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.

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.

Multigrid Numerical Solutions of Non-Isothermal Laminar Recirculating Flows

1999 ◽  
Author(s):  
Marcelo J. S. de Lemos ◽  
Maximilian S. Mesquita
Keyword(s):  
Numerical Solutions ◽  
Rectangular Domain ◽  
Optimal Number ◽  
Computational Effort ◽  
Temperature Fields ◽  
Two Dimensional ◽  
Discretization Scheme ◽  
Multigrid Algorithm ◽  
Finite Volume Discretization ◽  
Velocity And Temperature Fields

Abstract The present work investigates the efficiency of the multigrid numerical method applied to solve two-dimensional laminar velocity and temperature fields inside a rectangular domain. Numerical analysis is based on the finite volume discretization scheme applied to structured orthogonal regular meshes. Performance of the correction storage (CS) multigrid algorithm is compared for different inlet Reynolds number (Rein) and number of grids. Up to four grids were used for both V- and W-cycles. Simultaneous and uncoupled temperature-velocity solution schemes were also applied. Advantages in using more than one grid is discussed. Results further indicate an increase in the computational effort for higher Rein and an optimal number of relaxation sweeps for both V- and W-cycles.

Sensitivity of the Atlantic Thermohaline Circulation and Its Stability to Basin-Scale Variations in Vertical Mixing

2006 ◽  
Vol 19 (21) ◽  
pp. 5467-5478 ◽  
Author(s):  
Willem P. Sijp ◽  
Matthew H. England
Keyword(s):  
North Atlantic ◽  
Multiple Equilibria ◽  
Thermohaline Circulation ◽  
Vertical Mixing ◽  
Intermediate Water ◽  
Stable States ◽  
Atlantic Basin ◽  
Interhemispheric Competition ◽  
Basin Scale ◽  
The Stability

Abstract This study shows that a reduction in vertical mixing applied inside the Atlantic basin can drastically increase North Atlantic Deep Water (NADW) stability with respect to freshwater perturbations applied to the North Atlantic. This is contrary to the notion that the stability of the ocean’s thermohaline circulation simply scales with vertical mixing rates. An Antarctic Intermediate Water (AAIW) reverse cell, reliant upon upwelling of cold AAIW into the Atlantic thermocline, is found to be associated with stable states where NADW is collapsed. Transitions between NADW “on” and “off” states are characterized by interhemispheric competition between this AAIW cell and the NADW cell. In contrast to the AAIW reverse cell, NADW eventually upwells outside the Atlantic basin and is thus not as sensitive to changes in vertical mixing within the Atlantic. A reduction in vertical mixing in the Atlantic weakens the AAIW reverse cell, resulting in an enhanced stability of NADW formation. The results also suggest that the AAIW reverse cell is responsible for the stability of NADW collapsed states, and thereby plays a key role in maintaining multiple equilibria in the climate system. A global increase of vertical mixing in the model results in significantly enhanced NADW stability, as found in previous studies. However, an enhancement of vertical mixing applied only inside the Atlantic Ocean results in a reduction of NADW stability. It is concluded that the stability of NADW formation to freshwater perturbations depends critically on the basin-scale distribution of vertical mixing in the world’s oceans.

Analytical approximate solutions for two-dimensional viscous flow through expanding or contracting gaps with permeable walls

Open Physics ◽  
2009 ◽  
Vol 7 (4) ◽  
Author(s):  
Saeed Dinarvand ◽  
Mohammad Rashidi ◽  
Ahmad Doosthoseini
Keyword(s):  
Viscous Flow ◽  
Numerical Solutions ◽  
Biological Fluids ◽  
Approximate Solutions ◽  
Normal Pressure ◽  
Rectangular Domain ◽  
Permeable Walls ◽  
Homotopy Analysis ◽  
Flow Through ◽  
Expanding Or Contracting Walls

AbstractIn this paper, the problem of laminar, isothermal, incompressible and viscous flow in a rectangular domain bounded by two moving porous walls, which enable the fluid to enter or exit during successive expansions or contractions is solved analytically by using the homotopy analysis method (HAM). Graphical results are presented to investigate the influence of the nondimensional wall dilation rate α and permeation Reynolds number Re on the velocity, normal pressure distribution and wall shear stress. The obtained solutions, in comparison with the numerical solutions, demonstrate remarkable accuracy. The present problem for slowly expanding or contracting walls with weak permeability is a simple model for the transport of biological fluids through contracting or expanding vessels.

scholarly journals Multiple equilibria in a simple elastocapillary system

2012 ◽  
Vol 712 ◽  
pp. 273-294 ◽  
Author(s):  
Michele Taroni ◽  
Dominic Vella
Keyword(s):  
Multiple Equilibria ◽  
Microelectromechanical Systems ◽  
Numerical Solutions ◽  
Multiple Equilibrium ◽  
Type Model ◽  
Initial State ◽  
Micro Channels ◽  
Simple Model System ◽  
Stable Equilibria ◽  
The Stability

