Validity of sound-proof approaches in rapidly-rotating compressible convection: marginal stability versus turbulence

The validity of the anelastic approximation has recently been questioned in the regime of rapidly-rotating compressible convection in low Prandtl number fluids (Calkins, Julien and Marti, Proc. R. Soc. A, 2015, vol. 471, 20140689). Given the broad usage and the high computational efficiency of sound-proof approaches in this astrophysically relevant regime, this paper clarifies the conditions for a safe application. The potential of the alternative pseudo-incompressible approximation is investigated, which in contrast to the anelastic approximation is shown to never break down for predicting the point of marginal stability. Its accuracy, however, decreases close to the parameters corresponding to the failure of the anelastic approach, which is shown to occur when the sound-crossing time of the domain exceeds a rotation time scale, i.e. for rotational Mach numbers greater than one. Concerning the supercritical case, which is naturally characterised by smaller rotational Mach numbers, we find that the anelastic approximation does not show unphysical behaviour. Growth rates computed with the linearised anelastic equations converge toward the corresponding fully compressible values as the Rayleigh number increases. Likewise, our fully nonlinear turbulent simulations, produced with our fully compressible and anelastic models and carried out in a highly supercritical, rotating, compressible, low Prandtl number regime show good agreement. However, this nonlinear test example is for only a moderately low convective Rossby number of 0.14.     

Thermal convection in differentially rotating systems

Abstract

The onset of convection in a cylindrical fluid annulus is analyzed in the case when the cylindrical walls are rotating differentially, a temperature gradient in the radial direction is applied, and the centrifugal force dominates over gravity. The small gap approximation is used and no-slip conditions on the cylindrical walls are assumed. It is found that over a considerable range of the parameter space either convection rolls aligned with the axis of rotation or rolls in the perpendicular (azimuthal) direction are preferred. It is shown that by a suitable redefinition of parameters, results for finite amplitude Taylor vortices and for convection rolls in the presence of shear can be applied to the present problem. Weakly nonlinear results for transverse rolls in a Couette flow indicate the possibility of subcritical bifurcation for Prandtl numbers P less than 0.82. Heat and momentum transports are derived as functions of P and the problem of interaction between transverse and longitudinal rolls is considered. The relevance of the analysis for problems of convection in planetary and stellar atmospheres is briefly discussed.     

A nonlinear stability analysis of convection in a generalized incompressible fluid

Abstract

We establish a nonlinear stability result for convection in a generalized incompressible fluid. Both numerical calculations and an asymptotic analysis are carried out. The linear and nonlinear results are shown to be very close in both cases, implying that the region of possible subcritical instabilities is very small.

During this work I was supported by a research studentship awarded by the Science and Engineering Council of the United Kingdom.     

Layered convection with an interface at a depth of 1000 km: stability and generation of slab-like downwellings

We investigate the stability of hypothetical layered convection in the mantle and the mechanisms how the downwelling structures originating in the lower layer are generated. The stability is studied by means of numerical simulations of the double-diffusive convection in a 2D spherical model with radially dependent viscosity. We demonstrate that the stability of the layering strongly depends not only on the density contrast between the layers but also on the heating mode and the viscosity profile. In the case of the classical Boussinesq model with an internally heated lower layer, the density contrast of about 4% between the compositionally different materials is needed for the layered flow to be maintained. The inclusion of the adiabatic heating/cooling in the model reduces the temperature contrast between the two layers and, thus, enhances the stability of the layering. In this case, a density contrast of 2-3% is sufficient to preserve the layered convection on a time scale of billions of years. The generation of the downwelling structures in the lower layer occurs via mechanical or thermal coupling scenarios. If the viscosity dependent on depth and average temperature at each depth is considered, the low viscosity zone develops at a boundary between the two convecting layers which suppresses mechanical coupling. Then the downwelling structures originating in the lower layer develop beneath upper layer subductions, thus resembling continuous slab-like structures observed by seismic tomography.     

Instability of flows near the onset of convection in a rotating layer with stress-free horizontal boundaries

We investigate instability of convective flows of simple structure (rolls, standing and travelling waves) in a rotating layer with stress-free horizontal boundaries near the onset of convection. We show that the flows are always unstable to perturbations, which are linear combinations of large-scale modes and short-scale modes, whose wave numbers are close to those of the perturbed flows. Depending on asymptotic relations of small parameters α (the difference between the wave number of perturbed flows and the critical wave number for the onset of convection) and ε (ε² being the overcriticality and the perturbed flow amplitude being O(ε)), either small-angle or Eckhaus instability is prevailing. In the case of small-angle instability for rolls the largest growth rate scales as ε^8/5, in agreement with results of Cox and Matthews (Cox, S.M. and Matthews, P.C., Instability of rotating convection. J. Fluid. Mech., 2000, 403, 153–172) obtained for rolls with k = k _c . For waves, the largest growth rate is of the order ε^4/3. In the case of Eckhaus instability the growth rate is of the order of α².     

