Numerical Diffusion Research Articles

EDGE: a new approach to suppressing numerical diffusion in adaptive mesh simulations of galaxy formation

ABSTRACT We introduce a new method to mitigate numerical diffusion in adaptive mesh refinement (AMR) simulations of cosmological galaxy formation, and study its impact on a simulated dwarf galaxy as part of the ‘EDGE’ project. The target galaxy has a maximum circular velocity of $21\, \mathrm{km}\, \mathrm{s}^{-1}$ but evolves in a region that is moving at up to $90\, \mathrm{km}\, \mathrm{s}^{-1}$ relative to the hydrodynamic grid. In the absence of any mitigation, diffusion softens the filaments feeding our galaxy. As a result, gas is unphysically held in the circumgalactic medium around the galaxy for $320\, \mathrm{Myr}$, delaying the onset of star formation until cooling and collapse eventually triggers an initial starburst at z = 9. Using genetic modification, we produce ‘velocity-zeroed’ initial conditions in which the grid-relative streaming is strongly suppressed; by design, the change does not significantly modify the large-scale structure or dark matter accretion history. The resulting simulation recovers a more physical, gradual onset of star formation starting at z = 17. While the final stellar masses are nearly consistent ($4.8 \times 10^6\, \mathrm{M}_{\odot }$ and $4.4\times 10^6\, \mathrm{M}_{\odot }$ for unmodified and velocity-zeroed, respectively), the dynamical and morphological structure of the z = 0 dwarf galaxies are markedly different due to the contrasting histories. Our approach to diffusion suppression is suitable for any AMR zoom cosmological galaxy formation simulations, and is especially recommended for those of small galaxies at high redshift.

New strategies for vertical transport in chemistry transport models: application to the case of the Mount Etna eruption on 18 March 2012 with CHIMERE v2017r4

Abstract. Excessive numerical diffusion is one of the major limitations in the representation of long-range transport by chemistry transport models. In the present study, we focus on excessive diffusion in the vertical direction, which has been shown to be a major issue, and we explore three possible ways of addressing this problem: increasing the vertical resolution, using an advection scheme with anti-diffusive properties and more accurately representing the vertical wind. This study was carried out using the CHIMERE chemistry transport model for the 18 March 2012 eruption of Mount Etna, which released about 3 kt of sulfur dioxide into the atmosphere in a plume that was observed by satellite instruments (the Infrared Atmospheric Sounding Interferometer instrument, IASI, and the Ozone Monitoring Instrument, OMI) for several days. The change from the classical Van Leer (1977) scheme to the Després and Lagoutière (1999) anti-diffusive scheme in the vertical direction was shown to provide the largest improvement to model outputs in terms of preserving the thin plume emitted by the volcano. To a lesser extent, the improved representation of the vertical wind field was also shown to reduce plume dispersion. Both of these changes helped to reduce vertical diffusion in the model as much as a brute-force approach (increasing vertical resolution).

Applications of phase field fracture in modelling hydrogen assisted failures

The phase field fracture method has emerged as a promising computational tool for modelling a variety of problems including, since recently, hydrogen embrittlement and stress corrosion cracking. In this work, we demonstrate the potential of phase field-based multi-physics models in transforming the engineering assessment and design of structural components in hydrogen-containing environments. First, we present a theoretical and numerical framework coupling deformation, diffusion and fracture, which accounts for inertia effects. Several constitutive choices are considered for the crack density function, including choices with and without an elastic phase in the damage response. The material toughness is defined as a function of the hydrogen content using an atomistically-informed hydrogen degradation law. The model is numerically implemented in 2D and 3D using the finite element method. The resulting computational framework is used to address a number of case studies of particular engineering interest. These are intended to showcase the model capabilities in: (i) capturing complex fracture phenomena, such as dynamic crack branching or void-crack interactions, (ii) simulating standardised tests for critical components, such as bolts, and (iii) enabling simulation-based paradigms such as Virtual Testing or Digital Twins by coupling model predictions with inspection data of large-scale engineering components. The evolution of defects under in-service conditions can be predicted, up to the ultimate failure. By reproducing the precise geometry of the defects, as opposed to re-characterising them as sharp cracks, phase field modelling enables more realistic and effective structural integrity assessments.

