scholarly journals The fate of random initial vorticity distributions for two-dimensional Euler equations on a sphere

2015 ◽  
Vol 786 ◽  
pp. 1-4 ◽  
Author(s):  
Paul K. Newton
Keyword(s):  
Numerical Simulations ◽  
Euler Equations ◽  
Large Scale ◽  
Initial Conditions ◽  
Infinite Family ◽  
Small Scale ◽  
Two Dimensional ◽  
Random Initial Conditions ◽  
Long Time ◽  
Initial Vorticity

The paper by Dritschel et al. (J. Fluid Mech., vol. 783, 2015, pp. 1–22) describes the long-time behaviour of inviscid two-dimensional fluid dynamics on the surface of a sphere. At issue is whether the flow settles down to an equilibrium or whether, for generic (random) initial conditions, the long-time solution is periodic, quasi-periodic or chaotic. While it might be surprising that this issue is not settled in the literature, it is important to keep in mind that the Euler equations form a dissipationless Hamiltonian system, hence the set of equations only redistributes the initial vorticity, generating smaller and smaller scales, while keeping kinetic energy, angular impulse and an infinite family of vorticity moments (Casimirs) intact. While special solutions that never settle down to an equilibrium state can be constructed using point vortices, vortex patches and other distributions, the fate of random initial conditions is a trickier problem. Previous statistical theories indicate that the long-time state should be a stationary large-scale distribution of vorticity. By carrying out careful numerical simulations using two different methods, the authors make a compelling case that the generic long-time state resembles a large-scale oscillating quadrupolar vorticity field, surrounded by persistent small-scale vortices. While numerical simulations can never conclusively settle this issue, the results might help guide future theories that seek to prove the existence of such an interesting dynamical long-time state.

Vortices in oscillating spin-up

2007 ◽  
Vol 573 ◽  
pp. 339-369 ◽  
Author(s):  
M. G. WELLS ◽  
H. J. H. CLERCX ◽  
G. J. F. VAN HEIJST
Keyword(s):  
Numerical Simulations ◽  
Power Law ◽  
Large Scale ◽  
Laboratory Experiments ◽  
Passive Scalar ◽  
Decay Rates ◽  
Small Scale ◽  
Two Dimensional ◽  
Ekman Pumping ◽  
Viscous Boundary Layer

Laboratory experiments and numerical simulations of oscillating spin-up in a square tank have been conducted to investigate the production of small-scale vorticity near the no-slip sidewalls of the container and the formation and subsequent decay of wall-generated quasi-two-dimensional vortices. The flow is made quasi-two-dimensional by a steady background rotation, and a small sinusoidal perturbation to the background rotation leads to the periodic formation of eddies in the corners of the tank by the roll-up of vorticity generated along the sidewalls. When the oscillation period is greater than the time scale required to advect a full-grown corner vortex to approximately halfway along the sidewall, dipole structures are observed to form. These dipoles migrate away from the walls, and the interior of the tank is continually filled with new vortices. The average size of these vortices appears to be largely controlled by the initial formation mechanism. Their vorticity decays from interactions with other stronger vortices that strip off filaments of vorticity, and by Ekman pumping at the bottom of the tank. Subsequent interactions between the weaker ‘old’ vortices and the ‘young’ vortices result in the straining, and finally the destruction, of older vortices. This inhibits the formation of large-scale vortices with diameters comparable to the size of the container.The laboratory experiments revealed a k−5/3 power law of the energy spectrum for small-to-intermediate wavenumbers. Measurements of the intensity spectrum of a passive scalar were consistent with the Batchelor prediction of a k−1 power law at large wavenumbers. Two-dimensional numerical simulations, under similar conditions to those in the experiments (with weak Ekman decay), were also performed and the simultaneous presence of a k−5/3 and k−3−ζ (with 0 < ζ « 1) power spectrum is observed, with the transition occurring at the wavenumber at which vorticity is injected from the viscous boundary layer into the interior. For higher Ekman decay rates, steeper spectra are obtained for the large wavenumber range, with ζ = O(1) and proportional to the Ekman decay rate. Movies are available with the online version of the paper.

scholarly journals On the late-time behaviour of a bounded, inviscid two-dimensional flow

