Depth-dependent Viscosity Research Articles

Global nonlinear stability of bidispersive porous convection with throughflow and depth-dependent viscosity

The linear instability and the nonlinear stability analyses have been performed to examine the combined impact of a uniform vertical throughflow and a depth-dependent viscosity on bidispersive porous convection using the Darcy theory with a single temperature field. The validity of the principle of exchange of stability is proved. The eigenvalue problems resulting from both linear instability and nonlinear stability analyses with variable coefficients are numerically solved using the Chebyshev pseudo-spectral method. The equivalence of linear instability and nonlinear stability boundaries is established in the absence of throughflow, while in its presence, the subcritical instability is shown to be evident. The stability of the system is independent of the direction of throughflow in the case of constant viscosity, whereas upflow is found to be more stabilizing than downflow when the viscosity is varying with depth. While the viscosity parameter offers a destabilizing influence on the onset of convection in the absence of throughflow, it imparts both stabilizing and destabilizing effects on the same in its presence. The influence of the ratio of permeabilities and the interphase momentum transfer parameter is to make the system more stable. The findings of a mono-disperse porous medium are presented as a specific case within the broader context of this investigation.

Significance of geothermal viscosity for the magnetic fluid flow between co-rotating porous surfaces

Considering the Navier Stokes equations paired with the heat equation for ferrohydrodynamic flow of magnetic nanofluid amidst two rotating porous disks, nothing is known on the significance of geothermal viscosity for the magnetic fluid flow between co-rotating porous surfaces. The current analysis considers the effects of changing viscosity and thermal radiation. Additionally, the stretching and angular velocities of the two disks differ. The flow model’s estimated basic equations are transformed into nondimensional ordinary differential equations (ODEs) with the proper transformation before being numerically solved using Maple’s built-in BVP Midrich approach. For velocity and temperature fields, the effects of active flow parameters such as the depth-dependent viscosity variation parameter, ferro-hydrodynamics interaction parameter, Reynolds number, Prandtl number, Darcy number, Eckert number, and radiation parameter are discussed. It is worth noticing that the temperature field tends to decline as the Darcy parameter is higher estimated. It is worth concluding that there exists a high fluctuating viscosity around the lower disk, the magnitude of horizontal velocity decreased, but vertical velocity increased everywhere else. Furthermore, skin friction becomes more valuable at higher Reynolds numbers and porosity estimates.

Natural convection in a fluid saturating an anisotropic porous medium in LTNE: effect of depth-dependent viscosity

Thermal convection in a fluid saturating an anisotropic porous medium in local thermal nonequilibrium (LTNE) is investigated, with specific attention to the effect of variable viscosity on the onset of convection. Many fluids show a remarkable dependence of viscosity on temperature that cannot be neglected. For this reason, we take into account a fluid whose viscosity decreases exponentially with depth, according to Straughan (Acta Mech. 61:59–72, 1986), Torrance and Turcotte (J. Fluid Mech. 47(1):113–125, 1971). The novelty of this paper is to highlight how variable viscosity coupled with the LTNE assumption affects the onset of convection. A numerical procedure shows the destabilising effect of depth-dependent viscosity. Moreover, it comes out that the LTNE hypothesis makes the influence of viscosity more intense. Linear instability analysis of the conduction solution is carried out by means of the Chebyshev-tau method coupled to the QZ algorithm, which provides the critical Rayleigh number for the onset of convection in a straightforward way. The energy method is employed in order to study the nonlinear stability. The optimal result of coincidence between the linear instability threshold and the global nonlinear stability threshold is obtained. The influence of anisotropic permeability and conductivity, weighted conductivity ratio, and interaction coefficient on the onset of convection is highlighted.

