Two-dimensional magnetohydrodynamic turbulence in the limits of infinite and vanishing magnetic Prandtl number

Abstract

We study both theoretically and numerically two-dimensional magnetohydrodynamic turbulence at infinite and zero magnetic Prandtl number Pm (and the limits thereof), with an emphasis on solution regularity. For Pm= 0, both ∥ω∥2 and ∥j∥2, where ω and j are, respectively, the vorticity and current, are uniformly bounded. Furthermore, ∥∇j∥2 is integrable over [0, ∞). The uniform boundedness of ∥ω∥2 implies that in the presence of vanishingly small viscosity ν (i.e. in the limit Pm→0), the kinetic energy dissipation rate ν∥ω∥2 vanishes for all times t, including t= ∞. Furthermore, for sufficiently small Pm, this rate decreases linearly with Pm. This linear behaviour of ν∥ω∥ 2 is investigated and confirmed by high-resolution simulations with Pm in the range [1/ 64, 1] . Several criteria for solution regularity are established and numerically tested. As Pm is decreased from unity, the ratio ∥ω∥∞/∥ω∥ is observed to increase relatively slowly. This, together with the integrability of ∥∇j∥2, suggests global regularity for Pm= 0. When Pm= ∞, global regularity is secured when either ∥∇u∥ ∞/ ∥ω∥, where u is the fluid velocity, or ∥j∥ ∞/ ∥j∥ is bounded. The former is plausible given the presence of viscous effects for this case. Numerical results over the range Pm\in [1, 64] show that ∥∇u∥ ∞/ ∥ω∥ varies slightly (with similar behaviour for ∥j∥∞/∥j∥), thereby lending strong support for the possibility ∥∇u∥∞/∥ω∥