A benchmark comparison of numerical methods for infinite Prandtl number thermal convection in two-dimensional Cartesian geometry

Abstract

A comparison is made between seven different numerical methods for calculating two-dimensional thermal convection in an infinite Prandtl number fluid. Among the seven methods are finite difference and finite element techniques that have been used to model thermal convection in the Earth's mantle. We evaluate the performance of each method using a suite of four benchmark problems, ranging from steady-state convection to intrinsically time-dependent convection with recurring thermal boundary layer instabilities. These results can be used to determine the accuracy of other computational methods, and to assist in the development of new ones.     

Scroll bars are inner bank levees along meandering river bends

Scroll bars across a 65-km stretch of the Trinity River in Texas, USA were studied using LiDAR data as well as with a series of 11 trenches spread out across the survey area. We conclude that scroll bars are levees that are deposited along the inner banks of these meandering river bends. Scroll bar crests were found to have similar elevations to those of outer bank levee crests, implying that they are constructional features that create positive topographic relief above the elevation of the floodplain. Trenches reveal that scroll bars are built from reworked suspended sediment, with common ripple-scale cross stratification, planar laminations and muddy bioturbated layers – characteristics often associated with levee sedimentation in other systems. LiDAR observation of the erosion of scroll bars by bed material transport during flood implies that scroll bar spacing is an imperfect proxy for estimating overall channel migration rates. In addition, interspersed lenses of coarser sediment with dune-scale cross stratification represent the stratigraphic record of these erosional events and suggest that erosion of the channel-ward edge of the scroll bar is not uncommon. Preservation of scroll bars is unlikely, given that they are responsible for an average of only the uppermost 12% of the total inner bank relief. We suggest that misidentification of point bar lateral accretion surfaces as scroll bars is common and can lead to issues with reconstructing channel properties due to systematic differences between point bar and scroll bar planform geometries. © 2019 John Wiley & Sons, Ltd. © 2019 John Wiley & Sons, Ltd.     

Evolution of cutoffs across meander necks in Powder River,Montana, USA

Over a period of several decades, gullies have been observed in various stages of forming, growing and completing the cutoff of meander necks in Powder River. During one episode of overbank flow, water flowing over the down-stream bank of the neck forms a headcut. The headcut migrates up-valley, forming a gully in its wake, until it has traversed the entire neck, cutting off the meander. The river then follows the course of the gully, which is subsequently enlarged as the river develops its new channel. The complete process usually requires several episodes of high water: in only one of the five cases described herein was a meander cutoff initiated and completed during a single large flood. © 1998 John Wiley & Sons, Ltd.     

Quantifying nitrogen cycling beneath a meander of a low gradient,N‐impacted,agricultural stream using tracers and numerical modelling

In watersheds impacted by nitrate from agricultural fertilizers, nitrification and denitrification may be decoupled as denitrification in the hyporheic zone is not limited to naturally produced nitrate. While most hyporheic research focuses on the 1–2 m of sediment beneath the stream bed, there are a limited number of studies that quantify nitrogen (N) cycling at larger hyporheic scales (10s of metres to kms). We conducted an investigation to quantify N cycling through a single meander of a low gradient, meandering stream, draining an agricultural watershed. Chemistry (major ions and N species) and hydrologic data were collected from the stream and groundwater beneath the meander. Evidence indicates that nearly all the shallow groundwater flowing beneath the meander originates as stream water on the upgradient side of the meander, and returns to the stream on the downgradient side. We quantified the flux of water beneath the meander using a numerical model. The flux of N into and out of the meander was quantified by multiplying the concentration of the important N species (nitrate, ammonium, dissolved organic nitrogen (DON)) by the modelled water fluxes. The flux of N into the meander is dominated by nitrate, and the flux of N out of the meander is dominated by ammonium and DON. While stream nitrate varied seasonally, ammonium and DON beneath the meander were relatively constant throughout the year. When stream nitrate concentrations are high (>2 mg litre^?1), flow beneath the meander is a net sink for N as more N from nitrate in stream water is consumed than is produced as ammonium and DON. When stream nitrate concentrations are low (<2 mg litre^?1), the flux of N entering is less than exiting the meander. On an annual basis, the meander hyporheic flow serves as a net sink for N. Copyright © 2007 John Wiley & Sons, Ltd.     

A two‐dimensional morphodynamic model of gravel‐bed river with floodplain vegetation

Artificially straight river channels tend to be unstable, and ultimately develop into river meanders through bank erosion and point‐bar deposition. In this paper account is taken of the effects of riparian and floodplain vegetation on bank strength, floodplain flow resistance, shear stress partitioning, and bedload transport. This is incorporated into an existing 2D hydrodynamic‐morphological model. By applying the new model to an initially straight and single‐threaded channel, the way that its planform and cross‐sectional geometry evolve for different hydraulic and floodplain vegetation conditions is demonstrated. The results show the formation and upstream migration of gravel bars, confluence scouring and the development of meandering and braiding channel patterns. In cases where the channel becomes unstable, the instability grows out of bar formation. The resulting braiding patterns are similar to analytical results. The formation of a transition configuration requires a strong influence from vegetation. Copyright © 2010 John Wiley & Sons, Ltd.     

