The rotational quenching of the rotation-induced kinetic alpha-effect

Abstract

A theory of the non-diffusive anisotropic kinetic alpha-effect (“Γ-effect”) for densitystratified rotating turbulent fluids is developed. No limitations on the rotation rate are imposed and the fully nonlinear dependence of the Γ-effect on the angular velocity is studied. When the Coriolis number, ω? = 2τ ω, is small the dimensionless “dynamo number”, Cτ, characterising the power of the Γ-effect, grows with ω?. The dependence, however, reaches a maximum for ω? ～ 2. For still higher rotation rates C_Λ decreases as 1/ω?. In opposition, the corresponding number, C_x, of the hydromagnetic α² -dynamo problems remains finite for very large ω?. Hence, for fast rotation the hydrodynamic Γ-effect is small while the hydromagnetic α-effect remains large. In consequence, the large-scale magnetic and velocity structures are expected to be generated with roughly equal power in slowly rotating objects. In the rapid rotators, however, generation of the large-scale flows is problematic.     

Hydrodynamic helicity in Boussinesq-type models of the geodynamo

The generation of hydrodynamic helicity is considered in models of thermal convection of a rapidly rotating incompressible liquid (small Rossby numbers). It is shown that a system of cyclonic vortices arising in geodynamo models generates a large-scale hydrodynamic helicity. The spatial distribution of the helicity is analyzed as a function of the intensity of heat sources in models with various boundary conditions for the velocity field.     

Generation and maintenance of bisymmetric spiral magnetic fields and interaction with spiral density waves

Abstract

The paper consists of two parts. The first introduces the dynamo equation into a rotating gaseous disk of finite thickness and then searches for its solution for the generation and maintenance of large-scale bisymmetric spiral (BSS) magnetic fields. We determine numerically the dynamo strength and vertical thickness of the gaseous disk which are necessary for the BSS magnetic fields to rotate as a wave over large area of the disk.

Next we present linearized equations of motion for the self-gravitating disk gas under the Lorentz force due to the BSS magnetic fields. Since the angular velocity of the BSS field is very close to that of the spiral density wave, a nearly-resonant interaction is caused between these two waves to produce large-amplitude condensation of gas in a double-spiral way. The BSS magnetic field is considered as a promising agency to trigger and maintain the spiral density wave.     

Unipolar induction as the origin of generation of electric fields and currents in plasmaspheres of planets

We study the generation of plasmaspheric electric fields and currents caused by differential rotation of inhomogeneously conducting plasma envelope together with a magnetized planet (a planetary generator model). The solution of the first considered model problem describing a plasmasphere flow with a discontinuity in the angular rotation velocity reveals that regions of strongly non-uniform rotation of a plasma envelope play an essential role in the generation of currents in the plasmasphere. Within the framework of the second problem we analyze the planetary generator operation under the conditions of the Earth's plasmasphere. The approximation of an abrupt decrease of the effective conductivity at ionospheric heights is used. The calculation results compared to experimental data show that the considered mechanism of current generation may be important when analyzing current systems in lower layers of the Earth's ionosphere.     

Baroclinic,Kelvin and inertia-gravity waves in the barostrat instability experiment

The differentially heated rotating annulus is a laboratory experiment historically designed for modelling large-scale features of the mid-latitude atmosphere. In the present study, we investigate a modified version of the classic baroclinic experiment in which a juxtaposition of convective and motionless stratified layers is created by introducing a vertical salt stratification. The thermal convective motions are suppressed in a central region at mid-depth of the rotating tank, therefore double-diffusive convection rolls can develop only in thin layers located at top and bottom, where the salt stratification is weakest. For high enough rotation rates, the baroclinic instability destabilises the flow in the top and the bottom shallow convective layers, generating cyclonic and anticyclonic eddies separated by the stable stratified layer. Thanks to this alternation of layers resembling the convective and radiative layers of stars, the planetary’s atmospheric troposphere and stratosphere or turbulent layers at the sea surface above stratified waters, this new laboratory setup is of interest for both astrophysics and geophysical sciences. More specifically, it allows to study the exchange of momentum and energy between the layers, primarily by the propagation of internal gravity waves (IGW). PIV velocity maps are used to describe the wavy flow pattern at different heights. Using a co-rotating laser and camera, the wave field is well resolved and different wave types can be found: baroclinic waves, Kelvin and Poincaré type waves. The signature of small-scale IGW can also be observed attached to the baroclinic jet. The baroclinic waves occur at the thin convectively active layer at the surface and the bottom of the tank, though decoupled they show different manifestation of nonlinear interactions. The inertial Kelvin and Poincaré waves seem to be mechanically forced. The small-scale wave trains attached to the meandering jet point to an imbalance of the large-scale flow. For the first time, the simultaneous occurrence of different wave types is reported in detail for a differentially heated rotating annulus experiment.     

