Multiple equilibria in two-dimensional thermohaline circulation

1992 ◽  
Vol 241 ◽  
pp. 291-309 ◽  
Author(s):  
Paola Cessi ◽  
W. R. Young
Keyword(s):  
Multiple Equilibria ◽  
Thermohaline Circulation ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Amplitude Equation ◽  
Two Dimensional ◽  
Surface Salinity ◽  
Salinity Field ◽  
Small Perturbations ◽  
Variational Structure

As a model of the thermohaline circulation of the ocean we study the two-dimensional Boussinesq equations forced by prescribing the surface temperature and the surface salinity flux. We simplify the equations of motion using an expansion based on the small aspect ratio of the domain. The result is an amplitude equation governing the evolution of the depth averaged salinity field. This amplitude equation has multiple, linearly stable equilibria. The simplified dynamics has a Lyapunov functional and this variational structure permits a simple characterization of the relative stability of the alternative steady solutions.Even when the thermal and salinity surface forcing functions are symmetric about the equator there are asymmetric solutions, representing pole to pole circulations. These asymmetric solutions are stable to small perturbations and are always found in conjunction with symmetric solutions, also stable to small perturbations. Recent numerical solutions of the full two-dimensional equations have shown very similar flow patterns.

scholarly journals The Solution to the Task of Spreading the Non-Stationary Two-Dimensional Potential Flow (The Radial Source)

2020 ◽  
Vol 8 (2) ◽  
pp. 21-25
Author(s):  
Olga Burtseva ◽  
Viktor Kochanenko ◽  
Sergej Evtushenko ◽  
Anatoly Kondratenko
Keyword(s):  
Analytical Solution ◽  
Boundary Value Problem ◽  
Wave Front ◽  
Radial Flow ◽  
Equations Of Motion ◽  
Experimental Setup ◽  
Two Dimensional ◽  
Small Perturbations ◽  
Experimental Parameters ◽  
Good Agreement

The equations of motion of a non-stationary radial flow are derived, the boundary value problem is set, and its analytical solution is obtained. The solution of the problem in this paper is in good agreement with the experimental parameters obtained at the experimental setup for small perturbations. The equations for determining the height of the wave front that decreases downstream of the flow are obtained, and the instantaneous velocity of the wave front tends to zero.

scholarly journals PHASE PORTRAITS OF THE HENON-HEILES POTENTIAL

2018 ◽  
pp. 5-9
Author(s):  
E. Malkov ◽  
S. Momynov
Keyword(s):  
Numerical Integration ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Analytical Form ◽  
Phase Portraits ◽  
Two Dimensional ◽  
Poincaré Section ◽  
Dimensional Problem ◽  
Poincare Section

In this paper the Henon-Heiles potential is considered. In the second half of the 20th century, in astronomy the model of motion of stars in a cylindrically symmetric and time-independent potential was studied. Due to the symmetry of the potential, the three-dimensional problem reduces to a two-dimensional problem; nevertheless, finding the second integral of the obtained system in the analytical form turns out to be an unsolvable problem even for relatively simple polynomial potentials. In order to prove the existence of an unknown integral, the scientists Henon and Heiles carried out an analysis of research for trajectories in which the method of numerical integration of the equations of motion is used. The authors proposed the Hamiltonian of the system, which is fairly simple, which makes it easy to calculate trajectories, and is also complex enough that the resulting trajectories are far from trivial. At low energies, the Henon-Heiles system looks integrable, since independently of the initial conditions, the trajectories obtained with the help of numerical integration lie on two-dimensional surfaces, i.e. as if there existed a second independent integral. Equipotential curves, the momentum and coordinate dependences on time, and also the Poincaré section were obtained for this system. At the same time, with the increase in energy, many of these surfaces decay, which indicates the absence of the second integral. It is assumed that the obtained numerical results will serve as a basis for comparison with analytical solutions. Keywords: Henon-Heiles model, Poincaré section, numerical solutions.

