Restoring Boundary Conditions Research Articles

Method of restoring boundary conditions on the high-temperature sphere surface during its cooling

The paper describes the method of restoring heat flux values on the surface of high-temperature nickel ball 45 mm in diameter at process of its intensive cooling by water in 1D and 2D formulation.

Scaling of the strength of the meridional overturning with vertical diffusivity in an idealized global geometry

An important expression of the non-linear character of the ocean’s meridional overturning circulation (MOC) is the scaling of its amplitude with the magnitude of the vertical mixing coefficient (diffusivity) of heat and salt. This paper extends recent work that indicated that the Atlantic and Pacific MOC exhibit different scaling behaviour. An idealized two-basin model configuration is used to study the meridional overturning circulation under restoring boundary conditions. In particular, the effects of wind forcing and the choice of the parametrization of lateral mixing are examined.Without wind, the scaling is similar in the two basins and consistent with theoretical predictions, provided that the diffusivity is small enough. Towards higher diffusivities the scaling of the overturning diverges.With non-zero wind, the sensitivity is strongly determined by the choice of the lateral mixing scheme. In case of traditional horizontal diffusion, the scaling behaviour is asymmetric due to spurious diapycnal mixing.With the Gent-McWilliams parametrization, the scaling is symmetric for the lower range of diffusivities, where results agree partly with theoretical scaling relations. In all cases considered, the pycnocline depth has the same sensitivity to diffusivity in both basins, implying that there is no universally valid relation between overturning strength and pycnocline depth.

Restoring boundary conditions in the solidification of pure metals

The scope of the paper is an algorithm reconstructing the boundary conditions (heat flux and heat transfer coefficient) in the solidification of pure metals on the grounds of temperature measurements. For the verification of the algorithm experimental data derived in the course of the solidification of aluminum were used. An example of the application of the algorithm for designating the cooling conditions in continuous casting is provided, when the values of temperature at selected points on the boundary of the casting are known.

The connection between Labrador Sea buoyancy loss, deep western boundary current strength, and Gulf Stream path in an ocean circulation model

The sensitivity of the North Atlantic gyre circulation to high latitude buoyancy forcing is explored in a global, non-eddy resolving ocean general circulation model. Increased buoyancy forcing strengthens the deep western boundary current, the northern recirculation gyre, and the North Atlantic Current, which leads to a more realistic Gulf Stream path. High latitude density fluxes and surface water mass transformation are strongly dependent on the choice of sea ice and salinity restoring boundary conditions. Coupling the ocean model to a prognostic sea ice model results in much greater buoyancy loss in the Labrador Sea compared to simulations in which the ocean is forced by prescribed sea ice boundary conditions. A comparison of bulk flux forced hindcast simulations which differ only in their sea ice and salinity restoring forcings reveals the effects of a mixed thermohaline boundary condition transport feedback whereby small, positive temperature and salinity anomalies in subpolar regions are amplified when the gyre spins up as a result of increased buoyancy loss and convection. The primary buoyancy flux effects of the sea ice which cause the simulations to diverge are ice melt, which is less physical in the diagnostic sea ice model, and insulation of the ocean, which is less physical with the prognostic sea ice model. Increased salinity restoring ensures a more realistic net winter buoyancy loss in the Labrador Sea, but it is found that improvements in the Gulf Stream simulation can only be achieved with the excessive buoyancy loss associated with weak salinity restoring.

Restoring boundary conditions in heat conduction

The restoration of boundary conditions in one-dimensional transient inverse heat-conduction problems (IHCP) is described. In the formulation, the boundary conditions are represented by linear relations between the temperature and the heat flux, together with an initial condition as a function of space. The temperature inside the solution domain, together with the space or time-dependent ambient temperature of the environment surrounding the heat conductor, are found from additional boundary-temperature or average boundary-temperature measurements. Numerical results obtained using the boundary-element method are presented and discussed.

