The galactic dynamo: Axisymmetric and non-axisymmetric modes

Abstract

We discuss recent developments in the theory of large-scale magnetic structures in spiral galaxies. In addition to a review of galactic dynamo models developed for axisymmetric disks of variable thickness, we consider the possibility of dominance of non-axisymmetric magnetic modes in disks with weak deviations from axial symmetry. Difficulties of straightforward numerical simulation of galactic dynamos are discussed and asymptotic solutions of the dynamo equations relevant for galactic conditions are considered. Theoretical results are compared with observational data.     

A review of: “A perspective of physics,volume 2. Selections from 1977 comments on modern physics by sir Rudolf Peierls. ISBN 0 677 12400 7. Lib. Cong. Cat. Card 77-17657. xxxiv + 259 p. Price $30.00. Published by Gordon and Breach.”

Abstract

The thermally forced circulation of a stably stratified atmosphere in a valley is studied by aid of a simple numerical model. The model is based on the Boussinesq-equations for shallow convection. A diabatic heating is prescribed at slopes of the valley. To better understand the model's response to this heating the linearized basic equations are solved analytically and numerically for cases with highly idealized orography. The most conspicuous features of observed valley wind systems are represented in these solutions.

Next, numerical experiments with more complicated orography are described. The influence of the nonlinear terms and of the dissipative terms is considered. Various shapes of the valley and different localities of the heating are prescribed. It turns out that most of the computed features can be understood on the basis of the linear theory.     

Implementation of new time integration methods in POM

Instead of the standard leapfrog (SLF) scheme, an alternative leapfrog (ALF) scheme is used to solve the barotropic equations of the external mode in the Princeton Ocean Model (POM). The ALF scheme is modified in this study to deal with the nonlinear finite amplitude surface displacement. ALF has the advantage of improved numerical properties, longer time step relative to SLF, conservation of energy, and elimination of the Asselin filter. The numerical experiments of POM are implemented to show the above advantages. The split time stepping in 3D POM is found in this study to have numerical discrepancy due to the mismatched stepping between external and internal modes, and it results in a splitting error between the external and internal modes. A new split time stepping is therefore proposed. Numerical analysis indicates that there is no discrepancy with this split time stepping. The new split time stepping is implemented in the 3D POM. The numerical experiments demonstrate that the splitting error in POM can be reduced by three orders of magnitude relative to the original formulation, though the numerical error of the original formulation is already quite small.     

A practical numerical method for large strain liquefaction analysis of saturated soils

Accurate prediction of the liquefaction of saturated soils is based on strong coupling between the pore fluid phase and soil skeleton. A practical numerical method for large strain dynamic analysis of saturated soils is presented. The u–p formulation is used for the governing equations that describe the coupled problem in terms of soil skeleton displacement and excess pore pressure. A mixed finite element and finite difference scheme related to large strain analysis of saturated soils based on the updated Lagrangian method is given. The equilibrium equation of fluid-saturated soils is spatially discretized by the finite element method, whereas terms associated with excess pore pressure in the continuity equation are spatially discretized by the finite difference method. An effective cyclic elasto-plastic constitutive model is adopted to simulate the non-linear behavior of saturated soils under dynamic loading. Several numerical examples that include a saturated soil column and caisson-type quay wall are presented to verify the accuracy of the method and its usefulness and applicability to solutions of large strain liquefaction analysis of saturated soils in practical problems.     

A mathematical model for unsteady supercritical flow on a mobile sandy bed

ABSTRACT

Supercritical flow with sediment transport is a common phenomenon in steep rivers. This kind of flow presents features not present in rivers flowing in subcritical conditions. The development of a mathematical model to simulate supercritical flow with sediment transport in sandy rivers is described. The model is based on a numerical scheme for the integral version of the full governing equations and takes into account bottom configurations.     

An indirect boundary element method applied to simulate the seismic response of alluvial valleys for incident P,S and Rayleigh waves

A boundary integral formulation is presented and applied to model the ground motion on alluvial valleys under incident P, S and Rayleigh waves. It is based on integral representations for the diffracted and the refracted elastic waves using single-layer boundary sources. This approach is called indirect BEM in the literature as the sources' strengths should be obtained as an intermediate step. Boundary conditions lead to a system of integral equations for boundary sources. A discretization scheme based on the numerical and analytical integration of exact Green's functions for displacements and tractions is used. Various examples are given for two-dimensional problems of diffraction of elastic waves by soft elastic inclusion models of alluvial deposits in an elastic half-space. Results are displayed in both frequency and time domains. These results show the significant influence of locally generated surface waves in seismic response and suggest approximations of practical interest. For shallow alluvial valleys the response and its resonant frequencies are controlled by a coupling mechanism that involves both the simple one-dimensional shear beam model and the propagation of surface waves.     

