Periodic and aperiodic dynamo waves

Abstract

In order to show that aperiodic magnetic cycles, with Maunder minima, can occur naturally in nonlinear hydromagnetic dynamos, we have investigated a simple nonlinear model of an oscillatory stellar dynamo. The parametrized mean field equations in plane geometry have a Hopf bifurcation when the dynamo number D=1, leading to Parker's dynamo waves. Including the nonlinear interaction between the magnetic field and the velocity shear results in a system of seven coupled nonlinear differential equations. For D>1 there is an exact nonlinear solution, corresponding to periodic dynamo waves. In the regime described by a fifth order system of equations this solution remains stable for all D and the velocity shear is progressively reduced by the Lorentz force. In a regime described by a sixth order system, the solution becomes unstable and successive transitions lead to chaotic behaviour. Oscillations are aperiodic and modulated to give episodes of reduced activity.     

The unsteady plane flow of ice-sheets: A parabolic problem with two moving boundaries

Abstract

Finite difference algorithms have been developed to solve a one-dimensional non-linear parabolic equation with one or two moving boundaries and to analyse the unsteady plane flow of ice-sheets. They are designed to investigate the response of an ice-sheet to changes in climate, and to reconstruct climatic changes implied by past ice-sheet variations inferred from glacial geological data. Two algorithms are presented and compared. The first, a fixed domain method, replaces time as an independent variable with span. The grid interval in real space is kept constant, and thus the number of grid points changes with span. The second, a moving mesh method, retains time as one of the independent variables, but normalises the spatial variable relative to the span, which now enters the diffusion and advection coeficients in the parabolic equation for the surface profile.

Crank-Nicholson schemes for the solution of the equations are constructed, and iterative schemes for the solution of the resulting non-linear equations are considered.

Boundary (margin) motion is governed by the surface slope at the margin. Differentiation of the evolution equations results in an evolution equation for the margin slopes. It is shown that incorporation of this evolution equation, while not formally increasing the accuracy of the finite difference schemes, in practice increases accuracy of the solution.     

On the growth of disturbances to forced and dissipated barotropic flows

Abstract

We consider the growth of disturbances to large-scale zonally-asymmetric steady states in a truncated spectral model for forced and dissipated barotropic flow. A variant of the energy method is developed to optimize the instantaneous disturbance energy growth rate. The method involves solving a matrix eigenvalue problem amenable to standard numerical techniques. Two applications are discussed. (1) The global stability of a family of steady states is assessed in terms of the Ekman damping coefficient r. It is shown that monotonic global stability (i.e., every disturbances energy monotonically decays to zero) prevails when r≥r_c . (2) Initially fastest-growing disturbances are constructed in the r<r_c regime. Particular attention is paid to a subregion of the r<r_c regime where initially-growing disturbances exist despite stability with respect to normal modes. Nonlinear time-dependent simulations are performed in order to appraise the time evolution of various disturbances.     

A reduced model for nonlinear dispersive waves in a rotating environment

Abstract

The simplest model for geophysical flows is one layer of a constant density fluid with a free surface, where the fluid motions occur on a scale in which the Coriolis force is significant. In the linear shallow water limit, there are non-dispersive Kelvin waves, localized near a boundary or near the equator, and a large family of dispersive waves. We study weakly nonlinear and finite depth corrections to these waves, and derive a reduced system of equations governing the flow. For this system we find approximate solitary Kelvin waves, both for waves traveling along a boundary and along the equator. These waves induce jets perpendicular to their direction of propagation, which may have a role in mixing. We also derive an equivalent reduced system for the evolution of perturbations to a mean geostrophic flow.     

On open boundary conditions for three dimensional primitive equation ocean circulation models

Abstract

An open boundary condition is constructed for three dimensional primitive equation ocean circulation models. The boundary condition utilises dominant balances in the governing equations to assist calculations of variables at the boundary. The boundary condition can be used in two forms. Firstly as a passive one in which there is no forcing at the boundary and phenomena generated within the domain of interest can propagate outwards without distorting the interior. Secondly as an active condition where a model is forced by the boundary condition. Three simple idealised tests are performed to verify the open boundary condition, (1) a passive condition to test the outflow of free Kelvin waves, (2) an active condition during the spin up phase of an ocean, (3) finally an example of the use of the condition in a tropical ocean.     