A modified switching technique for advection and capturing of surfaces

An interface capturing scheme called modified switching technique for advection and capturing of surfaces (MSTACS) has been proposed. The proposed interface capturing scheme utilizes a basic switching technique for advection and capturing of surfaces (STACS) for solution of the volume fraction advection equation. MSTACS has been compared against three interface capturing schemes with the help of various two and three dimensional test cases. It is proved that MSTACS is able to capture sharp interfaces with minimum numerical diffusion over a wide range of Courant numbers. Further, the interface capturing capability of MSTACS is demonstrated by coupling it with the Navier-Stokes equations (NSE) to simulate three dimensional flow problems such as Rayleigh-Taylor instability and dam break with an obstacle.

Synergistic nanostructure and heterointerface design propelled ultra-efficient in-situ self-transformation of zinc-ion battery cathodes with favorable kinetics

Abstract In-situ self-transformation is proved to be an effective strategy to design high-performance cathodes for aqueous zinc-ion batteries (ZIBs). However, the inferior transformation efficiencies during phase transition limit its further application. Herein, a 3D spongy VO2-graphene (VO2-rG) precursor has been designed for achieving the ultra-efficient in-situ self-transformation process from VO2-rG into multifaceted V2O5·nH2O-graphene composite (VOH-rG). Benefiting from the highly conductive heterointerfaces, rich reaction sites and numerous ions diffusion channels of VO2-rG, almost 100% VO2 nanobelts are converted into VOH during the first charging with few side reactions, indicating a highly efficient transformation kinetics. This strategy enables structural modulation from micro-nano level to molecular level by integrating pre-inserted H2O molecules and constructing 3D porous heterogeneous architecture into the VOH-rG cathode simultaneously, leading to fast and enduring Zn2+ (de)intercalation kinetics. Consequently, the VOH-rG cathode exhibits high capacity of 466 mA h g−1 at 0.1 A g−1, superior rate performance (190 mA h g−1 even at 20 A g−1) and excellent cycling stability with 100% capacity retention over 5000 cycles. Moreover, the assembled VOH-rG//Zn flexible quasi-solid-state batteries also present impressive performance. Such an ultra-efficient in-situ self-transformation strategy would pave a new way to explore promising electrode materials for advanced energy storage.

Comparison of Eulerian Bin and Lagrangian Particle-Based Microphysics in Simulations of Nonprecipitating Cumulus

AbstractA single nonprecipitating cumulus congestus setup is applied to compare droplet spectra grown by the diffusion of water vapor in Eulerian bin and particle-based Lagrangian microphysics schemes. Bin microphysics represent droplet spectral evolution applying the spectral density function. In the Lagrangian microphysics, computational particles referred to as superdroplets are followed in time and space with each superdroplet representing a multiplicity of natural cloud droplets. The same cloud condensation nuclei (CCN) activation and identical representation of the droplet diffusional growth allow the comparison. The piggybacking method is used with the two schemes operating in a single simulation, one scheme driving the dynamics and the other one piggybacking the simulated flow. Piggybacking allows point-by-point comparison of droplet spectra predicted by the two schemes. The results show the impact of inherent limitations of the two microphysics simulation methods, numerical diffusion in the Eulerian scheme and a limited number of superdroplets in the Lagrangian scheme. Numerical diffusion in the Eulerian scheme results in a more dilution of the cloud upper half and thus smaller cloud droplet mean radius. The Lagrangian scheme typically has larger spatial fluctuations of droplet spectral properties. A significantly larger mean spectral width in the bin microphysics across the entire cloud depth is the largest difference between the two schemes. A fourfold increase of the number of superdroplets per grid volume and a twofold increase of the spectral resolution and thus the number of bins have small impact on the results and provide only minor changes to the comparison between simulated cloud properties.