AbstractWe consider the elastocapillary interaction of a liquid drop placed between two elastic beams, which are both clamped at one end to a rigid substrate. This is a simple model system relevant to the problem of surface-tension-induced collapse of flexible micro-channels that has been observed in the manufacture of microelectromechanical systems (MEMS). We determine the conditions under which the beams remain separated, touch at a point, or stick along a portion of their length. Surprisingly, we show that in many circumstances multiple equilibrium states are possible. We develop a lubrication-type model for the flow of liquid out of equilibrium and thereby investigate the stability of the multiple equilibria. We demonstrate that for given material properties two stable equilibria may exist, and show via numerical solutions of the dynamic model that it is the initial state of the system that determines which stable equilibrium is ultimately reached.

scholarly journals Ocean Gyres and Abrupt Change in the Thermohaline Circulation: A Conceptual Analysis

2005 ◽  
Vol 18 (13) ◽  
pp. 2403-2416 ◽  
Author(s):  
Hannah Longworth ◽  
Jochem Marotzke ◽  
Thomas F. Stocker
Keyword(s):  
High Latitude ◽  
General Circulation ◽  
Multiple Equilibria ◽  
Thermohaline Circulation ◽  
Abrupt Change ◽  
General Circulation Models ◽  
Water Vapor Transport ◽  
Bifurcation Structure ◽  
Conceptual Foundation ◽  
Freshwater Forcing

Abstract The implications are investigated of representing ocean gyre circulations by a diffusion term in the Stommel and Rooth box models of the thermohaline circulation (THC) in one and two hemispheres, respectively. The approach includes mostly analytical solution and study of the bifurcation structure, but also numerical integration and feedback analysis. Sufficient diffusion (gyre strength) eliminates multiple equilibria from either model, highlighting the need for accurate gyre circulation strength in general circulation models (GCMs) when considering the potential for abrupt climate change associated with THC shutdown. With diffusion, steady-state flow strength in the Rooth model depends on freshwater forcing (i.e., implied atmospheric water vapor transport) in both hemispheres, not only on that in the upwelling hemisphere, as in the nondiffusive case. With asymmetric freshwater forcing, two solutions (strong stable and weak unstable) are found with sinking in the hemisphere with stronger forcing and one solution with sinking in the other hemisphere. Under increased freshwater forcing the two solutions in the hemisphere with stronger forcing meet in a saddle-node bifurcation (if diffusion is sufficiently strong to prevent a subcritical Hopf bifurcation first), followed by flow reversal. Thus, the bifurcation structure with respect to freshwater forcing of the diffusive Rooth model of two-hemisphere THC is similar to that of the Stommel model of single-hemisphere THC, albeit with a very different dynamical interpretation. Gyre circulations stabilize high-latitude sinking in the Stommel model. In the Rooth model, gyre circulations only stabilize high-latitude sinking if the freshwater forcing is weaker in the sinking hemisphere than in the upwelling hemisphere, by an amount that increases with diffusion. The values of diffusion and freshwater forcing at which qualitative change in behavior occurs correspond to the range of the values used in and obtained with GCMs, suggesting that this analysis can provide a conceptual foundation for analyzing the stability of the interhemispheric THC, and also for the potential of the Atlantic THC to undergo abrupt change.

A Wind-Induced Thermohaline Circulation Hysteresis and Millennial Variability Regimes

2007 ◽  
Vol 37 (10) ◽  
pp. 2446-2457 ◽  
Author(s):  
Yosef Ashkenazy ◽  
Eli Tziperman
Keyword(s):  
Wind Stress ◽  
General Circulation ◽  
Multiple Equilibria ◽  
Thermohaline Circulation ◽  
Stress Amplitude ◽  
Circulation Model ◽  
Meridional Overturning Circulation ◽  
Box Model ◽  
Freshwater Flux ◽  
Relaxation Oscillations

Abstract The multiple equilibria of the thermohaline circulation (THC: used here in the sense of the meridional overturning circulation) as function of the surface freshwater flux has been studied intensively following a Stommel paper from 1961. It is shown here that multistability and hysteresis of the THC also exist when the wind stress amplitude is varied as a control parameter. Both the Massachusetts Institute of Technology ocean general circulation model (MITgcm) and a simple three-box model are used to study and explain different dynamical regimes of the THC and THC variability as a function of the wind stress amplitude. Starting with active winds and a thermally dominant thermohaline circulation state, the wind stress amplitude is slowly reduced to zero over a time period of ∼40 000 yr (40 kyr) and then increased again to its initial value over another ∼40 kyr. It is found that during the decreasing wind stress phase, the THC remains thermally dominant until very low wind stress amplitude at which pronounced Dansgaard–Oeschger-like THC relaxation oscillations are initiated. However, while the wind stress amplitude is increased, these relaxation oscillations are present up to significantly larger wind stress amplitude. The results of this study thus suggest that under the same wind stress amplitude, the THC can be either in a stable thermally dominant state or in a pronounced relaxation oscillations state. The simple box model analysis suggests that the observed hysteresis is due to the combination of the Stommel hysteresis and the Winton and Sarachik “deep decoupling” oscillations.

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