Advection Terms Research Articles

Mechanism of the Record-Breaking Heatwave Event Dynamics in South America in January 2022

Heatwaves in the Southern Hemisphere (SH) occur frequently but have received little attention over the years. This study presents a comprehensive analysis of a long-duration, wide-ranging, and high-intensity heatwave event in South America spanning from 9 to 16 January 2022. Before the heatwave occurred, the meridional sea surface temperature (SST) in the SH intensified due to the warming of the South Pacific, while the Southern Annular Mode (SAM) exhibited a positive phase. As a result, the intensified wave activities in the westerlies led to high-pressure anomalies in South America, which played a dominant role in the generation of the heatwave. The diagnostic analysis of thermodynamic equations in South America indicates that the temperature increase during the heatwave was primarily caused by the vertical advection term. In contrast, horizontal advection had a negative impact on surface warming. Additionally, the diabatic heating term associated with surface land types serves as a significant factor that cannot be disregarded. This study aims to deepen our understanding of the mechanisms underlying heatwave generation in South America, enabling the improved prediction of heatwaves and enhanced assessment of potential risks in the future.

The dissipative Generalized Hydrodynamic equations and their numerical solution

“Generalized Hydrodynamics” (GHD) stands for a model that describes one-dimensional integrable systems in quantum physics, such as ultra-cold atoms or spin chains. Mathematically, GHD corresponds to nonlinear equations of kinetic type, where the main unknown, a statistical distribution function f(t,z,θ), lives in a phase space which is constituted by a one-dimensional position variable z, and a one-dimensional “kinetic” variable θ, actually a wave-vector, called “rapidity”. Two key features of GHD equations are first a non-local and nonlinear coupling in the advection term, and second an infinite set of conserved quantities, which prevent the system from thermalizing. To go beyond this, we consider the dissipative GHD equations, which are obtained by supplementing the right-hand side of the GHD equations with a non-local and nonlinear diffusion operator or a Boltzmann-type collision integral. In this paper, we deal with new high-order numerical methods to efficiently solve these kinetic equations. In particular, we devise novel backward semi-Lagrangian methods for solving the advective part (the so-called Vlasov equation) by using a high-order time-Taylor series expansion for the advection fields, whose successive time derivatives are obtained by a recursive procedure. This high-order temporal approximation of the advection fields is used to design new implicit/explicit Runge–Kutta semi-Lagrangian methods, which are compared to Adams–Moulton semi-Lagrangian schemes. For solving the source terms, constituted by the diffusion and collision operators, we use and compare different numerical methods of the literature.

Analysis of travelling waves and propagating supports for a nonlinear model of flame propagation with a p-Laplacian operator and advection

In this paper, we propose a new model to characterize the behaviour of a flame driven by temperature and pressure variables. The model is formulated using a p-Laplacian operator, an advection term, and a nonlinear reaction (considering linear kinetics). First, the uniqueness and boundedness of the weak solutions are demonstrated. Subsequently, traveling wave solutions supported by the geometric perturbation theory are obtained. As a major outcome, minimum traveling wave speeds are shown to exist, for which the associated profiles of the solutions are purely monotonic with exponential behaviour. The assumptions considered in the analytical approach are further explored through a numerical assessment, and self-similar solutions are constructed to determine the evolution of the flame front in terms of the temperature and pressure variables.

Dynamics of a reaction-advection-diffusion model for cholera transmission with human behavior change

Cholera is a virulent intestinal infection caused by Vibrio cholerae strains. To explore multiple effects of spatial structure, seasonality and human behavior change on cholera transmission, we establish a reaction-advection-diffusion model with a general boundary condition at the downstream end of the river. We derive the basic reproduction number R0 and discuss its asymptotic behaviors. We further show that R0 is a strictly decreasing function of the bacterial loss c, which reveals that the bacterial loss at the downstream end of the river due to water flux reduces the risk of cholera transmission. Then we derive a threshold-type result in terms of R0. More precisely, (i) if R0<1, the disease-free periodic solution is globally attractive; (ii) if R0>1, the disease is uniformly persistent; (iii) if R0=1, the disease-free steady state is globally asymptotically stable in the absence of the advection term, in which the proof is quite challenging due to the introduction of human behavior change, involving some subtle inequalities and estimates. Numerically, an application is demonstrated by investigating the cholera outbreak in Somalia from 1995 to 2016.