Coupled Heat-Mass Transport Modelling of Radionuclide Migration from a Nuclear Waste Disposal Borehole

Disposal of radioactive waste originating from reprocessing of spent research reactor fuel typically includes stainless steel canisters with waste immobilised in a glass matrix. In a deep borehole disposal concept, waste packages could be stacked in a disposal zone at a depth of one to potentially several kilometres. This waste will generate heat for several hundreds of years. The influence of combining a natural geothermal gradient with heat from decaying nuclear waste on radionuclide transport from deep disposal boreholes is studied by implementing a coupled heat-solute mass transport modelling framework, subjected to depth-dependent temperature, pressure, and viscosity profiles. Several scenarios of waste-driven heat loads were investigated to test to what degree, if any, the additional heat affects radionuclide migration by generating convection-driven transport. Results show that the heat output and the calculated radioactivity at a hypothetical near-surface observation point are directly correlated; however, the overall impact of convection-driven transport is small due to the short duration (a few hundred years) of the heat load. Results further showed that the calculated radiation dose at the observation point was very sensitive to the magnitude of the effective diffusion parameter of the host rock. Coupled heat-solute mass transport models are necessary tools to identify influential processes regarding deep borehole disposal of heat-generating long-lived radioactive waste.

Wave-Modified Ekman Current Solutions for the Time-Dependent Vertical Eddy Viscosity

This study aimed to highlight a general lack of clarity regarding the scale of the temporal averaging implicit in Ekman-type models. Under the assumption of time and depth-dependent eddy viscosity, we present an analytical Fourier series solution for a wave-modified Ekman model. The depth dependence of eddy viscosity is based on the K-Profile Parameterization (KPP) scheme. The solution reproduces major characteristics of diurnal variation in ocean velocity and shear. Results show that the time variability in eddy viscosity leads to an enhanced mean current near-surface and a decrease in the effective eddy viscosity, which finally results in an intensified near-surface shear and wakes a low-level jet flow. Rectification values are dominated by the strength of diurnal mixing, and partly due to the nonlinear depth dependence of the eddy viscosity.

Polarization force and geothermal viscosity driven unsteady Bödewadt transport phenomenon over a ferrofluid saturated disk

This work aims to investigate mass transport phenomena on time dependent Bödewadt flow of magnetic Nanoliquid embedded in the porous medium with vertical polarization force and geothermal viscosity variations. The resultant nonlinear coupled system of partial differential equations is solved numerically. For thorough understanding of flow and transport phenomenon, a wider range of Prandtl number is taken to analyze the effects of various physical entities including polarization force, geothermal viscosity, permeability and rotation. Eventually, some new marvels are found. The diffusion rate enhances significantly for higher values of Prandtl, Schmidt numbers and the boundary layer thickness decimates with all parameters except depth dependent viscosity. Besides, frictions on the surface of the plate have also been computed and found them very high for all above mentioned entities. In nutshell, to have the realistic view, this investigation reveals that the polarization force and viscosity variation due to the temperature and depth have imperative role on the unsteady transport phenomenon in ferrofluid flow.

Instability threshold of a model for deep convection with depth-dependent viscosity

The effect of depth-dependent viscosity on the onset of convection in deep horizontal layers heated from below is investigated. In the Zeytounian deep convection model (1989), the depth-dependent viscosity is introduced. The instability threshold of the thermal conduction rest state, is evaluated (in the free-free case). It is obtained that: (1) the strong principle of exchange of stability holds; (2) the instability threshold depends – via a simple closed form(1.1)– on the viscosity law; (3) a fall in the instability threshold is driven by the depth-dependent viscosity. The action of (quadratic) polynomial and exponential depth-dependent viscosity on the instability threshold is evaluated. Although needs to be verified by experiments, the results obtained appear to be of interest not only for theoreticians but also for experimentalists.

Variable Viscosity Effects on Time Dependent Magnetic Nanofluid Flow past a Stretchable Rotating Plate

Abstract An attempt has been made to describe the effects of geothermal viscosity with viscous dissipation on the three dimensional time dependent boundary layer flow of magnetic nanofluids due to a stretchable rotating plate in the presence of a porous medium. The modelled governing time dependent equations are transformed a from boundary value problem to an initial value problem, and thereafter solved by a fourth order Runge-Kutta method in MATLAB with a shooting technique for the initial guess. The influences of mixed temperature, depth dependent viscosity, and the rotation strength parameter on the flow field and temperature field generated on the plate surface are investigated. The derived results show direct impact in the problems of heat transfer in high speed computer disks (Herrero et al. [1]) and turbine rotor systems (Owen and Rogers [2]).

