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 Incoming links: Electron magnetohydrodynamics

Critical balance is a scale-dependent ordering in which the wave-propagation time and nonlinear interaction time are comparable. For strong Alfvénic turbulence,

k_{∥} v_{A} \sim k_{⊥} δ u_{⊥}

v_{A} d_{i} k_{∥} k_{⊥}

and the nonlinear rate is

v_{A} d_{i} k_{⊥}^{2} δ B / B_{0}

, so critical balance instead gives

δ B / B_{0} \sim k_{∥} / k_{⊥}

. This assumption is an estimate of decorrelation dynamics, not an exact equality at every point.

An electron-magnetohydrodynamic cascade transfers magnetic fluctuations through the nonlinear electron magnetohydrodynamics induction equation. Writing

b_{l} = δ B_{l} / B_{0}

, its perpendicular nonlinear interaction time is

τ_{nl} \sim l_{⊥}^{2} / (v_{A} d_{i} b_{l})

. The anisotropic whistler wave time is

τ_{w} \sim l_{∥} l_{⊥} / (v_{A} d_{i})

. Scaling predictions depend on locality, phase decorrelation and the assumed balance of these times.

u_{e} = - J / (e n) = - c \nabla \times B / (4 π e n)

\partial_{t} B = - \frac{c}{4 π e n} \nabla \times [(\nabla \times B) \times B] .

For a uniform guide field, the solenoidal linear equation implies

ω^{2} = (c B_{0} /4 π e n)^{2} k_{∥}^{2} k^{2}

. With

v_{A} = B_{0} / 4 πn m_{i}

and

d_{i} = c m_{i} / (4 π e^{2} n)

c B_{0} / (4 π e n) = v_{A} d_{i}

For field amplitudes of order the guide field, the ion Lorentz-force response over time

τ

Δ u / v_{A} = O (v_{A} τ / l) ≪ 1

. The electron-current velocity is of order

v_{A} d_{i} / l

, larger than a bulk velocity of order

v_{A}

. Thus the leading Hall magnetohydrodynamics induction equation becomes the closed electron magnetohydrodynamics equation, while ions remain fixed to leading order on this fast scale. The first nonzero ion response need not have zero acceleration. This limit retains the fast branch of incompressible Hall-MHD wave dispersion, not the slow branch; very oblique modes must separately satisfy the fast-time ordering.

u_{e} = u - j / (e n)

j = c \nabla \times B / (4 π)

and

\partial_{t} B = \nabla \times [(u - c \nabla \times B / (4 π e n)) \times B]

. The ion velocity still satisfies the magnetohydrodynamic momentum equation. Neglecting that velocity gives electron magnetohydrodynamics, a further approximation rather than a synonym for the complete Hall system.

For a singly charged ion species in Gaussian units,

ω_{p i} = (4 πn e^{2} / m_{i})^{1/2}

d_{i} = c m_{i} / (4 πn e^{2})

. The ion skin depth marks a characteristic scale below which ion and electron motion can separate in a magnetized plasma. Together with the Alfvén speed, it satisfies