Hyporheic Exchange With and Without Traveling Surface Waves

Hyporheic exchange, the flow of surface water into and out of sediment substrates, play an important role in controlling temperature, pollutant, and dissolved oxygen levels in aquatic. A key driver for hyporheic flow is pressure variations over the water/sediment interface. Here, we compare and contrast, for a range flume conditions, hyporheic exchange in a flowing current with and without travelling surface waves. This is achieved experimentally by using a vertical array of electrical conductivity probes to track the movement of a conservative solute tracer in a gravel bed of a recirculating flume . In analyzing the experiment we fit a basic advection-dispersion model to the measured values of the solute uptake at our probe locations. This fitting shows, in the presence of waves, a significant (an order of magnitude or more) enhancement of the dispersion coefficient. Our hypothesis is that moving waves on the water surface drives a vertical pumping within the solute bed that, in turn, enhances the dispersion. This is confirmed by constructing a numerical simulation that directly accounts for the wave induced pressure fluctuations at the water/sediment interface and a vertically oscillating (pumping) velocity within the bed itself. We show that, on appropriate setting of the fluctuation amplitude (within the expected experimental range), predictions from the simulation exactly recover the fits to the experimental measurement

Chaos in a simple model of a delta network

The flux partitioning in delta networks controls how deltas build land and generate stratigraphy. Here, we study flux-partitioning dynamics in a delta network using a simple numerical model consisting of two orders of bifurcations. Previous work on single bifurcations has shown periodic behavior arising due to the interplay between channel deepening and downstream deposition. We find that coupling between upstream and downstream bifurcations can lead to chaos; despite its simplicity, our model generates surprisingly complex aperiodic yet bounded dynamics. Our model exhibits sensitive dependence on initial conditions, the hallmark signature of chaos, implying long-term unpredictability of delta networks. However, estimates of the predictability horizon suggest substantial room for improvement in delta-network modeling before fundamental limits on predictability are encountered. We also observe periodic windows, implying that a change in forcing (e.g., due to climate change) could cause a delta to switch from predictable to unpredictable or vice versa. We test our model by using it to generate stratigraphy; converting the temporal Lyapunov exponent to vertical distance using the mean sedimentation rate, we observe qualitatively realistic patterns such as upwards fining and scale-dependent compensation statistics, consistent with ancient and experimental systems. We suggest that chaotic behavior may be common in geomorphic systems and that it implies fundamental bounds on their predictability. We conclude that while delta “weather” (precise configuration) is unpredictable in the long-term, delta “climate” (statistical behavior) is predictable

A general non-Fourier Stefan problem formulation that accounts for memory effects

The Stefan problem is the classical model of a melting phase change. In heterogeneous systems, such phase changes can exhibit non-Fourier (anomalous) behaviors, where the advance of the melt interface does not follow the expected time scaling. These situations can be modeled by replacing the derivatives, in the governing partial differential equations, with fractional order derivatives. In particular, replacing the time derivatives leads to non-Fourier models that account for memory effects in the system. In this work, by using appropriate time convolution integrals, a general thermodynamic balance statement for melting phase problems, explicitly accounting for memory effects, is developed. From this balance, a gen- eral model formulation applicable to problems involving melting over a temperature range (i.e., a mushy region) is derived. A key component in this model is the representation of memory effects through the use of fractional derivative based constitutive models of the enthalpy and heat flux. On shrinking the mushy region to a single isotherm, a general sharp interface melting model is obtained. Here, in con- trast to the classic Stefan problem, the fractional derivatives induce a natural regularization, such that the constitutive models for enthalpy and heat flux are continuous at the melt interface; a result con- firmed through numerical simulation. To further support the theoretical findings, a physical example of a non-Fourier Stefan problem is presented. Overall the development and results in this paper underscore the importance of explicitly relating the development of fractional calculus models to the appropriate thermodynamic balance statements.Fil: Voller, Vaughan R.. University of Minnesota; Estados UnidosFil: Roscani, Sabrina Dina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Austral. Facultad de Ciencias Empresariales. Departamento de Matemáticas; Argentin

