Variation of Froude number versus depth during the passage of internal solitary waves from the in-situ observation and a numerical model

There are few studies on the variation of Froude number versus depth during the passage of internal solitary waves (ISWs). In this paper, according to the limited observational data of the ISWs near the Dongsha Islands in the northern South China Sea (SCS), it is found that, during the passage of an ISW, the local Froude number near the main thermocline gets maximum and the flow is supercritical, with the Froude number decreasing gradually upward and downward, whilst before and after the passage of the ISW, the local Froude number is less than 1 and the flow is subcritical in the entire water column. Since the observational current data are only available in the upper ocean, a two-dimensional continuously stratified nonlinear model is employed here to study the variation of Froude number versus depth during the passage of an ISW. The modeled ISW induced current field agrees well with that observed during the passage of the observational ISW. It is found that, even during the passage of the ISW, although the Froude number is larger than 1 in the upper layer, it is less than 1 in the lower layer, and the duration when the Froude number is larger than 1 also changes with depth. Thus, it is suggested that, once the Froude number near the main thermocline is larger than 1, the passage of ISWs could be identified. Moreover, it would be better to choose a sampling interval of no more than 1/4 passage period of the ISW during an in-situ observation of ISWs.     

Modelling the influence of small-scale effects upon the larger scale: an oceanographic challenge

The problem of resolving or parameterising small-scale processes in oceanographic models and the extent to which small-scale effects influence the large scale are briefly discussed and illustrated for a number of cases. For tides and surges in near-shore regions, the advantages of using a graded mesh to resolve coastal and estuarine small-scale features are demonstrated in terms of a west coast of Britain unstructured mesh model. The effect of mesh resolution upon the accuracy of the overall solution is illustrated in terms of a finite element model of the Irish Sea and Mersey estuary. For baroclinic motion at high Froude number, the effect of resolving small-scale topography within a non-hydrostatic model is illustrated in terms of tidally induced mixing at a single sill, or two closely spaced sills. The question of how to parameterise small-scale non-linear interaction processes that lead to significant mixing, in a form suitable for coarser grid hydrostatic models, is briefly considered. In addition, the importance of topographically induced mixing that occurs in the oceanic lateral boundary layer, namely, the shelf edge upon the large-scale ocean circulation is discussed together with the implications for coarse grid oceanic climate models. The use of unstructured grids in these models to enhance resolution in shelf-edge regions in a similar manner to that used in storm surge models to enhance near coastal resolution is suggested as a suitable “way forward” in large-scale ocean circulation modelling.     

The hydraulics of conduit flow in carbonate aquifers

Analysis of the morphology of dissolution-bedform assemblages and hydraulically-transported sediments found within conduits in carbonate aquifers in North America and the British Isles has allowed the hydraulic conditions under which conduit flows occur to be established. Mean values of flow velocity, boundary-shear stress, conduit Reynolds number, conduit Froude number, boundary friction factor, boundary roughness and flow power have been calculated. The values obtained are in agreement with other evidence in the conduits and are comparable to those obtained from other, similar, hydraulic systems.     

How valid are dynamic models of subduction and convection when plate motions are prescribed?

A detailed comparison between fully dynamic and kinematic plate formulations has been made in models of mantle convection. Plate velocity is computed self-consistently from fully dynamic plate models with temperature- and stress-dependent viscosity and preexisting mobile faults. In fully dynamic models, the flow is driven solely by internal buoyancy, while in kinematic models the flow is driven by a combination of the prescribed surface velocity and internal buoyancy. Only a temperature-dependent viscosity, close to the effective viscosity determined from the fully dynamic models, is used in the kinematic models. The two types of models give very similar temperature structures and slab evolutionary histories when the effective viscosity and surface velocity are nearly identical. In kinematic plate models, the additional work introduced by the prescribed velocity boundary condition is apparently dissipated within the lithosphere and has little influence on the convection under the lithosphere. In models with periodic lateral boundary conditions, slabs sink into the lower mantle at an oblique angle and this contrasts with the vertical sinking which occurs with reflecting boundary conditions. Models show that we can simulate fully dynamic models with kinematic models under either periodic boundary conditions or reflecting boundary conditions.     

