The high confinement mode (H-mode) is seen as the most promising operational regime for obtaining economically attractive fusion power plants based on the tokamak concept. This regime is characterized by the formation of a very steep plasma pressure profile at the plasma edge, which leads to repetitive instabilities known as 'edge localized modes' (ELMs). These instabilities have been identified as ideal magneto hydrodynamic (MHD) modes triggered by the increased plasma pressure gradient and/or current density in the plasma edge [1][2][3][4]. The crash of these modes releases a significant fraction of the Nuclear Fusion
Non-axisymmetric stationary magnetic perturbations lead to the formation of homoclinic tangles near the divertor magnetic saddle in tokamak discharges. These tangles intersect the divertor plates in static helical structures that delimit the regions reached by open magnetic field lines reaching the plasma column and leading the charged particles to the strike surfaces by parallel transport. In this article we introduce a non-axisymmetric rotating magnetic perturbation to model the time evolution of the three-dimensional magnetic field of a single-null DIII-D tokamak discharge developing a rotating tearing mode. The non-axiymmetric field is modeled using the magnetic signals to adjust the phases and currents of a set of internal filamentary currents that approximate the magnetic field in the plasma edge region. The stable and unstable manifolds of the asymmetric magnetic saddle are obtained through an adaptive calculation providing the cuts at a given poloidal plane and the strike surfaces. For the modeled shot, the experimental heat pattern and its time development are well described by the rotating unstable manifold, indicating the emergence of homoclinic lobes in a rotating frame due to the plasma instabilities.
In toroidally confined plasmas, the Grad-Shafranov equation, in general a non-linear PDE, describes the hydromagnetic equilibrium of the system. This equation becomes linear when the kinetic pressure is proportional to the poloidal magnetic flux and the squared poloidal current is a quadratic function of it. In this work, the eigenvalue of the associated homogeneous equation is related with the safety factor on the magnetic axis, the plasma beta and the Shafranov shift, then, the adjustable parameters of the particular solution are bounded through physical constrains. The poloidal magnetic flux becomes a linear superposition of independent solutions and its parameters are adjusted with a non-linear fitting algorithm. This method is used to find hydromagnetic equilibria with normal and reversed magnetic shear and defined values of the elongation, triangularity, aspect-ratio, and X-point(s). The resultant toroidal and poloidal beta, the safety factor at the 95% flux surface and the plasma current are in agreement with usual experimental values for high beta discharges and the model can be used locally to describe reversed magnetic shear equilibria.
In this work we introduce an exact calculation method and an approximation technique for tracing the invariant manifolds of unstable periodic orbits of planar maps. The exact method relies in an adaptive collocation procedure that prevents redundant calculations occurring in non-refinement approaches and the approximated method is based on an intuitive geometrical decomposition of the manifold in bare and fine details. The resulting approximated manifold is precise when compared to the exact manifold, and its calculation is computationally more efficient, making it ideal for mappings involving intensive calculations like numerical function inversion or the numerical integration of ODEs between crossings through a surface of section.
The changeover from normal to super diffusion in time dependent billiards is explained analytically. The unlimited energy growth for an ensemble of bouncing particles in time dependent billiards is obtained by means of a two dimensional mapping of the first and second moments of the velocity distribution function. We prove that for low initial velocities the mean velocity of the ensemble grows with exponent ∼ 1/2 of the number of collisions with the border, therefore exhibiting normal diffusion. Eventually, this regime changes to a faster growth characterized by an exponent ∼ 1 corresponding to super diffusion. For larger initial velocities, the temporary symmetry in the diffusion of velocities explains an initial plateau of the average velocity. PACS numbers: 05.45.-a, 05.45.Pq, 05.40.FbAs coined by Enrico Fermi [1] Fermi acceleration (FA) is a phenomenon where an ensemble of classical and non interacting particles acquires energy from repeated elastic collisions with a rigid and time varying boundary. It is typically observed in billiards [2-4] whose boundaries are moving in time [5][6][7][8][9]. If the motion of the boundary is random and the initial velocity is small enough [10], the growth of the average velocity is proportional to n 1/2 , with n denoting the number of collisions. If the initial velocity is larger, a plateau of constant velocity is observed in a plot V vs. n which is explained from the symmetry of the velocity diffusion [11]. The symmetry warrants that part of the ensemble grows and part of it decreases in such a way the growing parcel cancels the portion decreasing. As soon as such symmetry is broken the constant regime is changed to a regime of growth. For deterministic oscillations of the border, the scenario is different. Breathing oscillations preserve the shape but not the area of the billiard. It is known that the average velocity evolves in a sub-diffusive manner with a slope of the order of 1/6 [12,13]. For oscillations preserving the area but not the shape of the billiard there are two regimes of growth. For short time the diffusion of velocities is normal passing to super diffusion regime for large enough number of collisions [14]. This changeover is, so far, not yet explained and our contribution in this letter is to fill up this gap in the theory. This is achieved by studying the momenta of the velocity distribution function, noticing that the dynamical angular/time variables have an inhomogeneous distribution in phase space.The results presented in this letter are illustrated by a time-dependent oval-billiard [15] whose phase space is mixed when the boundary is static. The boundary of the billiard is written as R b (θ, t) = 1 + ǫ [1 + a cos(t)] cos(pθ) where R b is the radius of the boundary in polar coordinates, θ is the polar angle, ǫ controls the circle deformation, p > 0 is an integer number [16] given the shape of the boundary, t is the time and a is the amplitude of oscillation of the boundary. Figure 1 shows a typical scenario of the boundary and three collisions illustr...
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