Analysis of the shearing instability in nonlinear convection and magnetoconvection

Numerical experiments on two-dimensional convection with or without a vertical magnetic field reveal a bewildering variety of periodic and aperiodic oscillations. Steady rolls can develop a shearing instability, in which rolls turning over in one direction grow at the expense of rolls turning over in the other, resulting in a net shear across the layer. As the temperature difference across the fluid is increased, two-dimensional pulsating waves occur, in which the direction of shear alternates. We analyse the nonlinear dynamics of this behaviour by first constructing appropriate low-order sets of ordinary differential equations, which show the same behaviour, and then analysing the global bifurcations that lead to these oscillations by constructing one-dimensional return maps. We compare the behaviour of the partial differential equations, the models and the maps in systematic two-parameter studies of both the magnetic and the non-magnetic cases, emphasising how the symmetries of periodic solutions change as a result of global bifurcations. Much of the interesting behaviour is associated with a discontinuous change in the leading direction of a fixed point at a global bifurcation; this change occurs when the magnetic field is introduced

Altered plasticity of the parasympathetic innervation in the recovering rat submandibular gland following extensive atrophy

Adult rat submandibular glands have a rich autonomic innervation, with parasympathetic and sympathetic nerves working in synergy rather than antagonistically. Ligation of the secretory duct rapidly causes atrophy and the loss of most acini, which are the main target cell for parasympathetic nerves. Following deligation, there is a recovery of gland structure and function, as assessed by autonomimetic stimulation. This study examines whether the parasympathetic nerves reattach to new target cells to form functional neuro-effector junctions. Under recovery anaesthesia, the submandibular duct of adult male rats was ligated via an intra-oral approach to avoid damaging the chorda-lingual nerve. Four weeks later, rats were either killed or anaesthetized and the ligation clip removed. Following a further 8 weeks, both submandibular ducts were cannulated under terminal anaesthesia. Salivary flows were then stimulated electrically (chorda-lingual nerve at 2, 5 and 10 Hz) and subsequently by methacholine (whole-body infusion at two doses). Glands were excised, weighed and divided for further in vitro studies or fixed for histological examination. Ligation of ducts caused 75% loss of gland weight, with the loss of most acinar cells. Of the remaining acini, only 50% were innervated despite unchanged choline acetyltransferase activity, suggesting few parasympathetic nerves had died. Following deligation, submandibular glands recovered half their weight and had normal morphology. Salivary flows from both glands (per unit of gland tissue) were similar when evoked by methacholine but greater from the deligated glands when evoked by nerve stimulation. This suggests that parasympathetic nerves had reattached to new target cells in the recovered glands at a greater ratio than normal, confirming reinnervation of the regenerating gland

The inverse cascade and nonlinear alpha-effect in simulations of isotropic helical hydromagnetic turbulence

A numerical model of isotropic homogeneous turbulence with helical forcing is investigated. The resulting flow, which is essentially the prototype of the alpha^2 dynamo of mean-field dynamo theory, produces strong dynamo action with an additional large scale field on the scale of the box (at wavenumber k=1; forcing is at k=5). This large scale field is nearly force-free and exceeds the equipartition value. As the magnetic Reynolds number R_m increases, the saturation field strength and the growth rate of the dynamo increase. However, the time it takes to built up the large scale field from equipartition to its final super-equipartition value increases with magnetic Reynolds number. The large scale field generation can be identified as being due to nonlocal interactions originating from the forcing scale, which is characteristic of the alpha-effect. Both alpha and turbulent magnetic diffusivity eta_t are determined simultaneously using numerical experiments where the mean-field is modified artificially. Both quantities are quenched in a R_m-dependent fashion. The evolution of the energy of the mean field matches that predicted by an alpha^2 dynamo model with similar alpha and eta_t quenchings. For this model an analytic solution is given which matches the results of the simulations. The simulations are numerically robust in that the shape of the spectrum at large scales is unchanged when changing the resolution from 30^3 to 120^3 meshpoints, or when increasing the magnetic Prandtl number (viscosity/magnetic diffusivity) from 1 to 100. Increasing the forcing wavenumber to 30 (i.e. increasing the scale separation) makes the inverse cascade effect more pronounced, although it remains otherwise qualitatively unchanged.Comment: 21 pages, 26 figures, ApJ (accepted

Vicious walkers, friendly walkers and Young tableaux II: With a wall

We derive new results for the number of star and watermelon configurations of vicious walkers in the presence of an impenetrable wall by showing that these follow from standard results in the theory of Young tableaux, and combinatorial descriptions of symmetric functions. For the problem of $n$-friendly walkers, we derive exact asymptotics for the number of stars and watermelons both in the absence of a wall and in the presence of a wall.Comment: 35 pages, AmS-LaTeX; Definitions of n-friendly walkers clarified; the statement of Theorem 4 and its proof were correcte

