The stability of dissipative magnetohydrodynamic shear flow in a parallel magnetic field

Abstract

Strong decay bounds are obtained for linearized perturbations to an unbounded, plane Couette flow in a parallel magnetic field. Finite conductivity and molecular viscosity are found to be stabilizing. Those modes decaying most slowly have the form of rolls aligned with the shear flow. The non-aligned rolls decay at an enhanced rate. Stability bounds at finite amplitude are obtained for flows bounded in one direction using energy methods.     

Influence of condensation and latent heat release upon barotropic and baroclinic instabilities of vortices in a rotating shallow water f-plane model

Analysis of the influence of condensation and related latent heat release upon developing barotropic and baroclinic instabilities of large-scale low Rossby-number shielded vortices on the f-plane is performed within the moist-convective rotating shallow water model, in its barotropic (one-layer) and baroclinic (two-layer) versions. Numerical simulations with a high-resolution well-balanced finite-volume code, using a relaxation parameterisation for condensation, are made. Evolution of the instability in four different environments, with humidity (i) behaving as passive scalar, (ii) subject to condensation beyond a saturation threshold, (iii) subject to condensation and evaporation, with three different parameterisations of the latter, are inter-compared. The simulations are initialised with unstable modes determined from the detailed linear stability analysis in the “dry” version of the model. In a configuration corresponding to low-level mid-latitude atmospheric vortices, it is shown that the known scenario of evolution of barotropically unstable vortices, consisting in formation of a pair of dipoles (dipolar breakdown) is substantially modified by condensation and related moist convection, especially in the presence of surface evaporation. No enhancement of the instability due to precipitation was detected in this case. Cyclone-anticyclone asymmetry with respect to sensitivity to the moist effects is evidenced. It is shown that inertia-gravity wave emission during the vortex evolution is enhanced by the moist effects. In the baroclinic configuration corresponding to idealised cut-off lows in the atmosphere, it is shown that the azimuthal structure of the leading unstable mode is sensitive to the details of stratification. Scenarios of evolution are completely different for different azimuthal structures, one leading to dipolar breaking, and another to tripole formation. The effects of moisture considerably enhance the perturbations in the lower layer, especially in the tripole formation scenario.     

Nonlinear wave propagation on a sphere: Interaction between Rossby waves and gravity waves; stability of the Rossby waves

Abstract

An analytical spectral model of the barotropic divergent equations on a sphere is developed using the potential-stream function formulation and the normal modes as basic functions. Explicit expressions of the coefficients of nonlinear interaction are obtained in the asymptotic case of a slowly rotating sphere, i.e. when the normal modes can be expressed as single spherical harmonics.     

The influence of flow discharge variations on the morphodynamics of a diffluence–confluence unit on a large river

Bifurcations are key geomorphological nodes in anabranching and braided fluvial channels, controlling local bed morphology, the routing of sediment and water, and ultimately defining the stability of their associated diffluence–confluence unit. Recently, numerical modelling of bifurcations has focused on the relationship between flow conditions and the partitioning of sediment between the bifurcate channels. Herein, we report on field observations spanning September 2013 to July 2014 of the three‐dimensional flow structure, bed morphological change and partitioning of both flow discharge and suspended sediment through a large diffluence–confluence unit on the Mekong River, Cambodia, across a range of flow stages (from 13 500 to 27 000 m³ s^?1). Analysis of discharge and sediment load throughout the diffluence–confluence unit reveals that during the highest flows (Q = 27 000 m³ s^?1), the downstream island complex is a net sink of sediment (losing 2600 ± 2000 kg s^?1 between the diffluence and confluence), whereas during the rising limb (Q = 19 500 m³ s^?1) and falling limb flows (Q = 13 500 m³ s^?1) the sediment balance is in quasi‐equilibrium. We show that the discharge asymmetry of the bifurcation varies with discharge and highlight that the influence of upstream curvature‐induced water surface slope and bed morphological change may be first‐order controls on bifurcation configuration. Comparison of our field data to existing bifurcation stability diagrams reveals that during lower (rising and falling limb) flow the bifurcation may be classified as unstable, yet transitions to a stable condition at high flows. However, over the long term (1959–2013) aerial imagery reveals the diffluence–confluence unit to be fairly stable. We propose, therefore, that the long‐term stability of the bifurcation, as well as the larger channel planform and morphology of the diffluence–confluence unit, may be controlled by the dominant sediment transport regime of the system. © 2017 The Authors. Earth Surface Processes and Landforms published by John Wiley & Sons Ltd.     

