Topological groundwater hydrodynamics

Topological groundwater hydrodynamics is an emerging subdiscipline in the mechanics of fluids in porous media whose objective is to investigate the invariant geometric properties of subsurface flow and transport processes. In this paper, the topological characteristics of groundwater flows governed by the Darcy law are studied. It is demonstrated that: (i) the topological constraint of zero helicity density during flow is equivalent to the Darcy law; (ii) both steady and unsteady groundwater flows through aquifers whose hydraulic conductivity is an arbitrary scalar function of position and time are confined to surfaces on which the streamlines of the flow are geodesic curves; (iii) the surfaces to which the flows are confined either are flat or are tori; and (iv) chaotic streamlines are not possible for these flows, implying that they are inherently poorly mixing in advective solute transport.     

非对称型强飓风中的准平衡流特征分析

本文在论述飓风发生发展生命史过程中平衡、准平衡和非平衡态动力学特征的基础上,应用PV-ω方法,对具有非轴对称和长时间强度维持特征的飓风Bonnie（1998）进行了反演诊断分析,结果表明：基于非线性平衡模式的平衡流能够描述飓风水平涡旋场的基本特征,而加入准平衡ω方程得到的准平衡流能反映边界层入流、高层出流、眼墙区的剧烈倾斜上升运动和眼心区域的下沉运动.准平衡流描述了具有较长生命史组织化过程的强对流系统,而与其相联系的辐散运动与涡度同量级,证明了飓风准平衡流场具有涡散运动共存的特征,但在边界层顶的入流急流区和高层出流区仍存在高度非平衡态的超梯度流.利用反演的准平衡流场分析发现,当由环境风场低层到高层存在顺切变时,飓风内中尺度对流带移动方向的左侧,有利于强对流单体的发展和新对流单体的形成,右侧则相反,同时强气旋式旋转流场的作用,使得对流单体形成后随基本气流传播至对流减弱区,造成了飓风非对称结构的形成和维持.     

Shear flow driven magnetized planetary wave structures in the ionosphere

The generation and further linear and nonlinear dynamics of planetary ultra-low-frequency (ULF) waves are investigated in the rotating dissipative ionosphere in the presence of inhomogeneous zonal wind (shear flow). Planetary ULF magnetized Rossby type waves appear as a result of interaction of the medium with the spatially inhomogeneous geomagnetic field. An effective linear mechanism responsible for the intensification and mutual transformation of large scale magnetized Rossby type and small scale inertial waves is found. For shear flows, the operators of the linear problem are not self-conjugate, and therefore the eigenfunctions of the problem may not be orthogonal and can hardly be studied by the canonical modal approach. Hence, it becomes necessary to use the so-called nonmodal mathematical analysis. The nonmodal analysis shows that the transformation of wave disturbances in shear flows is due to the non-orthogonality of eigenfunctions of the problem in the conditions of linear dynamics. Using numerical modeling, the peculiar features of the interaction of waves with the background flow as well as the mutual transformation of wave disturbances are illustrated in the ionosphere. It has been shown that the shear flow driven wave perturbations effectively extract an energy of the shear flow increasing the own energy and amplitude. These perturbations undergo self-organization in the form of the nonlinear solitary vortex structures due to nonlinear twisting of the perturbation’s front. Depending on the features of the velocity profiles of the shear flows the nonlinear vortex structures can be either monopole vortices or vortex streets and vortex chains.     

Kinematic dynamo action in a sphere: Effects of periodic time-dependent flows on solutions with axial dipole symmetry

Choosing a simple class of flows, with characteristics that may be present in the Earth's core, we study the ability to generate a magnetic field when the flow is permitted to oscillate periodically in time. The flow characteristics are parameterised by D, representing a differential rotation, M, a meridional circulation, and C, a component characterising convective rolls. The dynamo action of all solutions with fixed parameters (steady flows) is known from earlier studies. Dynamo action is sensitive to these flow parameters and fails spectacularly for much of the parameter space where magnetic flux is concentrated into small regions, leading to high diffusion. In addition, steady flows generate only steady or regularly reversing oscillatory fields and cannot therefore reproduce irregular geomagnetic-type reversal behaviour. Oscillations of the flow are introduced by varying the flow parameters in time, defining a closed orbit in the space ( D,?M ). When the frequency of the oscillation is small, the net growth rate of the magnetic field over one period approaches the average of the growth rates for steady flows along the orbit. At increased frequency time-dependence appears to smooth out flux concentrations, often enhancing dynamo action. Dynamo action can be impaired, however, when flux concentrations of opposite signs occur close together as smoothing destroys the flux by cancellation. It is possible to produce geomagnetic-type reversals by making the orbit stray into a region where the steady flows generate oscillatory fields. In this case, however, dynamo action was not found to be enhanced by the time-dependence. A novel approach is being taken to solve the time-dependent eigenvalue problem where, by combining Floquet theory with a matrix-free Krylov-subspace method, we can avoid large memory requirements for storing the matrix required by the standard approach.     