2015 ◽  
Vol 783 ◽  
pp. 1-22 ◽  
Author(s):  
David G. Dritschel ◽  
Wanming Qi ◽  
J. B. Marston
Keyword(s):  
Large Scale ◽  
Point Vortex ◽  
Small Scale ◽  
Late Time ◽  
Two Dimensional ◽  
Time Behaviour ◽  
Vorticity Field ◽  
Dimensional Flow ◽  
Two Dimensional Flow ◽  
Initial Vorticity

Using complementary numerical approaches at high resolution, we study the late-time behaviour of an inviscid incompressible two-dimensional flow on the surface of a sphere. Starting from a random initial vorticity field comprised of a small set of intermediate-wavenumber spherical harmonics, we find that, contrary to the predictions of equilibrium statistical mechanics, the flow does not evolve into a large-scale steady state. Instead, significant unsteadiness persists, characterised by a population of persistent small-scale vortices interacting with a large-scale oscillating quadrupolar vorticity field. Moreover, the vorticity develops a stepped, staircase distribution, consisting of nearly homogeneous regions separated by sharp gradients. The persistence of unsteadiness is explained by a simple point-vortex model characterising the interactions between the four main vortices which emerge.

scholarly journals Toward understanding the multiscale spatial spectrum of solar convection

2012 ◽  
Vol 8 (S294) ◽  
pp. 361-363
Author(s):  
A. V. Getling ◽  
O. S. Mazhorova ◽  
O. V. Shcheritsa
Keyword(s):  
Convective Instability ◽  
Convective Flow ◽  
Large Scale ◽  
Fluid Layer ◽  
Boussinesq Equations ◽  
Spatial Spectrum ◽  
Small Scale ◽  
Two Dimensional ◽  
Solar Convection

AbstractConvection is simulated numerically based on two-dimensional Boussinesq equations for a fluid layer with a specially chosen stratification such that the convective instability is much stronger in a thin subsurface sublayer than in the remaining part of the layer. The developing convective flow has a small-scale component superposed onto a basic large-scale roll flow.

scholarly journals Finite Symmetries in Agent-Based Epidemic Models

2019 ◽  
Vol 24 (2) ◽  
pp. 44 ◽  
Author(s):  
Gilberto M. Nakamura ◽  
Ana Carolina P. Monteiro ◽  
George C. Cardoso ◽  
Alexandre S. Martinez
Keyword(s):  
Population Size ◽  
Transition Matrix ◽  
Large Scale ◽  
Initial Conditions ◽  
Disease Outbreaks ◽  
Epidemic Models ◽  
Small Scale ◽  
Stochastic Effects ◽  
Agent Based ◽  
Computational Performance

Predictive analysis of epidemics often depends on the initial conditions of the outbreak, the structure of the afflicted population, and population size. However, disease outbreaks are subjected to fluctuations that may shape the spreading process. Agent-based epidemic models mitigate the issue by using a transition matrix which replicates stochastic effects observed in real epidemics. They have met considerable numerical success to simulate small scale epidemics. The problem grows exponentially with population size, reducing the usability of agent-based models for large scale epidemics. Here, we present an algorithm that explores permutation symmetries to enhance the computational performance of agent-based epidemic models. Our findings bound the stochastic process to a single eigenvalue sector, scaling down the dimension of the transition matrix to o ( N 2 ) .

Community Replacement Pathways: What Do Fossil Sequences Reveal About Marine Ecosystem Transitions?

1990 ◽  
Vol 5 ◽  
pp. 262-272
Author(s):  
William Miller
Keyword(s):  
Fossil Record ◽  
Large Scale ◽  
Marine Ecosystem ◽  
Small Scale ◽  
Data Sets ◽  
Mass Extinctions ◽  
Ultimate Cause ◽  
Long Time ◽  
History Of ◽  
Time And Energy

Paleontologists have lavished much time and energy on description and explanation of large-scale patterns in the fossil record (e.g., mass extinctions, histories of monophyletic taxa, deployment of major biogeographic units), while paying comparatively little attention to biologic patterns preserved only in local stratigraphic sequences. Interpretation of the large-scale patterns will always be seen as the chief justification for the science of paleontology, but solving problems framed by long time spans and large areas is rife with tenuous inference and patterns are prone to varied interpretation by different investigators using virtually the same data sets (as in the controversy over ultimate cause of the terminal Cretaceous extinctions). In other words, the large-scale patterns in the history of life are the true philosophical property of paleontology, but there will always be serious problems in attempting to resolve processes that transpired over millions to hundreds-of-millions of years and encompassed vast areas of seafloor or landscape. By contrast, less spectacular and more commonplace changes in local habitats (often related to larger-scale events and cycles) and attendant biologic responses are closer to our direct experience of the living world and should be easier to interpret unequivocally. These small-scale responses are reflected in the fossil record at the scale of local outcrops.