Change in Concentration-Dependent Diffusion Activation Energy and Frequency Factor with Time: Identified by Numerical Analyses of Experimental Data

A newly developed fully explicit numerical diffusion model coupled with a forward simulation technique, to avoid flaws in conventional methods such as the Boltzmann–Matano, Sauer–Freise and Hall procedures, is used to compute the concentration-dependent activation energy and frequency factor for diffusion in two different alloy systems. In contrast to what has been commonly reported in the literature, the results show that the concentration dependence of activation energy and frequency factor can significantly change with time. In this situation, the use of concentration-dependent activation energy and frequency factor that does not vary with time to predict diffusion effect can lead to considerable error. Furthermore, when there is a time variation in the dependence of activation energy and frequency factor on concentration, a common expectation that diffusion-controlled kinetics would increase with an increase in temperature may not always apply for all time durations.

Hydraulic Transient Analysis of Sewer Pipe Systems Using a Non-Oscillatory Two-Component Pressure Approach

On the basis of the two-component pressure approach, we developed a numerical model to capture mixed transient flows in close conduit systems. To achieve this goal, an innovative Godunov finite-volume numerical scheme is proposed to suppress the spurious numerical oscillations occurring during rapid pipe pressurization. To dissipate the spurious numerical oscillations, we admit artificial numerical viscosity to the numerical scheme through applying a proposed Harten, Lax, and van Leer (HLL) Riemann solver for calculating the numerical fluxes at the computational cell interfaces. The proposed solver controls the magnitude of the numerical viscosity through adjusting the left and right wave velocities. A wave velocity calculator is proposed to optimally distribute the numerical viscosity over several computational cells around the computational cell in which the pressurization front is located. The proposed solver admits significant artificial numerical viscosity when the pipe pressurization is imminent and automatically reduces it in other places; in this way the numerical diffusion and data smearing is minimized. The validity of the proposed model is justified by the aid of several test cases in which the numerical results are compared with both experimental data and the results obtained from analytical methods. The results reveal that the proposed model succeeds in completely removing the spurious numerical oscillations, even when the pipe acoustic speed is over 1000 m/s. The numerical results also show that the model can successfully capture occurrence of negative pressures during the course of transient flow.

Verification of a Python-based TRANsport Simulation Environment for density-driven fluid flow and coupled transport of heat and chemical species

Abstract. Numerical simulation has become an inevitable tool for improving the understanding on coupled processes in the geological subsurface and its utilisation. However, most of the available open source and commercial modelling codes do not come with flexible chemical modules or simply do not offer a straight-forward way to couple third-party chemical libraries. For that reason, the simple and efficient TRANsport Simulation Environment (TRANSE) has been developed based on the Finite Difference Method in order to solve the density-driven formulation of the Darcy flow equation, coupled with the equations for transport of heat and chemical species. Simple explicit, weighted semi-implicit or fully-implicit numerical schemes are available for the solution of the system of partial differential equations, whereby the entire numerical code is composed of less than 1000 lines of Python code, only. A diffusive flux-corrected advection scheme can be employed in addition to pure upwinding to minimise numerical diffusion in advection-dominated transport problems. The objective of the present study is to verify the numerical code implementation by means of benchmarks for density-driven fluid flow and advection-dominated transport. In summary, TRANSE exhibits a very good agreement with established numerical simulation codes for the benchmarks investigated here. Consequently, its applicability to numerical density-driven flow and transport problems is proven. The main advantage of the presented numerical code is that the implementation of complex problem-specific couplings between flow, transport and chemical reactions becomes feasible without substantial investments in code development using a low-level programming language, but the easy-to-read and -learn Python programming language.

Design and Synthesis of Novel 2D Porous Zinc Oxide-Nickel Oxide Composite Nanosheets for Detecting Ethanol Vapor.

