Portion Of Phase Space Research Articles

The Stability of Linear Diffusion Acceleration Relative to CMFD

Coarse Mesh Finite Difference (CMFD) is a widely-used iterative acceleration method for neutron transport problems in which nonlinear terms are introduced in the derivation of the low-order CMFD diffusion equation. These terms, including the homogenized diffusion coefficient, the current coupling coefficients, and the multiplicative prolongation constant, are subject to numerical instability when a scalar flux estimate becomes sufficiently small or negative. In this paper, we use a suite of contrived problems to demonstrate the susceptibility of CMFD to failure for each of the vulnerable quantities of interest. Our results show that if a scalar flux estimate becomes negative in any portion of phase space, for any iterate, numerical instability can occur. Specifically, the number of outer iterations required for convergence of the CMFD-accelerated transport problem can increase dramatically, or worse, the iteration scheme can diverge. An alternative Linear Diffusion Acceleration (LDA) scheme addresses these issues by explicitly avoiding local nonlinearities. Our numerical results show that the rapid convergence of LDA is unaffected by the very small or negative scalar flux estimates that can adversely affect the performance of CMFD. Therefore, our results demonstrate that LDA is a robust alternative to CMFD for certain sensitive problems in which CMFD can exhibit reduced effectiveness or failure.

Restricted partition functions and inverse energy cascades in parity symmetry breaking flows

When the symmetries of homogenous isotropic turbulent flows are broken, different sets of modes with different physical roles emerge. In particular, choosing a forcing which puts more weight on one or the other of these sets may result in different statistics for the energy transfers. We use the general method of computing a partition function restricted to a portion of phase space to study analytically these different statistics. We illustrate this method in the case of parity symmetry breaking, measured by helicity. It is shown that when helicity is sign-definite at all scales, an inverse cascade is expected for the energy. When sign-definiteness is lost, even for a small set of modes, this cascade disappears and there is a sharp phase transition to the standard helical equipartition spectra.

Nonlocal electron heating at the Earth's bow shock and the role of the magnetically tangent point

The solar wind interaction with the terrestrial magnetosphere results in a standing bow shock. Due to the shock's collisionless nature, the plasma in the magnetosheath is far from equilibrium. The highly mobile magnetosheath electrons move quickly between different points on the shock surface. Consequently, portions of phase space are populated with electrons from distant locations. This distant influence renders the heating properties of the shock nonlocal. Previous work predicted that the electron temperature along magnetosheath magnetic field lines should be approximately constant and controlled by heating at the point where the field line first encounters the shock, that is, where the field line is tangential to the shock. This point sends shock‐heated electrons to other bow shock locations but does not receive such electrons from elsewhere. In this paper, we compare electron temperatures measured by spacecraft connected by a field line in the magnetosheath and find a close agreement as predicted, unless electron traversal of the magnetosheath becomes impeded due to geometry. We also study the relationship between shock heating and plasma conditions both locally and at the tangent point. We find that the observed electron heating correlates much better with conditions at the tangent point than with the local shock parameters. Therefore, the tangent point plays a critical role in the electron heating and hence the global nature of the shock's partition of incident solar wind energy.

Access to high beta advanced inductive plasmas at low injected torque

Recent experiments on DIII-D demonstrate that advanced inductive (AI) discharges with high equivalent normalized fusion gain can be accessed and sustained with very low amounts (∼1 N m) of externally injected torque, a level of torque that is anticipated to drive a similar amount of rotation as the beams on ITER, via simple consideration of the scaling of the moment of inertia and confinement time. The AI regime is typically characterized by high confinement, and high βN, allowing the possibility for high performance, high gain operation at reduced plasma current. Discharges achieved βN ∼ 3.1 with H98(y,2) ∼ 1 at q95 ∼ 4, and are sustained for the maximum duration of the counter neutral beams (NBs). In addition, plasmas using zero net NB torque from the startup all the way through to the high βN phase have been created. AI discharges are found to become increasingly susceptible to m/n = 2/1 neoclassical tearing modes as the torque is decreased, which if left unmitigated, generally slow and lock, terminating the high performance phase of the discharge. Access is not notably different whether one ramps the torque down at high βN, or ramps βN up at low torque. The use of electron cyclotron heating (ECH) and current drive proved to be an effective method of avoiding such modes, enabling stable operation at high beta and low torque, a portion of phase space that has otherwise been inaccessible. Thermal confinement is significantly reduced at low rotation, a result that is reproduced using the TGLF transport model. Although it is thought that stiffness is increased in regions of low magnetic shear, in these AI plasmas, the reduced confinement occurs at radii outside the low shear, and in fact, higher temperature gradients can be found in the low shear region at low rotation. Momentum transport is also larger at low rotation, but a significant intrinsic torque is measured that is consistent with a previous scaling considering the role of the turbulent Reynolds stress and thermal ion orbit loss. Although high normalized fusion performance has been achieved in these discharges, more detailed projections suggest that enhancement in the confinement needs to be realized in order to obtain a low current solution consistent with ITER Q = 10 performance, and this remains a future research challenge.

