The nonlinear spin-up of a stratified ocean

Abstract

The adjustment of a nonlinear, quasigeostrophic, stratified ocean to an impulsively applied wind stress is investigated under the assumption that barotropic advection of vortex tube length is the most important nonlinearity. The present study complements the steady state theories which have recently appeared, and extends earlier, dissipationless, linear models.

In terms of Sverdrup transport, the equation for baroclinic evolution is a forced advection-diffusion equation. Solutions of this equation subject to a “tilted disk” Ekman divergence are obtained analytically for the case of no diffusion and numerically otherwise. The similarity between the present equation and that of a forced barotropic fluid with bottom topography is shown.

Barotropic flow, which is assumed to mature instantly, can reverse the tendency for westward propagation, and thus produce regions of closed geostrophic contours. Inside these regions, dissipation, or equivalently the eddy field, plays a central role. We assume that eddy mixing effects a lateral, down-gradient diffusion of potential vorticity; hence, within the closed geostrophic contours, our model approaches a state of uniform potential vorticity. The solutions also extend the steady-state theories, which require weak diffusion, by demonstrating that homogenization occurs for moderately strong diffusion.

The evoiution of potential vorticity and the thermocline are examined, and it is shown that the adjustment time of the model is governed by dissipation, rather than baroclinic wave propagation as in linear theories. If dissipation is weak, spin-up of a nonlinear ocean may take several times that predicted by linear models, which agrees with analyses of eddy-resolving general circulation models. The inclusion of a western boundary current may accelerate this process, although dissipation will still play a central role.     

A comparison of the asymptotic and no-z approximations for galactic dynamos

Abstract

The asymptotic and the no-z approximation methods of solving the axisymmetric mean field αΩ dynamo equation in a galactic disc are compared. The behaviour of the solutions is explored in both the linear and nonlinear regimes for a variety of dynamo parameters and two different rotation curves. The solutions obtained from the two different approaches are found to be in good agreement.     

On steady and travelling waves over bottom topography in the model of homogeneous flow in a beta-plane channel

Abstract

A spectral low-order model is proposed in order to investigate some effects of bottom corrugation on the dynamics of forced and free Rossby waves. The analysis of the interaction between the waves and the topographic modes in the linear version of the model shows that the natural frequencies lie between the corresponding Rossby wave frequencies for a flat bottom and those applying in the “topographic limit” when the beta-effect is zero. There is a possibility of standing or eastward-travelling free waves when the integrated topograhic effect exceeds the planetary beta-effect.

The nonlinear interactions between forced waves in the presence of topography and the beta-effect give rise to a steady dynamical mode correlated to the topographic mode. The periodic solution that includes this steady wave is stable when the forcing field moves to the West with relatively large phase speed. The energy of this solution may be transferred to the steady zonal shear flow if the spatial scale of this zonal mode exceeds the scale of the directly forced large-scale dynamical mode.     

Large-scale vortices in helical turbulence of incompressible fluid

Abstract

The interaction of a mean flow with a random fluctuation field is considered. This interaction is described by the averaged Navier-Stokes equation in which terms nonlinear in the fluctuation field are expressed in terms of the mean flow and the statistical properties of the fluctuation field, which is assumed to be homogeneous, isotropic, and helical. Averaged equations are derived using a functional technique. These equations are solved for a mean background flow that depends linearly on the position vector. The solutions show that large-scale vortices may arise in this system.     

Linear theory of rotating fluids using spherical harmonics part I: Steady flows

Abstract

It is shown that a systematic development of physical quantities using spherical harmonics provides analytical solutions to a whole class of linear problems of rotating fluids.

These solutions are regular throughout the whole domain of the fluid and are not much affected by the equatorial singularity of steady boundary layers in spherical geometries.

A comparison between this method and the one based on boundary layer theory is carried out in the case of the steady spin-up of a fluid inside a sphere.     