The porous zinc oxide-nickel oxide (ZnO-NiO) composite nanosheets were synthesized via sputtering deposition of NiO thin film on the porous ZnO nanosheet templates. Various NiO film coverage sizes on porous ZnO nanosheet templates were achieved by changing NiO sputtering duration in this study. The microstructures of the porous ZnO-NiO composite nanosheets were investigated herein. The rugged surface feature of the porous ZnO-NiO composite nanosheets were formed and thicker NiO coverage layer narrowed the pore size on the ZnO nanosheet template. The gas sensors based on the porous ZnO-NiO composite nanosheets displayed higher sensing responses to ethanol vapor in comparison with the pristine ZnO template at the given target gas concentrations. Furthermore, the porous ZnO-NiO composite nanosheets with the suitable NiO coverage content demonstrated superior gas-sensing performance towards 50–750 ppm ethanol vapor. The observed ethanol vapor-sensing performance might be attributed to suitable ZnO/NiO heterojunction numbers and unique porous nanosheet structure with a high specific surface area, providing abundant active sites on the surface and numerous gas diffusion channels for the ethanol vapor molecules. This study demonstrated that coating of NiO on the porous ZnO nanosheet template with a suitable coverage size via sputtering deposition is a promising route to fabricate porous ZnO-NiO composite nanosheets with a high ethanol vapor sensing ability.

On the convergence of a finite volume method for the Navier–Stokes–Fourier system

Abstract The goal of the paper is to study the convergence of finite volume approximations of the Navier–Stokes–Fourier system describing the motion of compressible, viscous and heat-conducting fluids. The numerical flux uses upwinding with an additional numerical diffusion of order $\mathcal O(h^{ \varepsilon +1})$, $0<\varepsilon <1$. The approximate solutions are piecewise constant functions with respect to the underlying polygonal mesh. We show that the numerical solutions converge strongly to the classical solution as long as the latter exists. On the other hand, any uniformly bounded sequence of numerical solutions converges unconditionally to the classical solution of the Navier–Stokes–Fourier system without assuming a priori its existence. A similar unconditional convergence result is obtained for a sequence of numerical solutions with uniformly bounded densities and temperatures if the bulk viscosity vanishes.

Effects of terrain‐following vertical coordinates on simulation of stratus clouds in numerical weather prediction models

AbstractMany numerical weather prediction models employ terrain‐following vertical coordinates. As a consequence, over orography, flat tops of stratus clouds are intersected by the vertical coordinate surfaces. We conduct idealised two‐dimensional simulations of a stratus cloud with the COSMO model to study the effect of such sloping vertical coordinate surfaces. The evolution of the stratus cloud above a flat surface within a horizontally homogeneous atmosphere serves as a reference. During night‐time, the cloud thickens, driven by radiative cooling at the cloud top. Adding a sinusoidal perturbation to the vertical coordinate surfaces reduces the growth of the stratus cloud. With strong perturbations, the cloud starts to dissipate. The physical processes in the two simulations are identical, hence this behaviour is purely driven by numerical diffusion. The cloud is similarly thinned when sinusoidal orographic features are introduced. The reduction depends on the amplitude and wavelength of the perturbations and hills. Increasing the horizontal resolution partly mitigates the numerical diffusion. However, this is a very costly measure for an operational weather model. We suggest conducting further research on a new vertical coordinate with additional local smoothing of the orographic signal.

The potential for aqueous fluid-rock and silicate melt-rock interactions to re-equilibrate hydrogen in peridotite nominally anhydrous minerals

