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Critical balance is a scale-dependent ordering in which the wave-propagation time and nonlinear interaction time are comparable. For strong Alfvénic turbulence, $k_{\parallel }{v}_A\sim k_{\perp }\delta u_{\perp }$. For electron magnetohydrodynamics, the linear rate is $v_A{d}_i{k}_{\parallel }{k}_{\perp }$ and the nonlinear rate is $v_A{d}_i{k}_{\perp }^2\delta B/B_0$, so critical balance instead gives $\delta B/B_0\sim k_{\parallel }/k_{\perp }$. 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 $b_l=\delta B_l/B_0$, its perpendicular nonlinear interaction time is ${\tau }_{\mathrm{nl}}\sim l_{\perp }^2/(v_A{d}_i{b}_l)$. The anisotropic whistler wave time is ${\tau }_w\sim l_{\parallel }{l}_{\perp }/(v_A{d}_i)$. Scaling predictions depend on locality, phase decorrelation and the assumed balance of these times.

Electron magnetohydrodynamics neglects ion motion and electron inertia while retaining transport of the magnetic field by the electron current. At constant electron number density, ${\mathbf{u}}_e=-\mathbf{J}/(en)=-c\nabla \times \mathbf{B}/(4\pi en)$. Flux freezing in this velocity givesFor a uniform guide field, the solenoidal linear equation implies ${\omega }^2=(c{B}_0/4\pi en{)}^2{k}_{\parallel }^2{k}^2$. With $v_A=B_0/\sqrt{4\pi n{m}_i}$ and $d_i=c\sqrt{m_i/(4\pi e^{2}n)}$, the coefficient is $c{B}_0/(4\pi en)=v_A{d}_i$.

For field amplitudes of order the guide field, the ion Lorentz-force response over time $\tau$ is $\Delta u/v_A=O(v_A\tau /l)\ll 1$. 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.

Hall magnetohydrodynamics retains bulk ion inertia but freezes magnetic flux into the electron velocity ${\mathbf{u}}_e=\mathbf{u}-\mathbf{j}/(en)$. With constant density, negligible electrical resistivity and electron inertia, Gaussian units give $\mathbf{j}=c\nabla \times \mathbf{B}/(4\pi )$ and ${\partial }_t\mathbf{B}=\nabla \times [(\mathbf{u}-c\nabla \times \mathbf{B}/(4\pi en))\times \mathbf{B}]$. 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, ${\omega }_{pi}=(4\pi n{e}^2/m_i{)}^{1/2}$ is its plasma frequency and $d_i=c\sqrt{m_i/(4\pi n{e}^2)}$. 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 $v_A{d}_i=c{B}_0/(4\pi en)$, the coefficient of guide-field propagation in electron magnetohydrodynamics.

Use Gaussian units, uniform number density $n$, and ion mass density $\rho =n{m}_i$. Here $e>0$ is the magnitude of the elementary charge, as fixed by the printed current relation. Neglect electron inertia and displacement current. ThenFreezing the magnetic flux into this electron velocity, while retaining the ion momentum equation, gives the closed Hall magnetohydrodynamics systemThe pressure enforces the incompressible flow constraint; taking the divergence of momentum determines it up to the usual boundary-dependent constant. The induction equation preserves the solenoidal magnetic-field constraint. One can absorb $B^2/(8\pi )$ into magnetohydrodynamic total pressure, replacing the Lorentz term by magnetic tension. Keeping ion inertia here is important: electron magnetohydrodynamics with fixed ions will be a later limit, not the complete two-field system.

Write $ϵ=l/d_i\ll 1$ and $\delta =v_A\tau /l\ll 1$. For magnetic amplitudes of order $B_0$, ion acceleration is of order $v_A^2/l$. Thus its change over the fast time isThe ions are consequently fixed to leading order on this timescale. Their first nonzero response is driven by the momentum equation, with negligible self-advection for an initially resting ion flow; one need not assert that the dimensional Lorentz acceleration literally vanishes. In fast-time, Alfvén-velocity dimensionless variables, the ion equation reads ${\partial }_{\hat{t}}\hat{\mathbf{u}}=O(\delta )$, giving ${\partial }_{\hat{t}}\hat{\mathbf{u}}=0$ at leading order.

Meanwhile the electron-current velocity is $j/(en)\sim v_A{d}_i/l=v_A/ϵ$, larger than a bulk flow of order $v_A$. The ion-advection term in induction is smaller than Hall induction by $O(ϵ)$ even if that slow flow is retained. The leading magnetic equation is thereforewith stationary ions at leading order. This is the fast-time electron-MHD limit of Hall magnetohydrodynamics and is precisely electron magnetohydrodynamics.

The already obtained fast-wave eigenvector has $u/b=|q/\omega |\sim 1/(k{d}_i)\ll 1$, confirming that ion motion cannot affect its leading frequency. The closed magnetic equation therefore suffices to obtain that branch without using momentum to close it. Its timescale is ${\tau }_w\sim 1/(v_A{d}_i|k_{\parallel }|k)$; compatibility with ${\tau }_w\ll l/v_A$ requires $|k_{\parallel }|d_i\gg 1$ when $l\sim k^{-1}$. Arbitrarily small $k_{\parallel }$ can violate this fast-time assumption even though $k{d}_i\gg 1$. The slow branch does not obey the stated ordering and is not retained by the fixed-ion reduction.

