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.     

大气流场的拓扑结构

用简化的大气运动方程,定性分析了大气中几种常见天气系统流场的拓扑结构. 研究表明,气旋反气旋流场与中心点附近的流形相对应;长波演化形成阻塞高压流场,与鞍点和中心点从合并到分离的流形相对应;台风和龙卷风的流场与三维空间中鞍-焦点附近的流形相对应. 研究大气流场的拓扑结构,可以直观清晰地揭示大气运动的形态和形成机理,有助于认识大气运动的规律. 文中讨论基于一定假设,结果与实际大气有差异,因而具有局限性.     

Topological invariants of field lines rooted to planes

Abstract

Two open curves with fixed endpoints on a boundary surface can be topologically linked. However, the Gauss linkage integral applies only to closed curves and cannot measure their linkage. Here we employ the concept of relative helicity in order to define a linkage for open curves. For a magnetic field consisting of closed field lines, the magnetic helicity integral can be expressed as the sum of Gauss linkage integrals over pairs of lines. Relative helicity extends the helicity integral to volumes where field lines may cross the boundary surface. By analogy, linkages can be defined for open lines by requiring that their sum equal the relative helicity.

With this definition, the linkage of two lines which extend between two parallel planes simply equals the number of turns the lines take about each other. We obtain this result by first defining a gauge-invariant, one-dimensional helicity density, i.e. the relative helicity of an infinitesimally thin plane slab. This quantity has a physical interpretation in terms of the rate at which field lines lines wind about each other in the direction normal to the plane. A different method is employed for lines with both endpoints on one plane; this method expresses linkages in terms of a certain Gauss linkage integral plus a correction term. In general, the linkage number of two curves can be put in the form L=r + n, |r|≦1J2, where r depends only on the positions of the endpoints, and n is an integer which reflects the order of braiding of the curves.

Given fixed endpoints, the linkage numbers of a magnetic field are ideal magneto-hydrodynamic invariants. These numbers may be useful in the analysis of magnetic structures not bounded by magnetic surfaces, for example solar coronal fields rooted in the photosphere. Unfortunately, the set of linkage numbers for a field does not uniquely determine the field line topology. We briefly discuss the problem of providing a complete and economical classification of field topologies, using concepts from the theory of braid equivalence classes.     

On the Prognostic Efficiency of Topological Descriptors for Magnetograms of Active Regions

Geomagnetism and Aeronomy - Solar flare prediction remains an important practical task of space weather. An increase in the amount and quality of observational data and the development of...     

Topological diagnostics of the cyclic component of the time series associated with helium

Detection of the deterministic component from noised time series is a common procedure in the solar–terrestrial coupling problem when climate is modeled, solar activity is analyzed, or a signal associated with helium is extracted. Such series are mostly generated by the superposition of different processes for which the concept of a noise component cannot be determined formally. A method based on the combination of time-series topological embedding in Euclidean space and the identification of a persistent cycle by homology theory methods is proposed. The method application is demonstrated based on actual data.     

Topological structures of river networks and their regional-scale controls: A multivariate classification approach

Landscape evolution is governed by the interplay of uplift, climate, erosion, and the discontinuous pattern of sediment transfer from the proximal source of erosion to distal sedimentary sinks. The transfer of sediment through the catchment system is often referred to as a cascade, the pattern of which is modulated by the interaction of key network characteristics such as the distribution of transport capacity and resultant zones of sediment storage. An understanding of how sediment production is modulated through river networks with different topological structures at the associated timescales has remained elusive but presents significant implications for the knowledge of river response to disturbance events, and floodplain asset management. A multivariate method of identifying representative topological structures from a range of river networks is presented. Stream networks from 59 catchments in the South Island of New Zealand were extracted from a digital elevation model and their key topological parameters quantified. A principal component analysis was implemented to reduce these to two-dimensional axes that represent the magnitude of network branching and the topographic structure of each catchment, respectively. An agglomerative hierarchical clustering analysis revealed five network ‘types’, which are examined in terms of their internal structural characteristics and relationships to potential regional-scale controls. Implications for sediment transfer in these network ‘types’, and their use as representative networks for further analysis, are discussed. © 2020 The Authors. Earth Surface Processes and Landforms published by John Wiley & Sons Ltd