중해상도 전지구 해양대순환 모형의 상층 수온과 혼합층 깊이 모사 성능 평가

We investigated seasonal variations of the upper ocean temperature and the mixed layer depth (MLD) in an eddy-permitting global ocean general circulation model (OGCM) to assess the OGCM perfermance. The OGCM is based on the GFDL MOM3 which has a horizontal resolution of 0.5 degree and 30 vertical levels. The OGCM was integrated for 68 years using a monthly-mean climatological wind stress forcing. The model sea surface temperature (SST) and sea surface salinity were restored to the Levitus climatology with a time scale of 30 days. Annual-mean model SST shows a cold bias <TEX>$(<\;-2^{\circ}C)$</TEX> in the summer hemisphere and a warm bias <TEX>$(>\;1^{\circ}C)$</TEX> in the winter hemisphere mainly due to the restoring boundary condition of temperature. The model MLD captures well the observed features in most areas, with a slightly deep bias. However, in the Ross Sea and Weddell Sea, the model shows significantly deeper MLD than the climatology-mainly due to weak salinity stratifications in the model. For amplitude of seasonal variation, the model SST is smaller <TEX>$(1{\sim}3^{\circ}C)$</TEX> than the observation largely due to the restoring surface boundary condition while the model MLD has larger seasonal variation <TEX>$({\sim}50m)$</TEX>. It is suggested that for more realistic simulation of the upper ocean structure in the present eddy-permitting ocean model, more refinements in the surface boundary condition for the thermohaline forcing and parameterization for vertical mixing are required, together with the incorporation of a sea-ice model.

The Different Nature of the Interdecadal Variability of the Thermohaline Circulation under Mixed and Flux Boundary Conditions

Abstract The differences between the interdecadal variability under mixed and constant flux boundary conditions are investigated using a coarse-resolution ocean model in an idealized flat-bottom single-hemisphere basin. Objective features are determined that allow one type of oscillation to be distinguished versus the other. First, by performing a linear stability analysis of the steady state obtained under restoring boundary conditions, it is shown that the interdecadal variability under constant flux and mixed boundary conditions arises, respectively, from the instability of a linear mode around the mean stratification and circulation and from departure from the initial state. Based on the budgets of density variance, it is shown next that the two types of oscillations have different energy sources: Under the constant-flux boundary condition (the thermal mode), the downgradient meridional eddy heat flux in the western boundary current regions sustains interdecadal variability, whereas under mixed boundary conditions (the salinity mode), a positive feedback between convective adjustment and restoring surface heat flux is at the heart of the existence of the decadal oscillation. Furthermore, the positive correlations between temperature and salinity anomalies in the forcing layer are shown to dominate the forcing of density variance. In addition, the vertical structure of perturbations reveals vertical phase lags at different depths in all tracer fields under constant flux, while under mixed boundary conditions only the temperature anomalies show a strong dipolar structure. The authors propose that these differences will allow one to identify which type of oscillation, if any, is at play in the more exhaustive climate models.

An Adjoint Analysis of the Meridional Overturning Circulation in an Ocean Model

Abstract Using the adjoint of a fully three-dimensional primitive equation ocean model in an idealized geometry, spatial variations in the sensitivity to surface boundary forcing of the meridional overturning circulation’s strength are studied. Steady-state sensitivities to diapycnal mixing, wind stress, freshwater, and heat forcing are examined. Three different, commonly used, boundary-forcing scenarios are studied, both with and without wind forcing. Almost identical circulation is achieved in each scenario, but the sensitivity patterns show major (quantitative and qualitative) differences. Sensitivities to surface forcing and diapycnal mixing are substantially larger under mixed boundary conditions, in which fluxes of freshwater and heat are supplemented by a temperature relaxation term or under flux boundary conditions, in which climatological fluxes alone drive the circulation, than under restoring boundary conditions. The sensitivity pattern to diapycnal mixing, which peaks in the Tropics is similar both with and without wind forcing. Wind does, however, increase the sensitivity to diapycnal mixing in the regions of Ekman upwelling and decreases it in the regions of Ekman downwelling. Wind stress in the Southern Oceans plays a crucial role in restoring boundary conditions, but the effect is largely absent under mixed or flux boundary conditions. The results highlight how critical a careful formulation of the surface forcing terms is to ensuring a proper response to changes in forcing in ocean models.

