Trivial Meet and Join within the Lattice of Monotone Triangles

The lattice of monotone triangles $(\mathfrak{M}_n,\le)$ ordered by entry-wise comparisons is studied. Let $\tau_{\min}$ denote the unique minimal element in this lattice, and $\tau_{\max}$ the unique maximum. The number of $r$-tuples of monotone triangles $(\tau_1,\ldots,\tau_r)$ with minimal infimum $\tau_{\min}$ (maximal supremum $\tau_{\max}$, resp.) is shown to asymptotically approach $r|\mathfrak{M}_n|^{r-1}$ as $n \to \infty$. Thus, with high probability this event implies that one of the $\tau_i$ is $\tau_{\min}$ ($\tau_{\max}$, resp.). Higher-order error terms are also discussed.Comment: 15 page

An alternative approach to field-aligned coordinates for plasma turbulence simulations

Turbulence simulation codes can exploit the flute-like nature of plasma turbulence to reduce the effective number of degrees of freedom necessary to represent fluctuations. This can be achieved by employing magnetic coordinates of which one is aligned along the magnetic field. This work presents an approach in which the position along the field lines is identified by the toroidal angle, rather than the most commonly used poloidal angle. It will be shown that this approach has several advantages. Among these, periodicity in both angles is retained. This property allows moving to an equivalent representation in Fourier space with a reduced number of toroidal components. It will be shown how this duality can be exploited to transform conventional codes that use a spectral representation on the magnetic surface into codes with a field-aligned coordinate. It is also shown that the new approach can be generalised to get rid of magnetic coordinates in the poloidal plane altogether, for a large class of models. Tests are carried out by comparing the new approach with the conventional approach employing a uniform grid, for a basic ion temperature gradient (ITG) turbulence model implemented by the two corresponding versions of the ETAI3D code. These tests uncover an unexpected property of the model, that localized large parallel gradients can intermittently appear in the turbulent regime. This leaves open the question whether this is a general property of plasma turbulence, which may lead one to reconsider some of the usual assumptions on micro-turbulence dynamics.Comment: 19 pages (once in pdf format). 1 LaTeX file and 10 eps figures in the zip folde

NLO Leptoquark Production and Decay: The Narrow-Width Approximation and Beyond

We study the leptoquark model of Buchm\"uller, R\"uckl and Wyler, focusing on a particular type of scalar ($R_2$) and vector ($U_1$) leptoquark. The primary aim is to perform the calculations for leptoquark production and decay at next-to-leading order (NLO) to establish the importance of the NLO contributions and, in particular, to determine how effective the narrow-width-approximation (NWA) is at NLO. For both the scalar and vector leptoquarks it is found that the NLO contributions are large, with the larger corrections occurring for the case vector leptoquarks. For the scalar leptoquark it is found that the NWA provides a good approximation for determining the resonant peak, however the NWA is not as effective for the vector leptoquark. For both the scalar and vector leptoquarks there are large contributions away from the resonant peak, which are missing from the NWA results, and these make a significant difference to the total cross-section.Comment: 22 pages, 17 figure

On comparability of bigrassmannian permutations

Let Sn and Gn denote the respective sets of ordinary and bigrassmannian (BG) permutations of order n, and let (Gn,≤) denote the Bruhat ordering permutation poset. We study the restricted poset (Bn,≤), first providing a simple criterion for comparability. This criterion is used to show that that the poset is connected, to enumerate the saturated chains between elements, and to enumerate the number of maximal elements below r fixed elements. It also quickly produces formulas for β(ω) (α(ω), respectively), the number of BG permutations weakly below (weakly above, respectively) a fixed ω ∈ Bn, and is used to compute the Mo¨bius function on any interval in Bn. We then turn to a probabilistic study of β = β(ω) (α = α(ω) respectively) for the uniformly random ω ∈ Bn. We show that α and β are equidistributed, and that β is of the same order as its expectation with high probability, but fails to concentrate about its mean. This latter fact derives from the limiting distribution of β/n3. We also compute the probability that randomly chosen BG permutations form a 2- or 3-element multichain

Full-ff gyrokinetic simulation of turbulence in a helical open-field-line plasma

Curvature-driven turbulence in a helical open-field-line plasma is investigated using electrostatic five-dimensional gyrokinetic continuum simulations in an all-bad-curvature helical-slab geometry. Parameters for a National Spherical Torus Experiment scrape-off-layer plasma are used in the model. The formation and convective radial transport of plasma blobs is observed, and it is shown that the radial particle-transport levels are several times higher than diffusive Bohm-transport estimates. By reducing the strength of the poloidal magnetic field, the profile of the heat flux to the divertor plate is observed to broaden

Comparing linear ion-temperature-gradient-driven mode stability of the National Compact Stellarator Experiment and a shaped tokamak

One metric for comparing confinement properties of different magnetic fusion energy configurations is the linear critical gradient of drift wave modes. The critical gradient scale length determines the ratio of the core to pedestal temperature when a plasma is limited to marginal stability in the plasma core. The gyrokinetic turbulence code GS2 was used to calculate critical temperature gradients for the linear, collisionless ion temperature gradient (ITG) mode in the National Compact Stellarator Experiment (NCSX) and a prototypical shaped tokamak, based on the profiles of a JET H-mode shot and the stronger shaping of ARIES-AT. While a concern was that the narrow cross section of NCSX at some toroidal locations would result in steep gradients that drive instabilities more easily, it is found that other stabilizing effects of the stellarator configuration offset this so that the normalized critical gradients for NCSX are competitive with or even better than for the tokamak. For the adiabatic ITG mode, NCSX and the tokamak had similar critical gradients, though beyond marginal stability, NCSX had larger growth rates. However, for the kinetic ITG mode, NCSX had a higher critical gradient and lower growth rates until a/L_T is approximately 1.5 times a/L_{T,crit}, when it surpassed the tokamak's. A discussion of the results presented with respect to a/L_T vs. R/L_T is included.Comment: Accepted to Physics of Plasmas. 8 pages, 16 figure