In hydrodynamic (HD) and magnetohydrodynamic (MHD) turbulence an energy cascade can occur when there is a substantial range of scales—the inertial range (IR)—over which the energy transfer process is self-similar. An elementary question is the following: What are the minimum Reynolds numbers needed for a turbulent flow to have a substantial IR, that is, one of a given bandwidth. The effect of finite Reynolds numbers on the IR has been well studied in HD turbulence, which provides a roadmap for analogous study in MHD turbulence. To help identify the minimum Reynolds number associated with a substantial IR, we conduct a series of incompressible MHD turbulence simulations with Taylor Reynolds numbers ${\mathrm{Re}}_{\lambda }\approx 200\text{–}400$. These are analyzed in terms of the MHD third-order law, one of the few rigorous results concerning MHD turbulence energy transfer within a well-defined IR. Specifically, we use simulations and modelled structure functions to estimate the IR boundaries as a function of ${\mathrm{Re}}_{\lambda }$. Following studies in HD turbulence, a semiempirical power-law model is applied to structure function behavior, which allows us to extrapolate to ${\mathrm{Re}}_{\lambda }$ that are much higher than those accessible with experiments or simulations. The present results estimate the minimum Reynolds number needed to establish an IR spanning a specified number of decades. For MHD flows whose Reynolds numbers satisfy the criteria, this provides a justification for the use of the energy cascade scenario and the third-order law.

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• Energy cascade
• Magnetohydrodynamic turbulence
• Plasma turbulence
• Scaling laws of complex systems
• Turbulence