A Godunov-Type Scheme for Atmospheric Flows on Unstructured Grids: Euler and Navier-Stokes Equations

In recent years there has been a growing interest in using Godunov-type methods for atmospheric flow problems. Godunov's unique approach to numerical modeling of fluid flow is characterized by introducing physical reasoning in the development of the numerical scheme (van Leer, 1999). The construction of the scheme itself is based upon the physical phenomenon described by the equation sets. These finite volume discretizations are conservative and have the ability to resolve regions of steep gradients accurately, thus avoiding dispersion errors in the solution. Positivity of scalars (an important factor when considering the transport of microphysical quantities) is also guaranteed by applying the total variation diminishing condition appropriately. This paper describes the implementation of a Godunov-type finite volume scheme based on unstructured adaptive grids for simulating flows on the meso-, micro- and urban-scales. The Harten-Lax-van Leer-Contact (HLLC) approximate Riemann solver used to calculate the Godunov fluxes is described in detail. The higher-order spatial accuracy is achieved via gradient reconstruction techniques after van Leer and the total variation diminishing condition is enforced with the aid of slope-limiters. A multi-stage explicit Runge-Kutta time marching scheme is used for maintaining higher-order accuracy in time. The scheme is conservative and exhibits minimal numerical dispersion and diffusion. The subgrid scale diffusion in the model is parameterized via the Smagorinsky-Lilly turbulence closure. The scheme uses a non-staggered mesh arrangement of variables (all quantities are cell-centered) and requires no explicit filtering for stability. A comparison with exact solutions shows that the scheme can resolve the different types of wave structures admitted by the atmospheric flow equation set. A qualitative evaluation for an idealized test case of convection in a neutral atmosphere is also presented. The scheme was able to simulate the onset of Kelvin-Helmholtz type instability and shows promise in simulating atmospheric flows characterized by sharp gradients without using explicit filtering for numerical stability.     

A posteriori local error estimation and adaptive time-stepping for newmark integration in dynamic analysis

A simple a posteriori local error estimate for Newmark time integration schemes in dynamic analysis is presented, based on the concept of a so called ‘post-processing’ technique. In conjunction with the error estimate, an adaptive time-stepping algorithm is described, which adjusts the time step size so that the local error of each time step is within a prescribed error tolerance. Numerical examples given in the paper indicate that the error estimate is asymptotically convergent, computationally efficient and convenient, and the adaptive time-stepping scheme can predict a nearly optimal step size from time to time, thus making the numerical solution reliable in an efficient manner.     

Baroclinic instability in ocean currents

Abstract

Unstable waves in a western boundary current are investigated in a full three-dimensional, numerical model. A numerical integration is carried out which traces the evolution of a growing wave on an initially uniform current with vertical shear. As indicated in earlier analytic studies based on simpler 2-layer models (Orlanski, 1969) the current is baroclinically unstable for the observed parameter range of the Gulf Stream.

Large meanders of the jet in the western boundary current are noticeable within 10 days. Finite amplitude effects, which can be investigated by the numerical model, reduce the growth rate of the disturbance by nearly an order of magnitude compared to linear theory. Comparison with observations indicate that the meanders of the Florida Current between Miami and Hatteras are probably baroclinically unstable waves.     

Towards a realistic theory of the geodynamo

Abstract

The physics of the geodynamo is discussed. The main processes relevant for the buoyancy driven geodynamo are isolated. The successive stages of development of geodynamo theory are briefly described. The mechanism of local turbulence in the Earth's core is explained, and an estimate is presented of the turbulent transport of density inhomogeneities in the Earth's core. The significance of this turbulent transport to the geodynamo mechanism is stressed. The general scheme of the complete geodynamo theory of the future is outlined.     

Methods of analyzing radio polarization data

Abstract

Methods of estimating the strength and direction of galactic magnetic fields from radio polarization measurements are reviewed. Particular attention is paid to the analysis of the Faraday rotation in order to derive the large scale magnetic field structure. Ways in which an axisymmetric spiral field structure can be observationally distinguished from a bisymmetric spiral structure are described, as are the ways in which field symmetric with respect to the galactic plane can be distinguished from those that are antisymmetric.     

The Numerical Modeling of 3-D Elastic Wave Equation Using a High-Order,Staggered-Grid, Finite Difference Scheme

This article provides the application of the high-order, staggered-grid, finite-difference scheme to model elastic wave propagation in 3-D isotropic media. Here, we use second-order, temporal-and high-order spatial finite-difference formulations with a staggered grid for discretization of the 3-D elastic wave equations of motion. The set of absorbing boundary conditions based on paraxial approximations of 3-D elastic wave equations are applied to the numerical boundaries. The trial resuits for the salt model show that the numerical dispersion is decreased to a minimum extent, the accuracy high and diffracted waves abundant. It also shows that this method can be used for modeling wave propagation in complex media with the lateral variation of velocity.     

Galactic dynamo models without sharp boundaries

Abstract

A new numerical approach is introduced which allows investigation into the conditions for dynamogeneration of axisymmetric and non-axisymmetric large-scale magnetic field modes in galaxy models which are defined by axisymmetric distributions of the α-parameter, the angular velocity and the electrical conductivity. The velocity field is assumed to be localized, however, the common assumption of a sharp boundary of the conducting region is dropped.