Boundary-driven phase transitions in open driven systems with an umbilic point

Different phases in open driven systems are governed by either shocks or rarefaction waves. A presence of an isolated umbilic point in bidirectional systems of interacting particles stabilizes an unusual large scale excitation, an umbilic shock (U-shock). We show that in open systems the U-shock governs a large portion of phase space, and drives a new discontinuous transition between the two rarefaction-controlled phases. This is in contrast to strictly hyperbolic case where such a transition is always continuous. Also, we describe another robust phase which takes place at the phase governed by the U-shock, if the umbilic point is not isolated.

On the asymptotic behavior of average energy and enstrophy in 3D turbulent flows

Rigorous upper and lower bounds are proved for the Taylor and the Kolmogorov wavenumbers for the three-dimensional space periodic Navier–Stokes equations. Under the assumption that Kolmogorov’s two-thirds power law holds, the bounds sharpen to κ T ∼ Gr 1 / 4 and κ ϵ ∼ Gr 3 / 8 respectively, where Gr is the Grashof number. This provides a rigorous proof that the power law implies (1) the energy cascade, (2) Kolmogorov dissipation law, and (3) a connection between κ T and κ ϵ . The portion of phase space where a key a priori estimate on the nonlinear term is sharp is shown to be significant by means of a lower bound on any probability measure associated with an infinite-time average.

Singular Perturbation Analysis of the Fokker–Planck Equation: Kramers’ Underdamped Problem

The one-dimensional Fokker–Planck equation for a bistable potential is considered in the “underdamped” limit. Using singular perturbation analysis, an asymptotic series solution is constructed in the parameter associated with the small viscosity. This series solution is not uniformly valid: rather, it consists of four different types of asymptotic series valid in different time intervals and/or portions of phase space. By matching these series representations across overlapping regions, a reduced problem for the long term evolution of the distribution is extracted.

Union of chiral and heavy quark symmetries

We describe the chiral symmetric couplings of pions to heavy mesons ( B or D), valid in the portion of phase space where the pions have low momentum. In order to include consistently all low energy excitations, the vector mesons (B ∗ or D ∗) must appear explicitly in the effective lagrangian. The result is then invariant under both the chiral and heavy quark symmetries. We include matrix elements relevant for various weak decays.

The microscopic origin of cooperativity and its effect on long-lifetime kinetic modes for template-free RNA synthesis

We implement a model which describes the cooperative effects responsible for collective fluctuations on long lifetime modes in an open prebiotic chemical reactor. The model gives the correlation of spatially coupled subsystems, regarding them as information carriers subject to evolution equations of Eigen's type. We shall concentrate on the de novo(template-free) RNA synthetic apparatus which requires the Qβ-replicase as enzyme catalyst. The micsroscopic symmetry-breaking, given by a non-uniform distribution of RNA sequences, is linked in our model to the macroscopic symmetry-breaking, given by the irreversible amplification of template concentration. The connection between the two events is determined by the propagation of early intrinsic fluctuations along an attractive portion of phase space known as the centre manifold. The propagation leads in the final stage to the attractor which represents macsroscsoic symmetry-breaking. Experimental induction periods for the detection of amplification in template population are in agreement with theoretical calculations. In addition, enzyme and NTP (nucleoside triphosphate) dependence of induction periods are theoretically reproducible. An analysis of the first selfreplicating species unit has been assembled.

Almost periodic behavior of nonlinear waves

The motion of linearly arranged particles driven by nonlinear forces between nearest neighbors has been studied. Recent theoretical results concerning three nonlinear systems are reported here. Each of the systems discussed has an unusually large number of integrals, i.e., functionals conserved during motion; this fact might explain why some numerically computed trajectories of these systems seem to be confined to such an unexpectedly small portion of phase space. However, an additional mechanism must also be at work. The first of the systems discussed is Hamiltonian and completely integrable; accumulating evidence indicates that the second is, also. The Toda lattice, the Korteweg--de Vries equation, and the sine-Gordon equation are considered. (RWR)