Converging and diverging convection around axisymmetric magnetic flux tubes

A numerical model of idealized sunspots and pores is presented, where axisymmetric cylindrical domains are used with aspect ratios (radius versus depth) up to 4. The model contains a compressible plasma with density and temperature gradients simulating the upper layer of the Sun's convection zone. Non-linear magnetohydrodynamic equations are solved numerically and time-dependent solutions are obtained where the magnetic field is pushed to the centre of the domain by convection cells. This central magnetic flux bundle is maintained by an inner convection cell, situated next to it and with a flow such that there is an inflow at the top of the numerical domain towards the flux bundle. For aspect ratio 4, a large inner cell persists in time, but for lower aspect ratios it becomes highly time dependent. For aspect ratios 2 and 3 this inner convection cell is smaller, tends to be situated towards the top of the domain next to the flux bundle, and appears and disappears with time. When it is gone, the neighbouring cell (with an opposite sense of rotation, i.e. outflow at the top) pulls the magnetic field away from the central axis. As this happens a new inner cell forms with an inflow which pushes the magnetic field towards the centre. This suggests that to maintain their form, both pores and sunspots need a neighbouring convection cell with inflow at the top towards the magnetic flux bundle. This convection cell does not have to be at the top of the convection zone and could be underneath the penumbral structure around sunspots. For an aspect ratio of 1, there is not enough space in the numerical domain for magnetic flux and convection to separate. In this case the solution oscillates between two steady states: two dominant convection cells threaded by magnetic field and one dominant cell that pushes magnetic flux towards the central axis

Universal bifurcation property of two- or higher-dimensional dissipative systems in parameter space: Why does 1D symbolic dynamics work so well?

The universal bifurcation property of the H\'enon map in parameter space is studied with symbolic dynamics. The universal-$L$ region is defined to characterize the bifurcation universality. It is found that the universal-$L$ region for relative small $L$ is not restricted to very small $b$ values. These results show that it is also a universal phenomenon that universal sequences with short period can be found in many nonlinear dissipative systems.Comment: 10 pages, figures can be obtained from the author, will appeared in J. Phys.

Fluctuation dynamo and turbulent induction at low magnetic Prandtl numbers

This paper is a detailed report on a programme of simulations used to settle a long-standing issue in the dynamo theory and demonstrate that the fluctuation dynamo exists in the limit of large magnetic Reynolds number Rm>>1 and small magnetic Prandtl number Pm<<1. The dependence of the critical Rm_c vs. the hydrodynamic Reynolds number Re is obtained for 1<Re<6700. In the limit Pm<<1, Rm_c is ~3 times larger than for Pm>1. The stability curve Rm_c(Re) (and, it is argued, the nature of the dynamo) is substantially different from the case of the simulations and liquid-metal experiments with a mean flow. It is not as yet possible to determine numerically whether the growth rate is ~Rm^{1/2} in the limit Re>>Rm>>1, as should be the case if the dynamo is driven by the inertial-range motions. The magnetic-energy spectrum in the low-Pm regime is qualitatively different from the Pm>1 case and appears to develop a negative spectral slope, although current resolutions are insufficient to determine its asymptotic form. At 1<Rm<Rm_c, the magnetic fluctuations induced via the tangling by turbulence of a weak mean field are investigated and the possibility of a k^{-1} spectrum above the resistive scale is examined. At low Rm<1, the induced fluctuations are well described by the quasistatic approximation; the k^{-11/3} spectrum is confirmed for the first time in direct numerical simulations.Comment: IoP latex, 27 pages, 25 figures, 3 tables. Accepted by New J. Physic

Quasiperiodic waves at the onset of zero Prandtl number convection with rotation

We show the possibility of quasiperiodic waves at the onset of thermal convection in a thin horizontal layer of slowly rotating zero-Prandtl number Boussinesq fluid confined between stress-free conducting boundaries. Two independent frequencies emerge due to an interaction between a stationary instability and a self-tuned wavy instability in presence of coriolis force, if Taylor number is raised above a critical value. Constructing a dynamical system for the hydrodynamical problem, the competition between the interacting instabilities is analyzed. The forward bifurcation from the conductive state is self-tuned.Comment: 9 pages of text (LaTex), 5 figures (Jpeg format

The Chemical Properties of Milky Way and M31 Globular Clusters: I. A Comparative Study

A comparative analysis is performed between high-quality integrated spectra of 30 globular clusters in M31, 20 Milky Way clusters, and a sample of field and cluster elliptical galaxies. We find that the Lick CN indices in the M31 and Galactic clusters are enhanced relative to the bulges of the Milky Way, M31, and elliptical spheroids. Although not seen in the Lick CN indices, the near-UV cyanogen feature (3883 A) is strongly enhanced in M31 clustesr with respect to the Galactic globulars at metallicities, --1.5<[Fe/H]<--0.3. Carbon shows signs of varying amongst these two groups. For [Fe/H]>--0.8, we observe no siginificant differences in the Hdelta, Hgamma, or Hbeta indices between the M31 and Galactic globulars. The sample of ellipticals lies offset from the loci of all the globulars in the Cyanogen--[MgFe], and Balmer--[MgFe] planes. Six of the M31 cluster spectra appear young, and are projected onto the M31 disk. Population synthesis models suggest that these are metal-rich clusters with ages 100--800 Myr, metallicities --0.20 < [Fe/H] <0.35, and masses 0.7 -7.0x10^4 Msun. Two other young clusters are Hubble V in NGC 205, and an older (~3 Gyr) cluster ~7 kpc away from the plane of the disk. The six clusters projected onto the disk rotate in a similar fashion to the HI gas in M31, and three clusters exhibit thin disk kinematics (Morrison et al.). Dynamical masses and structural parameters are required for these objects to determine whether they are massive open clusters or globular clusters. If the latter, our findings suggest globular clusters may trace the build up of galaxy disks. In either case, we conclude that these clusters are part of a young, metal-rich disk cluster system in M31, possibly as young as 1 Gyr old.Comment: 52 pages, 14 figures, 8 tables, minor revisions in response to referee, conclusions remain the same. Scheduled to appear in the October 2004 issue of The Astronomical Journa