Critical balance (${\tau }_{\mathrm{nl}}\sim {\tau }_{\mathrm{wave}}$)

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.

Balance the inertial wave rate against horizontal nonlinear advection. The anisotropy $ϵ=|k_{\parallel }|/k_{\perp }$ is then of the same order as the Rossby number $k_{\perp }U/(2\Omega )$. Horizontal geostrophic flow gives kinematic pressure amplitude $\delta p\sim 2\Omega U/k_{\perp }\sim U^2/ϵ$. The first ageostrophic flow is $O(ϵU)$ and supplies the small horizontal divergence balancing $-{\partial }_z{u}_{\parallel }$. Rapid inertial oscillations are excluded by this slow-time ordering.