A new model of shoaling and breaking waves. Part 2. Run-up and two-dimensional waves

2019 ◽  
Vol 867 ◽  
pp. 146-194 ◽  
Author(s):  
G. L. Richard ◽  
A. Duran ◽  
B. Fabrèges
Keyword(s):  
Large Scale ◽  
Turbulent Viscosity ◽  
Breaking Waves ◽  
Small Scale ◽  
Two Dimensional ◽  
Numerical Resolution ◽  
Wide Range ◽  
Scale Turbulence ◽  
Run Up ◽  
Averaged Model

We derive a two-dimensional depth-averaged model for coastal waves with both dispersive and dissipative effects. A tensor quantity called enstrophy models the subdepth large-scale turbulence, including its anisotropic character, and is a source of vorticity of the average flow. The small-scale turbulence is modelled through a turbulent-viscosity hypothesis. This fully nonlinear model has equivalent dispersive properties to the Green–Naghdi equations and is treated, both for the optimization of these properties and for the numerical resolution, with the same techniques which are used for the Green–Naghdi system. The model equations are solved with a discontinuous Galerkin discretization based on a decoupling between the hyperbolic and non-hydrostatic parts of the system. The predictions of the model are compared to experimental data in a wide range of physical conditions. Simulations were run in one-dimensional and two-dimensional cases, including run-up and run-down on beaches, non-trivial topographies, wave trains over a bar or propagation around an island or a reef. A very good agreement is reached in every cases, validating the predictive empirical laws for the parameters of the model. These comparisons confirm the efficiency of the present strategy, highlighting the enstrophy as a robust and reliable tool to describe wave breaking even in a two-dimensional context. Compared with existing depth-averaged models, this approach is numerically robust and adds more physical effects without significant increase in numerical complexity.

Enhanced dissipation for quasi-geostrophic motion over small-scale topography

2000 ◽  
Vol 407 ◽  
pp. 105-122 ◽  
Author(s):  
JACQUES VANNESTE
Keyword(s):  
Large Scale ◽  
Spatial Scales ◽  
Additional Term ◽  
Equation Of Motion ◽  
Small Scale ◽  
Two Dimensional ◽  
Turbulent Dissipation ◽  
Dissipative Term ◽  
History Of ◽  
Large Scale Flow

The effect of a small-scale topography on large-scale, small-amplitude oceanic motion is analysed using a two-dimensional quasi-geostrophic model that includes free-surface and β effects, Ekman friction and viscous (or turbulent) dissipation. The topography is two-dimensional and periodic; its slope is assumed to be much larger than the ratio of the ocean depth to the Earth's radius. An averaged equation of motion is derived for flows with spatial scales that are much larger than the scale of the topography and either (i) much larger than or (ii) comparable to the radius of deformation. Compared to the standard quasi-geostrophic equation, this averaged equation contains an additional dissipative term that results from the interaction between topography and dissipation. In case (i) this term simply represents an additional Ekman friction, whereas in case (ii) it is given by an integral over the history of the large-scale flow. The properties of the additional term are studied in detail. For case (i) in particular, numerical calculations are employed to analyse the dependence of the additional Ekman friction on the structure of the topography and on the strength of the original dissipation mechanisms.