Effects of model resolution on salt transport through northern high-latitude passages and Atlantic meridional overturning circulation

We investigate the formation of dense water in the Greenland, Iceland, and Norwegian (GIN) Seas by focusing on salt balance between oceanic salt transport and sea surface flux. We conduct ocean general circulation model (OGCM) experiments under a sea surface salinity (SSS) restoring boundary condition with a horizontal resolution of 2° and those where the resolution is increased for northern high latitudes. A comparison of the salt transport at each passage and passive tracer distribution between the experiments reveals the following features: (1) in higher-resolution experiments, a stronger current at the Iceland–Scotland passage can carry saline water from the Atlantic Ocean to the GIN Seas and (2) higher resolution can represent the stronger West Spitsbergen Current that confines fresher water from the Arctic Ocean to the east coast of Greenland, thereby allowing saline water from the Atlantic Ocean to flow toward the Greenland Sea more efficiently. As the resolution increases, they contribute to a more realistic simulation of salinity distribution in the GIN Seas. These improvements are critically important for a realistic simulation of the Atlantic meridional overturning circulation (AMOC) when the model is forced without the SSS restoring condition. In OGCM experiments forced by climatological freshwater flux, the modeled AMOC in the experiment with the resolution of 2° is unrealistically weak because the inability to simulate the salt transport into the GIN Seas dramatically reduces the salinity in the GIN Seas and weakens deep water formation there. As the resolution increases, the AMOC becomes stronger due to a more realistic representation of ocean salt transport into the GIN Seas. These results demonstrate that the adoption of high resolution at least for the northern high latitudes facilitates realistic simulation of deep water formation in the GIN Seas and AMOC.

Thermohaline Circulation Stability: A Box Model Study. Part I: Uncoupled Model

Abstract A thorough analysis of the stability of the uncoupled Rooth interhemispheric three-box model of thermohaline circulation (THC) is presented. The model consists of a northern high-latitude box, a tropical box, and a southern high-latitude box, which correspond to the northern, tropical, and southern Atlantic Ocean, respectively. Restoring boundary conditions are adopted for the temperature variables, and flux boundary conditions are adopted for the salinity variables. This paper examines how the strength of THC changes when the system undergoes forcings that are analogous to those of global warming conditions by applying the equilibrium state perturbations to the moisture and heat fluxes into the three boxes. In each class of experiments, using suitably defined metrics, the authors determine the boundary dividing the set of forcing scenarios that lead the system to equilibria characterized by a THC pattern similar to the present one from those that drive the system to equilibria with a reversed THC. Fast increases in the moisture flux into the northern high-latitude box are more effective than slow increases in leading the THC to a breakdown, while the increases of moisture flux into the southern high-latitude box strongly inhibit the breakdown and can prevent it, as in the case of slow increases in the Northern Hemisphere. High rates of heat flux increase in the Northern Hemisphere destabilize the system more effectively than low ones; increases in the heat fluxes in the Southern Hemisphere tend to stabilize the system.

Reducing Errors in Temperature and Salinity in an Ocean Model Forced by Restoring Boundary Conditions*

Restoring boundary conditions, wherein the temperature and salinity are restored to surface target fields of temperature and salinity, are traditionally used for studies of the ocean circulation in ocean general circulation models. The canonical problem with these boundary conditions is that, when the target fields are chosen as the observed fields, accurate simulation of the surface fields of temperature and salinity would imply that the surface fluxes and therefore the ocean heat transports approach zero, a clearly unrealistic situation. It is clear that the target fields cannot be chosen as the observed fields. A simple but effective method of modifying conventional restoring boundary conditions is introduced, designed to keep the calculated values of surface temperature and salinity as close to observations as possible. The technique involves calculating the optimal target fields in the restoring boundary conditions by an iterative procedure. The method accounts for oceanic processes, such as advection and eddy mixing in the derivation of the new boundary conditions. A reduced version of this method is introduced that produces comparable results but offers greater simplicity in implementation. The simplicity of the method is particularly attractive in idealized studies, which often employ restoring surface boundary conditions. The success of the new method is, however, limited by several factors that cannot be easily compensated by the adjustment of the target profiles. These factors include inaccurate model dynamics, errors in the observations, and the too-simplified form of restoring surface boundary conditions themselves. The application of the method in this study with a coarse-resolution model leads to considerable improvements of the simulation of sea surface temperature (SST) and sea surface salinity (SSS). Both amplitude and phase of the annual cycle in SST greatly improve. The resulting magnitudes of surface heat and freshwater fluxes increase on average, and the meridional heat transport gets stronger. However, the fluxes in some regions remain unrealistic, notably the too-strong freshwater forcing of the western boundary currents in the Northern Hemisphere. Southern Ocean cooling and freshening are also likely to be too strong. The subsurface values of temperature improve greatly, proving that a large part of errors in the subsurface temperature distribution in our model can be corrected by reducing errors at the surface. In contrast, the reduction of errors in surface salinity fails to improve uniformly the simulated subsurface salinity values.

