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Register a new userThe unified ballooning theory with weak up-down asymmetric mode structure and the numerical studies
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A unified ballooning theory, constructed on the basis of two special theories [Y. Z. Zhang, S. M. Mahajan, X. D. Zhang, Phys. Fluids B4, 2729 (1992); Y. Z. Zhang, T. Xie, Nucl. Fusion & Plasma Phys. 33, 193 (2013)], shows that a weak up-down asymmetric mode structure is normally formed in an up-down symmetric equilibrium; the weak up-down asymmetry in mode structure is the manifestation of non-trivial higher order effects beyond the standard ballooning equation. It is shown that the asymmetric mode may have even higher growth rate than symmetric modes. Salient features of the theory are illustrated by investigating a fluid model for the ion temperature gradient (ITG) mode. The two dimensional (2D) analytical form of ITG mode, solved in ballooning representation, is then converted into the radial-poloidal space to provide the natural boundary condition for solving the 2D mathematical local eigenmode problem. We find the analytical expression of mode structure in good agreement with finite difference solution. This sets a reliable framework for quasi-linear computation.
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The existence of kinetic ballooning mode (KBM) high order (non-ground) eigenstates for tokamak plasmas with steep gradient is demonstrated via gyrokinetic electromagnetic eigenvalue solutions, which reveals that eigenmode parity transition is an intrinsic property of electromagnetic plasmas. The eigenstates with quantum number $l=0$ for ground state and $l=1,2,3ldots$ for non-ground states are found to coexist and the most unstable one can be the high order states ($l eq0$). The conventional KBM is the $l=0$ state. It is shown that the $l=1$ KBM has the same mode structure parity as the micro-tearing mode (MTM). In contrast to the MTM, the $l=1$ KBM can be driven by pressure gradient even without collisions and electron temperature gradient. The relevance between various eigenstates of KBM under steep gradient and edge plasma physics is discussed.
In large hot tokamaks like JET, the width of the reconnecting layer for resistive modes is determined by semi-collisional electron dynamics and is much less than the ion Larmor radius. Firstly a dispersion relation valid in this regime is derived which provides a unified description of drift-tearing modes, kinetic Alfven waves and the internal kink mode at low beta. Tearing mode stability is investigated analytically recovering the stabilising ion orbit effect, obtained previously by Cowley et al. [Phys. Fluids (29) 3230 1986], which implies large values of the tearing mode stability parameter Delta prime are required for instability. Secondly, at high beta it is shown that the tearing mode interacts with the kinetic Alfven wave and that there is an absolute stabilisation for all Delta prime due to the shielding effects of the electron temperature gradients, extending the result of Drake et. al [Phys. Fluids (26) 2509 1983] to large ion orbits. The nature of the transition between these two limits at finite values of beta is then elucidated. The low beta formalism is also relevant to the m=n=1 tearing mode and the dissipative internal kink mode, thus extending the work of Pegoraro et al. [Phys. Fluids B (1) 364 1989] to a more realistic electron model incorporating temperature perturbations, but then the smallness of the dissipative internal kink mode frequency is exploited to obtain a new dispersion relation valid at arbitrary beta. A diagram describing the stability of both the tearing mode and dissipative internal kink mode, in the space of Delta prime and beta, is obtained. The trajectory of the evolution of the current profile during a sawtooth period can be plotted in this diagram, providing a model for the triggering of a sawtooth crash.
Discovered by Christopher Alexander, living structure is a physical phenomenon, through which the quality of the built environment or artifacts can be judged objectively. It bears two distinguished properties just like a tree: far more small things than large ones across all scales, and more or less similar things on each scale. As a physical phenomenon, and mathematical concept, living structure is essentially empirical, discovered and developed from miniscule observation in nature- and human-made things, and it affects our daily lives in some substantial ways, such as where to put a table or a flower vase in a room, helping us to make beautiful things and environments. Living structure is not only empirical, but also philosophical and visionary, enabling us to see the world and space in more meaningful ways. This paper is intended to defend living structure as a physical phenomenon, clarifying some common questions and misgivings surrounding Alexanders design thoughts, such as the objective or structural nature of beauty, building styles advocated by Alexander, and mysterious nature of his concepts. We first illustrate living structure - essentially organized complexity, as advocated by the late Jane Jacobs (1916-2006) - that is governed by two fundamental laws (scaling law and Toblers law), and generated in some step by step fashion by two design principles (differentiation and adaptation) through the 15 structural properties. We then verify why living structure is primarily empirical, drawing evidence from Alexanders own work, as well as our case studies applied to the Earths surface including cities, streets, and buildings, and two logos. Before reaching conclusions, we concentrate on the most mysterious part of Alexanders work - the hypothesized I - as a substance that pervasively exists everywhere, in order to make better sense of living structure in our minds.
The role of anisotropic thermal diffusivity on tearing mode stability is analysed in general toroidal geometry. A dispersion relation linking the growth rate to the tearing mode stability parameter, Delta, is derived. By using a resistive MHD code, modified to include such thermal transport, to calculate tearing mode growth rates, the dispersion relation is employed to determine Delta in situations with finite plasma pressure that are stabilised by favourable average curvature in a simple resistive MHD model. We also demonstrate that the same code can also be used to calculate the basis-functions [C J Ham, et al, Plasma Phys. Control. Fusion 54 (2012) 105014] needed to construct Delta.
Binary black holes with spins that are aligned with the orbital angular momentum do not precess. However, post-Newtonian calculations predict that up-down binaries, in which the spin of the heavier (lighter) black hole is aligned (antialigned) with the orbital angular momentum, are unstable when the spins are slightly perturbed from perfect alignment. This instability provides a possible mechanism for the formation of precessing binaries in environments where sources are preferentially formed with (anti) aligned spins. In this paper, we present the first full numerical relativity simulations capturing this instability. These simulations span $sim 100$ orbits and $sim 3$-$5$ precession cycles before merger, making them some of the longest numerical relativity simulations to date. Initialized with a small perturbation of $1^{circ}$-$10^{circ}$, the instability causes a dramatic growth of the spin misalignments, which can reach $sim 90^{circ}$ near merger. We show that this leaves a strong imprint on the subdominant modes of the gravitational wave signal, which can potentially be used to distinguish up-down binaries from other sources. Finally, we show that post-Newtonian and effective-one-body approximants are able to reproduce the unstable dynamics of up-down binaries extracted from numerical relativity.
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