On the Origin of Perpendicular Cascades Among Nonlinear Alfvén Waves

We examine the effects of mildly oblique, low-frequency Alfvén waves on cross-field magnetic structures of varying scales. The magnetic structures correspond to the type of zero-frequency modes which can arise when ingoing and outgoing Alfvén waves interact with one another. Such modes are important to the develop of nearly perpendicular cascades in magnetohydrodynamic (MHD) turbulence. In linear theory, the structure remains static and the Alfvén wave generally refracts into the perpendicular direction and is resonantly absorbed. Resonant absorption is a pependicular cascade since small perpendicular wave features are generated. However, it differs from a turbulent cascade in that it is an phase-ordered process. Given a small wave amplitude, linear theory is valid when the structure's gradient scale is of order or larger than the Alfvén wavelength. However, when the gradient scale becomes smaller than the wavelength, nonlinear effects appear due to the Alfvén wave advecting and bending the magnetic structure. The magnetic structure acquires parallel scales and also a velocity field since moving the magnetic field of the structure induces an electric field and so also a velocity field. This renders the structure indistinct from a wave at a given monent although there is no net propagation. The Alfvén wave still undergoes resonant absorption but now without the refraction of the Alfvén wave and at a reduced rate. Instead of refraction, the Alfvén wave reflects and generates a backward Alfvén wave flux because advection of the structure gives Alfvén speed gradients along the background magnetic field. The reflection saturates when the wave flux in each direction along B₀ becomes equal. In this case, we have a mildly oblique Alfvén wave with the nonlinear capability of cascading power perpendicularly without changing its wave vector orientation and without its entrainment into the nearly perpendicular direction. We will also show that power accumulation along the advecting resonant field lines can be quenched by ion cyclotron resonant heating when small parallel scales develop in association with the advection. We relate this evolution to MHD simulations of isotropic spectra of Alfvén waves which develop nearly perpendicular cascades. We suggest that the cascade originates not simply from the development of zero-frequency modes but rather from the nonlinear modifications of resonant absorption.