AbstractHydrogen is a rapidly diffusing monovalent cation in nominally anhydrous minerals (NAMs, such as olivine, orthopyroxene, and clinopyroxene), which is potentially re-equilibrated during silicate melt-rock and aqueous fluid-rock interactions in massif and abyssal peridotites. We apply a 3D numerical diffusion modeling technique to provide first-order timescales of complete hydrogen re-equilibration in olivine, clinopyroxene, and orthopyroxene over the temperature range 600–1200 °C. Model crystals are 1–3 mm along the c-axis and utilize H+ diffusion coefficients appropriate for Fe-bearing systems. Two sets of models were run with different boundary compositions: (1) “low-H models” are constrained by mineral-melt equilibrium partitioning with a basaltic melt that has 0.75 wt% H2O, and (2) “high-H models,” which utilize the upper end of the estimated range of mantle water solubility for each phase. Both sets of models yield re-equilibration timescales that are identical and are fast for all phases at a given temperature. These timescales have strong log-linear trends as a function of temperature (R2 from 0.97 to 0.99) that can be used to calculate the expected re-equilibration time at a given temperature and grain size. At the high end of the model temperatures (1000–1200 °C), H+ completely re-equilibrates in olivine, orthopyroxene, and clinopyroxene within minutes to hours, consistent with previous studies. These short timescales indicate that xenolith NAM mantle water contents are likely to be overprinted prior to eruption. The models also resolve the decoupled water-trace element relationship in Southwest Indian Ridge peridotites, in which peridotite REE abundances are reproduced by partial melting models whereas the relatively high NAM H2O contents require later re-equilibration with melt.At temperatures of 600–800 °C, which correspond to conditions of hydrothermal alteration of pyroxene to amphibole and talc, H+ re-equilibration typically occurs over a range of timescales spanning days to years. These durations are well within existing estimates for the duration of fluid flow in oceanic hydrothermal systems, suggesting that peridotite NAM water contents are susceptible to diffusive overprinting during higher temperature hydrothermal alteration. Thus, diffusion during aqueous fluid-rock interactions may also explain NAM H2O contents that are too high to reflect residues of melting. These relatively short timescales at low temperatures suggest that the origin of water contents measured in peridotite NAMs requires additional constraints on sample petrogenesis, including petro-graphic and trace element analyses. Our 3D model results also hint that H+ may diffuse appreciably during peridotite serpentinization, but diffusion coefficients at low temperature are unconstrained and additional experimental investigations are needed.

Robust and efficient 3-D numerical model for the hydrodynamic simulation of tsunami wave on land

This study presents the development, refinement, and application of a 3-D multiphase flow model for the simulation of tsunami propagation on land and its hydrodynamic force on a coastal building. The model is based on the Volume/Surface Integrated Average-based Multi-Moment Method (VSIAM3) with improved accuracy by adopting the temporary moment (TM) method to update the non-normal flow variables. A third-order gradient approximation is chosen for the reconstruction of an interpolation profile in the CIP-CLS3 advection solver to further reduce numerical diffusion and suppress numerical oscillation. The moving interface between air–water is captured by using the volume of fluid (VOF) method. The Tangent of Hyperbola for Interface Capturing (THINC) scheme is used to reproduce a sharp and smear-less air–water interface. The model performance improved by as much as 16% in the estimation of impact force on the downstream wall in the dam-break flow benchmarking problem by refining the original VSIAM3 model. The robustness of the model is further demonstrated by the accurate reproduction of tsunami wave propagation speed and its hydrodynamic pressures on a single box-shaped obstacle. The wave-induced pressures on the front face of the obstacle are accurately reproduced, with a maximum of 10% difference compared with the hydraulic experimental results. Thus, the study shows that the overall robustness and hydrodynamic performance of the model can be improved by refining the original model.

Diffusion coefficients in nanoporous solids derived from membrane permeation measurements

Based on the molecular understanding of the interplay of diffusion and adsorption, new membrane materials can be developed and the operational conditions of gas separation membranes can be optimized. Therefore, numerous diffusion and adsorption studies are conducted to optimize membrane materials. However, in an opposite way, transport or Fickian diffusion coefficients DT can be derived from membrane permeation studies with surprising accuracy. From measuring the gas transport through nanoporous supported thin-layer membranes or through mixed matrix membranes with nanoporous fillers in a polymer matrix, the transport diffusion coefficients DT of gases in novel nanoporous materials such as zeolites, MOFs, COFs… can be estimated.