The self-similar,non-linear evolution of rotating magnetic flux ropes

We study, in the ideal MHD approximation, the non-linear evolution of cylindrical magnetic flux tubes differentially rotating about their symmetry axis. Our force balance consists of inertial terms, which include the centrifugal force, the gradient of the axial magnetic pressure, the magnetic pinch force and the gradient of the gas pressure. We employ the “separable” class of self-similar magnetic fields, defined recently. Taking the gas to be a polytrope, we reduce the problem to a single, ordinary differential equation for the evolution function. In general, two regimes of evolution are possible; expansion and oscillation. We investigate the specific effect rotation has on these two modes of evolution. We focus on critical values of the flux rope parameters and show that rotation can suppress the oscillatory mode. We estimate the critical value of the angular velocity _crit, above which the magnetic flux rope always expands, regardless of the value of the initial energy. Studying small-amplitude oscillations of the rope, we find that torsional oscillations are superimposed on the rotation and that they have a frequency equal to that of the radial oscillations. By setting the axial component of the magnetic field to zero, we study small-amplitude oscillations of a rigidly rotating pinch. We find that the frequency of oscillation is inversely proportional to the angular velocity of rotation ; the product being proportional to the inverse square of the Alfvén time. The period of large-amplitude oscillations of a rotating flux rope of low beta increases exponentially with the energy of the equivalent 1D oscillator. With respect to large-amplitude oscillations of a non-rotating flux rope, the only change brought about by rotation is to introduce a multiplicative factor greater than unity, which further increases the period. This multiplicative factor depends on the ratio of the azimuthal speed to the Alfvén speed. Finally, considering interplanetary magnetic clouds as cylindrical flux ropes, we inquire whether they rotate. We find that at 1 AU only a minority do. We discuss data on two magnetic clouds where we interpret the presence in each of vortical plasma motion about the symmetry axis as a sign of rotation. Our estimates for the angular velocities suggest that the parameters of the two magnetic clouds are below critical values. The two clouds differ in many respects (such as age, bulk flow speed, size, handedness of the magnetic field, etc.), and we find that their rotational parameters reflect some of these differences, particularly the difference in age. In both clouds, a rough estimate of the radial electric field in the rigidly rotating core, calculated in a non-rotating frame, yields values of the order mV m⁻¹.     

MHD nature of the origination,dynamics, geoeffectiveness,and disappearance of the four-sector structure of the solar magnetic field during the cycle 23 decline phase

The MHD nature of the origination, dynamics, geoeffectiveness, and disappearance of the four-sector structure of the solar magnetic field during the cycle 23 decline phase has been established. A prolonged ordered MHD process including the chain of the interrelated phenomena (unknown before this study), which begin and end in one of the main zones of active longitudes and are responsible for the above nature of the four-sector structure, has been detected as a result of the simulation of the large-scale open solar magnetic field and an analysis of the dynamics of this field fluxes. These phenomena are as follows: the extreme concentration of the photospheric field sources of the same sign in the zone of active longitudes; blocking of regular differential rotation by these sources; origination of a nonstationary MHD disturbance in the form of a four-sector structure, traveling in the direction of solar rotation at a nearly Alfvén velocity; upset of blocking, displacement of blocking sources from the zone, and their shearing motion relative to a traveling MHD disturbance; deceleration and dissipation of a four-sector MHD disturbance; and reconstruction of a bisector structure. The interactions during this process, which lasted from May 2004 to December 2005, were accompanied by the generation of an ordered succession of heliospheric and solar-terrestrial disturbances including the series of nine extrastorms that were observed from July 2004 to September 2005 and were the last storms in the finished cycle 23 of solar activity.     

