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**Example text**

2 (50a,b) These conserved quantities play important roles in the nonlinear statistical dynamics of the equation, as discussed in Sec. 4 (p. 206). If the terms explicitly involving δϕ are neglected, Eq. (49) becomes the 2D neutral-fluid Euler equation in the vorticity representation [Eq. (17) with µcl = 0; see also the guiding-center model (31)], which has been studied extensively. However, although this observation is instructive, such neglect is physically unjustified. The first δϕ term in Eq.

The diamagnetic term can be formed into a perfect y derivative whose integral vanishes under periodic boundary conditions. For the nonlinear term, (i) dx ϕV E · ∇ω = dx E · V E ω = 0 (the E × B drift does no work), and (ii) dx ωV E · ∇ω = dx ∇ · (V E 12 ω 2 ) = 0. Note that in the limit of small δ the TH invariant is Z ≈ E + W. 51 37 Krommes, 1992). The real utility of the HME lies in its clean description of the polarization-drift nonlinearity; it is an important limit to which more complete, possibly kinetic theories should reduce.

Since electron inertia is negligible, . a simplified electron momentum equation is 0 ≈ −ne eE − ∇ Pe − ne me νei u e , where P = nT ; temperature fluctuations and ion parallel motion are neglected for simplicity. This can be rewritten . 2 as u e = D ∇ (ϕ − ne ), where D = vte /νei is the classical parallel diffusion coefficient. Upon substituting for u e , one can reduce the electron continuity equation in the usual gyro-Bohm units . −1 to dδne /dt = α(δϕ − δne ) − κ ∂y δϕ, where α = −ωci D ∇2.