3-D Simulation of Tectonic Loading at Convergent Plate Boundary Zones: Internal Stress Fields in Northeast Japan

The evidence of east-west compression in northeast Japan has been reported by many investigators on the basis of geodetic, geologic and geomorphic data, but its origin still remains far from understood. In the present study we have proposed a mechanical model of tectonic loading at convergent plate boundary zones, and demonstrated its validity through the numerical simulation of internal stress fields in northeast Japan with realistic 3-D geometry of plate interfaces. At convergent plate boundary zones, in general, a part of plate convergence is consumed by steady slip along plate interfaces, and the remaining part by inelastic deformation (seismic faulting, aseismic faulting, and active folding) of overriding plates. Such a plate boundary process to be called ``partial collision' can be quantitatively described by introducing a collision rate defined as c = 1 − steady slip rate at plate interfaces/plate convergence rate. By this definition, we can simply represent the mechanical process of partial collision, which includes total subduction (c = 0) and total collision (c = 1) as two extreme cases, in terms of steady slip rates at plate interfaces. On the basis of elastic dislocation theory, first, we numerically computed the internal stress fields in northeast Japan produced by the total subduction of the Pacific plate beneath the North American plate, however the computed stress pattern was opposite in sense to observations. Then, we computed the internal stress fields by taking c = 0.1 on average, and succeeded in reproducing the observed east-west compression in northeast Japan. This indicates that the concept of partial collision is essential to understand the mechanism of intraplate tectonic loading.     

Improved bounds on linear instability of barotropic zonal flow within the shallow water equations

Here we develop mathematical results to describe the location of linear instability of a parallel mean flow within the framework of the shallow water equations; growth estimates of near neutral modes (for disturbances subcritical with respect to gravity wave speed) in the cases of non-rotating and rotating shallow water. The bottom topography is taken to be one-dimensional and the isobaths are parallel to the mean flow. In the case of a rotating fluid, the isobaths and the mean flow are assumed to be zonal. The flow is front-like: there is a monotonic increase of mean flow velocity. Our results show that for barotropic flows the location of instabilities will be a semi-ellipse region in the complex wave velocity plane, that is based on the wave-number, Froude number, and depth of the fluid layer. We also explore the instability region for the case of spatially unbounded mean velocity profiles for non-rotating shallow water.     

Sand wave formation by stationary waves

In this paper we develop a new hypothesis which relates the formation of sand waves in open flow to the effect of stationary (lee) waves on flow with smooth nonhomogeneous downstream depth or behind an obstacle. The lee wave moves upstream with the phase velocity equal to the absolute value of the flow velocity, hence the wave crest does not move. Results of our experiments show that sand waves on the flow bed appear only below the lee waves and characteristics of sand waves are determined by the wave properties. We investigated the nonlinear stationary waves for subcritical and supercritical Froude numbers. These results allow us to predict sand waves characteristics for a particular flow.     

Froude number effects on flow over topography

Abstract

The response or a depth independent two layer flow to an underlying topographic irregularity is studied for flows in which the square of the internal Froude number exceeds the Rossby number. Irrespective of the magnitude of the Rossby number, rotation is important for such flows. The flow generally adjusts so that the thickness of the lower layer is nearly constant. However small anomalies from the constant thickness are found to extend to very large distances from the topography when the Rossby number exceeds unity.     

Developing Potentiometric Surfaces and Flow Fields with a Head‐Specified MODFLOW Model