Non-linear hydrodynamic stability of oceanic flows

Abstract

In this paper an analytical method to study the hydrodynamic stability of simple barotropic, non-divergent flows is discussed. The method is based on the variational approach introduced by Arnold and derived from the Lyapunov stability criteria. In this context, the sufficient condition for the stability of a steady barotropic flow ψ(x,y) is obtained when dP(ψ)/dP_{ψ = ψ}, the derivative of the absolute vorticity P(ψ), is positive definite. In this case, we discuss the effect of higher derivatives dⁿP(ψ)/dψⁿψ_{ψ = ψ} on the non-linear stability. Then we show that some classical examples of oceanic non-divergent flows (i.e. lee waves downstream an Island, steady flows through a Strait, the Fofonoff gyre) are stable to finite-amplitude perturbations.     

A BRIEF ACCOUNT OF THE GEOGRAPHY OF THE CENTRAL OETZTAL ALPS WITH SPECIAL REFERENCE TO ITS GLACIOLOGY

Abstract

Beginning with an energy loss relation of the form S = aVⁿ, it is possible to describe a number of basic analytical types of flows. The derived general equation may be shown to give solutions for simple cases such as radial and vortex flows as well as reducing to Laplace's equation. Laminar flow is a special case with n = 1, and further solutions range from laminar to fully turbulent flow.     

Structure of numerically simulated katabatic and anabatic flows along steep slopes

Direct numerical simulation (DNS) is applied to investigate properties of katabatic and anabatic flows along thermally perturbed (in terms of surface buoyancy flux) sloping surfaces in the absence of rotation. Numerical experiments are conducted for homogeneous surface forcings over infinite planar slopes. The simulated flows are the turbulent analogs of the Prandtl (1942) one-dimensional laminar slope flow. The simulated flows achieve quasi-steady periodic regimes at large times, with turbulent fluctuations being modified by persistent low-frequency oscillatory motions with frequency equal to the product of the ambient buoyancy frequency and the sine of the slope angle. These oscillatory wave-type motions result from interactions between turbulence and ambient stable stratification despite the temporal constancy of the surface buoyant forcing. The structure of the mean-flow fields and turbulence statistics in simulated slope flows is analyzed. An integral dynamic similarity constraint for steady slope/wall flows forced by surface buoyancy flux is derived and quantitatively verified against the DNS data.     

Advection of passive scalars induced by a bay-trapped nonstationary vortex

A simple model of fluid particle advection induced by the interaction of a point vortex and incident plane flow occurring near a curved boundary is analyzed. The use of the curved boundary in this case is aimed at mimicking the geometry of an isolated bay of a circular shape. An introduction of such a boundary to the model results in the appearance of retention zones, where the vortex can be permanently trapped being either stationary or periodically oscillating. When stationary, it induces a steady velocity field that in turn ensures regular advection of nearby fluid particles. When the vortex oscillates periodically, the induced velocity field turns unsteady leading to the manifestation of chaotic advection of fluid particles. We show that the size of the fluid region engaged into chaotic advection increases almost monotonically with the increased magnitude of the vortex oscillations provided the magnitude remains relatively small. The monotonicity is accounted for the fact that the frequency of the vortex oscillations incommensurable with the proper frequency of fluid particle rotations in the steady state. Another point of interest is that it is demonstrated that bounded regions, in which the vortex may be trapped, can appear even at a significant distance from the bay. Making use of a Lagrangian indicator, examples of fluid particle advection induced by the periodic motion of the vortex inside the bay are adduced.     

Numerical Modeling of Wind-Induced Flows in Stratified Water Bodies

Two-dimensional (in the vertical plane) wind-induced flows in no-flow-through reservoirs are considered. A numerical algorithm in the flow function–vortex variables is proposed based on the equations of slow stratified flows in the Boussinesq and boundary layer approximations with variable coefficient of vertical turbulent exchange. An analytical solution is given for a simplified problem.     

Enhanced bed load sediment transport by unsteady flows in a degrading channel

Laboratory flume experiments were done to investigate bed load sediment transport by both steady and unsteady flows in a degrading channel. The bed, respectively composed of uniform sand, uniform gravel, or sand-gravel mixtures, always undergoes bulk degradation. It is found that both uniform and non-uniform bed load transport is enhanced greatly by unsteady flows as compared to their volume-equivalent steady flows. This enhancement effect is evaluated by means of an enhancement factor, which is shown to be larger with a coarser bed and lower discharges. Also, the fractional transport rates of gravel and sand in non-uniform sand-gravel mixtures are compared with their uniform counterparts under both steady and unsteady flows. The sand is found to be able to greatly promote the transport of gravel, whilst the gravel considerably hinders the transport of sand. Particularly, the promoting and hindering impacts are more pronounced at lower discharges and tend to be weakened by flow unsteadiness.     

