Magnetohydrodynamic waves and instabilities in rotating tokamak plasmas

J.W. Haverkort

Research output: ThesisPhd Thesis 2 (Research NOT TU/e / Graduation TU/e)

1069 Downloads (Pure)


The effect of toroidal rotation on the equilibrium and the waves and instabilities in tokamak plasmas was investigated within the framework of magnetohydrodynamics (MHD). A comprehensive analysis was performed of the effects of toroidal plasma rotation on the low-frequency MHD spectrum of a large aspect ratio, low-beta, tokamak plasma with a circular cross-section. Continuum modes, waves, and instabilities were investigated both analytically and numerically. We introduced an adiabatic constant of the equilibrium ¿e that may deviate from the dynamical adiabatic constant ¿. All analytical results in this thesis hold for general ¿e. Analytical expressions were obtained for the continuum frequencies ¿- and ¿+ corresponding to zonal flow modes and geodesic acoustic modes, respectively. When ¿e > ¿ the zonal flow modes acquire a finite Brunt-Väisälä frequency in the presence of toroidal rotation. For ¿e <¿ they become convectively unstable. The analytical results were successfully compared to the results from numerical simulations. When modes have a significant component normal to the magnetic surfaces, profile variation in this direction can influence their existence and stability. A stability criterion was derived for localized modes. The terms in this criterion were interpreted in terms of well-known hydrodynamic and MHD instabilities. A regime was identified in which toroidal rotation is stabilizing. This occurs when the flow shear is low enough to avoid the destabilizing Kelvin-Helmholtz effect but the rotation frequency squared decreases faster radially than the density so that the centrifugal Rayleigh-Taylor instability is avoided. The term that is responsible for this stabilization has the same form as that of the magnetorotational instability, but is of opposite sign and in a toroidal geometry is twice as large. It exists only for non-axisymmetric modes. The physical origin of this flow shear stabilization was elucidated as a Coriolis-pressure effect. Close to when the derived stability criterion predicts instability, stable global modes arise near the ¿+ frequency. These may be interpreted as the rotation-driven counterpart of beta-induced Alfvén eigenmodes. In the absence of resistivity, global but radially highly localized modes are present close to the ¿- frequency. These may be interpreted as the non-axisymmetric counterparts of the axisymmetric zonal flows. Close to when the stability criterion predicts stability, stable modes arise above the ¿+ continuum frequency when the magnetic shear is reversed. Toroidal rotation can make these reversed-shear Alfvén eigenmodes disappear through the Coriolispressure effect when they move at a lower velocity than the plasma or propagate in the opposite direction. The generality of the analysis also allows for the description of downwards cascading modes that are sometimes observed, reversed-shear modes above ¿-, and regular-shear modes below both ¿+ and ¿-. Finally, the full viscoresistive MHD equations were implemented in the reduced- MHD finite element code JOREK. With this code, it was found to be possible to simulate extremely anisotropic diffusion phenomena with high accuracy. This property seems to be associated with the use of a magnetic vector potential to describe the magnetic field. Numerical pollution of the components of this magnetic vector potential could be avoided by choosing a weak formulation in which the momentum equation is projected in the direction parallel to the magnetic field rather than in the toroidal direction. Conditions were derived to ensure regularity at the grid axis, where many nodes can come together in one point. The implementation of the equations was thoroughly tested using both analytical solutions and benchmarks with MHD instabilities. The linear growth rates of an internal kink mode, a tearing mode, and a ballooning mode were used. The tearing mode evolution was followed well into the nonlinear regime and the results were compared with those of the reduced MHD models. The implemented equilibrium description allows for toroidal rotation with arbitrary ¿e. This code will in the future allow further investigation of the nonlinear evolution of MHD phenomena, also in the presence of toroidal rotation.
Original languageEnglish
QualificationDoctor of Philosophy
Awarding Institution
  • Mathematics and Computer Science
  • Centrum voor Wiskunde en Informatica
  • Koren, Barry, Promotor
  • de Blank, Hugo J., Copromotor
Award date21 Mar 2013
Place of PublicationEindhoven
Print ISBNs978-90-386-3340-4
Publication statusPublished - 2013


Dive into the research topics of 'Magnetohydrodynamic waves and instabilities in rotating tokamak plasmas'. Together they form a unique fingerprint.

Cite this