Numerical simulations form an indispensable tool to understand the behavior of a hot plasma that is created inside a tokamak for providing nuclear fusion energy. Various aspects of tokamak plasmas have been successfully studied through the reduced magnetohydrodynamic (MHD) model. The need for more complete modeling through the full MHD equations is addressed here. Our computational method is presented along with measures against possible problems regarding pollution, stability, and regularity. The problem of ensuring continuity of solutions in the center of a polar grid is addressed in the context of a finite element discretization of the full MHD equations. A rigorous and generally applicable solution is proposed here. Useful analytical test cases are devised to verify the correct implementation of the momentum and induction equation, the hyperdiffusive terms, and the accuracy with which highly anisotropic diffusion can be simulated. A striking observation is that highly anisotropic diffusion can be treated with the same order of accuracy as isotropic diffusion, even on non-aligned grids, as long as these grids are generated with sufficient care. This property is shown to be associated with our use of a magnetic vector potential to describe the magnetic field.Several well-known instabilities are simulated to demonstrate the capabilities of the new method. The linear growth rate of an internal kink mode and a tearing mode are benchmarked against the results of a linear MHD code. The evolution of a tearing mode and the resulting magnetic islands are simulated well into the nonlinear regime. The results are compared with predictions from the reduced MHD model.Finally, a simulation of a ballooning mode illustrates the possibility to use our method as an ideal MHD method without the need to add any physical dissipation.@en

The continuous spectrum of analytical toroidally rotating magnetically confined plasma equilibria is investigated analytically and numerically. In the presence of purely toroidal flow, the ideal magnetohydrodynamic equations leave the freedom to specify which thermodynamic quantity is constant on the magnetic surfaces. Introducing a general parametrization of this quantity, analytical equilibrium solutions are derived that still posses this freedom. These equilibria and their spectral properties are shown to be ideally suited for testing numerical equilibrium and stability codes including toroidal rotation. Analytical expressions are derived for the low-frequency continuous Alfvén spectrum. These expressions still allow one to choose which quantity is constant on the magnetic surfaces of the equilibrium, thereby generalizing previous results. The centrifugal convective effect is shown to modify the lowest Alfvén continuum branch to a buoyancy frequency, or Brunt-Väisälä frequency. A comparison with numerical results for the case that the specific entropy, the temperature, or the density is constant on the magnetic surfaces yields excellent agreement, showing the usefulness of the derived expressions for the validation of numerical codes.@en

As a result of toroidal rotation, sequences of new global modes are predicted to arise in magnetically confined plasmas. The frequencies of these Alfvén modes lie inside gaps of the continuous magnetohydrodynamic (MHD) spectrum that are created or enlarged by toroidal flow. The numerically obtained results are compared with an analytical investigation, yielding a useful criterion for mode existence. Because of their low frequencies, these modes may be easily destabilized by energetic particles. Because of their sensitivity to the Mach number however, these modes can provide a valuable extension to MHD spectroscopy by giving information on the rotational velocity.@en

A radially decreasing toroidal rotation frequency can have a stabilizing effect on nonaxisymmetric magnetohydrodynamic (MHD) instabilities. We show that this is a consequence of the Coriolis effect that induces a restoring pressure gradient force when plasma is perturbed radially. In a rotating cylindrical plasma, this Coriolis-pressure effect is canceled by the centrifugal effect responsible for the magnetorotational instability. In a magnetically confined toroidal plasma, a large aspect ratio expansion shows that only half of the effect is canceled. This analytical result is confirmed by numerical computations. When the plasma rotates faster toroidally in the core than near the edge, the effect can contribute to the formation of transport barriers by stabilizing MHD instabilities.@en

The ideal magnetohydrodynamic stability is investigated of localized interchange modes in a large-aspect ratio tokamak plasma. The resulting stability criterion includes the effects of toroidal rotation and rotation shear and contains various well-known limiting cases. The analysis allows for a general adiabatic index, resulting in a stabilizing contribution from the convective effect. A further stabilizing effect from rotation exists when the angular frequency squared decreases radially more rapidly than the density. Flow shear, however, also decreases the stabilizing effect of magnetic shear through the Kelvin-Helmholtz mechanism. Numerical simulations reveal the merits and limitations of the performed local analysis.@en