Modifications to the K-Profile parameterization with nondiffusive fluxes for Langmuir turbulence

The K-profile parameterization (KPP) is a common method to model turbulent fluxes in regional and global oceanic models. Many versions of KPP exist in the oceanic sciences community and one of their main differences is how they take the effects of nonbreaking waves into account. Although there is qualitative consensus that nonbreaking waves enhance vertical mixing due to the ensuing Langmuir circulations, there is no consensus on the quantitative aspects and modeling approach. In this paper we use a recently-developed method to estimate both components of KPP (the diffusive term, usually called local, and the nondiffusive component, usually called nonlocal) based on numerically-simulated turbulent fluxes without any a priori assumptions about their scaling or their shape. Through this method we show that the cubic shape usually used in KPP is not optimal for wavy situation and propose new ones. Furthermore we show that the formulation for the nondiffusive fluxes, which currently only depend on the presence of surface buoyancy fluxes, should also take wave effects into account. Finally, we investigate how the application of these changes to KPP improves the representation of turbulent fluxes in a diagnostic approach when compared to previous models.

Diffusive–Nondiffusive Flux Decompositions in Atmospheric Boundary Layers

Eddy diffusivity models are a common method to parameterize turbulent fluxes in the atmospheric sciences community. However, their inability to handle convective boundary layers leads to the addition of a nondiffusive flux component (usually called nonlocal) alongside the original diffusive term (usually called local). Both components are often modeled for convective conditions based on the shape of the eddy diffusivity profile for neutral conditions. This assumption of shape is traditionally employed due to the difficulty of estimating both components based on numerically simulated turbulent fluxes without any a priori assumptions. In this manuscript we propose a novel method to avoid this issue and estimate both components from numerical simulations without having to assume any a priori shape or scaling for either. Our approach is based on optimizing results from a modeling perspective and taking as much advantage as possible from the diffusive term, thus maximizing the eddy diffusivity. We use our method to diagnostically investigate four different large-eddy simulations spanning different stability regimes, which reveal that nondiffusive fluxes are important even when trying to minimize them. Furthermore, the calculated profiles for both diffusive and nondiffusive fluxes suggest that their shapes change with stability, which is an effect that is not included in most models currently in use. Finally, we use our results to discuss modeling approaches and identify opportunities for improving current models.

Modeling Fluid Flow in Tight Unconventional Reservoirs: Micro/Nano Scale Approach

The need for a tool to predict transport phenomena in petroleum unconventional tight reservoirs is placing more stringent demand on establishing more realistic models beyond the currently used viscous and inertial dependent models. Since diffusion is the principal mechanism in tight unconventional reservoirs that take place in both Micro and Nano scales, a diffusive term was added to the diffusivity model that operates both viscous and inertial forces, introduced by (Belhaj, et al., 2003). This diffusive term is a modified Fick’s 1st Law. It counts for the flow velocity caused by the diffusion process. Using the three-term model as a rate equation, in addition to the continuity equation and the EOS, a new model (a form of PDE) has been developed. The new model works ideally in unconventional tight reservoirs where oil and/or gas flow. The model has been numerically solved and tested. A comprehensive parametric study has been conducted and revealed clear trends. It has been concluded that diffusion mechanism contribution to flow increases with low permeability of the medium and low viscosity of the flowing fluid. An index (a combination of permeability and viscosity) has been developed and used to verify the influence and impact of the diffusion forces.

Vectorial dispersive shock waves on an incoherent landscape

We study the impact of temporal randomness on the formation of vectorial dispersive shock-waves that emerge due to the interaction of a partially coherent probe wave co-propagating together with an orthogonally polarized intense short pulse. Experiments carried out in a normally dispersive optical fiber demonstrate that the lack of coherence of the probe landscape acts as a strong diffusive term, which is able to hamper or inhibit the vectorial shock formation.

Swell Propagation through Submesoscale Turbulence

The propagation of ocean swells from generating regions to remote coastlines is affected by submesoscale turbulence in the surface flow field. The presence of submesoscale velocity variations results in random scattering of wave rays. While the interactions with these flow fields are weak, cumulative effects over oceanic scales are significant and result in observable changes in the wave field. Using geometrical optics and statistical mechanics we derive a framework to express these scattering effects on the mean wave statistics directly in terms of the variance spectrum of the submesoscale current field. The theoretical results are presented in Lagrangian and Eulerian forms, where the latter takes the form of a radiative transport equation augmented with a diffusive term in directional space. The theoretical results are verified through Monte Carlo simulations with a geometrical optics model. We show that including submesoscale scattering on ocean wave evolution can explain observed delays in swell arrivals, accelerated wave height decay, and much larger directional spreading of the wave field than predicted by geometrical spreading alone.

Chebyshev Spectral Collocation Method for Population Balance Equation in Crystallization

The population balance equation (PBE) is the main governing equation for modeling dynamic crystallization behavior. In the view of mathematics, PBE is a convection–reaction equation whose strong hyperbolic property may challenge numerical methods. In order to weaken the hyperbolic property of PBE, a diffusive term was added in this work. Here, the Chebyshev spectral collocation method was introduced to solve the PBE and to achieve accurate crystal size distribution (CSD). Three numerical examples are presented, namely size-independent growth, size-dependent growth in a batch process, and with nucleation, and size-dependent growth in a continuous process. Through comparing the results with the numerical results obtained via the second-order upwind method and the HR-van method, the high accuracy of Chebyshev spectral collocation method was proven. Moreover, the diffusive term is also discussed in three numerical examples. The results show that, in the case of size-independent growth (PBE is a convection equation), the diffusive term should be added, and the coefficient of the diffusive term is recommended as 2G × 10−3 to G × 10−2, where G is the crystal growth rate.

Getting started with controlled-source electromagnetic 1D modeling

Forward modeling is an important part of understanding controlled-source electromagnetic (CSEM) responses. The diffusive term in the electromagnetic wave equation is dominant over the displacement term at these frequencies. It is the diffusive behavior that makes it difficult to imagine the actual propagation of the signal. An important tool in gaining experience therefore is forward modeling, and lots of it. The advantage of one-dimensional (1D) forward modeling, besides its speed, is to study isolated effects (see for instance Key, 2009): What is the influence of resistivity anisotropy, or of fine-scale resistivity variations? What is the influence of the airwave? With 1D modeling you can quickly study these effects in isolation before you go on to more complex models in higher dimensions. For an introduction to CSEM for hydrocarbon exploration see, for instance, Constable (2010).

A Self-Adaptive Numerical Method to Solve Convection-Dominated Diffusion Problems

Convection-dominated diffusion problems usually develop multiscaled solutions and adaptive mesh is popular to approach high resolution numerical solutions. Most adaptive mesh methods involve complex adaptive operations that not only increase algorithmic complexity but also may introduce numerical dissipation. Hence, it is motivated in this paper to develop an adaptive mesh method which is free from complex adaptive operations. The method is developed based on a range-discrete mesh, which is uniformly distributed in the value domain and has a desirable property of self-adaptivity in the spatial domain. To solve the time-dependent problem, movement of mesh points is tracked according to the governing equation, while their values are fixed. Adaptivity of the mesh points is automatically achieved during the course of solving the discretized equation. Moreover, a singular point resulting from a nonlinear diffusive term can be maintained by treating it as a special boundary condition. Serval numerical tests are performed. Residual errors are found to be independent of the magnitude of diffusive term. The proposed method can serve as a fast and accuracy tool for assessment of propagation of steep fronts in various flow problems.