On an enthalpy formulation for a sharp-interface memory-flux Stefan problem

Stefan melting problems involve the tracking of a sharp melt front during the heat conduction controlled melting of a solid. A feature of this problem is a jump discontinuity in the heat flux across the melt interface.Time fractional versions of this problem introduce fractional time derivatives into the governing equations. Starting from an appropriate thermodynamic balance statement, this paper develops a new sharp interface time fractional Stefan melting problem, the memory-enthalpy formulation. A mathematical analysis reveals that this formulation exhibits a natural regularization in that, unlike the classic Stefan problem, the flux is continuous across the melt interface. It is also shown how the memory-enthalpy formulation, along with previously reported time fractional Stefan problems based on a memory-flux, can be derived by starting from a generic continuity equation and melt front condition. The paper closes by mathematically proving that the memory-enthalpy fractional Stefan formulation is equivalent to the previous memory-flux formulations. A resultthat provides a thermodynamic consistent basis for a widely used and investigated class of time fractional (memory) Stefan problems.Fil: Roscani, Sabrina Dina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Austral. Facultad de Ciencias Empresariales. Departamento de Matemáticas; ArgentinaFil: Voller, Vaughan R.. University of Minnesota; Estados Unido

Evaluation of a Field Permeameter to Measure Saturated Hydraulic Conductivity of Base/Subgrade Materials

This report presents the results of a cooperative study on the field use of a permeameter, built by researchers at the Minnesota Department of Transportation (Mn/DOT) and the University of Minnesota, to estimate the saturated hydraulic conductivity of pavement base materials. Field measurements using the permeameter were taken on various highway construction projects, and researchers measured the saturated hydraulic conductivity of samples in the laboratory. Researchers also reviewed theories for converting a field-measured flow rate into a saturated hydraulic conductivity estimate. By numerical simulation and analysis of the field data, researchers determined an appropriate method for converting the Mn/DOT permeameter flow measurements into estimates of hydraulic conductivity. Variations between the field estimated and laboratory measured hydraulic conductivity are within one order of magnitude. Variations between the field estimate and numerical simulation, however, are much closer. The study found the Mn/DOT permeameter can be used to obtain a reliable estimate of the base hydraulic conductivity provided that the base layer is at least 15 cm (six inches) deep. When the base is to thin, permeameter readings are restricted to early infiltration times.Minnesota Department of TransportationClyne, Timothy R; Voller, Vaughan; Birgisson, Bjorn. (2001). Evaluation of a Field Permeameter to Measure Saturated Hydraulic Conductivity of Base/Subgrade Materials. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/737

Fractional Stefan problems exhibiting lumped and distributed latent-heat memory effects

We consider fractional Stefan melting problems which involve a memory of the latent-heat accumulation. We show that the manner in which the memory of the latent-heat accumulation is recorded depends on the assumed nature of the transition between the liquid and the solid phases. When a sharp interface between the liquid and the solid phases is assumed, the memory of the accumulation of the latent heat is "lumped" in the history of the speed of the interface. In contrast, when a diffuse interface is assumed, the memory of the accumulation is "distributed" throughout the liquid phase. By use of an example problem, we demonstrate that the equivalence of the sharp-and diffuse-interface models can only occur when there is no memory in the system. DOI: 10.1103/PhysRevE.87.04240

A Geometric Model for the Dynamics of a Fluvially Dominated Deltaic System Under Base-Level Change

We present a geometric model to study the role of base-level change in the dynamics of the alluvial-bedrock transition and shoreline positions in a fluvially dominated deltaic system. The domain of the problem is a sediment wedge in the long-profile cross-section. On assuming that the fluvial surface has a quadratic form, its evolution is determined by imposing an overall volume balance, and conditions for the elevations and slopes at the domain boundaries. This results in a coupled system, involving one ordinary differential equation and one non-linear equation. These equations are solved through an explicit Euler time stepping algorithm to predict the movement of the shoreline and alluvial-bedrock transition boundaries under a wide range of base-level change conditions. The mathematics of the approach are verified by comparing predictions from the geometric model with a closed form solution of a downslope gravity-driven transport model under the specific case of a square-root of time base-level change. Testing with more general base-level change scenarios reveals that this simple geometric mass balance is able to predict system dynamics that are fully consistent with both physical and numerical experiments. Moreover, model predictions under a base-level cycle (fall-rise) suggest a behavior where river incision occurs during the base-level rise stage, a predicted dynamic that has not been previously reported

Author's personal copy Morphology of a melt front under a condition of spatial varying latent heat ☆

A melting model with spatially varying latent heat (applicable to the formation of sedimentary basins) is numerically investigated. The geometry is a rectangular channel with linearly increasing latent heat in the length direction and a step contrast in slope in the width direction. The melt front is driven by a heat flux. After introducing suitable scaling, it is shown that the significant evolution of the shape of the melt front occurs within 5 widths of the flux boundary and that the melt-thickness (the downstream length between the most and least advanced parts of the melt front) scales with the log of the ratio of latent heat slopes