Spatial and temporal evolution of arc volcanism in the northeast Honshu and Izu‐Bonin Arcs: Evidence of small‐scale convection under the island arc?

Abstract   Arc volcanism of the past 10 my in the northeast Honshu and Izu-Bonin Arcs shows several notable features. In the northeast Honshu Arc, the spatial distribution of volcanism exhibits several clusters elongated nearly perpendicular to the arc and the possible migration of volcanism from the back-arc side to the volcanic front side, at least, during the past 5 my. The pattern of clusters seems to have flip-flopped around 5 Ma. In the Izu-Bonin Arc, there are a series of across-arc seamount chains, in which volcanic activity occurred from ca 17 Ma to ca 3 Ma, similar to the clusters of the northeast Honshu Arc, although the recent active rifting occurs almost parallel to the arc. On the basis of studies of numerical modeling, these features might be explained, at least qualitatively, by the small-scale convection under the island arc. Several inferences can be made from our modeling results for the tectonics of the Izu-Bonin Arc. The angle of dip of subducting plate in the Izu-Bonin Arc might have increased. This can explain the disappearance of volcanism along the seamount chains and the recent along-arc volcanism with narrow rifting. The trend of seamount chains, which is oblique to the arc, might not be their intrinsic feature but rather a result of the lateral movement of the back-arc region after their formation. These inferences can be tested by the future detailed morphological and chronological studies of the Izu-Bonin Arc.     

Transitions in the style of mantle convection at high Rayleigh numbers

The pattern and style of mantle convection govern the thermal evolution, internal dynamics, and large-scale surface deformation of the terrestrial planets. In order to characterize the nature of heat transport and convective behaviour at Rayleigh numbers, Ra, appropriate for planetary mantles (between 10⁴ and 10⁸), we perform a set of laboratory experiments. Convection is driven by a temperature gradient imposed between two rigid surfaces, and there is no internal heating. As the Rayleigh number is increased, two transitions in convective behaviour occur. First we observe a change from steady to time-dependent convection at Ra≈10⁵. A second transition occurs at higher Rayleigh numbers, Ra≈5×10⁶, with large-scale time-dependent flow being replaced by isolated rising and sinking plumes. Corresponding to the latter transition, the exponent β in the power law relating the Nusselt number Nu to the Rayleigh number (NuRa^β) is reduced. Both rising and sinking plumes always consist of plume heads followed by tails. There is no characteristic frequency for the formation of plumes.     

Hot springs mediate spatial exchange of heat and mass in the enclosed sediment domain: A stability perspective

In many natural environments, such as in underwater hot springs and hydrothermal vents, thermal gradients are accompanied with changes in the concentration of chemical compounds transported to the seawater, causing the so-called double-diffusive, mixed convection. To study the physical scenarios in such systems, a vertical channel filled with a porous medium saturated with saline water is considered. The motion in the sediment-filled channel is induced by two buoyancy forces and an external pressure gradient, similar to the situation in a vent with an upward flow direction. The fluid flow has been modeled by an extended Darcy model, and the flow instability mechanisms have been studied numerically. The linear stability analysis is performed considering a wide range of Darcy number (Da = 10⁻⁵ -10⁻⁸). The instability boundary curve showed three distinct dynamic regimes: (i) Rayleigh-Taylor (R-T), (ii) log-log non-linear variation, and (iii) log-log linear variation. The domain of different regimes were sensitive to external pressure gradient as well as permeability. Similar to cross-diffusive natural convection in pure viscous fluids, a linear relationship between logarithmic absolute values of critical thermal Rayleigh number (∣Ra_T∣) and solute Rayleigh number (Ra_C) is found in the third regime. Based on the permeability, for any solute Rayleigh number (Ra_C), there existed a minimum value of Reynolds number (Re), below which R-T type of instability appeared. Above this minimum value, the instability was due to two buoyancy forces, known as buoyant instability. Simulations of secondary flow via energy analysis demonstrated the development of complex dynamics at the critical state in all three regimes characterized by transition of multi to uni-cellular structures and vice verse.     