Warm Season Extreme Flood Events in the Midwestern US—Sources of Moisture and Physical Mechanisms

AbstractA comprehensive picture of the development of warm season extreme floods in the Midwestern US is presented. We first identify the climatological moisture sources for precipitation in the Midwest using the two‐layer dynamic recycling model (2L‐DRM) with ECMWF Reanalysis v5 (ERA5) data. Terrestrial sources supply most of the moisture for Midwestern US precipitation during the warm season, while oceanic sources dominate during the cold season. The Empirical Orthogonal Function (EOF) analysis is used to select extreme flood events characterized by both positive soil moisture and precipitation anomalies. During the warm season flood events, moisture coming from oceanic sources increases by more than 45% compared to the climatology. In addition, our results show that moisture can come from remote regions due to sustained anomalous circulation. Low‐level circulation anomalies associated with wave trains that traverse the continent enhance moisture contributions from terrestrial sources along narrow paths. However, moisture budget analysis reveals that the primary flood‐producing mechanism is the convergence of moisture due to intense circulation anomalies. Moisture advection and thermodynamic terms are responsible for flood termination. Our results suggest that knowledge of antecedent wet soil moisture conditions is unlikely to improve the predictability of flood‐producing storms.

Theory of porous media with the advection term and mass exchange between phases

Earlier modeling approaches in the field of mixtures assumed the solid phase of the mixture was the predominant constituent and simplified the constitutive laws for the fluid phase. In the field of mixtures, it has been argued that methods such as the theory of poroelasticity are incapable of accurately capturing mass exchange between phases and stress and strain fields. Mass exchange between phases and accurate stress and strain variations of each phase are indispensable to many fields of study, such as bone remodeling, fault movement in rocks, and shock waves in soils. This study presents an alternative method for capturing the stress and strain fields of mixtures, focusing on the advection term in the balance of linear momentum and mass exchange for each phase. In addition, this extended theory has the potential for measuring the shear stress field of a viscous fluid component, considering complex boundary conditions, and modeling chemical reactions between mixture phases. The capability of the model has been evaluated by analyzing 2D and 3D examples, including a realistic bone remodeling scenario. Calculations have shown a maximum difference of 20% in the displacement fields. Thus, results clearly demonstrate the importance of considering the advection term and mass exchange between phases, as its application causes significant variations in the displacement and stress fields of the mixture. The developed model of this study can be considered as a general theory, which yields the classic poroelastic model with certain manipulations.

On the evolution of scalar iso-surface area density in a turbulent mixing layer

In this study, the transport equation for scalar iso-surface area density ( $\varSigma$ ) in a turbulent, temporally developing mixing layer is examined. Exploring the spatial and temporal evolution of the terms in the $\varSigma$ transport equation is vital to improving our understanding of turbulent flows characterized by distinct interfaces, e.g. the flame surface or the turbulent/non-turbulent interface. Previous work reported by the authors identified that $\varSigma$ exhibits self-similar behaviour consistent with the development of the temporal mixing layer (Blakeley et al., J. Fluid Mech., vol. 951, 2022, A44). Accordingly, each of the terms in the $\varSigma$ transport equation is found to behave in a self-similar manner, though there are notable differences in the self-similar behaviours for each term. Based on the results presented herein, it is suggested that the rate of change of $\varSigma$ ( $\partial \varSigma /\partial t$ ) and the advection term scale with $h\lambda _\varPhi /\Delta U$ , where $h$ is the width of the mixing layer, $\lambda _\varPhi$ is the scalar Taylor length scale and $\Delta U$ is the velocity difference. The production and destruction terms are found to scale with an additional factor $({Re}\,Sc)^{1/2}$ . In contrast, the molecular diffusion term is found to scale with a factor $({Re}\,Sc)^{-1/2}$ compared to $\partial \varSigma /\partial t$ . Importantly, it is found that the difference between the production and destruction terms, or net surface ‘stretch’, scales with the same factor as $\partial \varSigma /\partial t$ and the advection term, which may have a significant impact on how the evolution of $\varSigma$ is understood and modelled in turbulent flows.