Scaling of viscous shear zones with depth-dependent viscosity and power-law stress–strain-rate dependence

Scaling of viscous shear zones with depth-dependent viscosity and power-law stress–strain-rate dependence

Mantle plume characteristics in three-dimensional depth- and temperature-dependent models

Numerical calculations in three-dimensional Cartesian geometry have been carried out. The aim of the paper is to investigate systematically the effect of Rayleigh number and temperature- and depth-dependent viscosity on the radius and temperature anomaly of mantle plumes and on surface observable characteristics like topography and geoid anomaly. As reference models depth-dependent models were chosen, then the addition of weak and stronger temperature-dependence was studied. The Rayleigh number changed between \(10^5\) and \(10^7\). If viscosity decreases with temperature the convection becomes more vigorous and thermally weaker plumes rise from the bottom boundary layer. The radius and temperature anomaly of the upwelling decreases with increasing temperature-dependence. Topographic uplift significantly decreases with increasing temperature-dependence. For weak depth-dependent models the geoid decreases together with the topography as Rayleigh number increases. However, strongly depth-depend models show negative geod anomaly above the hot plume. In case of temperature- and depth dependent models the geoid have a maximum in the function of Rayleigh number. This maximum appears at different Rayleigh number depending on the actual factors of temperature- and depth-dependence. The amplitudes of topographic uplift in the case of stronger temperature-dependence and high Rayleigh number are not far from the observed values, but the geoid signal is higher by a factor of about 2.

The influence of viscosity structure in the lithosphere on predictions from models of glacial isostatic adjustment

The thickness of the lithosphere inferred in most glacial isostatic adjustment (GIA) modelling studies tends to be significantly thinner than when found through seismic or thermal modelling studies. In those GIA studies, the lithosphere tends to be modelled as a plate of uniform and very high viscosity. We develop and test Earth models that include depth-dependent viscosity in the lithosphere to consider the implications for inferring lithospheric thickness from observed relative sea-level (RSL) changes. We find that when comparing predictions of RSL between the traditional plate lithosphere models and those with viscous structure, the latter produce RSL predictions that most closely resemble those from traditional models that are 10s of km thinner. The greatest sensitivity to this change in the Earth model is most evident in regions loaded by relatively small ice sheets such as the British Isles. We also find that the effective elastic thickness of the lithosphere models with viscous structure is time-dependent, with thinning by tens of kilometres over a timescale of ∼10kyr.

Instability in Porous Layers with Depth-Dependent Viscosity and Permeability

The paper is concerned with a multicomponent porous horizontal layer L with depth-dependent viscosity and permeability, heated from below and salted partly from below and partly from above. The instability of the thermal conduction solution, due to the growth of the gradient of temperature or of the gradient of chemical species salting L from above, is investigated.

Effects of wind input and wave dissipation formulations on the steady Ekman current solution

The effects of different wind input and wave dissipation formulations on the steady Ekman current solution are described. Two formulations are considered: one from the wave modeling (WAM) program proposed by Hasselmann and Komen and the other provided by Tsagareli and Babanin. The solution adopted for our study was presented by Song for the wave-modifi ed Ekman current model that included the Stokes drift, wind input, and wave dissipation with eddy viscosity increasing linearly with depth. Using the Combi spectrum with tail effects, the solutions are calculated using two formulations for wind input and wave dissipation, and compared. Differences in the results are not negligible. Furthermore, the solution presented by Song and Xu for the eddy viscosity formulated using the K-Profi le Parameterization scheme under wind input and wave dissipation given by Tsagareli and Babanin is compared with that obtained for a depth-dependent eddy viscosity. The solutions are further compared with the available well-known observational data. The result indicates that the Tsagareli and Babanin scheme is more suitable for use in the model when capillary waves are included, and the solution calculated using the K-Profi le Parameterization scheme agrees best with observations.

