Subcritical Turbulence Research Articles

Turbulent Transport in Tokamak Plasmas with Rotational Shear

Nonlinear gyrokinetic simulations are conducted to investigate turbulent transport in tokamak plasmas with rotational shear. At sufficiently large flow shears, linear instabilities are suppressed, but transiently growing modes drive subcritical turbulence whose amplitude increases with flow shear. This leads to a local minimum in the heat flux, indicating an optimal E×B shear value for plasma confinement. Local maxima in the momentum fluxes are observed, implying the possibility of bifurcations in the E×B shear. The critical temperature gradient for the onset of turbulence increases with flow shear at low flow shears; at higher flow shears, the dependence of heat flux on temperature gradient becomes less stiff. The turbulent Prandtl number is found to be largely independent of temperature and flow gradients, with a value close to unity.

Transport Bifurcation in a Rotating Tokamak Plasma

The effect of flow shear on turbulent transport in tokamaks is studied numerically in the experimentally relevant limit of zero magnetic shear. It is found that the plasma is linearly stable for all nonzero flow shear values, but that subcritical turbulence can be sustained nonlinearly at a wide range of temperature gradients. Flow shear increases the nonlinear temperature gradient threshold for turbulence but also increases the sensitivity of the heat flux to changes in the temperature gradient, except over a small range near the threshold where the sensitivity is decreased. A bifurcation in the equilibrium gradients is found: for a given input of heat, it is possible, by varying the applied torque, to trigger a transition to significantly higher temperature and flow gradients.

On the relevance of subcritical hydrodynamic turbulence to accretion disk transport

Hydrodynamic unstratified keplerian flows are known to be linearly stable at all Reynolds numbers, but may nevertheless become turbulent through nonlinear mechanisms. However, in the last ten years, conflicting points of view have appeared on this issue. We have revisited the problem through numerical simulations in the shearing sheet limit. It turns out that the effect of the Coriolis force in stabilizing the flow depends on whether the flow is cyclonic (cooperating shear and rotation vorticities) or anticyclonic (competing shear and rotation vorticities); keplerian flows are anticyclonic. We have obtained the following results: i/ The Coriolis force does not quench turbulence in subcritical flows; ii/ The resolution demand, when moving away from the marginal stability boundary, is much more severe for anticyclonic flows than for cyclonic ones. Presently available computer resources do not allow numerical codes to reach the keplerian regime. iii/ The efficiency of turbulent transport is directly correlated to the Reynolds number of transition to turbulence $Rg$, in such a way that the Shakura-Sunyaev parameter $\alpha\sim 1/Rg$. iv/ Even the most optimistic extrapolations of our numerical data show that subcritical turbulent transport would be too inefficient in keplerian flows by several orders of magnitude for astrophysical purposes. v/ Our results suggest that the data obtained for keplerian-like flows in a Taylor-Couette settings are largely affected by secondary flows, such as Ekman circulation.

Statistical Theory of Subcritically-Excited Strong Turbulence in Inhomogeneous Plasmas. II

A statistical description is developed for a self-sustained subcritical turbulence in inhomogeneous plasmas. Interchange mode in the presence of inhomogeneous magnetic field, which reveals a submarginal strong turbulence, is considered. Nonlinear dispersion relation is extended to a Langevin equation for a dressed test mode, in which nonlinear interactions are kept as renormalized drag and random self noise. Based upon the assumption that the random noise has a faster time scale, the solutions are obtained for the fluctuation level, decorrelation rate, auto- and cross-correlation functions and spectrum. They are expressed as nonlinear functions of non-equilibrium parameters like gradient. Extended fluctuation-dissipation theorem (Einstein relation) is described as statistical relations. Then the Langevin equation is reformulated into a Fokker-Planck equation of the probability distribution function. The steady state probability function is solved. Imposing the constraint of the self-noise, the power-law d...