Nonlinear stability of baroclinic flows over topography

Abstract

A new nonlinear stability criterion is derived for baroclinic flows over topography in spherical geometry. The stability of a wide class of exact three-dimensional nonlinear steady state solutions subject to arbitrary disturbances is established. The resonance condition, at the highest total wavenumber, for the steady state solutions and the stability criteria for baroclinic flow in the absence of topography provide the boundaries of the regions of stability in the presence of topography. The analogous results for flow on periodic or infinite beta planes incorporating non-orthogonal function large scale flows are also discussed.     

Kinematic dynamo action with helical symmetry in an unbounded fluid conductor: Part 2. Further development of an explicit solution for the prototype case of lortz

Abstract

An explicit example of a steady prototype Lortz dynamo is elaborated in terms of a previously derived illustrative, exact, closed form solution to the nonlinear dynamo equations. The eigenvalue character of the dynamo problem is now introduced which simplifies the solution. The magnetic field lines, which lie on circular cylinders, and velocity streamline pattern are then displayed and discussed. Analysis of the magnetic energy balance by way of the Poynting flux reveals the existence of a finite critical cylinder across which zero net magnetic energy flows, thereby proving that the material inside is a self-excited dynamo, despite the fact that the total magnetic energy is unbounded.     

Energy spectra associated with long's model for steady finite-amplitude flow over orography

Abstract

The process of wave steepening in Long's model of steady, two-dimensional stably stratified flow over orography is examined. Under conditions of the long-wave approximation, and constant values of the background static stability and basic flow, Long's equation is cast into the form of a nonlinear advection equation. Spectral properties of this latter equation, which could be useful for the interpretation of data analyses under mountain wave conditions, are presented. The principal features, that apply at the onset of convective instability (density constant with height), are:

i) a power spectrum for available potential energy that exhibits a minus eight-thirds decay, in terms of the vertical wavenumber k _z ^-;

ii) a rate of energy transfer across the spectrum that is inversely proportional to the wavenumber for large k _z ^-;

iii) an equipartition between the kinetic energy of the horizontal motion and the available potential energy, under the longwave approximation, although all the disturbance energy is kinetic at the point where convective instability is initiated. It is also shown that features i) and ii) apply to more general conditions that are appropriate to Long's model, not just the long-wave approximation. Application to fully turbulent flow or to conditions at the onset of shearing instability are not considered to be warranted, since the development only applies to conditions at the onset of convective instability.     

On the asymptotic behavior of a sirnple stochastic-dynamic system

Abstract

The investigation is concerned with the impact of initial uncertainties on predictions. The problem can be solved exactly for sufficiently simple non-linear systems where an exact solution to the deterministic problem is known. In this paper we shall use the advective equation as an example.

It is found that the behavior at large times of the system depends on the initial uncertainty and the nature of the probability density function.

In applications it is normally necessary to introduce a closure approximation because exact analytical solutions are unknown. Such a closure scheme based on the neglect of third and higher moments will be used in the example and solutions from the closure scheme will be compared with the exact solutions.

It is found that the asymptotic values of the uncertainty may be less than the initial uncertainty.     

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.     

MÉTHODE D'ÉTUDE DES NAPPES LIBRES DANS LES AQUIFÈRES ANISOTROPES

Abstract

The necessary and sufficient conditions for non-zero phase shift and non-zero attenuation in linear flood routing can be derived from the continuity equation alone and are found to depend on the existence of an imaginary part in the expression for frequency or in the expression for wave number. It is shown that in linear flood routing the phase lag between flow rate and area of flow is directly related to the attenuation per unit wave length. The effects of using various forms of the momentum equation, in addition to the continuity equation, are exemplified by deriving analytical expressions in terms of the frequency, both for attenuation per unit channel length and for phase shift, for the kinematic wave, the general diffusion analogy, and the complete St. Venant equation.     

