Extended Boussinesq Model Research Articles

Computing interfacial flows of viscous fluids

We present a novel technique for solving for the flow of a viscous multi-fluid system in which interfaces are present. The Navier-Stokes equations are used in their exact form, without the need for phase-field approximations. A Boussinesq-type approach is invoked, in which the component fluids of differing densities are represented by a single fluid in which the density changes smoothly but rapidly across each narrow interfacial zone. This formulation is illustrated in detail for classical planar Rayleigh-Taylor flow containing two horizontal layers of fluid, in which the upper fluid has greater density. The interface between the fluids is therefore unstable and overturns as time progresses. The results of this new approach are compared against the predictions of classical Boussinesq theory and a recent Extended Boussinesq model proposed by the authors. Finally, a comparison with the predictions of a smoothed-particle-hydrodynamics model gives strong support for this new technique.

Numerical study on the performance of semicircular and rectangular submerged breakwaters

A systematic numerical comparative study of the performance of semicircular and rectangular submerged breakwaters interacting with solitary waves is the basis of this paper. To accomplish this task, Nwogu\'s extended Boussinesq model equations are employed to simulate the interaction of the wave with breakwaters. The finite difference technique has been used to discretize the spatial terms while a fourth-order predictor-corrector method is employed for time discretization in our numerical model. The proposed computational scheme uses a staggered-grid system where the first-order spatial derivatives have been discretized with fourth-order accuracy. For validation purposes, five test cases are considered and numerical results have been successfully compared with the existing analytical and experimental results. The performances of the rectangular and semicircular breakwaters have been examined in terms of the wave reflection, transmission, and dissipation coefficients (RTD coefficients) denoted by K_R, K_T, K_D. The latter coefficient K_D emerges due to the non-energy conserving K_R and K_T,. Our computational results and graphical illustrations show that the rectangular breakwater has higher reflection coefficients than semicircular breakwater for a fixed crest height, but as the wave height increases, the two reflection coefficients approach each other. On the other hand, the rectangular breakwater has larger dissipation coefficients compared to that of the semicircular breakwater and the difference between them increases as the height of the crest increases. However, the transmission coefficient for the semicircular breakwater is greater than that of the rectangular breakwater and the difference in their transmission coefficients increases with the crest height. Quantitatively, for rectangular breakwaters the reflection coefficients K_R are 5-15% higher while the diffusion coefficients K_D are 3-23% higher than that for the semicircular breakwaters, respectively. The transmission coefficients K_T for rectangular breakwater shows the better performance up to 2.47% than that for the semicircular breakwaters. Based on our computational results, one may conclude that the rectangular breakwater has a better overall performance than the semicircular breakwater. Although the model equations are non-dissipative, the non-energy conserving transmission and reflection coefficients due to wave-breakwater interactions lead to dissipation type contribution.

EXTENDED BOUSSINESQ EQUATIONS FOR WAVES IN TWO POROUS LAYERS

In this study we continue the work of Vu et al. (2018) [Coastal Eng. 139, 85-97] to develop an extended Boussinesq model that predicts the propagation of water waves in two porous layers. The first and second layer can be a water layer or a porous layer. The inertial and drag resistances are considered in the developed model. After being successfully validated against the analytical solutions, the model is used to simulate waves propagating over a submerged triangular porous bar. The numerical results show good agreement with the physical experimental data of Hsiao et al. (1998) [Proc. Royal Society of London A 458, 1291-1322].

Extended Boussinesq equations for waves in porous media

We develop an extended Boussinesq model to predict the propagation of waves in porous media. The inertial and drag resistances are taken into account in the developed model. When these resistances are neglected, the developed equations are reduced to the extended Boussinesq equations of Madsen and Sørensen (1992) [Coastal Eng. 18, 183–204] for non-porous media The model can be applied for non-porous media as well as porous media because it does not need any matching condition at the interface between the porous and non-porous media. The developed model is verified by comparing numerical solutions to experimental data with several cases of solitary waves or sinusoidal waves interacting with a porous breakwater in one- and two-dimensional domains.

