On the edge of an inverse cascade

Abstract

We demonstrate that systems with a parameter-controlled inverse cascade can exhibit critical behavior for which at the critical value of the control parameter the inverse cascade stops. In the vicinity of such a critical point standard phenomenological estimates for the energy balance will fail since the energy flux towards large length scales becomes zero. We demonstrate these concepts using the computationally tractable model of two-dimensional magneto-hydrodynamics in a periodic box. In the absence of any external magnetic forcing the system reduces to hydrodynamic fluid turbulence with an inverse energy cascade. In the presence of strong magnetic forcing the system behaves as 2D magneto-hydrodynamic turbulence with forward energy cascade. As the amplitude of the magnetic forcing is varied a critical value is met for which the energy flux towards the large scales becomes zero. Close to this point the energy flux scales as a power law with the departure from the critical point and the normalized amplitude of the fluctuations diverges. Similar behavior is observed for the flux of the square vector potential for which no inverse flux is observed for weak magnetic forcing, while a finite inverse flux is observed for magnetic forcing above the critical point. We conjecture that this behavior is generic for systems of variable inverse cascade.In many dynamical systems in nature energy is transfered to smaller or to larger length scales by a mechanism known as forward or inverse cascade, respectively. In three-dimensional hydrodynamic (HD) turbulence energy cascades forward from large to small scales while in two-dimensional HD turbulence energy cascades inversely from small scales to large scales [1,2]. When the dissipation coefficients are very small (large Reynolds numbers) the rate that energy is dissipated ǫ equals the flux of energy Π E introduced by the cascade and thus this process is of fundamental interest for many fields (astrophysics, atmospheric sciences, industry, ect.). There are some examples, however, that have a mixed behavior such as fast rotating fluids, stratified flows, conducting fluids in the presence of strong magnetic fields, or flows in constrained geometry [3][4][5][6][7][8]. In these examples the injected energy cascades both forward and inversely in fractions that depend on the value of a control parameter µ (rotation rate/magnetic field/aspect ratio). In rotating flows, for example, when the rotation is weak the behavior of the flow is similar to isotropic turbulence and energy cascades forward. As the rotation rate is increased variations along the direction of rotation are suppressed and the flow starts to become quasi-2D. Eventually when rotation is strong enough the two-dimensional component of the flow dominates and energy starts to cascade inversely to the large scales. This dual cascade behavior is not restricted to quasi-2D flows neither to the cascade of energy. It is also observed in wave systems such as surface waves [9], elastic waves [10] and quantum fluids [11]. The varia...

Rotating Taylor–Green flow

“…At this state there is a very weak inverse energy transfer just sufficient to sustain the large-scale flow against viscosity. Aspects of such marginal states of inverse cascades have been recently investigated in more simplified models (Seshasayanan et al 2014).…”

“…This result implies that this state is unlikely to be described by an Ro U → 0 expansion as the eddy turnover time is the same order as the rotation period. It also implies that condensates saturate by reaching a state of marginal inverse cascade (Seshasayanan, Benavides & Alexakis 2014).…”

Third-order structure functions for isotropic turbulence with bidirectional energy transfer

“…To test our heuristic theory we performed numerical simulations of a 2D MHD turbulent flow in which the velocity v and the magnetic field B are coplanar. This is an ideal test system because Seshasayana et al (2014) found bidirectional energy transfer in this 2D system, and also its KHM equation has the generic form (2.1) with a third-order structure function vector (cf. Podesta 2008) defined by V = δu (δu·δu) + δu (δB·δB) − 2δB (δB·δu).…”

Inverse cascade of hybrid helicity inBΩ-MHD turbulence

“…In particular, − u,b /( + + − ) provides a measure of the strength of the inverse cascade [20]. Note that this mea- sure does not require a mechanism of inverse cascade driven by the total energy.…”