The possible anisotropy of the α-tensor is taken into account. The critical dynamo numbers (excitation conditions) for different modes are obtained by a direct method. The required steady states are attained by the use of an artificial non-linearity.

Initial test calculations demonstrate the efficacy of this new concept.     

Inversion of controlled‐source electromagnetic data using a model‐based approach

A two‐and‐half dimensional model‐based inversion algorithm for the reconstruction of geometry and conductivity of unknown regions using marine controlled‐source electromagnetic (CSEM) data is presented. In the model‐based inversion, the inversion domain is described by the so‐called regional conductivity model and both geometry and material parameters associated with this model are reconstructed in the inversion process. This method has the advantage of using a priori information such as the background conductivity distribution, structural information extracted from seismic and/or gravity measurements, and/or inversion results a priori derived from a pixel‐based inversion method. By incorporating this a priori information, the number of unknown parameters to be retrieved becomes significantly reduced. The inversion method is the regularized Gauss‐Newton minimization scheme. The robustness of the inversion is enhanced by adopting nonlinear constraints and applying a quadratic line search algorithm to the optimization process. We also introduce the adjoint formulation to calculate the Jacobian matrix with respect to the geometrical parameters. The model‐based inversion method is validated by using several numerical examples including the inversion of the Troll field data. These results show that the model‐based inversion method can quantitatively reconstruct the shapes and conductivities of reservoirs.     

Nonlinear wave propagation on a sphere: Interaction between Rossby waves and gravity waves; stability of the Rossby waves

Abstract

An analytical spectral model of the barotropic divergent equations on a sphere is developed using the potential-stream function formulation and the normal modes as basic functions. Explicit expressions of the coefficients of nonlinear interaction are obtained in the asymptotic case of a slowly rotating sphere, i.e. when the normal modes can be expressed as single spherical harmonics.     

A note on edge waves on a flat shelf

Abstract

A variational approximation to the dispersion relation for trapped waves on a flat shelf of depth h ₁, bounded internally by a vertical coast and externally by a semi-infinite ocean of depth h ₂>h ₁, is obtained through an integral-equation formulation that accounts for all of the non-propagated modes that are excited at the discontinuity in depth (the conventional formulation of the edge-wave problem allows only for the propagated mode on the shelf and the dominant, non-propagated mode in the deep water). Coriolis effects are neglected. The exact result in the limit ω² h ₂/g↓0 (ω = angular frequency) is obtained by conformal mapping and compared with the variational approximation, which proves to be quite accurate over the entire range 1>h ₂/h ₁>x. The effects of the higher-order, non-propagated modes are found to be small for the long waves observed over the Southern California shelf by Snodgrass, Munk and Miller (1962).     

The amplitude of turbulent shear flow

A new formulation of the problem of the statistical stability of fully turbulent shear flow is proposed, in which one seeks mean fields that bound the observed flow from the stable side. In the spirit of maximum transport theory, this formulation admits a larger set of “flows” than are dynamically possible. A sequence of constraints derived from the equations of motion can narrow this set, permitting at each step the determination of a “most stable” field free of any empirical elements. Turbulent channel flow is proposed as the first application and test of this quantitative theory. Past deductive theories for this flow, from “mean field” to “transport upper bounds,” are assessed. It is shown why these theories do not retain the significant destabilizing mechanisms of the actual flow. The implications for turbulent flow of recent work on the nonlinear and three-dimensional instability of laminar shearing flow are described. In first exploration of the “decoupled mean” stability theory proposed here, approximate analytical and numerical stability methods are used to find an amplitude and structure for the averaged flow propoerties. The quantitative results differ by considerably less than two from the observed values, providing an incentive for a more complete numerical study and for further constraints on the admitted class of flows. In the language now current for nonlinear stability theory, evidence is advanced here that anN-dimensional central manifold is adjacent to the realized turbulent flow, whereN has the largest possible value compatible with the dynamical relations.     

Solution of the nonlinear transport equation using modified Picard iteration

The transport and fate of reactive chemicals in groundwater is governed by equations which are often difficult to solve due to the nonlinear relationship between the solute concentrations for the liquid and solid phases. The nonlinearity may cause mass balance errors during the numerical simulation in addition to numerical errors for linear transport system. We have generalized the modified Picard iteration algorithm of Celia et al.⁵ for unsaturated flow to solve the nonlinear transport equation. Written in a ‘mixed-form’ formulation, the total solute concentration is expanded in a Taylor series with respect to the solution concentration to linearize the transport equation, which is then solved with a conventional finite element method. Numerical results of this mixed-form algorithm are compared with those obtained with the concentration-based scheme using conventional Picard iteration. In general, the new solver resulted in negligible mass balance errors (< ∥10⁻⁸∥%) and required less computational time than the conventional iteration scheme for the test examples, including transport involving highly nonlinear adsorption under steady-state as well as transient flow conditions. In contrast, mass balance errors resulting from the conventional Picard iteration method were higher than 10% for some highly nonlinear problems. Application of the modified Picard iteration scheme to solve the nonlinear transport equation may greatly reduce the mass balance errors and increase computational efficiency.