Hydrodynamical Modeling Of Oceanic Vortices

Mesoscale coherent vortices are numerous in the ocean.Though they possess various structures in temperature and salinity,they are all long-lived, fairly intense and mostly circular. Thephysical variable which best describes the rotation and the density anomaly associated with coherent vortices is potential vorticity. It is diagnostically related to velocity and pressure, when the vortex is stationary. Stationary vortices can be monopolar (circular or elliptical) or multipolar; their stability analysis shows thattransitions between the various stationary shapes are possible when they become unstable. But stable vortices can also undergo unsteady evolutions when perturbed by environmental effects, likelarge-scale shear or strain fields, -effect or topography. Changes in vortex shapes can also result from vortex interactions. such as the pairing, merger or vertical alignment of two vortices, which depend on their relative polarities and depths. Such interactions transfer energy and enstrophy between scales, and are essential in two-dimensional and in geostrophic turbulence. Finally, in relation with the observations, we describe a few mechanisms of vortex generation.     

Magnetized spherical body rotation in plasma

The problem of isotropic plasma flow near a rotating magnetized ideally conducting spherical body has been numerically solved. The methods of lines and shooting have been used to construct the numerical solution. The effect of super-rotation, when the angular velocity of plasma is higher than that of spherical body rotation, is observed near the surface of this body.     

Differential rotation set up by latitude-dependent heat transport

Abstract

Models of differentially rotating compressible deep spherical shells are computed according to the method of Belvedere and Paternò (1977): the heat transport is entirely convective, small-scale motions are parametrized by a thermal diffusivity and a kinematic viscosity, and the limit of slow rotation and large viscosity is considered.

In order to adapt the resulting differential rotation to the observed equatorial acceleration of the Sun, the heat transport must be more effective in the vicinity of the equator. In all models the latitude dependence of the transport coefficient induces meridional circulation in the form of a large cell, with rising material at high latitudes and sinking material near the equator. On top of this cell, one or two thin countercells develop in a minority of cases. Large pole-equator temperature differences and meridonal velocities at the surface are obtained when the Prandtl number is 1. But values of, say, 1/10 are sufficiently small to allow the models to be applied to the Sun. In general an angular velocity increasing with depth is found, and the surfaces of constant angular velocity are inclined towards greater depth and higher latitude.     

