In magnetized plasmas, a turbulent cascade occurs in phase space at scales smaller than the thermal Larmor radius (“sub-Larmor scales”) [Tatsuno et al., Phys. Rev. Lett. 103, 015003 (2009)]. When the turbulence is restricted to two spatial dimensions perpendicular to the background magnetic field, two independent cascades may take place simultaneously because of the presence of two collisionless invariants. In the present work, freely decaying turbulence of two-dimensional electrostatic gyrokinetics is investigated by means of phenomenological theory and direct numerical simulations. A dual cascade (forward and inverse cascades) is observed in velocity space as well as in position space, which we diagnose by means of nonlinear transfer functions for the collisionless invariants. We find that the turbulence tends to a time-asymptotic state, dominated by a single scale that grows in time. A theory of this asymptotic state is derived in the form of decay laws. Each case that we study falls into one of three regimes (weakly collisional, marginal, and strongly collisional), determined by a dimensionless number , a quantity analogous to the Reynolds number. The marginal state is marked by a critical number that is preserved in time. Turbulence initialized above this value become increasingly inertial in time, evolving toward larger and larger ; turbulence initialized below become more and more collisional, decaying to progressively smaller .
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Essential features of the collisionless dynamics may be described in a 3D phase space , however, we use four phase-space dimensions to account for the fact that collisions isotropize the velocity space.
Here, the Hankel transform is defined by the three-dimensional velocity integral. When there is no dependence on gyroangle and Maxwellian dependence on , we will obtain the standard Hankel transform with respect to times .
is the local version of D of Ref. 16: As the latter was defined on the Larmor scale, i.e., D is the same as for .