Stationary fronts of the thermohaline circulation in the low-aspect-ratio limit

1997 ◽  
Vol 349 ◽  
pp. 117-147 ◽  
Author(s):  
LAURENCE FLEURY ◽  
OLIVIER THUAL
Keyword(s):  
Multiple Equilibria ◽  
Thermohaline Circulation ◽  
Numerical Solutions ◽  
Rectangular Domain ◽  
Topological Classification ◽  
Boussinesq Model ◽  
Asymptotic Equation ◽  
Thermohaline Convection ◽  
Ratio Limit ◽  
Shallow Basin

A two-dimensional Boussinesq model of the thermohaline convection in a rectangular domain is forced at the top by a prescribed temperature and a prescribed salinity flux. The two forcings have opposite effects on the density field, which leads to the formation of fronts and multiple equilibria. Numerical results are interpreted through a comparison with the solutions of an asymptotic equation, derived in the limit of a shallow basin by Cessi & Young (1992). In order to explain the discrepancies between the numerical and the asymptotic solutions, we extend this asymptotic approach through a geometrical representation and a topological classification of the surface forcings. By comparing three forcings, we propose a global picture which gives clues to interpret the numerical solutions.

Two-dimensional leaps in near-critical flow over isolated orography

2005 ◽  
Vol 461 (2064) ◽  
pp. 3747-3763 ◽  
Author(s):  
E.R Johnson ◽  
G.G Vilenski
Keyword(s):  
Unsteady Flow ◽  
Equations Of Motion ◽  
Fold Increase ◽  
Flow Speed ◽  
Two Dimensional ◽  
Subcritical Flow ◽  
One Dimensional ◽  
Petviashvili Equation ◽  
Lee Wave ◽  
Supercritical Flows

This paper describes steady two-dimensional disturbances forced on the interface of a two-layer fluid by flow over an isolated obstacle. The oncoming flow speed is close to the linear longwave speed and the layer densities, layer depths and obstacle height are chosen so that the equations of motion reduce to the forced two-dimensional Korteweg–de Vries equation with cubic nonlinearity, i.e. the forced extended Kadomtsev–Petviashvili equation. The distinctive feature noted here is the appearance in the far lee-wave wake behind obstacles in subcritical flow of a ‘table-top’ wave extending almost one-dimensionally for many obstacles widths across the flow. Numerical integrations show that the most important parameter determining whether this wave appears is the departure from criticality, with the wave appearing in slightly subcritical flows but being destroyed by two-dimensional effects behind even quite long ridges in even moderately subcritical flow. The wave appears after the flow has passed through a transition from subcritical to supercritical over the obstacle and its leading and trailing edges resemble dissipationless leaps standing in supercritical flow. Two-dimensional steady supercritical flows are related to one-dimensional unsteady flows with time in the unsteady flow associated with a slow cross-stream variable in the two-dimensional flows. Thus the wide cross-stream extent of the table-top wave appears to derive from the combination of its occurrence in a supercritical region embedded in the subcritical flow and the propagation without change of form of table-top waves in one-dimensional unsteady flow. The table-top wave here is associated with a resonant steepening of the transition above the obstacle and a consequent twelve-fold increase in drag. Remarkably, the table-top wave is generated equally strongly and extends laterally equally as far behind an axisymmetric obstacle as behind a ridge and so leads to subcritical flows differing significantly from linear predictions.

scholarly journals Modeling Transient Flows in Heterogeneous Layered Porous Media Using the Space–Time Trefftz Method

2021 ◽  
Vol 11 (8) ◽  
pp. 3421
Author(s):  
Cheng-Yu Ku ◽  
Li-Dan Hong ◽  
Chih-Yu Liu ◽  
Jing-En Xiao ◽  
Wei-Po Huang
Keyword(s):  
Porous Media ◽  
Time Domain ◽  
Numerical Solutions ◽  
Space Time ◽  
Two Dimensional ◽  
Transient Flows ◽  
Layered Porous Media ◽  
Time Marching ◽  
Time Basis ◽  
Marching Scheme