Macroscale surface roughness and frictional resistance in overland flow

The hydraulics of overland flow on rough granular surfaces can be modelled and evaluated using the inundation ratio rather than the flow Reynolds number, as the primary dimensionless group determining the flow behaviour. The inundation ratio describes the average degree of submergence of the surface roughness and is used to distinguish three flow regimes representing partially inundated, marginally inundated and well-inundated surfaces. A heuristic physical model for the flow hydraulics in each regime demonstrates that the three states of flow are characterized by very different functional dependencies of frictional resistance on the scaled depth of flow. At partial inundation, flow resistance is associated with the drag force derived from individual roughness and therefore increases with depth and percentage cover. At marginal inundation, the size of the roughness elements relative to the depth of flow controls the degree of vertical mixing in the flow so that frictional resistance tends to decrease very rapidly with increasing depth of flow. Well-inundated flows are described using rough turbulent flow hydraulics previously developed for open channel flows. These flows exhibit a much more gradual decrease in frictional resistance with increasing depth than that observed during marginal inundation. A data set compiled from previously published studies of overland flow hydraulics is used to assess the functional dependence of frictional resistance on inundation ratio over a wide range of flow conditions. The data confirm the non-monotonic dependence predicted by the model and support the differentiation of three flow regimes based on the inundation ratio. Although the percentage cover and the surface slope may be of importance in addition to the inundation ratio in the partially and marginally inundated regimes, the Reynolds number appears to be of significance only in describing well-inundated flows at low to moderate Reynolds numbers. As these latter conditions are quite rare in natural environments, the inundation ratio rather than the Reynolds number should be used as the primary dimensionless group when evaluating the hydraulics of overland flow on rough surfaces. © 1997 by John Wiley & Sons, Ltd.     

Stability of the line preserving flows

We examine the equations that are used to describe flows which preserve field lines. We study what happens if we introduce perturbations to the governing equations. The stability of the line preserving flows in the case of the magneto-fluids permeated by magnetic fields is strictly connected to the non-null magnetic reconnection processes. In most of our study we use the Euler potential representation of the external magnetic field. We provide general expressions for the perturbations of the Euler potentials that describe the magnetic field. Similarly, we provide expressions for the case of steady flow as well as we obtain certain conditions required for the stability of the flow. In addition, for steady flows we formulate conditions under which the perturbations of the external field are negligible and the field may be described by its initial unperturbed form. Then we consider the flow equation that transforms quantities from the laboratory coordinate system to the related external field coordinate system. We introduce perturbations to the equation and obtain its simplified versions for the case of a steady flow. For a given system, use of this method allows us to simplify the considerations provided that some part of the system may be described as a perturbation. Next, to study regions favourable for the magnetic reconnection to occur we introduce a deviation vector to the basic line preserving flows condition equation. We provide expressions of the vector for some simplifying cases. This method allows us to examine if given perturbations either stabilise the system or induce magnetic reconnection. To illustrate some of our results we study two examples, namely a simple laboratory plasma flow and a simple planetary magnetosphere model.     

Dynamos on stream surfaces of a highly conducting fluid

Abstract

This paper discusses dynamo action in generalisations of the Ponomarenko dynamo at large magnetic Reynolds number. The original Ponomarenko dynamo consists of a spiralling flow in which the stream surfaces are concentric cylinders of circular cross section, and the flow depends only on distance from the axis in cylindrical polar coordinates.

In this study, the stream surfaces are allowed to be cylinders of arbitrary cross section, and the flow is only required to be independent of the coordinate along the cylinder axes. For smooth flows alpha and eddy diffusion effects are identified, in terms of the geometry of the stream surfaces, and asymptotic formulae for growth rates in the limit of large magnetic Reynolds number are obtained. Numerical support for these results is presented using direct simulation of dynamo action in selected flows at high conductivity. Finally the case is considered when in spherical polar coordinates the flow is independent of the azimuthal coordinate and the stream surfaces, which are tori, have arbitrary cross sections.     

Time-optimal path planning in dynamic flows using level set equations: theory and schemes

We develop an accurate partial differential equation-based methodology that predicts the time-optimal paths of autonomous vehicles navigating in any continuous, strong, and dynamic ocean currents, obviating the need for heuristics. The goal is to predict a sequence of steering directions so that vehicles can best utilize or avoid currents to minimize their travel time. Inspired by the level set method, we derive and demonstrate that a modified level set equation governs the time-optimal path in any continuous flow. We show that our algorithm is computationally efficient and apply it to a number of experiments. First, we validate our approach through a simple benchmark application in a Rankine vortex flow for which an analytical solution is available. Next, we apply our methodology to more complex, simulated flow fields such as unsteady double-gyre flows driven by wind stress and flows behind a circular island. These examples show that time-optimal paths for multiple vehicles can be planned even in the presence of complex flows in domains with obstacles. Finally, we present and support through illustrations several remarks that describe specific features of our methodology.     