Depth-dependent viscosity and mantle stress amplification: implications for the role of the asthenosphere in maintaining plate tectonics

Boundary layer theory is used to derive scaling relationships for plate stresses in a mantle convection system with a low-viscosity asthenosphere. The theory assumes a plate tectonic like mode of mantle convection with flow driven by an active upper boundary layer. The theory predicts that the confinement of horizontal mantle flow within a low-viscosity, sublithospheric channel can lead to an increase in plate stress compared to the case lacking a channel (even if the absolute viscosity of the sublithosphere mantle does not change between the two cases). The theory further predicts increasing shear stress with decreasing low-viscosity channel thickness. If the thickness of tectonic plates is determined dominantly by a dehydrated chemical lithosphere, then the plate normal stress is predicted to also increase with decreasing channel thickness. We use 3-D spherical shell simulations of mantle convection with temperature-, depth- and stress dependent rheology to test scaling trends. The simulations and theoretical scalings demonstrate that a low-viscosity layer (asthenosphere) can amplify convective stresses. If the level of convective stress plays a role in maintaining and/or reactivating plate boundaries, this suggests that a relatively thin low viscosity layer may help to maintain plate tectonics. The numerical simulations support this suggestion as they show that an increase in the thickness of a low viscosity channel can cause the system to transition from an active-lid mode of convection to a stagnant lid state. Collectively, the simulations and theoretical scalings lead to the conclusion that the role of the asthenosphere in maintaining plate tectonics does not come principally from a basal lubrication effect, associated with a low absolute asthenosphere viscosity, but, instead, from a mantle flow channelization effect, associated with a high viscosity contrast from the asthenosphere to the mantle below.

The signature of depth-dependent viscosity structure in post-seismic deformation

Geodetic observations of post-seismic ground motion reflect the integrated effects of several relaxation mechanisms. To evaluate the viscous relaxation component accurately the crustal viscosity structure beneath a region must be estimated. In this study, using a 3-D finite element model, we describe the viscoelastic relaxation that follows an instantaneous strike-slip faulting event in a crustal layer whose viscosity decreases exponentially with depth. At any surface observation point the depth-dependent viscosity (DDV) model displacement history closely fits the history predicted for a uniform viscosity (UNV) model. The difference can be minimized to obtain the UNV viscosity (ηu) that best fits the DDV model displacement at that location. The differences in displacement histories between DDV and best-fit UNV models are minor but depend on distance from fault, the viscosity gradient and the fault configuration. In the near field, the DDV model prediction can be well approximated by the UNV model, regardless of the viscosity gradient and fault configuration. On the other hand, in the far field, small differences between DDV model and best-fit UNV model become apparent: the displacements are controlled by greater viscosities in later phases, and the differences are greater for DDV models with greater viscosity gradient. The more important result we obtain is that: apparent viscosity ηu decreases with distance from the fault, and its rate of decrease is directly diagnostic of the vertical viscosity gradient. This result points to a practical method of analysing post-seismic ground-displacement data for the variation of viscosity in the ductile crust; the apparent crustal viscosity should be determined at a series of points at increasing distance from the fault. The variation of apparent viscosity with distance can then be used to directly infer the vertical gradient of viscosity in the crust.

Geoid and topography of Venus in various thermal convection models

Important though indirect information about the internal structure of Venus is provided by its topography and geoid. In the last decades this information has been used to constrain the Venus mantle viscosity structure and its dynamic regime. Recently, the geodynamic inversion of the Venus’ geoid and topography resulted in a group of best fitting viscosity profiles. We use these viscosity models here as an input to our mantle convection code. We carry out simulations of the Venus’ mantle evolution in a 3D spherical shell with depth dependent viscosity and check whether the character of the dynamic topography and the geoid represented by their power spectra fits the observed quantities. We compare the results with several other models obtained for different viscosity stratifications (constant, constant with highly viscous lithosphere, linear increase of viscosity). Further, we estimate the effect of other factors such as internal heating and varying Rayleigh number. We use a 2D spherical axisymmetric convection code to study the effect of lateral viscosity variations. In these 2D models we monitor the topography and the geoid developing above the axisymmetric plume and compare them with the observed elevations of Venus’ geoid and topography in several Regia. Though none of the models fits observed data perfectly, we can generally conclude, that the best fit between the observed and predicted quantities is reached for viscosity profiles with 200 km thick lithosphere followed by a gradual increase of viscosity with depth and with the upper mantle viscosity of 2 × 10 21 Pa s. For all viscosity profiles the predicted geoid and topography spectra match the observed ones only up to the degree 40, thus indicating other than dynamic origin of these quantities for higher degrees.

