arXiv subjects

Mark A. Taylor

Publications and source records attributed to Mark A. Taylor.

7 recordsLinked to original sources

arXiv2019

An energy consistent discretization of the nonhydrostatic equations in primitive variables

We derive a formulation of the nonhydrostatic equations in spherical geometry with a Lorenz staggered vertical discretization. The combination conserves a discrete energy in exact time integration when coupled with a mimetic horizontal discretization. The formulation is a version of Dubos and Tort (2014) rewritten in terms of primitive variables. It is valid for terrain following mass or height coordinates and for both Eulerian or vertically Lagrangian discretizations. The discretization relies on an extension to Simmons and Burridge (1981) vertical differencing which we show obeys a discrete derivative product rule. This product rule allows us to simplify the treatment of the vertical transport terms. Energy conservation is obtained via a term-by-term balance in the kinetic, internal and potential energy budgets, ensuring an energy-consistent discretization with no spurious sources of energy. We demonstrate convergence with respect to time truncation error in a spectral element code with a HEVI IMEX timestepping algorithm

math.NA↗

arXiv2018

Geometric Partitioning and Ordering Strategies for Task Mapping on Parallel Computers

We present a new method for mapping applications' MPI tasks to cores of a parallel computer such that applications' communication time is reduced. We address the case of sparse node allocation, where the nodes assigned to a job are not necessarily located in a contiguous block nor within close proximity to each other in the network, although our methods generalize to contiguous allocations as well. The goal is to assign tasks to cores so that interdependent tasks are performed by "nearby" cores, thus lowering the distance messages must travel, the amount of congestion in the network, and the overall cost of communication. Our new method applies a geometric partitioning algorithm to both the tasks and the processors, and assigns task parts to the corresponding processor parts. We also present a number of algorithmic optimizations that exploit specific features of the network or application. We show that, for the structured finite difference mini-application MiniGhost, our mapping methods reduced communication time up to 75% relative to MiniGhost's default mapping on 128K cores of a Cray XK7 with sparse allocation. For the atmospheric modeling code E3SM/HOMME, our methods reduced communication time up to 31% on 32K cores of an IBM BlueGene/Q with contiguous allocation.

cs.DC↗

arXiv2011

Low Rossby limiting dynamics for stably stratified flow with Finite Froude number

In this paper we explore the fast rotation, nonhydrostatic limit of the rotating and stratified Boussinesq equations. We derive new reduced equations for the slow dynamics that describe Taylor-Proudman flows. One new aspect of the dynamics is a decoupling of the horizontal kinetic energy, described by 2D Navier-Stokes, from new dynamics that describe the coupling of vertical kinetic energy and buoyancy. We support the theory with high resolution numerical simulations of the full Boussinesq equations that, in this limit, reveal the spontaneous formation of Taylor-Proudman columns and their dynamics.

nlin.CD↗

arXiv2005

Several new quadrature formulas for polynomial integration in the triangle

We present several new quadrature formulas in the triangle for exact integration of polynomials. The points were computed numerically with a cardinal function algorithm which imposes that the number of quadrature points $N$ be equal to the dimension of a lower dimensional polynomial space. Quadrature forumulas are presented for up to degree $d=25$, all which have positive weights and contain no points outside the triangle. Seven of these quadrature formulas improve on previously known results.

math.NA↗

arXiv2004

Isotropic third-order statistics in turbulence with helicity: the 2/15-law

The so-called 2/15-law for two-point, third-order velocity statistics in isotropic turbulence with helicity is computed for the first time from a direct numerical simulation of the Navier-Stokes equations in a 512^3 periodic domain. This law is a statement of helicity conservation in the inertial range, analogous to the benchmark Kolmogorov 4/5-law for energy conservation in high-Reynolds number turbulence. The appropriately normalized parity-breaking statistics, when measured in an arbitrary direction in the flow, disagree with the theoretical value of 2/15 predicted for isotropic turbulence. They are highly anisotropic and variable and remain so over a long times. We employ a recently developed technique to average over many directions and so recover the statistically isotropic component of the flow. The angle-averaged statistics achieve the 2/15 factor to within about 7% instantaneously and about 5% on average over time. The inertial- and viscous-range behavior of the helicity-dependent statistics and consequently the helicity flux, which appear in the 2/15-law, are shown to be more anisotropic and intermittent than the cor- responding energy-dependent reflection-symmetric structure functions, and the energy flux, which appear in the 4/5-law. This suggests that the Kolmogorov assumption of local isotropy at high Reynolds numbers needs to be modified for the helicity-dependent statistics investigated here.

nlin.CD↗

arXiv2004

Cascade time-scales for energy and helicity in homogeneous isotropic turbulence

We extend the Kolmogorov phenomenology for the scaling of energy spectra in high-Reynolds number turbulence, to explicitly include the effect of helicity. There exists a time-scale $τ_H$ for helicity transfer in homogeneous, isotropic turbulence with helicity. We arrive at this timescale using the phenomenological arguments used by Kraichnan to derive the timescale $τ_E$ for energy transfer (J. Fluid Mech. {\bf 47}, 525--535 (1971)). We show that in general $τ_H$ may not be neglected compared to $τ_E$, even for rather low relative helicity. We then deduce an inertial range joint cascade of energy and helicity in which the dynamics are dominated by $τ_E$ in the low wavenumbers with both energy and helicity spectra scaling as $k^{-5/3}$; and by $τ_H$ at larger wavenumbers with spectra scaling as $k^{-4/3}$. We demonstrate how, within this phenomenology, the commonly observed ``bottleneck'' in the energy spectrum might be explained. We derive a wavenumber $k_h$ which is less than the Kolmogorov dissipation wavenumber, at which both energy and helicity cascades terminate due to dissipation effects. Data from direct numerical simulations are used to check our predictions.

nlin.CD↗

arXiv2003

Recovering Isotropic Statistics in Turbulence Simulations: The Kolmogorov 4/5th-Law

One of the main benchmarks in direct numerical simulations of three-dimensional turbulence is the Kolmogorov 1941 prediction for third-order structure functions with homogeneous and isotropic statistics in the infinite-Reynolds number limit. Previous DNS techniques to obtain isotropic statistics have relied on time-averaging structure functions in a few directions over many eddy turnover times, using forcing schemes carefully constructed to generate isotropic data. Motivated by recent theoretical work which removes isotropy requirements by spherically averaging structure functions over all directions, we will present results which supplement long-time averaging by angle-averaging over up to 73 directions from a single flow snapshot. The directions are among those natural to a square computational grid, and are weighted to approximate the spherical average. The averaging process is cheap, and for the Kolmogorov 1941 4/5ths law, reasonable results can be obtained from a single snapshot of data. This procedure may be used to investigate the isotropic statistics of any quantity of interest.

nlin.CD↗