MHD Stability and Energy Principle for Two-Dimensional Equilibria without Assumption of Nested Magnetic Surfaces

Abandoning the assumption of nested magnetic surfaces in tokamak plasma expands the field of research and opens up new approaches for both theoretical and experimental plasma physics. The computer code KINX for calculations of the ideal MHD stability was developed for studies of doublet plasmas with two magnetic axes and using block-structured grids in each subdomain with nested magnetic surfaces. Then, the MHD_NX code on unstructured grids was developed to calculate the stability of two-dimensional equilibria with an arbitrary topology of magnetic surfaces. The study of equilibrium and stability of equilibrium configurations with toroidal current density reversal and axisymmetric n = 0 islands, which are associated with internal transport barrier and low current density at the magnetic axis, as well as with the operation of tokamaks in the alternating current regime, leads to more general issues of MHD stability of two-dimensional solutions of the Grad-Shafranov equations with islands under other types of symmetrychain of islands in helical symmetry and cylindrically symmetric m = 0 islands in configurations with the longitudinal field reversal. New ideal MHD unstable modes have been discovered for various types of two-dimensional island configurations. The energy principle with MHD-compatible boundary conditions at open magnetic field lines is necessary for the self-consistent stability analysis of divertor configurations in tokamaks with a finite current density at the separatrix, taking into account the plasma outside the separatrix. Several codes have been developed for calculations of plasma equilibrium and stability, taking into account the influence of currents outside the separatrix, which are ready for integration with other codes for edge plasma modeling.

Stability and toroidal rotation properties of highly shaped plasmas in the TCV tokamak

Andrea Scarabosio

Magnetohydrodynamic (MHD) instabilities and plasma rotation have various impacts on particle and thermal transport in toroidal plasmas. MHD instabilities degrade the confinement, limit the maximum achievable plasma pressure, and can lead to plasma disruptions. Plasma rotation is observed to have beneficial effects on global instabilities and improve confinement through the reduction of turbulent transport. Plasma rotation and stability are strongly coupled, influencing each other in a complicated fashion. The Tokamak à Configuration Variable (TCV), unique in its plasma shaping capabilities, is equipped with a flexible auxiliary heating system (ECRH) and several diagnostics allowing us to study the effect of the plasma shape on stability in a wide range of scenarios. TCV also provides for measurements of carbon toroidal velocity in the absence of external momentum input, an experimental condition poorly studied in the past but of major interest for an accurate prediction of the toroidal rotation in future large experiments. The work presented in this thesis may be divided into two parts. In the first part, we focus on plasma instabilities appearing in three different TCV scenarios. These instabilities limit the achievable maximum pressure and current density. New features and previously unnoticed dependencies are shown. In the second part of this thesis, we experimentally study the plasma rotation properties and their relation with the plasma parameters and MHD activity. The new insight on spontaneous rotation may help in constructing a complete model of plasma rotation in tokamaks. During the initial plasma current rise, edge MHD instabilities are commonly observed in most tokamaks when the edge safety factor qa approaches low rational values. These instabilities may lead to plasma disruptions and can effectively limit the edge safety factor to qa > 3, which is above the well-known operational limit qa ≥ 2. We report on a detailed analysis of the MHD modes leading to disruptions and show experimental evidence of the key role of mode coupling. The beneficial effect of plasma shaping on current rise experiments was early noted in the TCV operation. In this thesis we characterise the effect of the plasma shape on the instability, showing how plasma elongation and triangularity (positive and negative) act as stabilising factors. The complex shape dependence of such MHD modes is interpreted on the basis of the theory of coupled tearing modes. Three stabilising mechanisms linked with plasma shaping are shown to be important for TCV and likely to other tokamaks. Sawteeth, which are central relaxation oscillations common to most tokamak scenarios, also have a significant effect on central plasma parameters. In highly elongated TCV discharges heated with far off-axis ECRH, the sawtooth oscillations are observed to disappear and to be replaced by a continuous MHD mode resonant on the q = 1 surface. The combination of a flat current profile and small q = 1 radius determines the change in the plasma stability, which we study using ideal and resistive MHD models. Plasmas with internal transport barriers (ITBs) are routinely produced in TCV using high power ECRH and current drive. A variety of MHD activity is observed during these discharges including classical m/n = 3/1 and neoclassical m/n = 2/1 tearing modes, and more exotic q = 2 pseudo-sawteeth central relaxations. Disruptive modes are observed in ITB plasmas with strong reverse magnetic shear. In general, the MHD instabilities limit the maximum achievable plasma pressure. The experimental β-limit in this scenario is observed to depend strongly on the peaking of the electron pressure profile, in agreement with the ideal MHD theory. The properties of the toroidal carbon impurity rotation in Ohmic limited L-mode plasmas are studied in detail in stationary conditions. The dependence and scaling of the toroidal velocity with plasma parameters, such as the plasma current and density, are highlighted, as well as the effect of the sawtooth activity on the rotation profile. We show that the toroidal rotation in TCV is generally directed in the counter-current or electron diamagnetic drift direction. We also compare the experimental results with the neoclassical prediction for stationary toroidal rotation in absence of external momentum input. Angular momentum relaxations are observed in TCV Ohmic plasmas. Large magnetic islands cause strong losses of angular momentum, flattening the rotation profile. Once the MHD mode has disappeared, the stationary angular velocity profile is restored over a typical time scale, 100–200 ms, providing experimental evidence of the spontaneous torque spinning the plasma column. We show that we can reproduce the phenomena of "toroidal spin-up" and rotation inversion observed in TCV Ohmic plasmas with a phenomenological momentum transport model. We infer transport coefficients, such as the momentum diffusivity, and compare them with theoretical predictions.