A fictitious-play finite-difference method for linearly solvable mean field games

An iterative finite difference scheme for mean field games (MFGs) is proposed. The target MFGs are derived from control problems for multidimensional systems with advection terms. For such MFGs, linearization using the Cole-Hopf transformation and iterative computation using fictitious play are introduced. This leads to an implementation-friendly algorithm that iteratively solves explicit schemes. The convergence properties of the proposed scheme are mathematically proved by tracking the error of the variable through iterations. Numerical calculations show that the proposed method works stably for both one- and two-dimensional control problems.

Chimney‐Like Intense Pelagic Upwelling in the Center of Basin‐Scale Cyclonic Gyres in Large Lake Geneva

AbstractBasin‐scale quasi‐geostrophic gyres are common features of large lakes subject to Coriolis force. Cyclonic gyres are often characterized by dome‐shaped thermoclines that form due to pelagic upwelling that takes place in their center. At present, the dynamics of pelagic upwelling within the surface mixed layer (SML) of large lakes are poorly documented. A unique combination of high‐resolution 3D numerical modeling, satellite imagery and field observations allowed confirming, for the first time in a lake, the existence of intense pelagic upwelling in the center of cyclonic gyres under strong shallow (summer) and weak deep (winter) stratified conditions/thermocline. Field observations in Lake Geneva revealed that surprisingly intense upwelling from the thermocline to the SML and even to the lake surface occurred as chimney‐like structures of cold water within the SML, as confirmed by Advanced Very High‐Resolution Radiometer data. Results of a calibrated 3D numerical model suggest that the classical Ekman pumping mechanism cannot explain such pelagic upwelling. Analysis of the contribution of various terms in the vertically averaged momentum equation showed that the nonlinear (advective) term dominates, resulting in heterogeneous divergent flows within cyclonic gyres. The combination of nonlinear heterogeneous divergent flow and 3D ageostrophic strain caused by gyre distortion is responsible for the chimney‐like upwelling in the SML. The potential impact of such pelagic upwelling on long‐term observations at a measurement station in the center of Lake Geneva suggests that caution should be exercised when relying on limited (in space and/or time) profile measurements for monitoring and quantifying processes in large lakes.

A conservative network element method for diffusion-advection-reaction problems

We derive a conservative network element method for heterogeneous and anisotropic diffusion problems by modifying the non-conservative version, and extend it to the approximation of an additional advection term. The numerical scheme possesses the flux formulation reminiscent of classical finite volume methods. Its convergence is naturally governed by the network element theory. Numerical results illustrate the good behavior of the method even on distorted point clouds.

A very high order Flux Reconstruction (FR) method for the numerical simulation of 1-D compositional fluid flow model in petroleum reservoirs

Compositional reservoir simulation is an extremely important tool for modeling the fluid flow in complex petroleum reservoirs, as in cases where the reservoir fluid is volatile and composed of several pseudo-components with distinct characteristics, or reservoirs that require the use of enhanced oil recovery process. For these cases, simpler black-oil models are not suitable. The compositional model involves the solution of a large system of partial differential equations, that comprises the mass conservation, the Darcy’s law and the fugacity constraints. However, the high number of equations and constraints render the compositional problems extremely complex and computationally demanding solutions. To improve accuracy and reduce the high computational costs, one can use higher than first or even second-order methods to approximate the advective flux terms in the hyperbolic conservation laws that describe the multicomponent transport in the reservoir. Those very high-order schemes can replace low-order approaches, such as the First Order Upwind (FOU) method, which is traditionally used in commercial petroleum reservoir simulators (CMG, 2019; ECLIPSE, 2022), obtaining more accurate solutions with reduced computational cost. In this work, we consider a three-phase and isothermal fluid flow of water, oil and gas and assume that there is no mass transfer between the water and the hydrocarbon phases. Physical dispersion and capillary effects are neglected. To solve the system of Partial Differential Equations (PDEs), we used the classical IMPEC (Implicit Pressure Explicit Composition) approach and, to model the complex phase behavior, we used the Peng–Robinson equation of state (PR-EOS). The classical Two Point Flux Approximation (TPFA) method is used to spatially discretize the diffusion terms in the pressure equation. For the first time in literature the very high order Flux Reconstruction (FR) method is adapted for the numerical simulation of the compositional model in petroleum reservoirs. The FR method is a numerical scheme employed to discretize the hyperbolic conservation laws using general grids. To avoid spurious oscillations in the vicinity of solution discontinuities, the h-MLP (hierarchical Multi-dimensional Limiting Process) is applied in the reconstruction stage. For the time integration, we have used the third-order Runge–Kutta approach. To appraise the robustness of our formulation, we have solved some one-dimensional benchmark problems found in literature and compared the FR solution with the FOU method and with the second-order Monotonic Upstream-Centered Scheme for Conservation Laws (MUSCL) method. From our results, the FR method has shown improved accuracy, efficiency and capability of detecting sharper shocks when compared to the other methods that we have considered for the numerical modeling of the compositional flow in the present paper.