The Impact of Surface Flux Parameterizations on the Modeling of the North Atlantic Ocean

The response of an ocean general circulation model to several distinct parameterizations of the surface heat and freshwater fluxes, which differ primarily by their representation of the ocean–atmosphere feedbacks, is investigated in a realistic configuration for the North Atlantic Ocean. The impact of explicitly introducing oceanic information (climatological sea-surface temperature) into the computation of the heat flux through a Haney-type restoring boundary condition, as opposed to the case in which the flux is based on atmosphere-only climatologies and is computed with the full bulk formulation, is considered. The strong similarity between these two approaches is demonstrated, and the sources of possible differences are discussed. When restoring boundary conditions are applied to the surface salinity, however, an unphysical feedback mechanism is being introduced. The model's response to this restoring is contrasted to the response to a flux boundary condition that prescribes the freshwater flux derived from evaporation, precipitation, and river runoff climatologies (and therefore does not allow any feedback), as well as to the more realistic case in terms of the feedback parameterization, in which the dependence of evaporation on the model sea surface temperature is explicitly represented. Limited-area models introduce a further complicating factor for the thermodynamic adjustment, namely the representation of the oceanic heat and freshwater fluxes at the lateral boundaries. The degree to which the model solution is influenced by such fluxes, in combination with the different surface parameterizations, is also assessed. In all cases, the various components of the model's thermodynamic adjustment are considered, and the interdependence between the surface fluxes and the simulated sea surface temperature and surface salinity, their combined effect upon the ventilation of subsurface layers and production of different water masses, and their effect upon the simulated meridional heat and freshwater transports are analyzed.

Linear stability analysis of the three-dimensional thermally-driven ocean circulation: application to interdecadal oscillations

What can we learn from performing a linear stability analysis of the large-scale ocean circulation? Can we predict from the basic state the occurrence of interdecadal oscillations, such as might be found in a forward integration of the full equations of motion? If so, do the structure and period of the linearly unstable modes resemble those found in a forward integration? We pursue here a preliminary study of these questions for a case in idealized geometry, in which the full nonlinear behavior can also be explored through forward integrations. Specifically, we perform a three-dimensional linear stability analysis of the thermally-driven circulation of the planetary geostrophic equations. We examine the resulting eigenvalues and eigenfunctions, comparing them with the structure of the interdecadal oscillations found in the fully nonlinear model in various parameter regimes. We obtain a steady state by running the time-dependent, nonlinear model to equilibrium using restoring boundary conditions on surface temperature. If the surface heat fluxes are then diagnosed, and these values applied as constant flux boundary conditions, the nonlinear model switches into a state of perpetual, finite amplitude, interdecadal oscillations. We construct a linearized version of the model by empirically evaluating the tangent linear matrix at the steady state, under both restoring and constant-flux boundary conditions. An eigen-analysis shows there are no unstable eigenmodes of the linearized model with restoring conditions. In contrast, under constant flux conditions, we find a single unstable eigenmode that shows a striking resemblance to the fully-developed oscillations in terms of three-dimensional structure, period and growth rate. The mode may be damped through either surface restoring boundary conditions or sufficiently large horizontal tracer diffusion. The success of this simple numerical method in idealized geometry suggests applications in the study of the stability of the ocean circulation in more realistic configurations, and the possibility of predicting potential oceanic modes, even weakly damped, that might be excited by stochastic atmospheric forcing or mesoscale ocean eddies.

Simulation of ocean temperature change due to the opening of Drake Passage

An ocean general circulation model (GCM) is used to test the hypothesis that the opening of Drake Passage in the Oligocene may have lead to cooling in southern high latitudes. Results of previous studies show that no ocean models using restoring boundary conditions for the surface heat flux could reveal substantial southern high latitude cooling without prescribing atmospheric cooling a priori. To compute the surface heat flux that would be free from the restoring constraints we use another method, which is derived from an atmospheric energy balance model with lateral heat transport. In experiments conducted with both idealized and realistic topography the energy balance method produces substantial cooling not only in high latitudes but in deep waters as well.