Effects of sediment grain size and channel slope on the stability of river bifurcations

Channel bifurcations can be found in river network systems from high gradient gravel-bed rivers to fine-grained low gradient deltas. In these systems, bifurcations often evolve asymmetrically such that one downstream channel silts up and the other deepens and, in most cases, they eventually avulse. Past analytical and numerical studies showed that symmetric bifurcations are unstable in high and low Shields stress conditions resulting in asymmetric bifurcations and avulsion, while they can be stable in the mid-Shields range, but this range is smaller for larger width-to-depth ratio. Here, using a one-dimensional (1D) numerical model, we show that effects of sediment grain size and of channel slope are much larger than expected for low-gradient systems when a sediment transport relation is used that separates between bedload and suspended load transport. We found that the range of Shields stress conditions with unstable symmetric bifurcations expanded for lower channel slopes and for finer sediment. In high sediment mobility, suspended load increasingly dominates the sediment transport, which increases the sediment transport nonlinearity and lowers the relative influence of the stabilizing transverse bedslope-driven flux. Contrary to previous works, we found another stable symmetric solution in high Shields stress, but this only occurs in the systems with small width-to-depth ratio. This indicates that suspended load-dominated bifurcations of lowland rivers are more likely to develop into highly asymmetric channels than previously thought. This explains the tendency of channel avulsion observed in many systems.     

The Effect of Uniform Rotation on Convective Instability of a Mushy Layer During Binary Alloys Solidification

The linear stability analysis of interstitial liquid within the mushy layer relative to a steady basic state is performed in order to incorporate the effect of uniform rotation on the onset of convective instability whose existence was found by Anderson and Worster (1996). The full set of non–dimensional governing equations for the temperature field, the local solid fraction and the fluid velocity are reduced asymptotically. In particular, the limit of small dimensionless mushy layer thickness, the limit of small differences between the initial and eutectic compositions of the liquid and the limit of large Stefan number are considered. The square root of Taylor number is assumed to be of the order unity. These simplifying assumptions involved in our analysis lead to a much simplified model which can essentially be solved analytically.     

Evolution of a bifurcation in a meandering river with adjustable channel widths,Rhine delta apex,The Netherlands

Rivers may dramatically change course on a fluvial plain. Such an avulsion temporarily leads to two active channels connected at a bifurcation. Here we study the effect of dynamic meandering at the bifurcation and the effect of channel width adjustment to changing discharge in both downstream branches on the evolution of a bifurcation and coexisting channels. As an example, we reconstructed the last major avulsion at the Rhine delta apex. We combined historical and geological data to reconstruct a slowly developing avulsion process spanning 2000 years and involving channel width adjustment and meandering at the bifurcation. Based on earlier idealised models, we developed a one‐dimensional model for long‐term morphodynamic prediction of upstream channel and bifurcates connected at the bifurcation node. The model predicts flow and sediment partitioning at the node, including the effect of migrating meanders at the bifurcation and channel width adjustment. Bifurcate channel width adaptation to changing discharge partitioning dramatically slows the pacing of bifurcation evolution because the sediment balance for width adjustment and bed evolution are coupled. The model further shows that meandering at the bifurcation modulates channel abandonment or enlargement periodically. This explains hitherto unrecognised reactivation signals in the sedimentary record of the studied bifurcation meander belts, newly identified in our geological reconstruction. Historical maps show that bifurcation migration due to meander bend dynamics increases the bifurcation angle, which increases the rate of closure of one bifurcate. The combination of model and reconstruction identifies the relevant timescales for bifurcation evolution and avulsion duration. These are the time required to fill one downstream channel over one backwater length, the time to translate one meander wavelength downstream and, for strong river banks, the adaptation timescale to adjust channel width. The findings have relevance for all avulsions where channel width can adjust to changing discharge and where meandering occurs. Copyright © 2011 John Wiley & Sons, Ltd.     

Impact of the surface temperature and vertical shear of zonal wind on the dynamics of a simple two-layer model of the atmosphere

The paper presents and analyzes, from the point of view of smooth dynamic systems theory, a two-layer baroclinic model of the troposphere in geostrophic approximation. The model describes airflow in β-channel within the tropospheric part of the main Hadley circulation cell. It enables to obtain, after application of the Galerkin method, a fairly simple low-parametric dynamic system describing the phenomena of non-linear interactions, bifurcations and blocking in the atmosphere. This enables to take into consideration such basic factors influencing the atmospheric dynamics like the heat exchange within the surface, orography, vertical variability of zonal wind and hydrostatic stability. Impact of zonal thermal variability of the surface and vertical shear of zonal wind in the troposphere on the orographic bifurcation was investigated and the oscillation character in the dynamic system after Hopf bifurcation of the second kind was analyzed. Additionally, the model dynamics was investigated in conditions including momentum forcing in the upper and lower parts of the troposphere and excluding orographic interaction, as well as in the conditions of thermal interaction between the troposphere and the surface for the vertical shear of zonal wind in both tropospheric layers. Impact of the mean zonal wind in the troposphere on the properties of model dynamics was assessed. It was proved that zonally varied surface temperature and layered mean zonal wind in the atmosphere are the parameters that have basic influence on the model dynamics. They cause numerous bifurcations and strongly influence the periods of oscillations of the model variables. They are often Hopf bifurcations of the second kind during which tropospheric states fairly distant from the ones before the bifurcations are generated. This significantly influences the model predictability.     