EPFL2006

MHD stability limits in the TCV tokamak

Holger Reimerdes

Magnetohydrodynamic (MHD) instabilities can limit the performance and degrade the confinement of tokamak plasmas. The Tokamak à Configuration Variable (TCV), unique for its capability to produce a variety of poloidal plasma shapes, has been used to analyse various instabilities and compare their behaviour with theoretical predictions. These instabilities are perturbations of the magnetic field which usually extend to the plasma edge where they can be detected with magnetic pick-up coils as magnetic fluctuations. A spatially dense set of magnetic probes, installed inside the TCV vacuum vessel, allows for a fast observation of these fluctuations. The structure and temporal evolution of coherent modes is extracted using several numerical methods. In addition to the setup of the magnetic diagnostic and the implementation of analysis methods, the subject matter of this thesis focuses on four instabilities which impose local and global stability limits. Al1 of these instabilities are relevant for the operation of a fusion reactor and a profound understanding of their behaviour is required in order to optimise the performance of such a reactor. Sawteeth, which are central relaxation oscillations common to most standard tokamak scenarios, have a significant effect on central plasma parameters. In TCV, systematic scans of the plasma shape have revealed a strong dependence of their behaviour on elongation κ, and triangularity δ, with high κ and low δ leading to shorter sawteeth with smaller crashes. This shape dependence is increased by applying central electron cyclotron heating (ECH), which increases or decreases the sawtooth period, depending on the plasma shape. The response to additional heating power is determined by the role of ideal or resistive MHD in triggering the sawtooth crash. For plasma shapes where additional heating and consequently, a faster increase of the central pressure shortens the sawteeth, the low experimental limit of the pressure gradient within the q = 1 surface is consistent with ideal MHD predictions. The observed decrease of this limit with elongation is also in qualitative agreement with ideal MHD theory. Edge localised modes (ELMs), occurring in TCV Ohmic high-confinement mode discharges, were observed to be preceded by coherent magnetic oscillations. The precursors prior to small ELMs, believed to be of type III, and prior to larger ELMs, previously referred to as TCV large ELMs, show the same characteristics, which allows for an identification of both ELMs to be of type III according to the usual classification scheme. The detected poloidal and toroidal mode structures are consistent with a resonant flux surface close to the plasma edge. Unlike conventional MHD modes, these precursors start at a random toroidal location and then grow in amplitude and toroidal extent until they encompass the whole toroidal circumference. Thus, the asymmetry causing and maintaining the toroidal localisation of the ELM precursor, must be intrinsic to the plasma. Soft X-ray measurements show that the localised precursor always coincides with a central m = 1 mode, which can usually be associated with the sawtooth pre- or postcursor mode. A comparison of the phases indicates a correlation with the maximum of the central mode preceding the toroidal location of the ELM precursor and, therefore, a hitherto unobserved coupling between central modes and ELMs. Highly elongated plasmas promise several advantages, among them higher current and beta limits. During TCV experiments dedicated to an increasing of the plasma elongation, a new disruptive current limit, at values well below the conventional current limit corresponding to qa >2, was encountered for κ >2.3. This limit, which is preceded by a kink-type mode, is found to be consistent with ideal MHD stability calculations. The TCV observations, therefore, provide the first experimental confirmation of a deviation of the linear Troyon-scaling of the ideal beta limit with normalised current at high elongation, which was predicted over 10 years ago. In addition to the ideal beta limit, several other MHD events are observed in highly elongated plasmas. The axisymmetric mode causes vertical displacement events and thereby, imposes a lower current limit. This operational limit is in good agreement with theoretical predictions of the growth rate of the axisymmetric mode. Furthermore, minor disruption are occasionally observed. They are caused by bursts of tearing modes located close to the q = 1 surface, which are destabilised by flat central current profiles, typical for highly elongated plasmas. Neoclassical tearing modes (NTMs), which have been observed to limit the achievable beta in a number of tokamaks, arise from a helical perturbation of the bootstrap current caused by an existing seed island. Neoclassical m/n = 2/1 tearing modes have been identified in TCV discharges which are characterised by a low electron collisionality νe*, a medium ion collisionality ν*, a medium value of beta and strongly peaked pressure and current profiles. In contrast to other tokamak experiments, where sawteeth, fishbones or ELMs generate the seed island, the required seed island is provided by a conventional tearing mode. The island clearly shows a conventional and a neoclassical growth phase. The TCV results provide the first clear observations of such a trigger mechanism which could also explain the occurrence of "triggerless" NTMs observed in other experiments. The slowly growing seed island also allows a measurement of the critical seed island width ωcrit, which is observed to increase with increasing density. In conclusion, several local and a global stability limits are analysed. These instabilities can limit the pressure gradient and thereby, the performance of the plasma. The presented results reveal several previously unobserved features of commonly observed instabilities. Since the most of the new observations can be explained by theory, they improve the predictive capabilities with respect to new experiments. The experiments have also shown some new interactions among different instabilities, which add to the already crowded complexities of MHD phenomena in fusion plasmas.