STUDY ON RELATIONSHIP BETWEEN CONDENSED PARTICLES AND STRUCTURE OF CONDENSATION JET USING 2D IMAGE AND DISCRETE WAVELETS MULTIRESOLUTION

2006 ◽  
Vol 04 (02) ◽  
pp. 227-238
Author(s):  
MOTOAKI KIMURA ◽  
MASAHIRO TAKEI ◽  
YOSHIFURU SAITO ◽  
KIYOSHI HORII
Keyword(s):  
Wavelet Transforms ◽  
Large Scale ◽  
Particle Density ◽  
Small Scale ◽  
Discrete Wavelet ◽  
Discrete Wavelet Transforms ◽  
Two Dimensional ◽  
Condensation Zone ◽  
Two Dimensional Image ◽  
Discrete Wavelets

This paper describes the application of discrete wavelet transforms to the analysis of condensation jets in order to clarify the associated fluid and heat transfer phenomena. An experimentally-obtained, two-dimensional image of the condensation particle density around the jet was decomposed into 7 levels of resolution with their respective wavelengths. Based on the known physical characteristics of turbulent flow around the jet, levels 0 and 1 were shown to represent the large-scale components of the condensation particle density and the higher levels represent the small-scale components. From the wavelet-analyzed images, the width of the condensation zone was obtained and this compared well with the width inferred from temperature measurements. Thus, the method was verified and also provided data not available experimentally.

scholarly journals Thermomechanically coupled modelling for land-terminating glaciers: a comparison of two-dimensional, first-order and three-dimensional, full-Stokes approaches

2015 ◽  
Vol 61 (228) ◽  
pp. 702-712 ◽  
Author(s):  
Tong Zhang ◽  
Lili Ju ◽  
Wei Leng ◽  
Stephen Price ◽  
Max Gunzburger
Keyword(s):  
Initial Conditions ◽  
Three Dimensional ◽  
Integration Time ◽  
Two Dimensional ◽  
Shape Factors ◽  
Coupled Modelling ◽  
First Order ◽  
Glacier Changes ◽  
Basal Sliding ◽  
Long Time

AbstractFor many regions, glacier inaccessibility results in sparse geometric datasets for use as model initial conditions (e.g. along the central flowline only). In these cases, two-dimensional (2-D) flowline models are often used to study glacier dynamics. Here we systematically investigate the applicability of a 2-D, first-order Stokes approximation flowline model (FLM), modified by shape factors, for the simulation of land-terminating glaciers by comparing it with a 3-D, ‘full’-Stokes ice-flow model (FSM). Based on steady-state and transient, thermomechanically uncoupled and coupled computational experiments, we explore the sensitivities of the FLM and FSM to ice geometry, temperature and forward model integration time. We find that, compared to the FSM, the FLM generally produces slower horizontal velocities, due to simplifications inherent to the FLM and to the underestimation of the shape factor. For polythermal glaciers, those with temperate ice zones, or when basal sliding is important, we find significant differences between simulation results when using the FLM versus the FSM. Over time, initially small differences between the FLM and FSM become much larger, particularly near cold/temperate ice transition surfaces. Long time integrations further increase small initial differences between the two models. We conclude that the FLM should be applied with caution when modelling glacier changes under a warming climate or over long periods of time.

Mixing layer produced by a screen and its dependence on initial conditions

1984 ◽  
Vol 142 ◽  
pp. 217-231 ◽  
Author(s):  
Hakuro Oguchi ◽  
Osamu Inoue
Keyword(s):  
Large Scale ◽  
Initial Conditions ◽  
Mixing Layer ◽  
Mixing Layers ◽  
Laser Doppler Velocimeter ◽  
Small Scale ◽  
Wire Screen ◽  
Flow Features ◽  
Wire Method ◽  
Layer Flow

This paper aims to elucidate the structure of the turbulent mixing layers, especially, its dependence on initial disturbances. The mixing layers are produced by setting a woven-wire screen perpendicular to the freestream in the test section of a wind tunnel to obstruct part of the flow. Three kinds of model geometry are treated; these model screens produced mixing layers which may be regarded as the equivalents of the plane mixing layer and of two-dimensional and axisymmetric wakes issuing into ambient streams of higher velocity. The initial disturbances are imposed by installing thin rods of various sizes along the edge of the screen or at the origin of the mixing layer. Flow features are visualized by the smoke-wire method. Statistical quantities are measured by a laser-Doppler velocimeter. In all cases large-scale transverse vortices seem to persist, although comparatively small-scale vortices are superimposed on the flow field in the mixing layer. The mixing layers are in self-preserving state at least up to third-order moments, but the self-preserving state is different in each case. The growth rates of the mixing layer are shown to depend strongly on the initial disturbance imposed.

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