The influence of a fluid-porous interface on solar pond stability

The linear instability of the gradient zone of a solar pond containing a fluid-porous interface is investigated. It is found that the gradient zone can retain the same stability for lower values of the solute Rayleigh number with the introduction of a porous material compared with a purely fluid layer, whilst maintaining the same lower convective zone temperature.Interestingly, it is also shown that for certain parameter values the penetration of a porous medium into the gradient zone can cause the temperature of the lower convective zone to rise. However, for certain parameter ranges, when the fluid-porous interface is towards the top of the gradient zone, the solar pond can become highly unstable.     

The influence of litho-stratification on the infiltrating water front in a granite terrain

Abstract

Under a sustainable groundwater development and quality management project over a semi-arid granite terrain, several rainwater harvesting structures were proposed based on geophysical investigations in the study area. In order to validate the site suitability and check the efficacy of the recommended recharge structures in terms of source sustainability, certain hydrological tests, such as infiltration studies, were conducted within the recharge area of these structures. Infiltration rate estimation using a double-ring infiltrometer normally results in an exponential decay relationship with high initial infiltration rate followed by gradual reduction until, ultimately, a stabilized rate is attained. In this paper, we present the results of infiltration tests conducted using a double-ring infiltrometer (under constant head condition) within the tank bed area of a proposed percolation tank site. The results showed unique features in the infiltration rate at different time intervals within the total experimental period: a staircase-like evolution with intermittent stabilization for a short duration. Based on the integrated approach of geophysical investigations, hydrological tests, and supported by physical evidence, the infiltration behaviour observed over the study area was considered to be due to the lithological stratification with different permeability and textural conditions.

Citation Andrade, R. & Muralidharan, D. (2011) The influence of litho-stratification on the infiltrating water front in a granite terrain. Hydrol. Sci. J. 56(5), 907–915.     

Analysis of slip-softening instability on a fault with asperities

Slip-softening instability on a vertical strike-slip fault with asperities has been analysed. The fault strength is uniform in depth, but the strength is nonuniform in the strike direction, i.e., there are asperities on the fault. These asperities and other segments of the fault have the same type of constitutive law but different peak stresses. The material surrounding the fault is represented by elastic plates, of which the top and bottom surfaces are stress-free.We use a finite element method to study the evolution of theoretical displacement, stress and strain field with a growing displacement applied at the remote plate ends. The slip and frictional stress are obtained as part of the solution. We have compared the difference of theoretical displacement, strain field and the distribution of frictional stress on the fault between unstable and stable slip. In addition, we have studied the effect of size and strength of asperities on instability, and the softening behaviour of asperities before instability.We find that (1) the failure of the fault zone may be due to either dynamic instability or rapid quasistable slip. A general characteristic of unstable mode is that slippage, on some parts of asperities increases indefinitely for a small finite increase in remote imposed displacement until, immediately before the unstable slip; (2) the size and peak strength of asperities have a large effect on instability. Reducing the size and peak strength of asperities tends to replace inertially unstable deformation with stable deformation; (3) the location with maximum acceleration during unstable slip, as the plausible nucleating seismic source, is in asperities; (4) the shapes of the changes in theoretical stress and strain at a given location, caused by the nonlinear constitutive property of the fault, are all similar whether instability, happens or not. This fact suggests that the changes of peak type or bend type in crustal deformation are not required for earthquake instability.     

The convective stability of a thin viscous disc

Abstract

We prove that the presence of viscosity does not affect stability to axisymmetric convective modes of a thin differentially rotating disc with no thermal conduction but in which viscosity is taken fully into account. In such a case the Schwarzschild criterion is necessary and sufficient for convective stability to local perturbations. In the proof we use a general formulation of local stability analysis, which allows a rigorous demonstration. Restricted particular forms of the viscous stress tensor introduced in the modelling of thin accretion discs may lead to viscous overstabilities. The additional instability found by Elstner et al. (1989) and described by the authors as a correction to the Schwarzschild criterion is a manifestation of these. However, when viscosity is taken fully into account, such instabilities cannot be discussed within the framework of a local analysis, a fully global treatment being required.     

速度传感反馈地震计的参数稳定性分析

从理论上定量分析机械摆和环路滤波等开环诸参数变化对速度传感反馈地震计闭环参数的影响.研究指出,在环路设计中可半自由选取的开环阻尼取值对参数稳定性有重要影响.过大或过小的开环阻尼取值均可能造成参数稳定性急剧变差,影响地震计在只标不调的状况下长期连续稳定工作.     