In this study, we developed a novel boundary-type meshless approach for dealing with two-dimensional transient flows in heterogeneous layered porous media. The novelty of the proposed method is that we derived the Trefftz space–time basis function for the two-dimensional diffusion equation in layered porous media in the space–time domain. The continuity conditions at the interface of the subdomains were satisfied in terms of the domain decomposition method. Numerical solutions were approximated based on the superposition principle utilizing the space–time basis functions of the governing equation. Using the space–time collocation scheme, the numerical solutions of the problem were solved with boundary and initial data assigned on the space–time boundaries, which combined spatial and temporal discretizations in the space–time manifold. Accordingly, the transient flows through the heterogeneous layered porous media in the space–time domain could be solved without using a time-marching scheme. Numerical examples and a convergence analysis were carried out to validate the accuracy and the stability of the method. The results illustrate that an excellent agreement with the analytical solution was obtained. Additionally, the proposed method was relatively simple because we only needed to deal with the boundary data, even for the problems in the heterogeneous layered porous media. Finally, when compared with the conventional time-marching scheme, highly accurate solutions were obtained and the error accumulation from the time-marching scheme was avoided.

EQUIVALENCE OF TWO-DIMENSIONAL GRAVITIES

1990 ◽  
Vol 05 (16) ◽  
pp. 1251-1258 ◽  
Author(s):  
NOUREDDINE MOHAMMEDI
Keyword(s):  
Gauge Theory ◽  
Explicit Solution ◽  
Light Cone ◽  
Equations Of Motion ◽  
Two Dimensional ◽  
Induced Gravity ◽  
Light Cone Gauge ◽  
Dimensional Gravity ◽  
The Relationship ◽  
Chern Simons

We find the relationship between the Jackiw-Teitelboim model of two-dimensional gravity and the SL (2, R) induced gravity. These are shown to be related to a two-dimensional gauge theory obtained by dimensionally reducing the Chern-Simons action of the 2+1 dimensional gravity. We present an explicit solution to the equations of motion of the auxiliary field of the Jackiw-Teitelboim model in the light-cone gauge. A renormalization of the cosmological constant is also given.

Transient Two-Dimensional Heat Conduction Problem With Partial Heating Near Corners

2016 ◽  
Author(s):  
Robert L. McMasters ◽  
Filippo de Monte ◽  
James V. Beck ◽  
Donald E. Amos
Keyword(s):  
Heat Conduction ◽  
Numerical Integration ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Thermal Protection ◽  
Heat Conduction Problem ◽  
Separation Of Variables ◽  
Two Dimensional ◽  
Rectangular Region ◽  
Long Distance

This paper provides a solution for two-dimensional heating over a rectangular region on a homogeneous plate. It has application to verification of numerical conduction codes as well as direct application for heating and cooling of electronic equipment. Additionally, it can be applied as a direct solution for the inverse heat conduction problem, most notably used in thermal protection systems for re-entry vehicles. The solutions used in this work are generated using Green’s functions. Two approaches are used which provide solutions for either semi-infinite plates or finite plates with isothermal conditions which are located a long distance from the heating. The methods are both efficient numerically and have extreme accuracy, which can be used to provide additional solution verification. The solutions have components that are shown to have physical significance. The extremely precise nature of analytical solutions allows them to be used as prime standards for their respective transient conduction cases. This extreme precision also allows an accurate calculation of heat flux by finite differences between two points of very close proximity which would not be possible with numerical solutions. This is particularly useful near heated surfaces and near corners. Similarly, sensitivity coefficients for parameter estimation problems can be calculated with extreme precision using this same technique. Another contribution of these solutions is the insight that they can bring. Important dimensionless groups are identified and their influence can be more readily seen than with numerical results. For linear problems, basic heating elements on plates, for example, can be solved to aid in understanding more complex cases. Furthermore these basic solutions can be superimposed both in time and space to obtain solutions for numerous other problems. This paper provides an analytical two-dimensional, transient solution for heating over a rectangular region on a homogeneous square plate. Several methods are available for the solution of such problems. One of the most common is the separation of variables (SOV) method. In the standard implementation of the SOV method, convergence can be slow and accuracy lacking. Another method of generating a solution to this problem makes use of time-partitioning which can produce accurate results. However, numerical integration may be required in these cases, which, in some ways, negates the advantages offered by the analytical solutions. The method given herein requires no numerical integration; it also exhibits exponential series convergence and can provide excellent accuracy. The procedure involves the derivation of previously-unknown simpler forms for the summations, in some cases by virtue of the use of algebraic components. Also, a mathematical identity given in this paper can be used for a variety of related problems.

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