Investigating changes in an aquifer system often involves comparison of observed heads from different synoptic measurements, generally with potentiometric surfaces developed by hand or a statistical approach. Alternatively, head‐specified MODFLOW models, in which constant head cells simulate observed heads, generate gridded potentiometric surfaces that explicitly account for Darcy's Law and mass balance. We developed a transient head‐specified MODFLOW model for the stratified Cambrian‐Ordovician sandstone aquifer system of northeastern Illinois to analyze flow within its 275 m deep cone of depression. Potentiometric surfaces were developed using static heads from production wells regardless of open interval; hence assuming no vertical head difference. This assumption was tested against steady‐state, head‐specified models of each sandstone strata for 1980 and 2014. The results indicate that the original conceptual model was appropriate in 1980 but not 2014, where a vertical head difference had developed at the center of the cone of depression. For earlier years, when the head difference was minimal, the transient head‐specified model compared well with a traditional, flow‐specified model. In later years, the transient head‐specified model overestimated removal of water from storage. MODFLOW facilitates the development of a time‐series of potentiometric surfaces and can easily be modified to test the impacts of different conceptual models, such as assumptions on vertical head differences. For this study of a deep confined aquifer, MODFLOW also offers advantages in generating potentiometric surfaces and flow fields over statistical interpolation techniques, although future research is needed to assess its performance in other settings.     

Reactive transport in stratified flow fields with idealized heterogeneity

A two-dimensional equation governing the steady state spatial concentration distribution of a reactive constituent within a heterogeneous advective–dispersive flow field is solved analytically. The solution which is developed for the case of a single point source can be generalized to represent analogous situations with any number of separate point sources. A limiting case of special interest has a line source of constant concentration spanning the domain’s upstream boundary. The work has relevance for improving understanding of reactive transport within various kinds of advection-dominated natural or engineered environments including rivers and streams, and bioreactors such as treatment wetlands. Simulations are used to examine quantitatively the impact that transverse dispersion (deviations from purely stochastic-convective flow) can have on mean concentration decline in the direction of flow. Results support the contention that transverse mixing serves to enhance the overall rate of reaction in such systems.     

Convection with heat flux prescribed on the boundaries of the system. I. The effect of temperature dependence of material properties

Abstract

We study the bifurcation to steady two-dimensional convection with the heat flux prescribed on the fluid boundaries. The fluid is weakly non-Boussinesq on account of a slight temperature dependence of its material properties. Using expansions in the spirit of shallow water theory based on the preference for large horizontal scales in fixed flux convection, we derive an evolution equation for the horizontal structure of convective cells. In the steady state, this reduces to a simple nonlinear ordinary differential equation. When the horizontal scales of the cells exceed a certain critical size, the bifurcation to steady convection is subcritical and the degree of subcriticality increases with increasing cell size.     

Three-dimensional local grid refinement for block-centered finite-difference groundwater models using iteratively coupled shared nodes: a new method of interpolation and analysis of errors

This paper describes work that extends to three dimensions the two-dimensional local-grid refinement method for block-centered finite-difference groundwater models of Mehl and Hill [Development and evaluation of a local grid refinement method for block-centered finite-difference groundwater models using shared nodes. Adv Water Resour 2002;25(5):497–511]. In this approach, the (parent) finite-difference grid is discretized more finely within a (child) sub-region. The grid refinement method sequentially solves each grid and uses specified flux (parent) and specified head (child) boundary conditions to couple the grids. Iteration achieves convergence between heads and fluxes of both grids. Of most concern is how to interpolate heads onto the boundary of the child grid such that the physics of the parent-grid flow is retained in three dimensions. We develop a new two-step, “cage-shell” interpolation method based on the solution of the flow equation on the boundary of the child between nodes shared with the parent grid. Error analysis using a test case indicates that the shared-node local grid refinement method with cage-shell boundary head interpolation is accurate and robust, and the resulting code is used to investigate three-dimensional local grid refinement of stream-aquifer interactions. Results reveal that (1) the parent and child grids interact to shift the true head and flux solution to a different solution where the heads and fluxes of both grids are in equilibrium, (2) the locally refined model provided a solution for both heads and fluxes in the region of the refinement that was more accurate than a model without refinement only if iterations are performed so that both heads and fluxes are in equilibrium, and (3) the accuracy of the coupling is limited by the parent-grid size—a coarse parent grid limits correct representation of the hydraulics in the feedback from the child grid.     