Statistical Theory of Subcritically-Excited Strong Turbulence in Inhomogeneous Plasmas. I

A statistical description is developed for a self-sustained subcritical turbulence in inhomogeneous plasmas. Interchange mode in the presence of inhomogeneous magnetic field, which reveals a submarginal strong turbulence, is considered. Nonlinear dispersion relation is extended to a Langevin equation for a dressed test mode, in which nonlinear interactions are kept as renormalized drag and random self noise. Based upon the assumption that the random noise has a faster time scale, the solutions are obtained for the fluctuation level, decorrelation rate, auto- and cross-correlation functions and spectrum. They are expressed as nonlinear functions of non-equilibrium parameters like gradient. Extended fluctuation-dissipation theorem (Einstein relation) is described as statistical relations. Then the Langevin equation is reformulated into a Fokker-Planck equation of the probability distribution function. The steady state probability function is solved. Imposing the constraint of the self-noise, the power-law distribution with respect to the fluctuation amplitude is obtained in the tail of the distribution function.

Theory of fully developed turbulence in buoyancy-driven fluids and pressure-gradient-driven plasmas

A new theoretical method is presented to analyse turbulence and associated transport in far-non-equilibrium fluids and plasmas. First, direct nonlinear interactions with background turbulence are renormalized into nonlinear dielectric form. The relation between the turbulent intensity spectra of energy and temperature, E( k) and , and the nonlinear transfer rates (dielectric) of momentum and energy, and , are obtained as recurrent formulae of integral equations. Second, nonlinear marginal stability conditions are examined by introduction of dressed test mode analysis. Solutions have a power law which are analogous to critical exponents in renormalization group theory. The same paradigm is first applied to neutral fluids to recover conventional results. For two-dimensional (2D) buoyancy-driven turbulence, where a supercritical turbulence appears, spectral forms of E( k), and , are obtained in the energy containing range (or subrange). ( is the temperature gradient and g is the gravity.) The relation between global turbulent transport coefficients, such as and , and nonlinear transfer rates and is obtained. A global spatial structure of the turbulent fluid, which is consistent with the spectrum, is solved. The Nusselt number is obtained as and the relations and are obtained. ( is the Rayleigh number and is the critical Rayleigh number.) To turbulence in the magnetized plasma, where a subcritical turbulence appears, this paradigm is applied. The combination of the pressure gradient and magnetic field gradient , characterizes the non-equilibrium form of the plasma. The spectral intensity of the fluctuating fields for the potential, current and pressure and the nonlinear transfer rates are first obtained (symbols and correspond to the ion viscosity, current diffusivity and heat diffusivity, respectively) in the energy containing range. The turbulence level, W, and the transport coefficients, , are derived as and . The dissipation balance is also examined. These analyses demonstrate that this method is applicable to both the supercritical and subcritical turbulences in neutral fluid and plasma turbulences, i.e. in systems which are far from the thermal equilibrium.

Dynamic structure in self-sustained turbulence

A dynamical equation for self-sustained and pressure-driven turbulence in toroidal plasmas is derived. The growth rate of the dressed-test mode, which belongs to the subcritical turbulence, is obtained as a function of the turbulent transport coefficient. In the limit of a low fluctuation level, the mode has nonlinear instability and shows explosive growth. The growth rate vanishes when the driven transport reaches a stationary-turbulent level. The stationary solution is thermodynamically stable. The characteristic time, by which the stationary and self-sustained turbulence are established, scales with the ion-sound transit time and is accelerated by bad magnetic curvature. The influence of the pressure gradient as well as the radial electric field inhomogeneity are quantified.

Self-sustained plasma turbulence due to current diffusion

Plasma turbulence and anomalous transport by the electrostatic current diffusive interchange mode are studied by the nonlinear simulation based on the magnetohydrodynamic model. The turbulence is found to have a typical characteristic of subcritical turbulence. The saturation level, as a function of the pressure gradient ∇p, is confirmed to scale like ‖∇p‖3/2. This nature holds independent of the ratio ‖∇p‖/‖∇pc‖ where ‖∇pc‖, is a critical pressure gradient against linear instability. The turbulence-driven transport is also evaluated. The simulation result confirms the theoretical prediction, which is based on the self-sustained turbulence, with respect to the nonlinear growth and damping. Both the normal cascade and inverse cascade are essential in establishing the stationary turbulent state.

Subcritical magnetohydrodynamic turbulence.

Numerical simulations of a simple model for ideal magnetohydrodynamic modes have shown stationary turbulence and transport at $\ensuremath{\beta}$ values substantially below the critical $\ensuremath{\beta}$ required for linear instability. The phenomena appears to be similar to subcritical turbulence in plane-parallel pipe flow.