EXTENDED BOUSSINESQ EQUATIONS FOR WAVES IN POROUS MEDIA: DERIVATION OF GOVERNING EQUATIONS AND GENERATION OF WAVES INTERNALLY

In this study we develop a new extended Boussinesq model that predicts the propagation of water waves in porous media. The inertial and drag resistances are taken account into the model in which the results are the same with the extended Boussinesq equations of Madsen and Sorensen (1992) when these resistances are removed. The developed model introduces its simplicity in solving the matching conditions at the permeable breakwater interfaces. The whole computational domain can be involved by specifying the porosity equal to unity outside the breakwater and to a value below unity inside the breakwater. There is no need for using any matching conditions at the interface. Furthermore, the applications of this current developed model are also extended to the cases that waves propagate inside and/or over a porous layer. For verification of the developed model, the internal generation of wave technique is applied to simulate sinusoidal and cnoidal waves propagating inside porous media in shallow and deep waters and nonlinear cnoidal waves interacting with porous breakwater. Numerical results give a good agreement with analytical solutions. Transformation of solitary waves to porous breakwater is also carried out. Refraction and transmission of solitary waves to the porous breakwater are well captured and verified by available physical experimental data

Numerical study of coastal sandbar migration, by hydro-morphodynamical coupling

We present a numerical model based on the hydro-morphodynamical coupling to study coastal sandbar migration. In order to improve both nonlinear and dispersive wave processes in relatively shallow water, we developed a finite element model based on the Legendre polynomials and on the Extended Boussinesq model. This model reproduces the propagation of wave trains with a high degree of accuracy on a greater range of depths than the standard Boussinesq models. We also implemented the Total Variation Diminishing schemes to improve the quality of the computed hydrodynamic fields, especially in areas where sharp flow gradients occurred. The coupled morpho-hydrodynamical model is then used to simulate the migration of real sandbars observed at Rousty beach (Mediterranean French coast). For verification the model results are compared with field measurements obtained from a small-scale field campaign carried out over two years at Rousty beach, and the results of this comparison are thoroughly discussed and analyzed.

Influences of lower-mantle properties on the formation of asthenosphere in oceanic upper mantle

Asthenosphere is a venerable concept based on geological intuition of Reginald Daly nearly 100 years ago. There have been various explanations for the existence of the asthenosphere. The concept of a plume-fed asthenosphere has been around for a few years due to the ideas put forth by Yamamoto et al.. Using a two-dimensional Cartesian code based on finite-volume method, we have investigated the influences of lower-mantle physical properties on the formation of a low-viscosity zone in the oceanic upper mantle in regions close to a large mantle upwelling. The rheological law is Newtonian and depends on both temperature and depth. An extended-Boussinesq model is assumed for the energetics and the olivine to spinel, the spinel to perovskite and perovskite to post-perovskite (ppv) phase transitions are considered. We have compared the differences in the behavior of hot upwellings passing through the transition zone in the mid-mantle for a variety of models, starting with constant physical properties in the lower-mantle and culminating with complex models which have the post-perovskite phase transition and depth-dependent coefficient of thermal expansion and thermal conductivity. We found that the formation of the asthenosphere in the upper mantle in the vicinity of large upwellings is facilitated in models where both depth-dependent thermal expansivity and conductivity are included. Models with constant thermal expansivity and thermal conductivity do not produce a hot low-viscosity zone, resembling the asthenosphere. We have also studied the influences of a cylindrical model and found similar results as the Cartesian model with the important difference that upper-mantle temperatures were much cooler than the Cartesian model by about 600 to 700 K. Our findings argue for the potentially important role played by lower-mantle material properties on the development of a plume-fed asthenosphere in the oceanic upper mantle.

Mars mantle convection: Influence of phase transitions with core cooling

The influence of phase transitions in the Martian mantle, the two exothermic phases, olivine to β‐spinel and β‐toγ‐spinel transitions, and the endothermic spinel to perovskite phase transition, has been studied with an axisymmetric spherical shell model. An extended Boussinesq model with a temperature and depth dependent viscosity, including the decay of radioactive elements and the cooling of the core has been used with two different sizes of the Martian core as we do not know its composition. In the case of a small core, the endothermic phase transition tends to inhibit the convective flow. Decreasing in depth with the core cooling, that phase transition can disappear after about 1 billion years, inducing a sudden increase of the heat flux out of the core who could explain a core dynamo reactivation. In the present work, we conclude that it is difficult to rejuvenate a dynamo, even in the presence of a thin perovskite layer at the bottom of the mantle. Core–mantle heat flux evolution is very sensitive to the onset of the whole mantle convection, which depends on initial conditions, but the endothermic phase prevents the core from cooling and allows melt generation in hot plumes. In large core case, uncertainties on the iron content of the mantle allow to compare models with two exothermic phase transitions or only a single thicker one. Our results demonstrate that these both configurations have same influence on the mantle convection and no degree one convection can be obtained as the core is cooling.

Modelling of Non-Linear Waves by An Extended Boussinesq Model

The main limitation of the most common form of the equations of Boussinesq is that they are valid only for relatively shallow depth of water. In the present study, an innovative approach (h − s) based on the finite elements method is presented to improve the dispersion relationship. This approach, based on the so-called extended Boussinesq model, reproduces with a high degree of accuracy the propagation of waves processes on a greater range of depths than the standard Boussinesq models.

