Linearized Equations and the Ideal MHD Force Operator

The ideal MHD equations are Since we are interested in the behavior of a system when it is perturbed only slightly from its equilibrium state, we write all dependent variables in the form where f ₀ is the value in the equilibrium state (i.e., the solution of Eqs. (20.1, 20.2, 20.3, 20.4) when ∂/∂t = 0) and f ₁ is a small perturbation, i.e., ∣f ₁/f ₀∣≪1. When we substitute the ansatz (20.5) into Eqs. (20.1, 20.2, 20.3, 20.4), the nonlinear terms (like V·∇V and V × B, for example) will behave as since the product (u ₁/u ₀)(υ ₁/υ ₀)is much, much less than either u ₁/u ₀ or υ ₁/υ ₀. The resulting equations will be linear in the perturbed quantities f ₁; not surprisingly, the formal process is called linearization.

Be wise. Linearize.

Ed Greitzer