A class of analytic solutions of the magnetic force-free field equations

Abstract

Formation of electric current sheets in the corona is thought to play an important role in solar flares, prominences and coronal heating. It is therefore of great interest to identify magnetic field geometries whose evolution leads to variations in B over small length-scales. This paper considers a uniform field B ₀[zcirc], line-tied to rigid plates z = ±l, which are then subject to in-plane displacements modeling the effect of photospheric motion. The force-free field equations are formulated in terms of field-line displacements, and when the imposed plate motion is a linear function of position, these reduce to a 4 × 4 system of nonlinear, second-order ordinary differential equations. Simple analytic solutions are derived for the cases of plate rotation and shear, which both tend to form singularities in certain parameter limits. In the case of plate shear there are two solution branches—a simple example of non-uniqueness.     

A wave theory for free spreading of buoyant fluids in a two-layer system

Abstract

A wave theory based on the non-linear shallow water equations for a two-layer fluid is constructed. The initial value conditions for the equations are the same as for the sudden release of a buoyant fluid. The theory has been tested in a series of experiments of lock-exchange type. Good agreement was found between prediction and experimental data for large times.     

Hydrodynamische Methoden in der Ozeanographie: Beiträge zur Physik der Bewegungsvorgänge im Meere

Summary The subject is treated, how far the events of motion in the sea can be reproduced by application of methods, which are based on the hydrodynamical differential equations. In particular comparisons between observed and computed sea-levels for the tides in estuaries and in the North-sea are worked out. Furtheron, a method is communicated, which is giving the shape of the sea-surface, when distribution of density is known, without using a layer of no motion. The result shows a remarkable agreement with the topography of the sea-surface in the South-Atlantic given byA. Defant.

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Aus einem Vortrag gehalten am 3. April 1959 auf der 7. Allgem. Versammlung der Società Italiana di Geofisica e Meteorologia (Genova: 3.–5. April 1959).     

Role of diffusive overturning in nonlinear axisymmetric convection in a differentially heated rotating annulus

Abstract

The “viscous overturning” mechanism, described in its simplest form by the linearized instability theory of the previous paper, is discussed in relation to certain numerical solutions recently obtained by G. P. Williams for steady thermally driven axisymmetric convective flow of water (Prandtl number = 7) in a rotating annulus differentially heated in the horizontal, in the “upper symmetric regime” parameter range. Viscous overturning plays an important and clearly identifiable role in the flows A3B, A4 and A5, which have free‐slip side walls and top surface, and a less clearcut role in A3 and B2, for which only the top surface is free. The discussion leads to various predictions about annulus flows not yet studied in detail.     

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.     

Inertial effects in an oceanic circulation model

Abstract

The flow in a mechanically driven thin barotropic rotating fluid system is analysed. The linear theory of Baker and Robinson (1969) is modified and extended into the non-linear regime.

An internal parameter, the “local Rossby number”, is indicative of the onset of nonlinear effects. If this parameter is 0(1) then inertial effects are as important as Coriolis accelerations in the interior of the transport-turning western boundary layer and both of its Ekman layers. The inertial effects in the Ekman layers, ignored in previous explorations of non-linear wind driven oceanic circulation, are retained here and calculated using an approximation of the Oseen type. The circulation problem is reduced to a system of scalar equations in only two independent variables; the system is valid for non-small local Rossby number provided only that the approximate total vorticity is positive.

To complete the solution for small Rossby number a boundary condition for the inertially induced transport is needed. It is found by examining the dynamics controlling this additional transport from the western boundary layer as the transport recirculates through the rest of the ocean basin. The strong constraint of total recirculation within the western boundary layer (zero net inertial transport) is derived.

The calculated primary inertial effects are in agreement with the observations of the laboratory model of Baker and Robinson (1969).

The analysis indicates the extent to which three-dimensional non-linear circulation can be reduced to a two dimensional problem.     

Hamiltonian description of almost geostrophic flow

Abstract

Geostrophic flow in the theory of a shallow rotating fluid is exactly analogous to the drift approximation in a strongly magnetized electrostatic plasma. This analogy is developed and exhibited in detailed to derive equations for the slow nearly geostrophic motion. The key ingredient in the theory is the isolation, to whatever order in Rossby number desired, of the fast motion near the inertial frequency. One of the remaining degrees of freedom represents a new approximate constant of the motion for nearly geostrophic flow. This is the analogue of the familiar magnetic moment adiabatic invariant in the plasma problem.

The procedure is a Rossby number expansion of the Hamiltonian for the fluid expressed in Lagrangian, rather than Eulerian variables. The fundamental Poisson brackets of the theory are not expanded so desirable properties such as energy conservation are maintained throughout.     