The importance of radiative heat transfer on superplumes in the lower mantle with the new post-perovskite phase change

A new post-perovskite phase has been identified recently from both X-ray observations and first-principles calculations. This novel phase change occurs at a depth about 200 km above the core–mantle boundary for temperatures of around 2800 K. Using a two-dimensional, extended Boussinesq model, we have studied the dynamical consequences of this deep-mantle phase transition on mantle convection with particular emphasis on the effects on lower mantle plume structures. We have employed a depth-dependent viscosity with a viscosity maximum in the mid-lower mantle and two phase transitions, one at 670 km depth and the other at a depth of 2650 km. The phase transition at 670 km is endothermic, while we have examined both exothermic and endothermic possibilities for the new phase transition. Our results show the following situations favorable for the development of superplumes: (1) endothermic phase transition for both constant and radiative thermal conductivities and (2) exothermic phase transition and radiative thermal conductivity. Smaller unstable plumes are found for exothermic phase transition and constant thermal conductivity. Extremely high lateral temperature increases, exceeding 1500 degrees, are found inside the lower mantle plume for constant thermal conductivity. This lateral temperature contrast is mitigated by the presence of radiative thermal conductivity. In order for superplumes to prevail in the lower mantle with a deep exothermic phase change, we must invoke some form of radiative thermal conductivity for stabilizing and promoting large upwellings with a width exceeding at least 500 km. Strong lateral heterogeneities in both the seismic velocity and density fields can be produced in the plumes and the surrounding mantle for this new phase transition with a large Clapeyron slope, close to 10 MPa/K.

Incorporation of Wind Effects Into Boussinesq Wave Models

Recent advances in the Boussinesq modeling of nearshore hydrodynamics offer a platform for the study of wind effects on wave transformation and breaking-generated nearshore circulation. The paper documents: (1) the new parameterization of the momentum flux transferred from the wind to surface gravity waves in the coastal region on the basis of the field observations; (2) the implementation of the parameterized wind stress into phase-resolving Boussinesq wave models; (3) the development of empirical breaking criteria with the wind effect based on the existing laboratory data; and (4) the tests of the extended Boussinesq model against field observations and empirical results with respect to wind drag coefficients over shoaling waves and wave growth on a shallow lake. Fairly good agreement between the model results and measurements is observed. The methodology for the parameterization of the wind stress as a function of wave steepness and wind speed as well as the extended Boussinesq model incorporating the wind forcing can be used as a tool to improve our understanding of wind effects on nearshore wave propagation and horizontal circulation.

Linear analysis of a new type of extended Boussinesq model

An extended Boussinesq model is developed using a mathematical artifice that decomposes the velocity on a series of functions on the water depth. The aim of this paper is to present the theoretical bases of this finite elements model and the different series of functions available in this model with their respective physical range of validity. We then examine its linear frequency behaviour, its ability to reproduce the vertical variations of velocity and also its linear shoaling behaviour. A special attention is paid to the Legendre polynomials series. One of the advantages of this model is to implicitly improve its behaviour without any addition of higher order derivative terms and without any calibration parameters to fit theoretical curves. This improvement is made only by adding extra functions on the chosen series. Another advantage of this model is to implicitly include slope effect and evanescent modes. It can therefore deal with propagation of waves over relatively steep slopes.

Boussinesq Modeling of Wave Transformation, Breaking, and Runup. II: 2D

In this paper, we focus on the implementation and verification of an extended Boussinesq model for surf zone hydrodynamics in two horizontal dimensions. The time-domain numerical model is based on the fully nonlinear Boussinesq equations. As described in Part I of this two-part paper, the energy dissipation due to wave breaking is modeled by introducing an eddy viscosity term into the momentum equations, with the viscosity strongly localized on the front face of the breaking waves. Wave runup on the beach is simulated using a permeable-seabed technique. We apply the model to simulate two laboratory experiments in large wave basins. They are wave transformation and breaking over a submerged circular shoal and solitary wave runup on a conical island. Satisfactory agreement is found between the numerical results and the laboratory measurements.

Boussinesq Modeling of Wave Transformation, Breaking, and Runup. I: 1D

In this paper, we focus on the implementation and verification of an extended Boussinesq model for surf zone hydrodynamics in two horizontal dimensions. The time-domain numerical model is based on the fully nonlinear Boussinesq equations. As described in Part I of this two-part paper, the energy dissipation due to wave breaking is modeled by introducing an eddy viscosity term into the momentum equations, with the viscosity strongly localized on the front face of the breaking waves. Wave runup on the beach is simulated using a permeable-seabed technique. We apply the model to simulate two laboratory experiments in large wave basins. They are wave transformation and breaking over a submerged circular shoal and solitary wave runup on a conical island. Satisfactory agreement is found between the numerical results and the laboratory measurements.

