es equations; Extreme Behaviour; Variational methods; Vortex Flows

Introduction {#sec:intro}
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The objective of this investigation is to study three-dimensional (3D) flows of viscous incompressible fluids which are constructed to exhibit extreme growth of enstrophy. It is motivated by the question whether the solutions to the 3D incompressible Navier-Stokes system on unbounded or periodic domains corresponding to smooth initial data may develop a singularity in finite time [@d09]. By formation of a “singularity” we mean the situation when some norms of the solution corresponding to smooth initial data have become unbounded after a finite time. This so-called “blow-up problem” is one of the key open questions in mathematical fluid mechanics and, in fact, its importance for mathematics in general has been recognized by the Clay Mathematics Institute as one of its “millennium problems” [@f00]. Questions concerning global-in-time existence of smooth solutions remain open also for a number of other flow models including the 3D Euler equations [@gbk08] and some of the “active scalar” equations [@k10].

While the blow-up problem is fundamentally a question in mathematical analysis, a lot of computational studies have been carried out since the mid-’90s in order to shed light on the hydrodynamic mechanisms which might lead to singularity formation in finite time. Given that such flows evolving near the edge of regularity involve formation of very small flow structures, these computations typically require the use of state-of-the-art computational resources available at a given time. The computational studies focused on the possibility of finite-time blow-up in the 3D Navier-Stokes and/or Euler system include [@bmonmu83; @ps90; @b91; @k93; @p01; @bk08; @oc08; @o08; @ghdg08; @gbk08; @h09; @opc12; @bb12; @opmc14], all of which considered problems defined on domains periodic in all three dimensions. Recent investigations by [@dggkpv13; @k13; @gdgkpv14; @k13b] focused on the time evolution of vo