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We introduce an active vector system, which generalises both the 3D Euler equations and the electron–magnetohydrodynamic equations (E–MHD). We may as well view the system as singularised systems for the 3D Euler equations, in which case the equations of (E–MHD) correspond to the order two more singular one than the 3D Euler equations. The generalised surface quasi-geostrophic equation (gSQG) can be also embedded into a special case of our system when the unknown functions are constant in one coordinate direction. We investigate some basic properties of this system as well as the conservation laws. In the case when the system corresponds up to order one more singular than the 3D Euler equations, we prove local well-posedness in the standard Sobolev spaces. The proof crucially depends on a sharp commutator estimate similar to the one used for (gSQG) in Chae et al (2012 Commun. Pure Appl. Math. 65 1037–66). Since the system covers many areas of both physically and mathematically interesting cases, one can expect that there are various related problems to be investigated, parts of which are discussed here.

Vortex thinning is one of the main mechanisms of two-dimensional turbulence. By direct numerical simulation to the two-dimensional Navier–Stokes equations with small-scale forcing and large-scale damping, Xiao et al (2009 J. Fluid Mech. 619 1–44) found an evidence that inverse energy cascade may proceed with the vortex thinning mechanism. On the other hand, Alexakis and Doering (2006 Phys. Lett. A 359 652–657) calculated the upper bound of the bulk averaged enstrophy dissipation rate of the steady-state two dimensional turbulence. In this paper, we show that vortex thinning induces enhanced dissipation with strictly slower vanishing order of the enstrophy dissipation than Re−1.

The generalized Constantin–Lax–Majda (gCLM) equation was introduced to model the competing effects of advection and vortex stretching in hydrodynamics. Recent investigations revealed possible connections with the two-dimensional turbulence. With this connection in mind, we consider the steady problem for the viscous gCLM equations on T : a v ω x − v x ω = ν Δ ω + f , v = ( − Δ ) − 1 2 ω , where a ∈ R is the parameter measuring the relative strength between advection and stretching, ν > 0 is the viscosity constant, and f is a given O(1)-forcing independent of ν. For some range of parameters, we establish existence and uniqueness of stationary solutions. We then numerically investigate the behaviour of solutions in the vanishing viscosity limit, where bifurcations appear, and new solutions emerge. When the parameter a is away from [−1/2, 1], we verify that there is convergence towards smooth stationary solutions for the corresponding inviscid equation. Moreover, we analyse the inviscid limit in the fractionally dissipative case, as well as the behaviour of singular limiting solutions.