Paleosalinity: confidence limits and future applications

Error propagation is investigated through the only commonly applied technique for reconstruction of paleosalinity, based on oxygen isotope residuals. Although this technique is known to be fairly inaccurate, there remained a need for thorough assessment of confidence limits in a variety of global settings. Here, it is found that a theoretical minimum error bar of between ±0.3‰ and 0.7‰ applies to paleosalinity estimates, and that real-life practical limits are rarely better than ±0.6‰. Best results are expected in the latitudinal ranges >50°N and >70°S, taking care to avoid the sea–ice margin. Confidence intervals at low and mid-latitudes (including marginal seas) are very wide, at least ±1.8‰ for the Mediterranean and ±2.2‰ for the equatorial Atlantic. It appears that oxygen isotope distributions are best used in their own right to investigate circulation and advective processes in coupled ocean–atmosphere–cryosphere models, instead of calibrating them to paleosalinity to define restoring boundary conditions. However, isotope residual-based paleosalinity maps from large numbers of (regionally averaged) studies in the above-mentioned latitudinal bands may help in defining initial conditions and first-order validation arguments for models.

Interdecadal Variability of the Thermohaline Circulation in Box-Ocean Models Forced by Fixed Surface Fluxes

Intrinsic modes of decadal variability are analyzed using box-geometry ocean models forced by constant surface fluxes. An extensive parameter sensitivity analysis of the oscillatory behavior is carried out with respect to the spherical/Cartesian geometry, the b effect, the Coriolis parameter, the parameterization of momentum dissipation and associated boundary conditions, the vertical and horizontal diffusivities, the convective adjustment algorithm, the horizontal and vertical model resolution, the forcing amplitude, and the basin width. The oscillations stand out as a robust geostrophic feature whose amplitude is mainly controlled by the horizontal diffusivity. Unsuccessful attempts to reproduce the variability in zonally averaged 2D models suggest that the 3D adjustment processes are necessary. However, the b effect is not necessary for decadal variability to occur, and therefore classical Rossby waves play no fundamental role in the mechanism. Various experiments with different geometry and forcing are conducted and do not support the necessity of viscous numerical boundary waves or any boundary in sustaining the oscillations. The models show two types of oscillatory behavior: 1) temperature anomalies propagating geostrophically westward in the eastward jet (northern part of the basin) and inducing an opposite anomaly in their wake and 2) stationary temperature anomalies in the northwest quarter that respond to the western boundary current transport changes and reinforce geostrophically this change until the opposite temperature anomaly built on the east finally reverses the meridional overturning anomaly. The analysis of the transition from steady to oscillatory states (using heat fluxes diagnosed at the equilibrium under restoring boundary conditions) and the comparison of the variability under various forcing fields suggest that the oscillations are triggered in the regions of strongest surface cooling. Finally, a simple box-model analogy is proposed that captures the crucial phase shift between meridional overturning and north‐south density gradient anomalies on these decadal timescales.

On the influence of the parameterization of lateral boundary layers on the thermohaline circulation in coarse-resolution ocean models

Because of the first order geostrophic balance in the ocean interior, the parameterization of lateral boundary layers has more influence than the parameterization of viscosity on the thermohaline overturning and the deep water properties in coarse-resolution ocean circulation models. Different formulations of momentum dissipation and associated boundary conditions are implemented within a planetary-geostrophic ocean circulation model for a Cartesian coordinate, flat-bottomed, β-plane, with restoring boundary conditions for the surface density and zero wind stress. Traditional Laplacian friction with a no-slip boundary condition produces an interior circulation in good agreement with geostrophy and the Sverdrup balance, but generates very large vertical (diapycnal) transports at lateral boundaries, especially upwelling in the western boundary current and downwelling in the northeast corner. The meridional and zonal overturning are thus enhanced, but drive to depth surface waters that are not as cold as the ones in the deep convection regions. Rayleigh friction with various frictional closures for the alongshore velocities within a no-normal-flow boundary condition framework efficiently reduces the diapycnal vertical transports along the boundaries, by allowing horizontal recirculation of geostrophic currents impinging into coasts. Hence, these parameterizations induce weaker overturnings, with colder deep water and a sharper thermocline resulting in higher poleward heat transports. We suggest that the upwelling along the boundaries is a consequence of the coarse-resolution dynamics and not only horizontal diffusion (termed the Veronis effect, horizontal diffusion produces large diapycnal fluxes once the isopycnals are tilted by coastal upwellings). Alternative parameterizations for the lateral boundary layers reduce this effect without the need for rotating the mixing tensor along isopycnals. This model comparison proves the need to clearly assess the extent of the diapycnal upwelling in the western boundary currents and to develop physically-based parameterizations of lateral boundary layers in order to improve coarse-resolution OGCMs.