Taylor columns in horizontally sheared flow

Abstract

Solutions of the steady, inviscid, non-linear equations for the conservation of potential vorticity are presented for linearly sheared geostrophic flow over a right circular cylinder. The indeterminancy introduced by the presence of closed streamline regions is removed by requiring that the steady flow retains above topography a given fraction of that fluid initially present there, assuming the flow to have been started from rest. Those solutions which retain the largest fraction in uniform and negatively sheared streams satisfy the Ingersoll (1969) criterion (that, in the limit of vanishingly small viscosity, closed streamline regions are stagnant) and so are unaffected by Ekman pumping. These flows are set up on the advection time scale. In positively sheared flows the maximum retention solutions do not satisfy the Ingersoll criterion and thus would be slowly spun down on the far longer viscous spin-up time.

For arbitrary isolated topography, both the partial retention and Ingersoll problems are reduced to a one-dimensional non-linear integral equation and the solution of the Ingersoll problem obtained in the limit of strong positive shear. The stagnant region is symmetric about the zero velocity line and extends to infinity in the streamwise direction. Its cross-stream width is proportional to the rotation rate and fractional height occupied by the obstacle and inversely proportional to the strength of the shear, decreasing inversely as the square of distance upstream and downstream.     

Convection with heat flux prescribed on the boundaries of the system. I. The effect of temperature dependence of material properties

Abstract

We study the bifurcation to steady two-dimensional convection with the heat flux prescribed on the fluid boundaries. The fluid is weakly non-Boussinesq on account of a slight temperature dependence of its material properties. Using expansions in the spirit of shallow water theory based on the preference for large horizontal scales in fixed flux convection, we derive an evolution equation for the horizontal structure of convective cells. In the steady state, this reduces to a simple nonlinear ordinary differential equation. When the horizontal scales of the cells exceed a certain critical size, the bifurcation to steady convection is subcritical and the degree of subcriticality increases with increasing cell size.     

Magnetoconvection

Abstract

Cowling investigated the effect of an imposed magnetic field on convection in order to explain the origin of sunspots. After summarizing the classical linear theory of Boussinesq magnetoconvection, this review proceeds to more recent nonlinear results. Weakly nonlinear theory is used to establish the relevant bifurcation structure, which involves steady, oscillatory and chaotic solutions. Behaviour found in numerical experiments can then be related to these analytical results. Thereafter, attention is focused on the astrophysically relevant problem of fully compressible magnetoconvection. Steady two-dimensional nonlinear solutions show two important effects: stratification introduces an asymmetry between rising and falling fluid, while compressibility leads to evacuated magnetic flux sheets. Time-dependent behaviour includes transitions between standing waves and travelling waves, as well as changes in horizontal scale, leading to the development of more complicated spatial structures. Work on three-dimensional models, which is now in progress, will lead to a better understanding of the structure of a sunspot.     

Finite prandtl number convection in spherical shells

Abstract

Finite amplitude convection in spherical shells with spherically symmetric gravity and heat source distribution is considered. The nonlinear problem of three-dimensional convection in shells with stress-free and isothermal boundaries is solved by expanding the dependent variables in terms of powers of the amplitude of convection. The preferred mode of convection is determined by a stability analysis in which arbitrary infinitesimal disturbances are superimposed on the steady solutions. The shell is assumed to be thick and only shells for which the ratio ζ of outer radius to inner radius is 2 or 3 are considered. Three cases, two of which lead to a self adjoint problem, are treated in this paper. The stable solutions are found to be l=2 modes for ζ=3 where l is the degree of the spherical harmonics and an l=3 non-axisymmetric mode which exhibits the symmetry of a tetrahedron for ζ=2. These stable solutions transport the maximum amount of heat. The Prandtl number dependence of the heat transport is computed for the various solutions analyzed in the paper.     

A note on the howard-Malkus-Whitehead floating heat sources

Abstract

Some deficiencies in a recent paper by Howard, Malkus and Whitehead are examined. The problem is reformulated in terms of an integro-differential equation, from which both asymptotic and numerical solutions are obtained.     