Put $a=c/(4\pi en)$ and retain terms linear in $\delta \mathbf{B}$. The electron magnetohydrodynamics induction equation becomesHere the curl of $(\nabla \times \delta \mathbf{B})\times B_0\hat{\mathbf{z}}$ reduces to $B_0{\partial }_z\nabla \times \delta \mathbf{B}$, since the background magnetic field is constant and the divergence of a curl vanishes. For a plane wave $\delta \mathbf{B}=\mathbf{b}{e}^{i(\mathbf{k}\cdot \mathbf{r}-\omega t)}$, the linear equation and the solenoidal constraint areOn the transverse plane, the operator $i\mathbf{k}\times$ has eigenvalues $\pm k$ by the helicity decomposition of a transverse Fourier mode. Equivalently, squaring the equation and using the vector triple product gives ${\omega }^2=a^2{B}_0^2{k}_{\parallel }^2{k}^2$. Since $a{B}_0=v_A{d}_i$, where $v_A$ is the Alfvén speed and $d_i$ the ion skin depth,The two branches have opposite circular polarizations. The specified electron-only induction model describes a whistler wave; the paper calls these branches kinetic Alfvén waves. The following cascade calculation uses the specified model and its dispersion relation, without adding the pressure response needed for the usual kinetic Alfvén interpretation.

At fixed number density $n$, the electron current givesThe second equality uses Ampère's law without displacement current and stationary ions. A perturbation with characteristic perpendicular wave number $k_{\perp }$ therefore produces $\delta u_e\sim \alpha k_{\perp }\delta B$. Its nonlinear magnetic-advection rate isFor the anisotropic electron magnetohydrodynamics wave, $k\simeq k_{\perp }$ and $|\omega |\sim v_A{d}_i|k_{\parallel }|k_{\perp }$. Equating this to the nonlinear rate by critical balance givesThis is small-amplitude but strong turbulence: a small perturbation can have order-one nonlinear distortion over one wave period because its perpendicular gradients are much larger than its parallel gradients.

Put $b=\delta B_{\parallel }/B_0$ and ${\mathbf{b}}_{\perp }=\delta {\mathbf{B}}_{\perp }/B_0=\hat{\mathbf{z}}\times {\nabla }_{\perp }\Psi /v_A$. Define the two-dimensional Poisson bracket byThen, to the retained anisotropic order, the derivative along the perturbed magnetic field isThe omitted $b{\partial }_z$ term is smaller by $ϵ$ than either retained term. The nonlinear field-line-bending term is of the same order as ${\partial }_z$, which is why it must remain.

Use ${\mathbf{u}}_e=-\alpha \nabla \times \mathbf{B}$ with $\alpha B_0=v_A{d}_i$. The leading parallel and perpendicular electron velocities areIn detail, $\nabla \times (B_{0}b\hat{\mathbf{z}})=-B_0\hat{\mathbf{z}}\times {\nabla }_{\perp }b$, while the parallel curl of $B_0\hat{\mathbf{z}}\times {\nabla }_{\perp }\Psi /v_A$ is $B_0{\nabla }_{\perp }^2\Psi /v_A$. The perpendicular curl involving ${\partial }_z\delta {\mathbf{B}}_{\perp }$ is order-smaller. Exact electron incompressibility follows from the divergence of a curl; its small velocity correction supplies any divergence omitted by this leading representation.

The electron magnetohydrodynamics equation is frozen-in induction,because both $\mathbf{B}$ and ${\mathbf{u}}_e$ are solenoidal. Keeping its leading order-${ϵ}^2$ perpendicular terms givesLet $X_f=\hat{\mathbf{z}}\times {\nabla }_{\perp }f=(-f_y,f_x)$. Expanding the two components gives $(X_f\cdot \nabla )X_g-(X_g\cdot \nabla )X_f=X_{\{f,g\}}$. Substitution therefore reduces the perpendicular equation toThe perpendicular-constant freedom in $\Psi$ has no magnetic effect and can be fixed so that

The parallel induction component at the same order isThe advection term vanishes since ${\mathbf{u}}_{e\perp }\cdot {\nabla }_{\perp }b=v_A{d}_i\{b,b\}=0$. ThusTogether these are both required reduced electron magnetohydrodynamics equations. Terms such as $u_{e\parallel }{\partial }_{z}b$ and $b{\partial }_z{\mathbf{u}}_e$ are order-${ϵ}^3$ and consistently absent.

Linearization removes the Poisson bracket from ${\nabla }_{\parallel }$. For Fourier perturbations proportional to $e^{i\mathbf{k}\cdot \mathbf{x}-i\omega t}$, the two reduced equations becomeEliminating either amplitude givesSince $k=(k_{\perp }^2+k_{\parallel }^2{)}^{1/2}=k_{\perp }[1+O({ϵ}^2)]$, this is the specified dispersion relation to the accuracy of the anisotropic reduction. The reduced equations cannot retain the subleading correction replacing $k_{\perp }$ by the full $k$.

The full electron magnetohydrodynamics equation checks that correction directly. Its linearization is ${\partial }_t\delta \mathbf{B}=-\alpha B_0{\partial }_z\nabla \times \delta \mathbf{B}$. HenceApplying the cross-product operator twice and using $\mathbf{k}\cdot \delta \mathbf{B}=0$ givesThus the full model gives the stated relation exactly and the reduced model gives its leading anisotropic limit, with all signs and normalization factors consistent.

In the inertia-free, constant-density electron magnetohydrodynamics model with uniform guide field $B_0\hat{\mathbf{z}}$, a transverse magnetic plane wave obeys $\omega \mathbf{b}=i{v}_A{d}_i{k}_{\parallel }\mathbf{k}\times \mathbf{b}$. The helicity decomposition of a transverse Fourier mode gives the displayed dispersive branches with opposite circular polarizations. This electron-current mode is called a whistler wave; see Lyutikov's analysis of electron magnetohydrodynamics. Its pressure-free induction equation should not be identified with a general kinetic Alfvén model.