Letwhere and . Here, is the safety factor profile. [See Equation (3.2).] Moreover, the superscript indicates a quantity that is first order in our ordering scheme. [Zeroth order terms are left without superscripts, whereas second order terms are given the superscript (2).]
It follows thatfor any . Furthermore,
We can automatically satisfy Equation (4.32) by writingIt follows that where [see Equations (3.1) and (3.2)]
Equations (4.33) and (4.41) givewhere
Equations (4.34) and (4.41) yield
Let us writeNote that the ordering of the plasma pressure adopted here is somewhat different to that adopted in Section 3.3. In fact, in Section 3.3, was assumed to be second order. Here, we are assuming that has a spatially constant component that is zeroth order, and a spatially varying component that is first order. This high pressure ordering is merely an artifice to aid the extraction of the compressible-Alfvén (i.e., fast magnetosonic) wave from the system of equations [3,4]. In fact, the ordering ensures that the compressible-Alfvén wave has a substantially different phase velocity than the shear-Alfvén and slow magnetosonic waves .
Equations (4.28)–(4.31), (4.41), (4.46), and (4.50) yieldHere, the additional factor involving in Equation (4.53) is needed because the plasma flow is slightly compressible. In fact, the velocity fields , , , and all have normalized divergences that are second order.
Evaluating the normalized drift-MHD fluid equations, (4.25)–(4.27), up to second order, we obtain
To first order, Equations (4.55) and (4.56) both yieldwhich is simply an expression of lowest-order equilibrium force balance [3,7].
Taking the scalar product of Equation (4.55) with annihilates the first-order terms, leavingMoreover, taking the scalar product of Equation (4.56) with annihilates the first-order terms, leaving Furthermore, taking the scalar product of the curl of Equation (4.55) with annihilates the first-order terms, leaving where Finally, taking the scalar product of the curl of Equation (4.56) with annihilates the first-order terms, leaving