Broken ergodicity in magnetohydrodynamic turbulence

Turbulent magnetofluids appear in various geophysical and astrophysical contexts, in phenomena associated with planets, stars, galaxies and the universe itself. In many cases, large-scale magnetic fields are observed, though a better knowledge of magnetofluid turbulence is needed to more fully understand the dynamo processes that produce them. One approach is to develop the statistical mechanics of ideal (i.e. non-dissipative), incompressible, homogeneous magnetohydrodynamic (MHD) turbulence, known as “absolute equilibrium ensemble” theory, as far as possible by studying model systems with the goal of finding those aspects that survive the introduction of viscosity and resistivity. Here, we review the progress that has been made in this direction. We examine both three-dimensional (3-D) and two-dimensional (2-D) model systems based on discrete Fourier representations. The basic equations are those of incompressible MHD and may include the effects of rotation and/or a mean magnetic field B _o. Statistical predictions are that Fourier coefficients of the velocity and magnetic field are zero-mean random variables. However, this is not the case, in general, for we observe non-ergodic behavior in very long time computer simulations of ideal turbulence: low wavenumber Fourier modes that have relatively large means and small standard deviations, i.e. coherent structure. In particular, ergodicity appears strongly broken when B _o?=?0 and weakly broken when B _o?≠?0. Broken ergodicity in MHD turbulence is explained by an eigenanalysis of modal covariance matrices. This produces a set of modal eigenvalues inversely proportional to the expected energy of their associated eigenvariables. A large disparity in eigenvalues within the same mode (identified by wavevector k ) can occur at low values of wavenumber k?=?| k |, especially when B _o?=?0. This disparity breaks the ergodicity of eigenvariables with smallest eigenvalues (largest energies). This leads to coherent structure in models of ideal homogeneous MHD turbulence, which can occur at lowest values of wavenumber k for 3-D cases, and at either lowest or highest k for ideal 2-D magnetofluids. These ideal results appear relevant for unforced, decaying MHD turbulence, so that broken ergodicity effects in MHD turbulence survive dissipation. In comparison, we will also examine ideal hydrodynamic (HD) turbulence, which, in the 3-D case, will be seen to differ fundamentally from ideal MHD turbulence in that coherent structure due to broken ergodicity can only occur at maximum k in numerical simulations. However, a nonzero viscosity eliminates this ideal 3-D HD structure, so that unforced, decaying 3-D HD turbulence is expected to be ergodic. In summary, broken ergodicity in MHD turbulence leads to energetic, large-scale, quasistationary magnetic fields (coherent structures) in numerical models of bounded, turbulent magnetofluids. Thus, broken ergodicity provides a large-scale dynamo mechanism within computer models of homogeneous MHD turbulence. These results may help us to better understand the origin of global magnetic fields in astrophysical and geophysical objects.     

Small-scale multi-axial hybrid simulation of a shear-critical reinforced concrete frame

This study presents a numerical multi-scale simulation framework which is extended to accommodate hybrid simulation (numerical-experimental integration). The framework is enhanced with a standardized data exchange format and connected to a generalized controller interface program which facilitates communication with various types of laboratory equipment and testing configurations. A small-scale experimental program was conducted using a six degree-of-freedom hydraulic testing equipment to verify the proposed framework and provide additional data for small-scale testing of shearcritical reinforced concrete structures. The specimens were tested in a multi-axial hybrid simulation manner under a reversed cyclic loading condition simulating earthquake forces. The physical models were 1/3.23-scale representations of a beam and two columns. A mixed-type modelling technique was employed to analyze the remainder of the structures. The hybrid simulation results were compared against those obtained from a large-scale test and finite element analyses. The study found that if precautions are taken in preparing model materials and if the shear-related mechanisms are accurately considered in the numerical model, small-scale hybrid simulations can adequately simulate the behaviour of shear-critical structures. Although the findings of the study are promising, to draw general conclusions additional test data are required.     

Buoyancy-driven perturbations in a rapidly rotating,electrically conducting fluid: part I – flow and magnetic field

The steady velocity, perturbation pressure and perturbation magnetic field, driven by an isolated buoyant parcel of Gaussian shape in a rapidly rotating, unconfined, incompressible electrically conducting fluid in the presence of an imposed uniform magnetic field, are obtained by means of the Fourier transform in the limit of small Ekman number. Lorentz and inertial forces are neglected. The solution requires at most evaluation of a single integral and is found in closed form in some spatial regions. The solution has structure on two disparate scales: on the scale of the buoyant parcel and on the scale of the Taylor column, which is elongated in the direction of the rotation axis. The detailed structures of the flow and pressure depend linearly on the relative orientation of gravity and rotation, with the solution for arbitrary orientation being a linear combination of two limiting cases in which these vectors are colinear (polar case) and perpendicular (equatorial case). The perturbation magnetic field depends additionally on the relative orientation of the imposed magnetic field, and three limiting cases of interest are presented in which gravity and rotation are colinear (polar–toroidal case), gravity and imposed field are colinear (equatorial–radial case) and all three are mutually perpendicular (equatorial–toroidal case). Visualization and analysis of the velocity and perturbation magnetic field vectors are facilitated by dividing these vector fields into geostrophic and ageostrophic protions. In all cases, the geostrophic and ageostrophic portions have different structure on the Taylor-column scale. The buoyancy force is balanced by a pressure force in the polar case and by a flux of momentum in the equatorial case. The pressure force and momentum flux do not decay in strength with increasing axial distance. Far from the parcel, the axial mass flux varies as the inverse one-third power of distance from the parcel. The velocity has a single geostrophic vortex in the polar case and two vortices in the equatorial case. The perturbation magnetic field has two, four and one geostrophic vortices in the polar–toroidal, equatorial–radial and equatorial–toroidal cases, respectively. To facilitate comparison of the present results with numerical simulations carried out in a finite domain, a set of boundary conditions are developed, with may be applied at a finite distance from the parcel.     