Convective dynamos with intermediate and strong fields

Abstract

The weak-field Benard-type dynamo treated by Soward is considered here at higher levels of the induced magnetic field. Two sources of instability are found to occur in the intermediate field regime M ～ T ^1/12, where M and T are the Hartmann and Taylor numbers. On the time scale of magnetic diffusion, solutions may blow up in finite time owing to destabilization of the convection by the magnetic field. On a faster time scale a dynamic instability related to MAC-wave instability can also occur. It is therefore concluded that the asymptotic structure of this dynamo is unstable to virtual increases in the magnetic field energy.

In an attempt to model stabilization of the dynamo in a strong-field regime we consider two approximations. In the first, a truncated expansion in three-dimensional plane waves is studied numerically. A second approach utilizes an ad hoc set of ordinary differential equations which contains many of the features of convection dynamos at all field energies. Both of these models exhibit temporal intermittency of the dynamo effect.     

Nonlinear internal waves in the upper atmosphere

To investigate the mechanism of mixing in oscillatory doubly diffusive (ODD) convection, we truncate the horizontal modal expansion of the Boussinesq equations to obtain a simplified model of the process. In the astrophysically interesting case with low Prandtl number (traditionally called semiconvection), large-scale shears are generated as in ordinary thermal convection. The interplay between the shear and the oscillatory convection produces intermittent overturning of the fluid with significant mixing. By contrast, in the parameter regime appropriate to sea water, large-scale flows are not generated by the convection. However, if such flows are imposed externally, intermittent overturning with enhanced mixing is observed.     

Stability of the subseismic wave equation for the Earth's fluid core

Abstract

The effects of compressibility on the stability of internal oscillations in the Earth's fluid core are examined in the context of the subseismic approximation for the equations of motion describing a rotating, stratified, self-gravitating, compressible fluid in a thick shell. It is shown that in the case of a bounded fluid the results are closely analogous to those derived under the Boussinesq approximation.     

Metamodel-based design of alluvial channels at incipient motion subjected to seepage

Abstract

The design of an alluvial channel affected by seepage requires information about five basic parameters: particle size, water depth, energy slope, seepage velocity, and average velocity. The conventional approach to predicting the incipient motion in an alluvial channel cannot be applied in the case of a channel affected by seepage. Metamodelling techniques are nowadays widely used in engineering design to simulate a complex system. Here, a metamodel is described which employs the radial-basis function (RBF) network to predict the seepage velocity and energy slope based on experimental data under incipient motion conditions. It was found that the model fits experimental data very well and provides predictions for the design. With the help of the metamodel generated by the RBF network, design curves based on the RBF metamodel are presented for use in designing an alluvial channel when it is affected by seepage.

Citation Kumar, B., Sreenivasulu, G. & Ramakrishna Rao, A. (2010) Metamodel-based design of alluvial channels at incipient motion subjected to seepage. Hydrol. Sci. J. 55(3), 459–466.     

Distinguishing the effects of climate on discharge in a tropical river highly impacted by large dams

Abstract

The hydrological regime of a river is defined by variables or representative curves that in turn have characteristics related to fluctuations in flow rates resulting from climate variability. Distinguishing between the causes of streamflow variations, i.e. those resulting from human intervention in the watershed and those due to climate variability, is not trivial. To discriminate the alterations resulting from climate variation from those due to regulation by dams, a reference hydrological regime was established using the classification of events based on mean annual streamflow anomalies and inferred climatic conditions. The applicability of this approach was demonstrated by analysis of the streamflow duration curves. An assessment of the hydrological regime in the lower reaches of the São Francisco River, Brazil, after the implementation of hydropower plants showed that the operation of the dams has been responsible for 59% of the hydrological changes, while the climate (in driest conditions) has contributed to 41% of the total changes.

Editor Z.W. Kundzewicz

Citation Genz, F. and Luz, L.D., 2012. Distinguishing the effects of climate on discharge in a tropical river highly impacted by large dams. Hydrological Sciences Journal, 57 (5), 1020–1034.     

Travelling wave convection in a rotating layer

Abstract

Small amplitude two-dimensional Boussinesq convection in a plane layer with stress-free boundaries rotating uniformly about the vertical is studied. A horizontally unbounded layer is modelled by periodic boundary conditions. When the centrifugal force is balanced by an appropriate pressure gradient the resulting equations are translation invariant, and overstable convection can take the form of travelling waves. In the Prandtl number regime 0.53 < [sgrave] < 0.68 such solutions are preferred over the more usual standing waves. For [sgrave] < 0.53, travelling waves are stable provided the Taylor number is sufficiently large.