嘉陵江古河曲发育的古地理环境重建

嘉陵江古河道的遥感影像轮廓反映了自由河曲的河流形态信息,分析嘉陵江古河曲的发育条件对其区域古地理环境的重建和恢复具有重要的地貌学研究意义.本文在综合前人关于自由河曲的理论与实际案例研究成果的基础上,总结了自由河曲的发育条件.据此,结合野外调查资料,认为嘉陵江古河曲形成时,该区应为平坦开阔的冲积平原,河流纵比降较比现在更小;河流两岸为具有“二元结构”的松散沉积物.与长江下荆江河段的地理环境类似.且“三水汇流”(嘉陵江与渠江、涪江)格局已经存在.嘉陵江由自由河曲转为深切河曲是由于区域地质环境(或地质事件)的改变所引起的.     

Dynamic Structure of the Kuroshio South of Kyushu in Relation to the Kuroshio Path Variations

The variation of velocity and potential vorticity (PV) of the Kuroshio at the PN line in the East China Sea and the TK line across the Tokara Strait were examined in relation to the path variations of the Kuroshio in the southern region of Japan, using quarterly data from a conductivity-temperature-depth profiler and a shipboard acoustic Doppler current profiler during 1987–97. At the PN line the Kuroshio has a single stable current core located over the continental slope and a significant maximum of PV located just onshore of the current axis in the middle part of the main pycnocline. On the other hand, the Kuroshio at the TK line has double current cores over the two gaps in the Tokara Strait; the northern core has a much larger velocity than the southern core on average during periods of the large meander of the Kuroshio, while the difference in strength between the double cores is small during the non-large-meander (NLM) period. At the TK line, PV in the middle pycnocline is variable; it is small and nearly uniform throughout the section for 40% of the total observations, while it has a significant maximum near the northern core for 30% and two maxima corresponding to the double current cores for 23%. The small, nearly uniform PV occurs predominantly during the NLM period, and is closely related to the generation of the small meander of the Kuroshio southeast of Kyushu. This revised version was published online in July 2006 with corrections to the Cover Date.     

Rates,distributions and mechanisms of change in meander morphology over decadal timescales,River Dane,UK

This paper analyses types and rates of change in river meander morphology and the links between mechanisms of change and emergent behaviour of planform morphology. It uses evidence of four dates of aerial photography combined with annual field mapping and ground photography to examine the morphological changes and mechanisms of change in a series of bends on an active meandering river, the River Dane in NW England, over a 25 year period. This unique data set allows insight into the spatial and temporal variability of bank line movement and component processes. Bank lines were mapped photogrametrically from air photos of 1984, 1996, 2001 and 2007 and the digitised courses compared in ArcGIS to produce calculations of erosional and depositional areas and rates. Most bends exhibit morphological change that largely follows the autogenic sequence, identified in qualitative models of meander development, from low sinuosity curves through simple symmetric and asymmetric bends to compound forms with lobe development in the apex region. Rates of erosion and bankline movement increase through this sequence until the compound phase. Relationships of amounts of movement to various curvature measures of bend morphology are complex. Several new loops, distinct from compound bend behaviour, have developed during the study period in formerly straight sections. Mechanisms of morphological change are illustrated for four types of bends: new, rapid growth bend; sharp‐angled bend with mid‐channel bar development; symmetric migrating bend; and simple to compound bend development. The changes take place in phases that are not simply related to discharge but to inherent sequences and feedbacks in development of bars and bend morphology and timescales for these are identified. Overall, emergent behaviour of systematic planform change, moderated by channel confinement and boundary features, is produced from spatially and temporally varied channel processes. Copyright © 2010 John Wiley & Sons, Ltd.     

Lithologic control of bedrock meander dimensions in the appalachian valley and ridge province: A comment

Braun's finding that the wavelength of bedrock meanders decreases as rock hardness increases is reconciled with Hack's and Tinkler's finding that meander wavelength is greater in bedrock than alluvial channels. In populations of alluvial or bedrock channels, bank resistance controls meander wavelength via its influence on channel shape and is negatively correlated with meander wavelength. In contrast, in mixed populations of alluvial and bedrock channels, bank resistance controls meander wavelength through its effect on the recurrence interval of the channel-forming discharge and channel size and is positively correlated with meander wavelength.     

Complexity, self-organisation and variation in behaviour in meandering rivers

River meanders are natural features on the surface of Earth that present some degree of regularity of form. They range from being highly dynamic to being stable under present conditions. Conventional theory is that meanders develop to an equilibrium form which is related to discharge and sediment load. Other research has demonstrated that many highly active meanders exhibit a continuous evolution over time and a non-linearity in rate of development. Ideas of autogenesis and of self-organised criticality as being an explanation of some meander changes have been proposed. In this paper data from rivers around the world are examined for further evidence of autogenic, self-organised or non-linear behaviour through analysis of change in sinuosity over time for reaches and change in individual bend form, particularly bend curvature and bend elongation. Some examples do exhibit trends of increasing sinuosity over time and a few show limits from which large decreases occur. Several case studies show non-linearity of behaviour and increasing complexity of form. Other case studies, however, do not exhibit such trends. Phase space plots are used to help uncover emergent behaviour but show a variety of patterns. The example of a reach in which multiple cut-offs occurred is analysed for mechanisms of self-organisation of the planform and in the pool-riffle pattern. Riffles are more closely spaced and also more transient in the more rapidly changing and higher sinuosity parts of the channel. Hypothetical trajectories of different meander behaviour, including for bedrock meanders, are plotted but the challenge remains to uncover the conditions for occurrence and for divergence of tendencies to stability and instability. Identification of attractors and phase space of behaviour of different meandering systems offer the potential for application to sustainable channel management.