Entanglement of helicity and energy in kinetic Alfvén wave/whistler turbulence

The role of magnetic helicity is investigated in kinetic Alfvén wave and oblique whistler turbulence in presence of a relatively intense external magnetic fieldb0e∥. In this situation, turbulence is strongly anisotropic and the fluid equations describing both regimes are the reduced electron magnetohydrodynamics (REMHD) whose derivation, originally made from the gyrokinetic theory, is also obtained here from compressible Hall magnetohydrodynamics (MHD). We use the asymptotic equations derived by Galtier and Bhattacharjee (2003Phys. Plasmas10, 3065–3076) to study the REMHD dynamics in the weak turbulence regime. The analysis is focused on the magnetic helicity equation for which we obtain the exact solutions: they correspond to the entanglement relation,n + ñ= −6, wherenandñare the power law indices of the perpendicular (tob0) wave number magnetic energy and helicity spectra, respectively. Therefore, the spectra derived in the past from the energy equation only, namelyn= −2.5 andñ= −3.5, are not the unique solutions to this problem but rather characterize the direct energy cascade. The solutionñ= −3 is a limit imposed by the locality condition; it is also the constant helicity flux solution obtained heuristically. The results obtained offer a new paradigm to understand solar wind turbulence at sub-ion scales where it is often observed that −3 <n< −2.5.