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...
Using a large number of numerical simulations we examine the steady state of rotating turbulent flows in triple periodic domains, varying the Rossby number Ro (that measures the inverse rotation rate) and the Reynolds number Re (that measures the strength of turbulence). The examined flows are sustained by either a helical or a nonhelical Roberts force, that is invariant along the axis of rotation. The forcing acts at a wavenumber k f such that k f L = 4, where 2πL is the size of the domain. Different flow behaviours were obtained as the parameters are varied. Above a critical rotation rate the flow becomes quasi two dimensional and transfers energy to the largest scales of the system forming large coherent structures known as condensates. We examine the behaviour of these condensates and their scaling properties close and away from this critical rotation rate. Close to the the critical rotation rate the system transitions supercritically to the condensate state displaying a bimodal behaviour oscillating randomly between an incoherent-turbulent state and a condensate state. Away from the critical rotation rate, it is shown that two distinct mechanisms can saturate the growth of the large scale energy. The first mechanism is due to viscous forces and is similar to the saturation mechanism observed for the inverse cascade in two-dimensional flows. The second mechanism is independent of viscosity and relies on the breaking of the twodimensionalization condition of the rotating flow. The two mechanisms predict different scaling with respect to the control parameters of the system (Rossby and Reynolds), which are tested with the present results of the numerical simulations. A phase space diagram in the Re, Ro parameter plane is sketched.
We investigate the critical transition from an inverse cascade of energy to a forward energy cascade in a two-dimensional magnetohydrodynamic flow as the ratio of magnetic to mechanical forcing amplitude is varied. It is found that the critical transition is the result of two competing processes. The first process is due to hydrodynamic interactions and cascades the energy to the large scales. The second process couples small-scale magnetic fields to large-scale flows, transferring the energy back to the small scales via a nonlocal mechanism. At marginality the two cascades are both present and cancel each other. The phase space diagram of the transition is sketched.
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