Implicit-explicit-compact methods for advection diffusion reaction equations

We provide a comparison of the dispersion properties, specifically the time-amplification factor, the scaled group velocity, the error in the phase speed and scaled numerical diffusion coefficient of four spatiotemporal discretization schemes utilized for solving a linear, one-dimensional (1D) as well as a linear/nonlinear, two-dimensional (2D) advection diffusion reaction (ADR) equation: (a) Explicit (RK2) temporal integration combined with the Optimal Upwind Compact Scheme (or OUCS3, J. Comp. Phys., 192, pg. 677–694 (2003)) and the central difference scheme (CD2), (b) Fully implicit mid-point rule for time integration coupled with the OUCS3 and the Lele’s compact scheme (J. Comp. Phys., 103, pg. 16–42 (1992)), (c) Implicit (mid-point rule)-explicit (RK2) or IMEX time integration blended with OUCS3 and Lele’s compact scheme (where the IMEX time integration follows the same ideology as introduced by Ascher et al., SIAM J. Numer. Anal., 32(3), pg. 797–823 (1995)), and (d) IMEX (mid-point/RK2) time integration melded with the New Combined Compact Difference scheme (or NCCD scheme, J. Comp. Phys., 228, pg. 6150–6168 (2009)). Analysis reveals the superior resolution features of the IMEX-OUCS3-Lele scheme and the IMEX-NCCD scheme including an enhanced region of asymptotic stability (a region with numerical amplification factor less than unity), a diminished region of spurious propagation characteristics (or a region of negative scaled group velocity) and a smaller region of nonzero phase speed error. In particular, the IMEX-NCCD scheme captures the correct propagation feature (or positive scaled group velocity) in the largest possible region in the wavenumber-Courant-Friedrichs-Lewy (CFL) number, parameter space, in comparison with the other three numerical methods. The in silico experiments investigating the role of q−waves in the numerical solution of the linear, 1D ADR equation divulge excellent Dispersion Relation Preservation (DRP) properties of the IMEX-NCCD scheme including minimal dissipation via implicit filtering and negligible unphysical oscillations (or Gibbs’ phenomenon) on coarser grids. The numerical solution of the 2D viscous Burgers’ Equation underline the supremacy of the IMEX method with regard to handling the ‘stiff’ derivative terms in contrast with a fully explicit time integration method and lower computational time versus with a fully implicit scheme. The DRP resolution of the IMEX-NCCD scheme is further benchmarked by solving the classical two-dimensional (2D), Patlak-Keller-Segel (PKS) nonlinear parabolic model. Numerical results reveal that the spiky structure of the solution is oscillation free and, when compared with the Explicit-OUCS3-CD2 method, the solution is better resolved by the IMEX-NCCD method.

Drop Size Distribution Broadening Mechanisms in a Bin Microphysics Eulerian Model

Abstract In this study, processes that broaden drop size distributions (DSDs) in Eulerian models with two-moment bin microphysics are analyzed. Numerous tests are performed to isolate the effects of different physical mechanisms that broaden DSDs in two- and three-dimensional Weather Research and Forecasting Model simulations of an idealized ice-free cumulus cloud. Sensitivity of these effects to modifying horizontal and vertical model grid spacings is also examined. As expected, collision–coalescence is a key process broadening the modeled DSDs. In-cloud droplet activation also contributes substantially to DSD broadening, whereas evaporation has only a minor effect and sedimentation has little effect. Cloud dilution (mixing of cloud-free and cloudy air) also broadens the DSDs considerably, whether or not it is accompanied by evaporation. This mechanism involves the reduction of droplet concentration from dilution along the cloud’s lateral edges, leading to locally high supersaturation and enhanced drop growth when this air is subsequently lifted in the updraft. DSD broadening ensues when the DSDs are mixed with those from the cloud core. Decreasing the horizontal and vertical model grid spacings from 100 to 30 m has limited impact on the DSDs. However, when these physical broadening mechanisms (in-cloud activation, collision–coalescence, dilution, etc.) are turned off, there is a reduction of DSD width by up to ~20%–50% when the vertical grid spacing is decreased from 100 to 30 m, consistent with effects of artificial broadening from vertical numerical diffusion. Nonetheless, this artificial numerical broadening appears to be relatively unimportant overall for DSD broadening when physically based broadening mechanisms in the model are included for this cumulus case.