Parameterization of density-driven downsloping flow for a coarse-resolution ocean model in z-coordinate

In the World Ocean, densest waters found on the continental shelves induce density drivendownsloping currents that can influence the deep ocean water masses properties. This processis poorly represented in z-coordinate ocean models, especially in Ocean General CirculationModel (OGCM) with coarse resolution in both horizontal and vertical directions. Consequently,continental shelves appear to be too isolated from the open ocean, whereas the density remainstoo low in the deep ocean. This study presents a simple parameterization of downsloping flowdesigned for z-coordinate, coarse resolution ocean model. At the shelf break, when the densityon the shelf is higher than that in the neighbouring deep water column, a downsloping currentis set up. This current is linearly related to the horizontal density gradient between the twoadjacent boxes, using a prescribed coefficient. For simplicity, a uniform value of the coefficientis used here, although it should ideally vary in space. From the shelf, the downsloping flow isassumed to go downward along the slope until it reaches a level of equal density. An upwardreturn flow of equal magnitude maintains the conservation of mass. This parameterization hasbeen implemented in an OGCM and two experiments, with and without this scheme, have beenintegrated until equilibrium using restoring boundary conditions. The impact of the downslopingparameterization on the global ocean is dominated by the improvement of the Antarcticbottom water circulation and water mass properties. The parameterization increases the densityof the deep ocean and tends to reduce the intensity and depth of the North Atlantic deep watercirculation, which is in better agreement with observations. As a result of a higher exchangewith the open ocean, the properties of continental shelf waters are also improved, with a markedreduction of the Antarctic shelves salinities. Therefore, this simple parameterization leads to asignificant improvement of the model results, at little computational cost.

Effects of convection instability due to incompatibility between ocean dynamics and surface forcings

Abstract. The study demonstrates that an incompatibility between a surface temperature climatology and a given ocean model, into which the climatology is assimilated via Haney restoration, can cause model ocean climate drift and interdecadal oscillations when the ocean is switched to a weaker restoration. This is made using an idealized Atlantic Ocean model driven by thermal and wind forcing only. Initially, the temperature climatology is forcefully assimilated into the model, and an implied heat flux field is diagnosed. During this stage any incompatibility is suppressed. The restoring boundary condition is then switched to a new forcing consisting of a part of the diagnosed flux and a part of the restoring forcing in such a way that at the moment of the switching the heat flux is identical to that prior to the switching. Under this new forcing condition, the incompatibility becomes manifest, causing changes in convection patterns, and producing drift and interdecadal oscillations. The mechanisms are described.

Effects of convection instability due to incompatibility between ocean dynamics and surface forcings

The study demonstrates that an incompatibility between a surface temperature climatology and a given ocean model, into which the climatology is assimilated via Haney restoration, can cause model ocean climate drift and interdecadal oscillations when the ocean is switched to a weaker restoration. This is made using an idealized Atlantic Ocean model driven by thermal and wind forcing only. Initially, the temperature climatology is forcefully assimilated into the model, and an implied heat flux field is diagnosed. During this stage any incompatibility is suppressed. The restoring boundary condition is then switched to a new forcing consisting of a part of the diagnosed flux and a part of the restoring forcing in such a way that at the moment of the switching the heat flux is identical to that prior to the switching. Under this new forcing condition, the incompatibility becomes manifest, causing changes in convection patterns, and producing drift and interdecadal oscillations. The mechanisms are described.