Test of scale modelling of sediment transport in steady unidirectional flow

In two steady uniform flows at different physical scales in a small open channel, with variables characterizing flow, sediment, and fluid adjusted for dynamic similitude by means of four dimensionless modelling parameters (a Reynolds number, a Froude number, a density ratio, and a length ratio), measured frequency distributions of height, spacing, and migration rate of current ripples were almost identical when scaled, thus verifying that exact Reynolds-Froude modelling of loose-sediment transport is valid and workable. Modelling should be valid as well for a wide range of other transport conditions in the same kind of flow, because no additional kinds of forces or effects would be present in transport of loose grains in modes other than as ripples. In scaled-down modelling, a scale ratio of 2.5 is attainable without recourse to exotic fluids by use of water at 85°C to model natural flows at 10°C.     

Intraplate seismicity in the western Bohemian Massif (central Europe): A possible correlation with a paleoplate junction

Locations of the Eger Rift, Cheb Basin, Quaternary volcanoes, crustal earthquake swarms and exhalation centers of CO₂ and ³He of mantle origin correlate with the tectonic fabric of the mantle lithosphere modelled from seismic anisotropy. We suggest that positions of the seismic and volcanic phenomena, as well as of the Cenozoic sedimentary basins, correlate with a “triple junction” of three mantle lithospheres distinguished by different orientations of their tectonic fabric consistent within each unit. The three mantle domains most probably belong to the originally separated microcontinents – the Saxothuringian, Teplá-Barrandian and Moldanubian – assembled during the Variscan orogeny. Cenozoic extension reactivated the junction and locally thinned the crust and mantle lithosphere. The rigid part of the crust, characterized by the presence of earthquake foci, decoupled near the junction from the mantle probably during the Variscan. The boundaries (transitions) of three mantle domains provided open pathways for Quaternary volcanism and the ascent of ³He- and CO₂-rich fluids released from the asthenosphere. The deepest earthquakes, interpreted as an upper limit of the brittle–ductile transition in the crust, are shallower above the junction of the mantle blocks (at about 12 km) than above the more stable Saxothuringian mantle lithosphere (at about 20 km), probably due to a higher heat flow and presence of fluids.     

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.     

Longitudinal velocity patterns and bed morphology interaction in a rill

Present‐day understanding of rill dynamics is hampered by a lack of detailed data on velocity distributions in rills. The latter are dif?cult to collect with traditional techniques due to the very low water depths and the relatively high ?ow velocities in rills. The objectives of this paper were to investigate the feasibility of miniaturized acoustic Doppler velocimeter (mADV) measurements in rill ?ow and to explore longitudinal variations in ?ow velocities and their relationship with rill bed morphology. Detailed data on longitudinal ?ow velocity were required to achieve these objectives. A 1·8 m long rill was formed freely in a ?ume at 5° slope and 0·001 m³ s^?1 discharge. Rill topography was characterized by an alternation of steps and pools. The ?ume surface was then ?xed to preserve rill roughness. A topographical scanning of the entire ?ume surface was made. Velocity was measured with a mADV along the rill, and at different depths. Flow depth in a longitudinal direction was also measured using an elevation gauge. A strong relationship exists between rill topography and ?ow hydraulics. Over steps, ?ow was unidirectional and rapidly accelerating until a threshold Froude number (Fn) value between 1·3 and 1·7 was reached and a hydraulic jump occurred leading to the formation of a pool. In the pool, the ?ow pattern was multidirectional and complex. The ?ow was subcritical when leaving the pool and accelerated over the next step until the threshold Froude number value was again reached. Energy loss in the rill was concentrated in the pools, mainly due to the action of a hydraulic jump. This mechanism of energy dissipation appeared to be an essential factor in rill formation and bedform evolution. Copyright © 2004 John Wiley & Sons, Ltd.     

Correction of Discretization Errors Simulated at Supply Wells

Many hydrogeology problems require predictions of hydraulic heads in a supply well. In most cases, the regional hydraulic response to groundwater withdrawal is best approximated using a numerical model; however, simulated hydraulic heads at supply wells are subject to errors associated with model discretization and well loss. An approach for correcting the simulated head at a pumping node is described here. The approach corrects for errors associated with model discretization and can incorporate the user's knowledge of well loss. The approach is model independent, can be applied to finite difference or finite element models, and allows the numerical model to remain somewhat coarsely discretized and therefore numerically efficient. Because the correction is implemented external to the numerical model, one important benefit of this approach is that a response matrix, reduced model approach can be supported even when nonlinear well loss is considered.     

