model. Our model is very similar
to the so-called *four-field model* derived by Hazeltine, et alia [7]. Equation (4.69) is the *generalized
Ohm's law* that governs the time evolution of the (normalized) helical magnetic flux, . [Note that this should not be confused with the equilibrium
poloidal flux defined in Equation (2.124).] Equation (4.70) is the *energy equation* that
governs the time evolution of the (normalized) perturbed total plasma pressure, . Equation (4.71) is the *ion parallel vorticity equation* that governs the time evolution of the (normalized) ion parallel vorticity, .
Finally, Equation (4.72) is the
*ion parallel equation of motion* that governs the time evolution of the (normalized) ion parallel velocity, .

Strictly speaking, the reduced resonant response model
derived in this chapter is only valid when the ion gyro-radius is smaller than the typical radial variation lengthscale in the resonant layer.
If this is not the case then it is necessary to adopt a so-called *gyro-fluid* approach in which the ion response is
averaged over the ion gyro-orbits [1,6,11].

Finally, the crucial element of the reduction process, by which our original drift-MHD response model is converted into a
reduced drift-MHD model, is the set of ordering assumptions that render
a second-order quantity while leaving
a first-order quantity. This ordering implies that the MHD fluid is
*incompressible* to lowest order. (See Section 3.6.) It should be noted, however, that the small, but finite, compressibility of the
MHD fluid has a significant influence on the form of the energy equation (4.70).