On the normal mode instability of harmonic waves on a sphere

Abstract

The normal mode instability of harmonic waves in an ideal incompressible fluid on a rotating sphere is analytically studied. By the harmonic wave is meant a Legendrepolynomial flow αP_n(μ) (n ≥ 1) and steady Rossby-Haurwitz wave of set F ₁ ⊕ H_n where H_n is the subspace of homogeneous spherical polynomials of the degree n(n ≥ 2), and F ₁ is the one-dimensional subspace generated by the Legendre-polynomial P₁(μ). A necessary condition for the normal mode instability of the harmonic wave is obtained. By this condition, Fjörtoft's (1953) average spectral number of the amplitude of each unstable mode must be equal to . It is noted that flow αP_n (μ) is Liapunov (and hence, exponentially and algebraically) stable to all the disturbances whose zonal wavenumber m satisfies condition |m| ≥ n. The bounds of the growth rate of unstable normal modes are estimated as well. It is also shown that the amplitude of each unstable, decaying or non-stationary mode is orthogonal to the harmonic wave.

The new instability condition can be useful in the search of unstable perturbations to a harmonic wave and on trials of numerical stability study algorithms. For a Legendre-polynomial flow, it complements Kuo's (1949) condition in the sense that while the latter is related to the basic flow structure; the former characterizes the structure of a growing perturbation.     

The effect of gravity on the stability of a line-tied coronal magnetohydrostatic equilibrium

Abstract

The magnetohydrodynamic stability of a class of magnetohydrostatic equilibria is investigated. The effect of gravity is included as well as the stabilising influence of the dense photospheric line-tying.

Although the two-dimensional equilibria exhibit a catastrophe point, when the ratio of plasma pressure to magnetic pressure exceeds a critical value, arcade structures, with both footpoints connected to the photosphere, become unstable to three-dimensional disturbances before the catastrophe point is reached.

Numerical results for field lines that are open into the solar corona suggest that they are completely stable. Although there is no definite proof of stability, this would allow the point of non-equilibrium to be reached.     

The existence of internal solitary waves in a two-fluid system near the KdV limit

Abstract

Two fluid layers of constant density lying one over the other on top of a rigid horizontal lower boundary with either a free upper surface or a rigid upper boundary can support solitary waves. The existence of a unique branch of such waves emanating from the horizontal flow at a critical speed U _? is demonstrated in both cases by use of the Nash—Moser implicit function theorem. These results complement the global results of Amick and Turner (1986) and are analogous to the work of Friedrichs and Hyers (1954) and Beale (1977) for surface waves. It is also noted that the most obvious variational principle which characterizes these waves as constrained extremals (Benjamin, 1984) is of indefinite type, having a Hessian with infinitely many positive and infinitely many negative eigenvalues.     

The influence of subsurface moisture on rill system evolution

Rill development studies have focused almost exclusively on surface erosion processes and critical threshold hydraulic conditions. Characteristic rill features, such as arcuate headcuts and knickpoints, are morphologically similar to the ‘theatre-headed’ valleys which have been associated with ‘sapping’ processes at various scales. This paper reports on laboratory experiments designed to identify linkages between surface flow hydraulics, subsurface moisture conditions and rill development. Experiments were carried out in a 16·57 m² flume under simulated rainfall with soil samples up to 0·15 m depth in which moisture conditions were monitored by miniature time-domain reflectometer probes. Tests showed complex responses in which some rill incision reflected surface flow conditions, but major rill system development with markedly enhanced sediment yield was closely associated with high soil moisture contents. It was not possible to measure seepage forces directly, but calculation and observation indicate that these were less important than reduction in soil strength with saturation, which resulted in increased effective runoff erosivity. This caused concentrated undercutting along the water table at rill walls, while slightly stronger surface layers above the water table formed microscarps. These retreated along the water table into interrill surfaces, producing residual pediment transport slopes. The microscarps eventually disappeared when the water table reached the surface, eliminating differential soil strength. The experiments showed complex relationships between surface and subsurface erosional processes in evolving rill systems, strongly influenced by soil moisture dynamics. The very small topographic and hydraulic head amplitudes indicate that seepage forces and ‘sapping’ were minimal. The dominant effect of soil moisture was reduction of soil strength with saturation, and increased runoff entrainment. Experimental conditions were not unusual, either for agricultural fields or natural hillslopes, and the intricate interrelationship of surface and subsurface erosion processes observed is probably not uncommon. Attempts to link specific morphologic features at rill scale to dominance of surface or subsurface processes alone are therefore unlikely to be successful or reliable. © 1998 John Wiley & Sons, Ltd.