Sensitivity analysis of 2D steady-state shallow water flow. Application to free surface flow model calibration

This paper presents the analytical properties of the solutions of the sensitivity equations for steady-state, two-dimensional shallow water flow. These analytical properties are used to provide guidelines for model calibration and validation. The sensitivity of the water depth/level and that of the longitudinal unit discharge are shown to contain redundant information. Under subcritical conditions, the sensitivities of the flow variables are shown to obey an anisotropic elliptic equation. The main directions of the contour lines for water depth and the longitudinal unit discharge sensitivity are parallel and perpendicular to the flow, while they are diagonal to the flow for the transverse unit discharge sensitivity. Moreover, the sensitivity for all three variables extends farther in the transverse direction than in the longitudinal direction, the anisotropy ratio being a function of the sole Froude number. For supercritical flow, the sensitivity obeys an anisotropic hyperbolic equation. These findings are confirmed by application examples on idealized and real-world simulations. The sensitivities to the geometry, friction coefficient or model boundary conditions are shown to behave in different ways, thus providing different types of information for model calibration and validation.     

磁层顶边界区剪切流MHD不稳定性的非线性演化

本文利用MHD二维不可压模式,研究了地球磁层顶边界区剪切流引起的Kelvin-Helmholtz（K-H）不稳定性问题,得到了一个新的非线性微分方程组.理论和数值分析表明:该问题的非线性演化对初值非常敏感,而且在雷诺数和磁雷诺数给定的条件下,Alfven马赫数（M_A）对K-H不稳定性的非线性演化起决定性作用.这组方程蕴含几个吸引子,如不动点,极限环和奇异吸引子等,这体现了磁层顶非线性系统的复杂性.文中还发现背景磁场在磁层顶K-H不稳定性的非线性演化过程中起很重要的作用.     

The impact of flow discharge on the hydraulic characteristics of headcut erosion processes in the gully region of the Loess Plateau

Headcut erosion is associated with major hydraulic changes induced by the gully head of concentrated flow. However, the variation in the hydraulic characteristics of the headcut erosion process is still not clear in the gully region of the Loess Plateau. A series of rainfall combined scouring experiments (flow discharges ranging from 3.6 to 7.2 m³ hr⁻¹, with 0.8 mm min⁻¹ rainfall intensity) were conducted on experimental plots to clarify the variation in the hydraulic parameters induced by gully head and erosion processes under different flow discharges. The results showed that concentrated flows in the catchment area and gully bed were turbulent (Reynolds number ranging from 1,876 to 6,693) and transformed between supercritical and subcritical (Froude number ranging from 0.96 to 3.73). The hydraulic parameters, such as the flow velocity, Reynolds number, shear stress, stream power, Darcy–Weisbach friction factor, and unit stream power in the catchment area were 0.45–0.59 m s⁻¹, 2086–6693, 1.96–5.33 Pa, 0.89–2.86 W m⁻², 0.08–0.16, and 0.023–0.031 m s⁻¹, respectively. When the concentrated flows dropped from the gully head, the hydraulic parameters in the gully bed decreased by 3.39–26.07%, 1.49–29.99%, 65.19–67.14%, 67.25–74.96%, 28.53–61.31%, and 67.82–77.14%, respectively, which contributed to the flow energy consumption at the gully head. As flow discharge increased, Reynolds number, shear stress, and stream power increased, while flow velocity, Froude number, unit stream power, and Darcy–Weisbach friction factor did not. The flow energy consumption at the gully head was 9.66–10.13, 13.25–13.74, 15.68–16.41, and 19.28–20.25 J s⁻¹, respectively, under different flow discharges and accounted for 60.58–68.50% of the flow energy consumption of the experimental plots. Generally, the sediment discharges increased rapidly at the initial stage, then increased slowly, and finally reached a steady state condition, which showed a significant declining logarithmic trend with experimental duration (P<.01) and increased with increasing flow discharge. Accordingly, the flow energy consumption was significantly correlated with the sediment yield. These findings could improve our understanding of the hydraulic properties and flow energy characteristics of headcut erosion.