14 research outputs found
Asymptotics of conduction velocity restitution in models of electrical excitation in the heart
We extend a non-Tikhonov asymptotic embedding, proposed earlier, for calculation of conduction velocity restitution curves in ionic models of cardiac excitability. Conduction velocity restitution is the simplest non-trivial spatially extended problem in excitable media, and in the case of cardiac tissue it is an important tool for prediction of cardiac arrhythmias and fibrillation. An idealized conduction velocity restitution curve requires solving a non-linear eigenvalue problem with periodic boundary conditions, which in the cardiac case is very stiff and calls for the use of asymptotic methods. We compare asymptotics of restitution curves in four examples, two generic excitable media models, and two ionic cardiac models. The generic models include the classical FitzHughāNagumo model and its variation by Barkley. They are treated with standard singular perturbation techniques. The ionic models include a simplified ācaricatureā of Noble (J. Physiol. Lond. 160:317ā352, 1962) model and Beeler and Reuter (J. Physiol. Lond. 268:177ā210, 1977) model, which lead to non-Tikhonov problems where known asymptotic results do not apply. The Caricature Noble model is considered with particular care to demonstrate the well-posedness of the corresponding boundary-value problem. The developed method for calculation of conduction velocity restitution is then applied to the BeelerāReuter model. We discuss new mathematical features appearing in cardiac ionic models and possible applications of the developed method
Problems of astrophysical turbulent convection: thermal convection in a layer without boundaries
Thermal convection in ļ¬uid layers heated from below are usually realized experimentally as well as treated theoretically with ļ¬xed boundaries on which conditions for the
temperature and the velocity ļ¬eld are prescribed. The thermal and velocity boundary
layers attached to the upper and lower boundaries determine to a large extent the properties of turbulent convection at high Rayleigh numbers. Fixed boundaries are often absent
in natural realizations of thermal convection. This paper studies the properties of convection driven by a planar heat source below a cooling source of equal size immersed in an
otherwise stably stratiļ¬ed ļ¬uid layer are studied in this paper. Unavoidable boundaries
do not inļ¬uence the convection ļ¬ow since they are separated from the active convection
layer by nearly motionless stably stratiļ¬ed regions. The onset of convection occurs in an
inner unstably stratiļ¬ed region where the mean temperature gradient is reversed. But
the region of a reversed horizontally averaged temperature gradient disappears at higher
amplitudes of convection such that the vertical derivative of the mean temperature no
longer changes its sig
Turbulent 3D MHD dynamo model in spherical shells: regular oscillations of the dipolar field
We report the results of three-dimensional numerical simulations of convection-driven dynamos in relatively thin rotating spherical shells that show a transition from an strong non-oscillatory dipolar magnetic field to a weaker regularly oscillating dipolar field. The transition is induced primarily by the effects a stress-free boundary condition. The variation of the inner to outer radius ratio is found to have a less important effect
Some Unusual Properties of Turbulent Convection and Dynamos in Rotating Spherical Shells
The dynamics of convecting fluids in rotating spherical shells is governed at
Prandtl numbers of the order unity by the interaction between differential
rotation and roll-like convection eddies. While the differential rotation is
driven by the Reynolds stresses of the eddies, its shearing action inhibits
convection and causes phenomena such as localized convection and turbulent
relaxation oscillations. The response of the system is enriched in the case of
dynamo action. Lorentz forces may brake either entirely or partially the
geostrophic differential rotation and give rise to two rather different dynamo
states. Bistability of turbulent dynamos exists for magnetic Prandtl numbers of
the order unity. While the ratios between mean magnetic and kinetic energies
differ by a factor of 5 or more for the two dynamo states, the mean convective
heat transports are nearly the same. They are much larger than in the absence
of a magnetic field.Comment: To appear in Procs. IUTAM Symposium on Turbulence in the Atmosphere
and Oceans, 08-7 = GA.06-0
Toroidal flux oscillation as possible cause of geomagnetic excursions and reversals
It is proposed that convection driven dynamos operating in planetary cores
could be oscillatory even when the oscillations are not directly noticeable
from the outside. Examples of dynamo simulations are pointed out that exhibit
oscillations in the structure of the azimuthally averaged toroidal magnetic
flux while the mean poloidal field shows only variations in its amplitude. In
the case of the geomagnetic field, global excursions may be associated with
these oscillations. Long period dynamo simulations indicate that the
oscillations may cause reversals once in a while. No special attempt has been
made to use most realistic parameter values. Nevertheless some similarities
between the simulations and the paleomagnetic record can be pointed out.Comment: Published in PEP
Analytically Solvable Asymptotic Model of Atrial Excitability
We report a three-variable simplified model of excitation fronts in human
atrial tissue. The model is derived by novel asymptotic techniques \new{from
the biophysically realistic model of Courtemanche et al (1998) in extension of
our previous similar models. An iterative analytical solution of the model is
presented which is in excellent quantitative agreement with the realistic
model. It opens new possibilities for analytical studies as well as for
efficient numerical simulation of this and other cardiac models of similar
structure
The mathematics of fluid dynamos
In this article we discuss ļ¬uid dynamos and how they generate the magnetic ļ¬elds of astrophysical objects such as the Earth. As an illustration, we sketch one well-known mathematical model of Earth's ļ¬uid dynamo and comment on some of its typical features and applications. The exposition will beaccessible to undergraduate students in Science and Mathematics with some knowledge in Diļ¬erential Equations and Physics
Magneto-inertial convection in rotating fluid spheres
The onset of convection in the form of magneto-inertial waves in a rotating fluid sphere permeated by a constant axial electric current is studied through a perturbation analysis. Explicit expressions for the dependence of the Rayleigh number on the azimuthal wavenumber are derived in the limit of high thermal diffusivity. Results for the cases of thermally infinitely conducting and of nearly thermally insulating boundaries are obtained
Proceedings of the 2013 UK national conference on geophysical, astrophysical and industrial magnetohydrodynamics at Glasgow
This document presents an accurate record of the scientific programme of the 2013 UK National Conference on Geophysical, Astrophysical and Industrial Magnetohydrodynamics held on 23rd and 24th of May 2013 at the School of Mathematics and Statistics of the University of Glasgow. The abstracts of presented contributions are listed in chronological order and a full list of participants and their affiliations is included