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  • 1.
    Mirza, Ahmed A.
    et al.
    KTH, School of Electrical Engineering (EES), Fusion Plasma Physics.
    Scheffel, Jan
    KTH, School of Electrical Engineering (EES), Fusion Plasma Physics.
    Numerical study of thermal conductivity effects on stability of the reversed-field pinch2011In: 38th EPS Conference on Plasma Physics, Strasbourg, France, 27 June – 1 July 2011, 2011Conference paper (Refereed)
  • 2.
    Mirza, Ahmed Akram
    KTH, School of Electrical Engineering (EES), Fusion Plasma Physics.
    Pressure driven instabilities in the reversed-field pinch: numerical and theoretical studies2013Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    According to classical linearized resistive magnetohydrodynamics theory, pressuredriven modes are unstable in the reversed-field pinch (RFP) due to unfavorable magnetic field line curvature. The result is based on the assumption of an adiabatic energy equation where anisotropic thermal conduction effects are ignored as compared to convection and compression. In this thesis the effects of heat conduction in the energy equation have been studied. We have examined these effects on the linear stability of pressure-driven resistive modes using boundary value theory (Δ´ ) and a novel initial-value full resistive MHD code employing the Generalized Weighted Residual Method (GWRM). In the Δ´ method, a shooting technique is employed by integrating from the resistive layer to boundaries. The GWRM method, on the other hand, is a time-spectral Galerkin method in which the fully linearized MHD equations are solved. For detailed computations, efficiency requires the temporal and spatial domains to be divided into subdomains. For this purpose, a number of challenging test cases including linearized ideal MHD equations are treated.

    Numerical and analytical investigations of equilibria reveal that thermal conduction effects are not stabilizing for reactor relevant values of Lundquist number, S0, and normalized pressure, βθ, for tearing-stable plasmas. These studies show that growth rate scales as  γ~_ S0−1/5 , which is weaker than for the adiabatic case, γ~_ S0−1/3.

    A numerical study of optimized confinement for an advanced RFP scenario including ohmic heating and heat conduction, is also part of this thesis. The fully nonlinear resistive MHD code DEBSP has been employed. We have identified, using both Δ´ and GWRM methods, that the observed crash of the high confinement is caused by resistive, pressure-driven modes.

  • 3.
    MIrza, Ahmed Akram
    et al.
    KTH, School of Electrical Engineering (EES), Fusion Plasma Physics.
    Scheffel, Jan
    KTH, School of Electrical Engineering (EES), Fusion Plasma Physics.
    Johnson, Thomas
    KTH, School of Electrical Engineering (EES), Fusion Plasma Physics.
    Effect of thermal conduction on pressure-driven modes in the reversed-field pinch2012In: Nuclear Fusion, ISSN 0029-5515, E-ISSN 1741-4326, Vol. 52, no 12, p. 123012-Article in journal (Refereed)
    Abstract [en]

    Classical linearized resistive magnetohydrodynamic (MHD) stability theory predicts unstable pressure-driven modes even at low plasma beta values for the reversed-field pinch (RFP) because of its unfavourable curvature and strong poloidal magnetic field. These resistive g-modes undermine energy confinement and are detrimental to the RFP reactor potential. In the analysis, one aspect is common, which is the usage of the adiabatic energy equation, ignoring the contribution due to thermal conduction effects. However, in recent analysis, stabilization of pressure-driven modes is demonstrated through inclusion of thermal conductivity. In this paper, we compare the results obtained from both classical and thermal conduction modified boundary layer stability analysis with those from a time-spectral resistive linearized MHD code. Ohmic heating and thermal conduction effects are included in the calculations. We have found that thermal conduction effects stabilize pressure-driven resistive g-modes only for very low values of plasma beta. In addition, analytical and numerical investigation of the equilibrium reveal that, for reactor relevant values of S-0 and tearing stable plasmas, the scaling gamma similar to S-0(-1/5) for the growth rate of these modes is weaker than that for the adiabatic case gamma similar to S-0(-1/3).

