Hamiltonian formalism in the problem on the modification of the current density profile during the development of tearin
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Hamiltonian Formalism in the Problem on the Modification of the Current Density Profile during the Development of Tearing Instability in a Tokamak A. A. Skovoroda National Research Center Kurchatov Institute, Moscow, 123182 Russia e-mail: [email protected] Received July 11, 2016; in final form, October 13, 2016
Abstract—The formation of a magnetic island as a result of tearing instability can be interpreted as the bifurcation of an axisymmetric equilibrium configuration at which nested magnetic surfaces are preserved. The modification of the current density profile due to such bifurcation is studied using the Hamiltonian formalism. In the case of a long narrow island, the gradient profile changes to a profile with an extremum on the axis of the magnetic island. DOI: 10.1134/S1063780X17040122
1. INTRODUCTION The formation of magnetic islands is a common phenomenon in tokamaks. The islands arise spontaneously due to tearing instability or are excited by external helical magnetic perturbations that are in resonance with the rotational transformation of the field lines. The magnetic islands substantially affect plasma confinement, as well as plasma stability and rotation. At present, the control of magnetic islands in tokamaks is provided by special systems controlling the plasma parameters [1]. This requires deeper understanding of the microscopic plasma dynamics of magnetic islands. In spite of the progressively improving spatial and time resolution of diagnostic systems, the main information is provided by theoretical and numerical methods. Tearing instability in a tokamak was discovered long ago [2]. In its nature, this instability is characterized by a rather complicated nonlinearity. Although it is obvious that the nonlinearity is caused by a change in the profile of the longitudinal current density during the formation and evolution of magnetic islands, the nonlinear theory capable of describing saturation of tearing modes, including neoclassical ones, is in the stage of development to date [3–12]. In this work, we consider classical tearing instability. This magnetohydrodynamic (MHD) mode corresponds to a growing global magnetic perturbation with a nonzero B r component on the resonance magnetic surface q ( rres ) = m n . The criterion for the onset of this instability is traditionally represented as the positivity of the jump in the logarithmic derivative
rres +ε
∂ ln B r ⎞ Δ ' = lim ε→0 ⎛⎜ ≥ 0 on the resonance sur⎟ ⎝ ∂ r ⎠ rres −ε face. The value of Δ' in a tokamak is proportional to the amount of free energy associated with the global longitudinal density current profile [7]. This energy is released via the reconnection of field lines in the finite-resistivity plasma resonance layer. The grounds of the tearing instability theory were laid in the 1970s in a large number of works, among which those by Rosenbluth [2, 4] and Rutherford [5] are acknowledged as basic ones. In [5], it was shown for the first time that, due to the quasilinear modification of the current density profile in the vicinity of
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