Stability of buoyancy opposed mixed convection in a vertical channel and its dependence on permeability

Non-Darcy mixed convective flow of water due to external pressure gradient and buoyancy opposed forces are considered in a vertical channel filled with porous medium, which can be either isotropic or anisotropic. The linear theory of stability analysis has been used to numerically investigate the dependence of the transition behavior of the fully developed basic flow on the permeability of the medium. Numerical experiments indicate that mainly two main instability modes appear: Rayleigh–Taylor (R–T) and buoyant instability. For Darcy numbers (Da) ?10⁻⁹, R–T instability dominates within the entire Reynolds number (Re) range considered here. It was also found that for the same Re, the fully developed base flow is highly unstable (stable) for porous media with high (low) permeability. Further, it was seen that the disturbance isotherm cells migrate from the channel walls toward the centerline when permeability is reduced. Reducing the permeability by one order of magnitude (corresponding to a decrease of Darcy number from 10⁻⁶ to 10⁻⁷) increases base flow stability approximately 20-fold. For higher Reynolds numbers, buoyant, mixed and shear instability of the basic flow were found when Da was increased from 10⁻⁷ to 10⁻³. However, for cases in which permeability and porosity behaved as suggested by Carman–Kozeny relation (CKR), buoyant stability was the only mode of instability. Critical values of the Rayleigh (Ra) and Darcy (Da) numbers in the R–T mode of instability were related to each other by the hyperbolic function RaDa = −2.465.     

Plasma convection at high latitudes using the EISCAT VHF and ESR incoherent scatter radars

The recent availability of substantial data sets taken by the EISCAT Svalbard Radar allows several important tests to be made on the determination of convection patterns from incoherent scatter radar results. During one 30-h period, the Svalbard Radar made 15 min scans combining local field aligned observations with two, low elevation positions selected to intersect the two beams of the Common Programme Four experiment being simultaneously conducted by the EISCAT VHF radar at Troms. The common volume results from the two radars are compared. The plasma convection velocities determined independently by the two radars are shown to agree very closely and the combined three-dimensional velocity data used to test the common assumption of negligible field-aligned flow in this regime.     

Application of fluid mechanic principles to the study of geodynamic processes at trench-arc-back arc systems

Geological processes at trench-arc-back arc systems are some of the most complex tectonic processes in need of study. A large amount of data based on geological and geophysical observations has been accumulated. To synthesize these data, mathematical models have proven to be very useful. Because geological processes are directly related to the dynamical behavior of the solid earth, many of them can be investigated in terms of fluid flow models. Some applications of fluid flow principles in studying tectonic processes at convergent plate boundaries are discussed in this paper. When mantle processes are modeled as convective systems, they are found to have direct implications for the determination of slab dip angles. Additionally, they can also account for the high heat flows in the back arc basins, provide a mechanism for back arc opening, and resolve the question whether subducted oceanic crust can reach melting at shallow depth for island arc magma generation. Besides mantle processes, flow models can also be used to study surface processes. A simple one-parameter plane flow theory is used to model the evolution of trench geometries. This model is able to fit simulataneously the trench curvature and the differential paleomagnetic rotation between volcanic islands for the Mariana. Despite the simplicity of many of these models, their ability to synthesize geological and geophysical data at convergent plate boundaries is quite remarkable.     

Treatment of Synthetic Olefin Plant Wastewater at Various Salt Concentrations in a Membrane Bioreactor

The objective of this study was to investigate the effect of salt concentration on performance of a membrane bioreactor (MBR) for treating an olefin plant wastewater. For this purpose, a lab‐scale submerged MBR with a flat‐sheet ultrafiltration membrane was used for treatment of synthetic wastewater according to oxidation and neutralization unit of olefin plant. The synthetic wastewater was adjusted to have 500 mg/L chemical oxygen demand (COD). Trials on different concentrations of sodium sulfate (Na₂SO₄) (0–20 000 ppm) in the feed were conducted under aerobic conditions in the MBR. The results showed that increasing the salt concentrations causes an increase in the effluent COD, phenol, and oil concentrations. These results are due to reduction of the membrane filtration efficiency and also decline in the microbial activity that it is indicated by decreasing the sOUR in MBR. But in all the trials, the effluent COD and oil concentration was well within the local discharge limit of 100 and 10 mg/L, respectively. These results indicate that the MBR system is highly efficient for treating the olefin plant wastewater, and although high salt concentrations decreased organic contaminant removal rates in the MBR, the effluent still met the discharge limits for treating the olefin plant wastewater.     

干湿循环下含盐量变化对遗址土剪切强度影响的试验研究

中国西北干旱和半干旱地区集中降雨和强烈蒸发的特殊气候环境加剧了对土遗址的破坏。以掺入不同含量Na2SO4和NaCl的重塑土样为研究对象,通过室内模拟开放系统下干湿循环过程,并采用声波测试,反映含盐类遗址土在干湿循环下的劣化过程;测试多次循环后的遗址土剪切强度随含盐量的变化规律,并从盐类性质及波速两方面分析了干湿循环对遗址土剪切强度的作用机理。结果表明:在含水率一定的情况下纵波波速随干湿循环次数的增加而减小;干湿循环后的土体黏聚力随着盐分含量的增加呈下降趋势,内摩擦角则出现先增大后减小的现象;纵波波速与黏聚力随着含盐量的增长整体变化趋势有较好的相关性,与内摩擦角的相关性则表现为纵波波速的间断性和内摩擦角阶段性的契合。