  • 4.
    Scheffel, Jan
    et al.
    KTH, School of Electrical Engineering (EES), Fusion Plasma Physics.
    MIrza, Ahmed A.
    KTH, School of Electrical Engineering (EES), Fusion Plasma Physics.
    Application of the Generalized Weighted Residual Method to stability problems within ideal and resistive MHD2010In: 52nd Annual Meeting of the APS Division of Plasma Physics, Chicago, Illinios, USA 8-12 November, 2010, 2010Conference paper (Refereed)
    Abstract [en]

    Initial-value stability and transport problems formulated in resistive MHD usually require extensive computations using a very large number of time steps. Although spectral methods are used for the spatial domains, finite steps are traditionally used for the temporal domain with resulting constraints in terms of CFL-like stability conditions for explicit and accuracy-related issues for implicit methods. The Generalized Weighted Residual Method (GWRM) alleviates these problems by representing the time domain in the form of a Chebyshev series. The solution is obtained as an approximate semi-analytical expression through solving a global system of algebraic equations for the expansion coefficients, valid for all time, spatial and physical parameter domains. We demonstrate solutions in terms of eigenvalues and eigenfunctions for the z-pinch, using the linearized ideal MHD equations. Including resistivity, results for resistive g-modes of the reversed-field pinch are also presented. 

  • 5.
    Scheffel, Jan
    et al.
    KTH, School of Electrical Engineering (EES), Fusion Plasma Physics.
    MIrza, Ahmed A.
    KTH, School of Electrical Engineering (EES), Fusion Plasma Physics.
    Pressure driven resistive modes in the advanced RFP2008In: 35th EPS Conference on Plasma Physics 2008, EPS 2008 - Europhysics Conference Abstracts: Volume 32, Issue 3, 2008, p. 2014-2017Conference paper (Refereed)
  • 6.
    Scheffel, Jan
    et al.
    KTH, School of Electrical Engineering (EES), Fusion Plasma Physics.
    MIrza, Ahmed A.
    KTH, School of Electrical Engineering (EES), Fusion Plasma Physics.
    Resistive g-modes and RFP confinement2009In: 51st Annual Meeting of the APS Division of Plasma Physics, Atlanta, Georgia, USA  2-6 November, 2009, 2009Conference paper (Refereed)
    Abstract [en]

    The role of pressure driven resistive modes in the reversed-field pinch remains unclear. It was early shown that unstable resistive g-modes would always exist for an inwardly directed pressure gradient. It now appears that pressure profile smoothing, due to incluson of heat conductivity terms in the energy equation, enables completey stable RFP states at moderate plasma beta. These calculations, apart from being restricted to linearized perturbations, suffer from the use of rather forced scalings, thus their accuracy can be questioned. Also, they have so far only been applied to conventional RFP states, where confinement-limiting tearing fluctuations maintain the reversed axial magnetic field. In the advanced RFP, current profile control has largely eliminated current driven tering modes. Fully nonlinear, numerical studies have shown that energy confinement and poloidal beta increase substantially, but that weak residual modes usually remain. The nature of these residual modes, which limit energy confinement, is studied using a novel semi-analytical, spectral scheme for solving the resistive MHD equations; the generalized weighted residual method (GWRM). Results from the analysis as well as comparisons with the competing linear resistive g-mode theories will be presented.

  • 7.
    Scheffel, Jan
    et al.
    KTH, School of Electrical Engineering (EES), Fusion Plasma Physics. KTH, School of Electrical Engineering (EES), Centres, Alfvén Laboratory Centre for Space and Fusion Plasma Physics.
    MIrza, Ahmed A.
    KTH, School of Electrical Engineering (EES), Fusion Plasma Physics. KTH, School of Electrical Engineering (EES), Centres, Alfvén Laboratory Centre for Space and Fusion Plasma Physics.
    Resistive pressure driven RFP modes are not removed by heat conduction effects2012In: 39th EPS Conference on Plasma Physics 2012, EPS 2012 and the 16th International Congress on Plasma Physics: Volume 3, 2012, 2012, p. 1690-1693Conference paper (Refereed)
    Abstract [en]

    During the last decade it has been shown theoretically, numerically and experimentally that current driven, resistive tearing modes can be significantly suppressed in the reversed-field pinch (RFP). In these advanced scenarios, the confinement time can be enhanced by a factor 5-10. Pressure driven resistive instabilities (g-modes) still stand in the way, however, for high RFP confinement. Classical theory [1] shows that the unfavourable RFP curvature inevitably leads to unacceptably large linear growth rates even at high Lundquist numbers. Later theory [2] demonstrates, however, that the classical assumption of adiabatic plasma energy dynamics is inaccurate. The reason is that anomalously large experimental perpendicular heat conduction, together with strong parallel heat conduction, to a certain extent outbalance the pressure terms of the plasma energy equation. Resulting resistive length scales appear to extend the resistive layer at the resonance to allow for fully stable, finite beta RFP configurations. In the present work we show theoretically that the latter result is limited to low beta only and that it scales unfavourably with Lundquist number. Numerical solution, using a novel time-spectral method [3] of the linearised resistive MHD initial-value equations including heat conduction, ohmic heating and resistivity, supports the analytical results

