Mohammad Nadeem

Computation of electric and magnetic stimulation in human head using the 3-D impedance method

A comparative, computational study of the modeling of transcranial magnetic stimulation (TMS) and electroconvulsive therapy (ECT) is presented using a human head model. The magnetic fields from a typical TMS coil of figure-eight type is modeled using the Biot-Savart law. The TMS coil is placed in a position used clinically for treatment of depression. Induced current densities and electric field distributions are calculated in the model using the impedance method. The calculations are made using driving currents and wave forms typical in the clinical setting. The obtained results are compared and contrasted with the corresponding ECT results. In the ECT case, a uniform current density is injected on one side of the head and extracted from the equal area on the opposite side of the head. The area of the injected currents corresponds to the electrode placement used in the clinic. The currents and electric fields, thus, produced within the model are computed using the same three-dimensional impedance method as used for the TMS case. The ECT calculations are made using currents and wave forms typical in the clinic. The electrical tissue properties are obtained from a 4-Cole-Cole model. The numerical results obtained are shown on a two-dimenaional cross section of the model. In this study, we find that the current densities and electric fields in the ECT case are stronger and deeper penetrating than the corresponding TMS quantities but both methods show biologically interesting current levels deep inside the brain.

Local magnetic shear and drift waves in stellarators

A study of the effect of local magnetic shear on the drift wave stability is presented. The eigenvalue problem for the drift wave equation is solved numerically in fully three-dimensional stellarator plasma using the ballooning mode formalism. It is found that negative local magnetic shear has a stabilizing effect on the drift wave instability and positive local shear is destabilizing. This is in agreement with the effect of negative global magnetic shear in tokamaks and also agrees with the simple estimates. As a consequence the highly unstable modes found on a specific magnetic surface are localized in the region of positive local magnetic shear.

A comparison of drift wave stability in stellarator and tokamak geometry

The influence of plasma geometry on the linear stability of electrostatic ion-temperature-gradient driven drift modes (ITG modes) is investigated. An advanced fluid model is used for the ions together with Boltzmann distributed electrons. The derived eigenvalue equation is solved numerically. A comparison is made between an H – 1NF [Fusion Technol. 17, 123 (1990)] like stellarator equilibrium, a numerical tokamak equilibrium and the analytical s - alpha equilibrium. The numerical and the analytical tokamak are found to be in good agreement in the low inverse aspect ratio limit. The growth rates of the tokamak and stellarator are comparable whereas the modulus of the real frequency is substantially larger in the stellarator. The threshold in Ln/LT for the stellarator is found to be somewhat larger. In addition, a stronger stabilization of the ITG mode growth is found for large L n / R in the stellarator case.

Ion-temperature-gradient modes in stellarator geometry

The ion-temperature-gradient (ITG)-driven drift mode is studied in three-dimensional stellarator geometry using a two-fluid reactive model in the electrostatic limit. The model includes first-order FLR effect in the presence of parallel ion dynamics and using the Boltzmann distribution for the electrons. The resulting eigenvalue is solved numerically using the ballooning mode theory. The results are contrasted with the corresponding tokamak results with positive shear. In stellarators, the level of the maximum growth rate of the ITG mode is found to be smaller and the threshold (ηi2.2) is somewhat higher. The effects of small and large temperature ratios and density gradients are found to be stabilizing on electrostatic ITG modes in stellarators.

Drift waves in stellarator geometry

Drift waves are investigated in a real three-dimensional stellarator geometry. A linear system, based on the cold ion fluid model and a ballooning mode formalism, is solved numerically in the geometry of the stellarator H1-NF. The spectra of stable and unstable modes, as well as localization, are discussed. The dependence of the spectrum of the unstable modes on the wavevector, plasma density variation, and the location in the plasma is presented.

Unstable ion-temperature-gradient modes in the Wendelstein 7-X stellarator configuration

The linear stability of the ion-temperature-gradient modes (ITG) in the electrostatic limit is examined in the short wavelength region by using a two fluid reactive model in fully three-dimensional Wendelstein 7-X (W7-X) stellarator [G. Grieger et al., Plasma Physics and Controlled Nuclear Fusion Research, 1990 (International Atomic Energy Agency, Vienna, 1991), Vol. 3, p. 525] geometry. The spectrum of stable and unstable modes and their real frequencies and eigenfunctions are calculated. The effects of density gradients, temperature gradients, temperature ratios, wavevector, ballooning angle, curvature and local magnetic shear on the ITG mode are also investigated. The frequency and growth rate of the most unstable ITG mode is calculated and visualized for a specific magnetic flux surface. For the equilibrium under investigation both localized and extended eigenmodes are found. The effect of small and large temperature ratios, small and large density gradients as well as large local magnetic shear are a...

Approximate localization and dispersion curves of drift waves in straight stellarators

Weakly and strongly localized drift waves are discussed within the framework of a helically symmetric stellarator model, using the ballooning mode formalism. A theory based on the Mathieu equation is compared with numerical solutions indicating the region of validity of the model.