Linear spin-up in a sliced cylinder

Abstract

spin-up and spin-down in a circular tank with a uniformly sloping bottom are studied experimentally and numerically for small values of the relative change in the angular velocity of the tank. Generally, the initial single-cell flow evolves into a number of smaller vortices. The evolution is compared with an analytical model based on an expansion of the flow field in linear Rossby waves (Pedlosky and Greenspan, 1967). Although it is possible to tune the experimental parameters in such a way that agreement with the theory is found, in most cases the experiments show shedding of vortices in the initial stage of the spin-up or spin-down, a phenomenon not described by the analytical model. Nonetheless, in such cases the analytical model still accounts for other observations: the alternating generation of cyclonic and anticyclonic vortices in the eastern part of the tank and their subsequent westward motion.     

Zonal jets and cyclone–anticyclone asymmetry in decaying rotating turbulence: laboratory experiments and numerical simulations

The problem of zonal jet formation and cyclone–anticyclone asymmetry in decaying rotating turbulence is addressed using both laboratory experiments and numerical simulations with a high-resolution shallow water model in a spherical geometry. Experiments are performed at different Rossby and Froude numbers and applying a rigid wall as meridional boundary in the numerical scheme to mimic the experimental apparatus. The formation of a zonally banded flow pattern, i.e. meridionally confined easterly/westerly jets, has observed; both experimental and numerical results confirmed that this tendency is favoured by high-planetary vorticity gradients. Also, in the experiments characterized by large rotation speeds and small Rossby deformation radius, an initial symmetric distribution of relative vorticity is found to evolve towards a dominance of anticyclonic structures, indicating a breaking of the cyclone–anticyclone symmetry. This aspect has deepened by numerically analysing the sensitivity of the temporal variations of the asymmetry index with respect to the position of the meridional confinement as well as the effect of relaxing the divergence of the fluid (i.e. non-divergent case) to zero. Results suggested that experiments characterized by the higher rotation speed and the lower fluid thickness are better reproduced by a divergent model with a high-latitude meridional boundary.     

On the nonlinear evolution of columnar vortices in a rotating environment

We examine the three-dimensional, nonlinear evolution of columnar vortices in a rotating environment. As the initial vorticity distribution, a wavetrain of finite amplitude Kelvin-Helmholtz vortices in shear is employed. Through direct numerical simulation of the Navier-Stokes equations we seek to better understand the process of maturation of the various three-dimensional modes of instability to which such vortical flows are subject, especially those which exist as a consequence of the action of the Coriolis force. In the absence of rotational influence, we thereby demonstrate that the nonlinear evolution of columnar vortices is most strongly controlled by one or the other of two mechanisms. One mechanism of instability is identifiable as a so-called elliptical instability, which promotes the initial bending of vortex tubes in a sinusoidal fashion, while the other is a hyperbolic mode, which is responsible for the development of streamwise vortex streaks in the "braids" between adjacent vortex cores. In the rotating case, anticyclonic vortices are strongly destabilized by weak background rotation, while rapid rotation stabilizes both the cyclones and anticyclones. The strong anticyclones are subject to two distinct forms of instability, namely a Coriolis force modified elliptical instability and an inertial (centrifugal) instability. The former instability is very similar to the nonrotating form of the elliptical instability as it promotes bending of vortex tubes, while the latter instability grows on the edge of the vortex core and generates streaks of vorticity, which surround the vortex core itself. These results of direct numerical simulation fully verify the results of previous linear stability analyses. Taken together, they provide a simple explanation for the broken symmetry that is often observed to be characteristic of the von Karman vortex streets that develop in the atmospheric lee of oceanic islands.     