  • 8.
    Scheffel, Jan
    et al.
    KTH, School of Electrical Engineering (EES), Fusion Plasma Physics.
    MIrza, Ahmed A.
    KTH, School of Electrical Engineering (EES), Fusion Plasma Physics.
    Thermal conductivity effects on resistive g-mode stability of the RFP2011In: 53rd Annual Meeting of the APS Division of Plasma Physics, Salt Lake City, Utah, USA 14-18 November, 2011, 2011Conference paper (Refereed)
    Abstract [en]

    Tearing modes presently dominate fluctuations in the reversed- field pinch (RFP). Using current profile control techniques, the tearing modes can be removed experimentally. Pressure driven resistive g-modes remain for all equilibria, however, according to classical theory. In the tokamak these modes can be eliminated by curvature effects. Resistive g-modes may cause modest global energy confinement and severly limit the reactor potential of the RFP. Work by Bruno et al, where the energy equation has been supplemented by heat conduction terms, appear to show that heat conduction smoothens pressure gradient effects and stabilises resistive g-modes at low beta. On the other hand, fully numerical studies including heat conduction effects as well as experimental work identify resistive g-mode activity. In this work, we present a detailed computational analysis of linear resistive g-mode stability with and without heat conductivity effects. Both traditional delta prime analysis and a fully resistive code, based on the novel Generalized Weighted Residual Method (GWRM), are used.

  • 9.
    Scheffel, Jan
    et al.
    KTH, School of Electrical Engineering (EES), Fusion Plasma Physics.
    Mirza, Ahmed A.
    KTH, School of Electrical Engineering (EES), Fusion Plasma Physics.
    Time-spectral solution of initial-value problems – subdomain approach2012In: American Journal of Computational Mathematics, ISSN 2161-1211, Vol. 2, no 2, p. 72-81Article in journal (Refereed)
    Abstract [en]

    Temporal and spatial subdomain techniques are proposed for a time-spectral method for solution of initial-value problems. The spectral method, called the generalized weighted residual method (GWRM), is a generalization of weighted residual methods to the time and parameter domains [1]. A semi-analytical Chebyshev polynomial ansatz is employed, and the problem reduces to determine the coefficients of the ansatz from linear or nonlinear algebraic systems of equations. In order to avoid large memory storage and computational cost, it is preferable to subdivide the temporal and spatial domains into subdomains. Methods and examples of this article demonstrate how this can be achieved. 

  • 10.
    Scheffel, Jan
    et al.
    KTH, School of Electrical Engineering (EES), Fusion Plasma Physics.
    Schnack, Dalton D.
    University of Wisconsin.
    MIrza, Ahmed A.
    KTH, School of Electrical Engineering (EES), Fusion Plasma Physics.
    Static current profile control and RFP confinement2013In: Nuclear Fusion, ISSN 0029-5515, E-ISSN 1741-4326, Vol. 53, no 11, p. 113007-Article in journal (Refereed)
    Abstract [en]

    Static current profile control (CPC) is shown numerically to substantially enhance plasma confinement in the reversed-field pinch (RFP). By suitable application of an auxiliary electric field and adjustment of its internal location, width and amplitude, strongly decreased levels of dynamo fluctuations are obtained. The simulations are performed using a fully non-linear, resistive magnetohydrodynamic model, including the effects of ohmic heating as well as parallel and perpendicular heat conduction along stochastic field lines. The importance of controlling the parallel current profile in the core plasma to minimize the effects of tearing modes on confinement is thus confirmed. A near three-fold increase in energy confinement is found and poloidal plasma beta increases by 30% from 0.20 to 0.27. The edge heat flux is reduced to a third of that of the conventional RFP. The high-confinement phase is interrupted here by a crash, characterized by a rapid decrease in confinement. A detailed study of the crash phase is carried out by the standard Delta' theory and a fully resistive linearized time-spectral method; the generalized weighted residual method. The analysis suggests that the instability is caused by pressure-driven, resistive g-modes. Inclusion of anisotropic thermal conduction reduces the linear growth rates. As compared with our earlier numerical studies of CPC in the RFP, employing feedback control, the present static control scheme should be more easily implemented experimentally.

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