New hybrid compact schemes for structured irregular meshes using Birkhoff polynomial basis

This paper discusses the formulation of a class of hybrid compact schemes for structured irregular meshes. First, we derived three- and five-point compact schemes with required boundary closures for non-periodic problems. Second, we devised hybrid compact schemes from the explicit representation of the first- and second-order derivatives on irregular staggered and collocated meshes, respectively. For staggered meshes, hybrid interpolation schemes are also formulated. The main advantage of the hybrid formulation is that it requires boundary closures only for the compact scheme for first-order derivatives on the collocated mesh. Resolution and added numerical diffusion properties of hybrid schemes are assessed by using global spectral analysis (Sengupta et al., 2003). Direct numerical simulations of 2D dispersive shallow water and incompressible Navier-Stokes equations at different Reynolds numbers with fourth-order RK4 time integration method are considered to validate the accuracy and efficiency of new schemes. Numerical solutions are also compared with results reported in the literature.

Identifying the key challenges for fog and low stratus forecasting in complex terrain

Forecasting fog and low stratus (FLS) accurately poses a challenge to current numerical weather prediction models, despite many advancements in recent years. We present a novel method to quantify FLS extent bias by comparing forecasts with satellite observations. Evaluating a four‐month period, we show that COSMO‐1, the MeteoSwiss high‐resolution operational model, exhibits a considerable negative FLS bias during wintertime. To study the cause, we conduct a series of sensitivity experiments for a representative case study, where COSMO‐1 dissipated extensive FLS erroneously. Replacing the one‐moment bulk microphysics parameterisation scheme by a two‐moment scheme, as well as increasing the number of vertical levels, did not show any improvements. The FLS dissipation was delayed (but not prevented) by decreasing the lower bound imposed on the turbulent diffusion coefficients from 0.4 to 0.01 m2·s−1, or by reducing horizontal grid spacing from 1.1 km to 550 m. Additionally, simulations at 1.1‐km grid spacing with smoothed orography led to more extensive FLS than the same simulations without smoothed orography. An analysis of the cloud water budget revealed that the model's advection scheme is causing a loss of liquid water content near the cloud top. A simulation with an alternative terrain‐following coordinate system, in which the vertical coordinates are quasihorizontal near the cloud top, reduced the loss of cloud water through advection and improved the evolution of FLS in the case study. In combination, our findings suggest that the advection scheme exhibits numerical diffusion, which promotes spurious mixing in the vertical of cloudy and adjacent cloud‐free grid cells in terrain‐following vertical coordinates; this process can become the root cause for too rapid dissipation of FLS during nighttime in complex terrain.

Stress Formation During In - Ga Interdiffusion in Thin-Film CuIn1−xGaxSe2 Absorber Layers Leads to Stable Ga Gradients

To optimize the opto-electronic properties of compound semiconductors, a detailed understanding and control of compositional gradients forming during their synthesis is crucial. A common fabrication process for $\mathrm{Cu}(\mathrm{In},\mathrm{Ga}){\mathrm{Se}}_{2}$ (CIGS) thin-film solar cells uses annealing at high temperatures, which---contrary to what could be expected from simple Fickian diffusion---results in the formation of steep and stable $\mathrm{Ga}$ gradients, deviating from the optimal $\mathrm{Ga}$ profile for high-efficiency CIGS absorbers. Here, we show that the formation of elastic stresses inside the material during the interdiffusion can have a profound effect on the final $\mathrm{Ga}$ distribution, resulting in persistent $\mathrm{Ga}$ gradients inside CIGS absorber layers. A comparison of numerical diffusion and stress-formation calculations with real-time synchroton-based energy-dispersive x-ray diffraction data acquired in-situ during selenization of CIGS thin films demonstrates that the model can reproduce the stagnation of $\mathrm{In}$-$\mathrm{Ga}$ interdiffusion. We discuss that a detailed understanding of the interplay between stress and diffusion processes in thin films may open alternative fabrication strategies for generating desired stable compositional gradients to improve the opto-electronic properties of compound semiconductors, such as chalcopyrite, kesterite, and perovskite solar cell absorbers for solar cells.