Superconcentration of photospheric sources of the large-scale open solar magnetic field in the main zone of active longitudes, differential rotation blocking, and origination of the four-sector structure. 1. The field and solar activity dynamics

The phenomena of superconcentration of the large-scale field photospheric sources in the main zone of active longitudes, blocking of regular differential rotation by these sources, and origination of the four-sector structure of the solar magnetic field during the decline phase of cycle 23 have been considered in more detail and taking into account the polar correction. It has been indicated that superconcentration was formed due to the penetration of photospheric sources into the zone from the western surroundings of this zone and owing to the generation of the large-scale field in the zone itself. The dynamics of a blocking-induced complex MHD disturbance with reflected from the zone and reconnecting photospheric sources of negative and positive polarity, respectively, and the transformation of the bisector structure into the four-sector one have been considered. It has been indicated that the dynamics of this MHD disturbance was responsible for that of associated solar activity: the generation of sunspot groups, appearance of flares, and, finally, origination of a powerful heliospheric storm and the solar-terrestrial extrastorm of July 22–27, 2004.     

Problems of simulation of large,long-lived vortices in the atmospheres of the giant planets (jupiter,saturn, neptune)

Large, long-lived vortices are abundant in the atmospheres of the giant planets. Some of them survive a few orders of magnitude longer than the dispersive linear Rossby wave packets, e.g. the Great Red Spot (GRS), Little Red Spot (LRS) and White Ovals (WO) of Jupiter, Big Bertha, Brown Spot and Anne's Spot of Saturn, the Great Dark Spot (GDS) of Neptune, etc. Nonlinear effects which prevent their dispersion spreading are the main subject of our consideration. Particular emphasis is placed on determining the dynamical processes which may explain the remarkable properties of observed vortices such as anticyclonic rotation in preference to cyclonic one and the uniqueness of the GRS, the largest coherent vortex, along the perimeter of Jupiter at corresponding latitude.We review recent experimental and theoretical studies of steadily translating solitary Rossby vortices (anticyclones) in a rotating shallow fluid. Two-dimensional monopolar solitary vortices trap fluid which is transported westward. These dualistic structures appear to be vortices, on the one hand, and solitary waves, on the other hand. Owing to the presence of the trapped fluid, such solitary structures collide inelastically and have a memory of the initial disturbance which is responsible for the formation of the structure. As a consequence, they have no definite relationship between the amplitude and characteristic size. Their vortical properties are connected with geostrophic advection of local vorticity. Their solitary properties (nonspreading and stationary translation) are due to a balance between Rossby wave dispersion and nonlinear effects which allow the anticyclones, with an elevation of a free surface, to propagate faster than the linear waves, without a resonance with linear waves, i.e. without wave radiation. On the other hand, cyclones, with a depression of a free surface, are dispersive and nonstationary features. This asymmetry in dispersion-nonlinear properties of cyclones and anticyclones is thought to be one of the essential reasons for the observed predominance of anticyclones among the long-lived vortices in the atmospheres of the giant planets and also among the intrathermocline oceanic eddies.The effects of shear flows and differences between the properties of monopolar vortices in planetary flows and various laboratory experiments are discussed. General geostrophic (GG) theory of Rossby vortices is presented. It differs essentially from the traditional quasi-geostrophic (QG) and intermediate-geostrophic (IG) approximations by the account of (i) all scales between the deformation radius and the planetary scale and (ii) the arbitrary amplitudes of vortices. It is shown that, unlike QG- and IG-models, the GG-model allows for explaining the mentioned cyclonic-anticyclonic asymmetry not only in planetary flows, but also in laboratory modeling with vessels of near paraboloidal form.     

Shear instability of differential rotation in stars

Abstract

The two-dimensional (horizontal) shear instability of a differentially rotating star is examined. A solar-type rotation law is investigated. and it is found that for equatorial accelerations there is instability when there is a difference of 29% between the angular velocity of the equator and the poles.