Critical Gradient Length Research Articles

Critical temperature gradient length signatures in heat wave propagation across internal transport barriers in the Joint European Torus

New results on electron heat wave propagation using ion cyclotron resonance heating power modulation in the Joint European Torus (JET) [P. H. Rebut et al., Nucl. Fusion 25, 1011 (1985)] plasmas characterized by internal transport barriers (ITBs) are presented. The heat wave generated outside the ITB, and traveling across it, always experiences a strong damping in the ITB layer, demonstrating a low level of transport and loss of stiffness. In some cases, however, the heat wave is strongly inflated in the region just outside the ITB, showing features of convective-like behavior. In other cases, a second maximum in the perturbation amplitude is generated close to the ITB foot. Such peculiar types of behavior can be explained on the basis of the existence of a critical temperature gradient length for the onset of turbulent transport. Convective-like features appear close to the threshold (i.e., just outside the ITB foot) when the value of the threshold is sufficiently high, with a good match with the theoretical predictions for the trapped electron mode threshold. The appearance of a second maximum is due to the oscillation of the temperature profile across the threshold in the case of a weak ITB. Simulations with an empirical critical gradient length model and with the theory based GLF23 [R. E. Waltz et al., Phys. Plasmas, 4, 2482 (1997)] model are presented. The difference with respect to previous results of cold pulse propagation across JET ITBs is also discussed.

Investigation of electron heat pinch in ASDEX-Upgrade

The existence of an electron heat pinch has been experimentally investigated in ASDEX-Upgrade in plasmas with strong off-axis electron cyclotron heating (ECH). Localization of 1.6 MW of ECH power at ρdep = 0.65 in plasmas with low Ohmic power (<100 kW) still resulted in peaked electron temperature (Te) profiles at densities ne0 ∼ 2.7 × 1019 m−3. Two out of four gyrotrons were modulated in order to study convective effects also on the Te transient response at different frequencies. Peculiar profiles of the modulation amplitude have been measured, very flat in the region just inside ρdep (0.3 < ρtor < 0.65) or even increasing rather than decreasing away from ρdep on the inner side. The phase profiles instead regularly indicate the propagation of the heat wave in both directions away from ρdep. Power balance analysis shows that the heat flux in the core region is very close to zero, but uncertainties do not allow a clear determination of its sign. Modelling steady-state and modulation results with an empirical model featuring a diffusive term characterized by a critical gradient length (LTe = −Te/∇Te) behaviour and a constant Gaussian shaped heat pinch profile allows good reproduction of the data when the region just inside ρdep is close to the critical threshold and oscillates around it, and a small convective term U ∼ 1–2 m s−1 is present in the same region. At higher density the Te profile becomes flatter and the amplitude profiles recover the usual shape with stronger decay inside with respect to outside ρdep. In this case no heat pinch is needed and the plasma goes below the critical threshold inside ρdep. The results are consistent with the theoretical prediction of a turbulence regulated by a threshold in R/LTe and a small turbulence generated electron heat pinch term. However initial attempts to perform first principle-based simulations using the quasi-linear electrostatic drift wave Weiland model have failed to reproduce the experimental results.

Electron transport and the critical temperature gradient

Tore Supra electron thermal fluxes, analyzed over a range of heating powers and plasma densities, are shown to vary parametrically according to the small-scale electron temperature gradient (ETG) model, rather than the ion inertial scale electrostatic gyro-Bohm model. Steady-state power balance analysis and time-varying interpretative transport simulations, performed on the Tore Supra Fast Wave Electron Heating database, validate the ETG thermal flux-versus-gradient relation and the existence of a critical electron temperature gradient. The critical gradient length R/Lc and the parametric dependence of the thermal flux, qe(ne,Te,∇Te,q,s), agree well with the ETG model.

A continuum one-dimensional SOC model for thermal transport in tokamaks

Based on the now well-known and experimentally observed critical gradient length (LTe) in tokamaks, we present a continuum one-dimensional model for explaining self organized heat transport in tokamaks. The key parameters of this model include a hysteresis parameter, which ensures that the switching of the heat transport coefficient χ upwards and downwards takes place at two different values of R/LTe. Extensive numerical simulations of this model reproduce many features of present day tokamaks, such as sub-marginal temperature profiles, intermittent transport events, 1/f scaling of the frequency spectra, propagating fronts, etc. This model utilizes a minimal set of phenomenological parameters, which may be determined from experiments and/or simulations. Analytical and physical understanding of the observed features has also been attempted.

Chapter 5: Core Energy Transport in Conventional Scenarios in ASDEX Upgrade

The studies carried out in ASDEX Upgrade on transport in conventional scenarios are presented. The well-known property of tokamak temperature profiles being resilient is investigated in and interpreted, for both ions and electrons, as due to the existence of an inverse critical gradient length below which transport is low and above which it increases. Experiments in H-mode with different heating power deposition profiles were carried out. Simulation results of a variety of H-mode plasmas with three different transport models based on the physics assumptions that include the existence of such a threshold confirm this hypothesis. However, the profiles are not extremely stiff and can significantly deviate from the critical value. Electron heat transport was investigated in various experiments using electron cyclotron heating combining steady-state and power modulation. A variation of the electron heat flux while keeping the edge flux constant allows measurement of the threshold and the properties of electron transport. These resilience properties lead to a correlation between core and edge and to a dependence of global confinement on the pedestal energy. This is quantified in the analyses of a database that yield expressions linking edge and global confinement.