Rossby wave resonance in the presence of a nonlinear critical layer

Abstract

The behavior of Rossby waves on a shear flow in the presence of a nonlinear critical layer is studied, with particular emphasis on the role played by the critical layer in a Rossby wave resonance mechanism. Previous steady analyses are extended to the resonant case and it is found that the forced wave dominates the solution, provided the flow configuration is not resonant for the higher harmonics induced by the critical layer. Numerical simulations for the forced initial value problem show that the solution evolves towards the analysed steady state when conditions are resonant for the forced wave, and demonstrate some of the complications that arise when they are resonant for higher harmonics. In relating the initial value and steady problems, it is argued that the time dependent solution does not require the large mean flow distortion that Haberman (1972) found to be necessary outside the critical layer in the steady case.     

UNSTEADY MOVEMENT OF FRESH WATER IN THICK UNCONFINED SALINE AQUIFERS

Abstract

Fresh-water lenses are formed in unconfined saline aquifers in response to deep percolation from rainfall, artificial recharge, and seepage from irrigation waters and/or in response to injecting fresh water through vertical or horizontal wells. An approximate differential equation is derived in terms of the depth of the fresh-salt water interface below the initial position of the saline-water table. This equation is analogous to that of the ground-water motion in two dimensions. The wealth of knowledge available from solving the latter equation is used to obtain approximate expressions for the movement of the fresh-salt water interface in several flow systems wherein this interface does not reach the bottom of the aquifer. These approximate solutions as well as others for related quantities of interest may afford useful tools for rationally planning the extraction of usable waters from such flow systems.     

THEORETICAL APPROACH TO THE SOLUTION OF THE INFILTRATION PROBLEM

ABSTRACT

The one-dimensional transient downward entry of water in unsaturated soils is investigated theoretically. The mathematical equation describing the infiltration process is derived by combining Darcy's dynamic equation of motion with the continuity and thermodynamic state equations adjusted for the unsaturated flow conditions. The resulting equation together with the corresponding initial and boundary conditions constitues a mathematical initial boundary value problem requiring the solution of a nonlinear partial differential equation of the parabolic type. The volumetric water content is taken as the dependent variable and the time and the position along the vertical direction are taken as the independent variables. The governing equation is of such nature that a solution exists for t > 0 and is uniquely determined if two relationships are defined, together with the specified state of the system, at the initial time t = 0 and at the two boundaries. The two required relations are those of pressure versus permeability and pressure versus volumetric water content.

Since the partial differential equation has strong non-linear terms, a discrete solution is obtained by approximating the derivatives with finite-differences at discrete mesh points in the solution domain and integrated for the corresponding initial and boundary conditions. The use of an implicit difference scheme is employed in order to generate a system of simultaneous non-linear equations that has to be solved for each time increment. For n mesh points the two boundary conditions provide two equations and the repetition of the recurrence formula provides n—2 equations, the total being n equations for each time increment. The solution of the system is obtained by matrix inversion and particularly with a back-substitution technique. The FORTRAN statements used for obtaining the solution with an electronic digital computer (IBM 704) are presented together with the input data.

Analysis of the errors involved in the numerical solution is made and the stability and convergence of the solution of the approximate difference equation to that of the differential equation is investigated. The method applied is that of making a Fourier series expansion of a whole line of errors and then following the progress of the general term of the series expansion and also the behavior of each constituent harmonic. The errors (forming a continuous function of points in an abstract Banach space) are represented by vectors with the Fourier coefficients constituting a second Banach space. The amplification factor of the difference equation is shown to be always less than unity which guarantees the stability of the employed implicit recurrence scheme.

Experiments conducted on a vertical column packed uniformly with very fine sand, show a satisfactory agreement between the theoretically and experimentally obtained values. Many experimental results are shown in an attempt to explain the infiltration phenomenon with emphasis on the shape and movement of the wet front, and the effects of the degree of compaction, initial water content and deaired water on the infiltration rate.