Reconciled bedload sediment transport rates in ephemeral and perennial rivers

It has been thought for some time that bedload sediment transport rates may differ markedly in ephemeral and perennial rivers and, supporting this thought, there has been observation of very high rates of bedload transport by flash floods in the ephemeral river Nahal Yatir. However, until now, there has been no quantitative model resolving the observation, nor a theory capable of explaining why bedload transport rates by unsteady flash floods can be reasonably well described by bedload transport capacity formulae initially derived for steady flows. Here a time scale analysis of bedload transport is presented as pertaining to Nahal Yatir, which demonstrates that bedload transport can adapt sufficiently rapidly to capacity determined exclusively by local flow regime, and accordingly the transport capacity formulations developed for steady flows can be applied even under unsteady flows such as flash floods. Complementing the time scale analysis, a series of computational exercises using a coupled shallow water hydrodynamic model are shown to adequately resolve the observation of the very high rates of bedload transport by flash floods in Nahal Yatir. While bedload transport rates in ephemeral and perennial rivers differ remarkably when evaluated against a pure flow parameter such as specific stream power, they are essentially reconciled if assessed with a physically sensible parameter incorporating not only the flow regime but also the sediment particle size. The present finding underpins the practice of fluvial geomorphologists relating measured bedload transport to local flow and sediment characteristics only, irrespective of whether the flow is unsteady or steady. Copyright © 2010 John Wiley & Sons, Ltd.     

Some properties of flows in unsaturated soils with an exponential dependence of the hydraulic conductivity upon the pressure head

A systematic development of the consequences of an exponential dependence of the hydraulic conductivity upon the pressure head is presented. Alternative expressions for the flux are discussed in detail. For steady flows, partial differential equations in terms of the matric flux potential, the pressure head, and the total head are derived. For steady, plane and axially symmetric flows, partial differential equations for the stream function are given. A theoretical basis for the construction of viscous flow analogs for steady, plane and axially symmetric flows is also presented.     

Some aspects of vortices in rotating stratified fluids

Summary Ideas concerning the overturning of unstably stratified, rotating fluids are explored using potential vorticity.A set of equations governing axi-symmetric flow in a quasi-Boussinesq system are found based on the gradient wind approximation, and a transformation analogous to that developed byHoskins [6] is used.The time-development of a linear, thermally unstable vortex under the action of Ekman pumping is studied with these equations. The changing radial scale during amplification of the vortex is well represented.Finally, some exact steady vortex states for stably stratified fluids are found and their possible relevance to atmospheric vortices is discussed.     

Nonlinear stability of baroclinic flows over topography

Abstract

A new nonlinear stability criterion is derived for baroclinic flows over topography in spherical geometry. The stability of a wide class of exact three-dimensional nonlinear steady state solutions subject to arbitrary disturbances is established. The resonance condition, at the highest total wavenumber, for the steady state solutions and the stability criteria for baroclinic flow in the absence of topography provide the boundaries of the regions of stability in the presence of topography. The analogous results for flow on periodic or infinite beta planes incorporating non-orthogonal function large scale flows are also discussed.     

Experimental investigation on scour topography around high-rise structure foundations

The current study aims to investigate the characteristics of scour topography around High-Rise Structure Foundations(HRSFs)via physical modeling tests.Clear-water scour tests with a uniform non-cohesive bed are modeled under the action of unidirectional steady flows.Time variations of the erosion and deposition topography are measured.The results show that deposition downstream of the first dune behind the HRSF is not located on the centerline of the wake.The deposition pattern indicates that a long steady wake region exists behind the permeable foundation.The scour depth around an HRSF is much less than that around a monopile because of the structural permeability,which gives rise to the bleed flow and a weakened downflow and horseshoe vortex.Additionally,the asymmetry of the HRSF affects the scour rate but not the final equilibrium scour depth.The average scour slope decreases along the direction of the flow.On the contrary,the scour radial distance increases along the direction of the flow,with the average value changing from 1.36De to 2.35De(where De is the equivalent diameter of the foundation).Furthermore,the scour hole around the HRSF is serrated rather than smooth owing to the presence of multiple piles.Empirical formulae are suggested for estimating the evolution of scour depth and volume.These laboratory experiments provide reference information for relevant numerical modeling studies and can be applied to guide engineering designs in an ocean area.