Dynamical effects of a temperature- and pressure-dependent lower-mantle rheology on the interaction of upwellings with the transition zone

We have examined the consequences of a temperature- and pressure-dependent lower-mantle viscosity on the interaction of upwellings with the two major phase transitions. The viscosity used is only temperature-dependent in the upper-mantle but assumes both temperature- and pressure dependences in the lower-mantle. The activation enthalpy increases with depth dependence scaled after experimental melting curves of lower-mantle components. We have employed a two-dimensional cartesian box with an aspect-ratio of five and a mantle layer thickness of 2900 km, using an extended Boussinesq model with a depth-dependent thermal expansivity. Latent heat release and viscous dissipation have been included in the temperature equation. We have considered models both without and with a viscosity jump (a factor of 8) across the 660 km boundary. The range of averaged Rayleigh numbers (Ra) considered lies between 2 × 10 6 and 10 7. A major effect of this temperature- and pressure-dependent rheology is the stabilization of the upwelling plumes and the increase of viscous heating in the bottom portion of these plumes. The impingement of these heated plumes at the endothermic phase boundary causes the development of secondary plumes to rise into the upper-mantle. Shear-heating together with the latent heat release produces large temperature increases in the plumes passing through the transition zone, which may cause melting in the deep upper-mantle. Although the average Rayleigh number in the case of temperature- and pressure-dependent viscosity is around a factor of 5 lower than in a case with purely temperature-dependent viscosity, the resulting flow is only slightly less layered, which is contrary to the earlier finding of Christensen and Yuen (1985). The tendencies for the development of secondary plumes and high temperatures inside the upwellings are enhanced by the viscosity jump at the 660 km boundary. In this situation few nearly steady, vigorous upwellings exist in the lower mantle, while in the upper mantle vigorous time-dependent convection takes place. For an averaged Rayleigh number of 10 7, the system becomes layered with many thin plumes appearing in the lower mantle.

Phase transitions in the Martian mantle: Implications for the planet's volcanic history

The influences of phase transitions on Martian mantle convection and melting in the Martian mantle have been studied with an axisymmetric spherical‐shell model. An extended Boussinesq model in which viscous and adiabatic heatings are included has been used. There are depth dependences in the thermal expansivity and gravity, which, taken together, decrease by a factor of between 2 and 3 across the Martian mantle. The two destabilizing exothermic phase transitions, olivine to β‐spinel and β‐ and γ‐spinel transitions, above the Martian core‐mantle boundary (CMB) accelerate the mantle flow and result in an amplification and superheating of plumes in Mars. The additional consideration of the endothermic phase transition, spinel to perovskite transition, which was only likely present in the early evolution of the planet, has little influence on Martian mantle convection in this model. Strong localized viscous heating is generated near the CMB and underneath the lithosphere because here the flow bends over sharply. A possible volcanic evolution of Mars can be derived from a comparison of the temperature fields with the solidus of anhydrous peridotite if the cooling of the Martian mantle is taken into account. In the early evolution the mantle was molten to a high degree along the plume axis, which possibly resulted in a strong differentiation of the mantle. As the planet cooled down, the region of melt generation receded to where the maximum of viscous heating occurred: at the CMB and immediately underneath the lithosphere. Upon further cooling, the deep‐mantle melt source region became subsolidus. The most recent volcanism on Mars was most likely generated at a shallow depth below the lithosphere.

Multiple phase transitions and the style of mantle convection

Multiple phase transitions, based on experimental and theoretical results, have been incorporated into an extended Boussinesq model of mantle convection. A formulation is used which facilitates the numerical computations of convection with multiple phase transitions and a triple point near the 670 km discontinuity. The propensity for layered convection is enhanced by including the additional phase transitions. The location of the triple point of the spinel to perovskite phase transitions plays an important role in controlling the fate of the ascending hot plumes. A plume with a core temperature above this triple point temperature T c tends to pass easily through the phase boundary at 670 km. Plumes with a core temperature lower than T c are blocked by the multiple phase transitions. Both depth-dependent thermal expansivity and internal heating increase the tendency of the convective system to layering. The dynamical time-scales in the transition zone are much shorter than those of the top and bottom boundary layers. Simulations with increasing Rayleigh number (Ra) from 10 6 to 5 × 10 7 show unambiguously the increased tendency toward layering with larger Ra. In the early Earth, when Ra was higher and internal heating was several times stronger, layered convection might have been the preferred mode of mantle convection. With time, as both Ra and radioactive heating decreased, the style of mantle convection would have become less layered.