Rational surface

About: Rational surface is a research topic. Over the lifetime, 436 publications have been published within this topic receiving 8328 citations.

Rational surfaces associated with affine root systems and geometry of the Painlevé equations

Abstract: We present a geometric approach to the theory of Painleve equations based on rational surfaces Our starting point is a compact smooth rational surface X which has a unique anti-canonical divisor D of canonical type We classify all such surfaces X To each X, there corresponds a root subsystem of E (1) 8 inside the Picard lattice of X We realize the action of the corresponding affine Weyl group as the Cremona action on a family of these surfaces We show that the translation part of the affine Weyl group gives rise to discrete Painleve equations, and that the above action constitutes their group of symmetries by Backlund transformations The six Painleve differential equations appear as degenerate cases of this construction In the latter context, X is Okamoto's space of initial conditions and D is the pole divisor of the symplectic form defining the Hamiltonian structure

Interaction of tearing modes with external structures in cylindrical geometry (plasma)

Abstract: A basic theoretical framework is developed for the investigation of tearing mode interactions in cylindrical geometry. A set of equations describing the coupled evolution of the amplitude and phase of each mode in the plasma is obtained by combining electromagnetic and fluid flow information. Two interactions are investigated in detail as examples. The first example considered is the slowing down of a rotating magnetic island interacting with a resistive wall. Under certain conditions bifurcated steady state solutions are obtained, allowing the system to make sudden irreversible transitions from high rotation to low rotation states as the interaction strength is gradually increased, and vice versa. The second example considered is the interaction of a rotating tearing mode with a static external magnetic perturbation. In general, a rotating island is stabilized to some extent by the interaction, whereas a locked island is destabilized. In fact, a rotating island of sufficiently small saturated width can be completely stabilized. However, once the island width becomes too large, conventional mode locking occurs prior to complete stabilization. The interaction with a tearing-stable plasma initially gives rise to a modification of the bulk plasma rotation, with little magnetic reconnection induced at the rational surface. However, once a critical perturbation field strength is exceeded, there is a sudden change in the plasma rotation as a locked island is induced at the rational surface, with no rotating magnetic precursor. The implications of these results for typical ohmically heated tokamaks are evaluated. The comparatively slow mode rotation in large tokamaks renders such devices particularly sensitive to error-field induced locked modes, and the collapse of mode rotation due to wall interactions

Plasma equilibrium with rational magnetic surfaces

Abstract: The self‐consistent classical plasma equilibrium with diffusion is studied in a toroidal geometry having a sheared magnetic field. Near each rational surface it is found that the pressure gradient is zero unless the Fourier component of 1/B2, which resonates with that surface, vanishes. Despite the resonances, the overall plasma confinement is, in practice, only slightly modified by the rational surfaces.

Bifurcated states of a rotating tokamak plasma in the presence of a static error-field

Abstract: The bifurcated states of a rotating tokamak plasma in the presence of a static, resonant, error-field are strongly analogous to the bifurcated states of a conventional induction motor. The two plasma states are the “unreconnected” state, in which the plasma rotates and error-field-driven magnetic reconnection is suppressed, and the “fully reconnected” state, in which the plasma rotation at the rational surface is arrested and driven magnetic reconnection proceeds without hindrance. The response regime of a rotating tokamak plasma in the vicinity of the rational surface to a static, resonant, error-field is determined by three parameters: the normalized plasma viscosity, P, the normalized plasma rotation, Q0, and the normalized plasma resistivity, R. There are 11 distinguishable response regimes. The extents of these regimes are calculated in P–Q0–R space. In addition, an expression for the critical error-field amplitude required to trigger a bifurcation from the “unreconnected” to the “fully reconnected” ...

The interaction of resonant magnetic perturbations with rotating plasmas

Abstract: The penetration of a helical magnetic perturbation into a rotating tokamak plasma is investigated. In the linear regime, it is found that unless the frequency of the imposed perturbation matches closely to one of the natural mode frequencies, reconnection at the rational surface is suppressed by a large factor. In order to deal with the problem in the nonlinear regime a theory of propagating, constant‐ψ magnetic islands is developed. This theory is valid provided the island width greatly exceeds any microscopic scale length (but still remains small compared with the minor radius), and the magnetic Reynolds number of the plasma is sufficiently large. An island width evolution equation is obtained which, in addition to the usual Rutherford term, contains a stabilizing term due ultimately to the inertia of the plasma flow pattern set up around the propagating island. A complete solution is presented for the case where the island and its associated flow pattern are steady. In the nonlinear regime, a fairly sh...