Analysis of Saturation Scalings and Time-Dependent Behavior in Ion Temperature Gradient Turbulence
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Saturation (graph theory)
Temperature Gradient
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We demonstrate that the scaling properties of slab ion and electron temperature gradient driven turbulence may be derived by dimensional analysis of a drift kinetic system with one kinetic species. These properties have previously been observed in gyrokinetic simulations of turbulence in magnetic fusion devices.
Slab
Electron temperature
Gyrokinetics
Temperature Gradient
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The developed kinetic theory for the stability of a magnetic-field-aligned (parallel) shear flow with inhomogeneous ion temperature [Mikhailenko et al., Phys. Plasmas 21, 072117 (2014)] predicted that a kinetic instability arises from the coupled reinforcing action of the flow velocity shear and ion temperature gradient in the cases where comparable ion and electron temperatures exist. In the present paper, the nonlinear theory was developed for the instability caused by the combined effects of ion-temperature-gradient and shear-flow (ITG–SF). The level of the electrostatic turbulence is determined for the saturation state of the instability on the basis of the nonlinear dispersion equation, which accounts for a nonlinear scattering of ions by the developed turbulence in a sheared flow. The renormalized quasilinear equation for the ion distribution function, which accounts for the turbulent scattering of ions by ITG–SF driven turbulence, was derived and employed for the estimation of the turbulent ion viscosity, the anomalous ion thermal conductivity, and anomalous ion heating rate at the saturation state of the instability.
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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.
Magnetohydrodynamic Turbulence
Pressure gradient
Turbulence Modeling
Gyrokinetics
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The emerging understanding of instability-driven plasma-turbulence saturation in terms of energy transfer to stable modes in the same scale range as the instability is employed to derive a saturation theory for the toroidal branch of ion temperature gradient turbulence that provides the scaling of turbulence and zonal flow levels for all physical parameters. The theory is based on the eigenmode decomposition of a nonlinear fluid model, which is subjected to a statistical closure and simplified via an ordering expansion consistent with zonal-flow catalyzed energy transfer from the unstable mode to the stable mode at large scale. Solution of the closed energy balance equations yields a turbulence level that is proportional to the ratio of the zonal flow damping rate and the inverse of the triplet correlation time of the zonal-flow catalyzed wavenumber triplet interaction. The zonal flow energy is proportional to the ratio of the growth rate and the inverse triplet correlation time. The saturation scalings are applied to the ion heat flux, showing that it has a factor proportional to the quasilinear heat flux and a correction factor that includes the inverse of the triplet correlation time and a reduction due to the stable mode.
Wavenumber
Saturation (graph theory)
Zonal flow (plasma)
Energy flux
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Based on first principle gyrokinetic calculations, a zonal flow generation mechanism in the slab electron temperature gradient driven (ETG) turbulence with weak magnetic shear is identified as self-organization via the turbulent spectral cascade in the two dimensional rotating fluid turbulence. The inverse energy cascade and the scaling of a zonal flow wavenumber, which is consistent with the Rhines scale length, are confirmed. An impact of the scaling, which depends on the density gradient, on the turbulent structure and transport is demonstrated for the slab ETG turbulence.
Energy cascade
Wavenumber
Temperature Gradient
Electron temperature
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We propose a quantitative model of ion temperature gradient driven turbulence in toroidal magnetized plasmas. In this model, the turbulence is regulated by zonal flows, i.e. mode saturation occurs by a zonal-flow-mediated energy cascade ('shearing'), and zonal flow amplitude is controlled by nonlinear decay. Our model is tested in detail against numerical simulations to confirm that both its assumptions and predictions are satisfied. Key results include (1) a sensitivity of the nonlinear zonal flow response to the energy content of the linear instability, (2) a persistence of zonal-flow-regulated saturation at high temperature gradients, (3) a physical explanation of the nonlinear saturation process in terms of secondary and tertiary instabilities, and (4) dependence of heat flux in terms of dimensionless parameters.
Energy cascade
Dimensionless quantity
Saturation (graph theory)
Zonal flow (plasma)
Shearing (physics)
Temperature Gradient
Gyrokinetics
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The turbulence-induced ion banana polarization current associated with steep ion temperature gradients is explored as a possible mechanism for generating poloidal momentum at the tokamak edge. In the light of a recently developed two-dimensional turbulence theory, one can obtain a simple closed expression relating this current (determined by turbulence levels) to the derivatives of the poloidal rotation speed. A self-consistent system, then, emerges, if we balance the turbulence-induced poloidal momentum with that dissipated by viscosity. Under suitable conditions this system may show a bifurcation controlled by a parameter dependent on temperature gradients. Both the bifurcation point, and the shear layer width are predicted for a prescribed flow in terms of a scale characterizing the nonlinearity of viscosity. The crucial relevance of the flow parity with the turbulence scenario is analyzed.
Turbulence Modeling
Zonal flow (plasma)
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A self-consistent theory for the interaction between electron temperature gradient (ETG) and drift-ion temperature gradient (DITG) turbulence is presented. Random shear suppression of ETG turbulence by DITG modes is studied, as well as the back-reaction of the ETG modes on the DITG turbulence via stresses. It is found that ETG dynamics can be sensitive to shearing by short-wavelength DITG modes. DITG modulations of the electron temperature gradient are also shown to be quite significant. Conversely, the back-reaction of the ETG on the DITG turbulence is found to be weak. The importance of different interactions is quantified via scalings which sensitively depend upon the electron–ion mass ratio. The findings are used to motivate a discussion of the development of a “super-grid” model for the effects of DITG turbulence on the ETG turbulence.
Shearing (physics)
Temperature Gradient
Wave turbulence
Electron temperature
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It is found in collisionless Electron Temperature Gradient (ETG) turbulence simulations that, while zonal flows are weak at early times, the zonal flows continue to grow algebraically (proportional to time). These fine‐scale zonal flows have a radial wave number such that krρi > 1 and krρe < 1. Eventually, the zonal flows grow to a level that suppresses the turbulence due to ExB shearing. The final electron energy flux is found to be relatively low. These conclusions are based on particle convergence studies with adiabatic ion electrostatic flux‐tube gyrokinetic δf particle simulations run for long times. The Rosenbluth‐Hinton random walk mechanism is given as an explanation for the long time build up of the zonal flow in ETG turbulence and it is shown that the generation is (k⊥ρe)2 smaller than for isomorphic Ion Temperature Gradient (ITG) problem. This mechanism for zonal flow generation here is different than the modulational instability mechanism for ITG turbulence. These results are important because previous results indicated zonal flows were unimportant for ETG turbulence. Weak collisional damping of the zonal flow is also shown to be a n important effect.
Zonal flow (plasma)
Electron temperature
Temperature Gradient
Flux tube
Shearing (physics)
Modulational Instability
Wave turbulence
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The effect of changes in plasma parameters, that are characteristic near or at an L-H transition in fusion edge plasmas, on fluctuation correlation lengths are analysed by means of drift-Alfvén turbulence computations.Scalings by density gradient length, collisionality, plasma beta, and by an imposed shear flow are considered.It is found that strongly sheared flows lead to the appearence of long-range correlations in electrostatic potential fluctuations parallel and perpendicular to the magnetic field.
Collisionality
Zonal flow (plasma)
BETA (programming language)
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