Plume-induced geoid anomalies from 2D axi-symmetric temperature- and pressure-dependent mantle convection models

A collection of numerical simulations of 2D axi-symmetric thermal convection is presented here. The aim is to investigate the shape of geoid anomalies and dynamic topography above a plume. The simulation is based on the Boussinesq approximation and infinite Prandtl number and is carried out in the spherical shell with strongly temperature- and depth-dependent Arrhenius-type viscosity. According to the Arrhenius law, plume models with purely depth-dependent rheology are unphysical and should be taken with care. The strongly coupled temperature- and depth-dependent viscosity enables us to better understand the plume's behavior inside the Earth.The topography and geoid anomalies produced from plumes are sensitive to rheology of the mantle and rheology of the plume; both have effects on shape and amplitude of the geoid anomalies. We determined different categories of the geoid which are related to various rheology. Depth-dependent viscosity models show a geoid with a negative sign above the plume, and temperature-dependent viscosity models depict a bell-shaped geoid. We identified different behaviors in the combined model with temperature- and pressure-dependent viscosity.

An analytic model of convection in a system with layered viscosity and plates

SUMMARY We present an analytic boundary layer model for thermal convection with a finite-strength plate and depth-dependent viscosity. The model permits solutions in which convective flow rates in the mantle exceed the plate velocity. The energy balance equations for the lithosphere and convective cell are distinct and the model yields the plate velocity, plate thickness and heat flow, as well as the laterally averaged horizontal flow profile for a convective cell. We demonstrate, by inspection of the lithospheric energy balance, that the dominant plate driving force depends not only on the material properties of the mantle and lithosphere, but also on the velocity of the plate. Multiple solutions are possible with three solution branches for the plate velocity representing three distinct modes of thermal convection. The branch of solutions with the largest plate velocity corresponds to the classic boundary layer solution, with a plate velocity approximately equal to the maximum mantle velocity. This branch reproduces the classic convective scaling laws for an isoviscous fluid and its dynamics are controlled by the mantle material properties. The branch of solutions with intermediate plate velocity represents a convective cell with a sluggish-lid and a plate velocity that is less than the underlying mantle velocity. The dynamics for this solution branch depend on the material properties of both the mantle and the lithosphere. Finally, the lower branch of solutions for the plate velocity yields a convective solution in which the plate velocity is determined entirely by the local dynamics of a thick and strong (relative to the mantle) lithosphere. The dynamics of this solution branch are independent of the material properties of the mantle and depend entirely on the properties of the lithosphere. The introduction of an asthenosphere or low viscosity layer (LVL) beneath the plate can significantly alter the dynamics of the system by affecting plate–mantle coupling. Lowering the LVL viscosity, relative to the lower mantle, can promote plate motion by providing a lubricating layer. However, a very low viscosity LVL beneath a strong plate can decouple plate and mantle and inhibit plate motion, producing a solution with a slow moving plate and a channelized flow in the LVL. Thus, a LVL can sometimes inhibit rather than promote plate tectonics.

Small-scale convection beneath oceans and continents

Small-scale convection supplies heat flow of ∼17 mW m−2 to the base of stable continents where xenolith studies resolve the geotherm. However, effects of small-scale convection are difficult to resolve in ocean basins. On first pass, most seafloor appears to subside to an asymptote compatible with ∼40 mW m−2 convective heat flow. These common regions are tracked by hotspots so uplift associated with ponded mantle material is an attractive alternative. Unaffected seafloor in the North and South Atlantic continues to subside with the square root of age as expected from pure conduction. The theory of stagnant-lid convection provides good scaling relationships for heat flow. For linear viscosity, heat flow is proportional to the underlying “half-space” viscosity to the −1/3 power and the temperature to change viscosity by a factor of e to the 4/3 power. The formalism is easily modified to represent convection beneath a lid of highly viscous and buoyant cratonal lithosphere and to represent transient convection beneath thickening oceanic lithosphere. Asthenospheric mantle with linear, strongly temperature-dependent, and weakly depth-dependent viscosity is compatible with both oceanic and continental data. More complicated rheology may allow vigorous small-scale convection under most but not all old ocean basins. Still viable hypotheses require poorly understood global features, including lateral variations of asthenospheric temperature. Seismological studies have the potential to resolve the lithosphere-asthenosphere boundary, including local variations of its depth associated with small-scale convection.

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.