Statistical analysis of internal transport barriers in JET

The potential role of magnetic shear, rational safety factor surfaces and shearing rate on the confinement is investigated on the basis of a database analysis of JET internal transport barriers (ITBs). The ITB is quantified using the JET ITB criterion (Tresset G et al 2002 Nucl. Fusion 42 520). The relation between the shearing rate and the magnetic shear, already established for positive magnetic shear barrier (Tala T J J et al 2001 Plasma Phys. Control. Fusion 43 507), is investigated at low and negative magnetic shear, where the threshold in shearing rate is found to be lower than for positive magnetic shear barriers. Defining the ITB from the departure from profile stiffness is found to be consistent with the database results, and the critical gradient length is found to be minimum at low magnetic shear. Finally, ITBs are found to be localized in the vicinity of integer safety factor surfaces in positive magnetic shear plasmas, whereas no correlation is found with integer values when the barrier is localized in negative magnetic shear regions.

Analytical solutions for the propagation of heat pulses with temperature gradient length-dependent diffusion coefficient

An original method is presented to solve the linearized heat transport equation for a general class of χ models, of the form χ = χ0TμG(∇T/T), where χ0 is a constant and G an arbitrary function which depends only on ∇T/T. These solutions allow one to address the propagation of heat and cold pulses in a tokamak plasma. By looking for solutions of the problem as functions of ∇T/T instead of the usual radial coordinates, the linearized transport equation is reformulated into a much simpler second-order differential equation. In this new form, it may be solved directly, if an analytical solution of the problem exists. The calculations are carried out both in slab and cylindrical geometry, and can be generalized to other geometries. For slab geometry, the exact solutions of the linearized transport equation are found in the case G = (|∇T|/T)α. An approximate solution is given in the case with critical gradient length G = (|∇T|/T)α(|∇T|/T−κ)β, valid for strong profile stiffness (∇T/T≈κ). In cylindrical geometry, the Wentzel–Kramers–Brillouin solutions are found in the case G = (|∇T|/T)α. These solutions are validated by comparison with numerical simulations. Their dependence on the modulation frequency and the model parameters is investigated. Finally, a method is proposed to identify the model parameters (e.g. χ0, α and μ) which best fit given temperature modulation data, as an original application of these analytical calculations to experimental transport studies.

Modelling of ECH modulation experiments in ASDEX Upgrade with an empirical critical temperature gradient length transport model

The relevance of the assumption of critical gradient length in electron heat transport is tested against the results of electron cyclotron heating (ECH) modulation experiments carried out on ASDEX Upgrade, using time-dependent transport simulations. A general χe model based on a ∇Te/Te threshold is proposed, which can provide two different heat pulse (HP) propagation regimes, with respect to the power balance (PB) diffusion coefficient χePB: χeHP = χePB occurs if ∇Te/Te<(∇Te/Te)c, while χeHP>χePB if ∇Te/Te⩾(∇Te/Te)c. By a careful choice of the model parameters, the experimental Te response to ECH modulations is simulated with excellent accuracy, and the transition between the two propagation regimes is, without ambiguity, ascribed to a ∇Te/Te threshold. Our findings are consistent with theoretical expectations, as well as with experimental observations made on TORE SUPRA. This global consistency shows the relevance of the main physical factors involved in our model, in particular the assumption of critical temperature gradient length.

Experimental evidence for gradient length-driven electron transport in tokamaks.

Energy transport by the electrons in a tokamak is examined in steady-state and power modulation experiments using electron cyclotron heating. The results are consistent with the assumption that temperature profiles are limited by a critical gradient length, leading to "stiff" profiles. The modulation experiments show that the stiffness factor increases with temperature. They strongly suggest that turbulence driven by the electron temperature gradient may be a dominant mechanism of electron transport. Although possibly not universal, these results are valid under various plasma conditions.

Desity Clamping and Power Deposition Profile in rf Heated Plasma in Toroidal Helical Systems

Radial transport driven by rf heating in toroidal helical systems with the large helical ripple is studied. The cyclotron resonance heating changes the pitch angle of resonating particles, so that the transition from the transit orbit to the trapped one occurs. The deviation of the average minor radius of the particle orbit from the magnetic surface strongly depends on the pitch angle of the particle velocity. Owing to this dependence, the slight change in the pitch angle by heating can generate a large step in the radial location. The approximate forms of the driven radial flow, which is rf-power-dependent, and the deformation of the heating profile are obtained. It is found that the deposition profile in this configuration is hardly made sharper than a critical gradient length. The injection of the heating power from the inside of the torus is preferable to improve the heating efficiency and to overcome the density clamping.