EPFL2001

Ideal MHD stability of tokamak plasmas with moderate and low aspect ratio

The need of durable and abundant energy sources for future ages stimulates the studies of thermonuclear energy sources, based on hot plasma confinement by magnetic fields. The most developed concept of hot plasma trap is the tokamak, where the plasma confinement is obtained by a combination of external magnetic fields with the magnetic field of the current flowing in the plasma torus. The stability of the tokamak plasma is the main subject of the present work. The hot plasma is approximated by the model of the ideal magnetohydrodynamics (ideal MHD) as a superconductive liquid. Being relatively simple, this model describes basic plasma stability properties and establishes necessary stability conditions. The analytical ideal MHD theory is well developed, but some assumptions, required for analytical treatment may not be valid for the plasmas of modern tokamaks and for future tokamak-based reactors. To circumvent this numerical codes have been created. These codes are free from such limitations, but they are not as convenient in use as analytical formulae. In the present work the validity of the analytical approach for the conditions of tokamaks like TCV and MAST is examined in comparison with numerical code predictions by studying the dependence of the ideal MHD stability on plasma toroidicity and shape parameters. The experimental study of the plasma dependence on triangularity, carried out on the TCV tokamak, is consistent with the results of the numerical calculations. A new formula, describing the ideal MHD stability dependence on plasma toroidicity and shape parameters is proposed for use in modern tokamaks and future reactors. This formula could be used instead of analytical expansions, which are not valid in such conditions. The ideal MHD stability of highly elongated TCV plasmas has been studied using numerical codes and the optimum plasma shape, which allows higher plasma performance, was found. Experimental data on the high elongation plasmas in TCV are consistent with the numerical predictions. Advanced tokamak plasma configurations, which provide better plasma properties, are amongst the main goals of the TCV tokamak research activity. The ideal MHD stability analysis of such plasmas, using numerical codes, can be useful for optimization of plasma parameters, and designing new experiments with improved plasma performance. Reversed shear plasmas with internal transport barrier were analyzed and the influence of the plasma pressure and current profiles on the ideal MHD stability of these plasmas was examined in detail. By fine tuning of the electron cyclotron heating and current drive system of TCV it was found that it might be possible to improve the plasma performance in reversed shear plasmas, by creating the optimal current and pressure profiles.

EPFL2005