Double Hopf bifurcation induced by gestation and memory delays in a spatial model with directed movement

We narrow our focus on double Hopf bifurcations triggered by gestation and memory delays in a spatial model with tactical directed movement in this letter. We generalize the calculation algorithm of normal form concerning the double Hopf bifurcation to a spatial model with different delays in both the reaction and advection terms. We then divide the dynamics around the double Hopf bifurcation points into 6 categories and thus we observe that stable periodic orbits with different spatial modes can coexist.

Stochastic differential equation modelling of cancer cell migration and tissue invasion

Invasion of the surrounding tissue is a key aspect of cancer growth and spread involving a coordinated effort between cell migration and matrix degradation, and has been the subject of mathematical modelling for almost 30 years. In this current paper we address a long-standing question in the field of cancer cell migration modelling. Namely, identify the migratory pattern and spread of individual cancer cells, or small clusters of cancer cells, when the macroscopic evolution of the cancer cell colony is dictated by a specific partial differential equation (PDE). We show that the usual heuristic understanding of the diffusion and advection terms of the PDE being one-to-one responsible for the random and biased motion of the solitary cancer cells, respectively, is not precise. On the contrary, we show that the drift term of the correct stochastic differential equation scheme that dictates the individual cancer cell migration, should account also for the divergence of the diffusion of the PDE. We support our claims with a number of numerical experiments and computational simulations.

A reduced-order model based on integrated radial basis functions with partition of unity method for option pricing under jump–diffusion models

The current research aims to develop a fast, stable and efficient numerical procedure for solving option pricing problems in jump–diffusion models. A backward partial integro-differential equation (PIDE) with diffusion and advection terms was investigated. Up to the best knowledge of the authors, some special numerical methods and strategies must be selected to solve advection–diffusion problems with reliable stability and accuracy. For the mentioned aims, the first- and second-order derivatives are approximated by integrated radial basis function based on partition of unity method. The IRBF-PU method is local mesh-free method that provides high order accurate result and is flexible for PDEs problems with sufficiently smooth initial conditions and also has a moderate condition number. In particular, we highlight European and American style put options, whose underlying asset follows a jump–diffusion model. For the distribution of the jumps, the Merton and Kou models are studied. Furthermore, the main model is classified in advection-x-diffusion category. As a result, we must increase the number of collocation points as well as the time steps to arrive at the final time. This procedure lengthens the execution time. To address this issue, we use the proper orthogonal decomposition (POD) method to reduce the size of the final algebraic system of equations. This numerical procedure is known as the proper orthogonal decomposition-IRBF-PU method (POD-IRBF-PU). The presented computational results (including the computation of option Greeks) and comparisons with other competing approaches suggest that the IRBF-PU and POD-IRBF-PU methods are efficient and reliable numerical methods to solve elliptic and parabolic PIDEs arising from applied areas such as financial engineering.

Interannual variations of heat budget in the lower layer of the eastern Ross Sea shelf and the forcing mechanisms in the Southern Ocean State Estimate

AbstractThe temporal variation of heat budget in the lower layer of the eastern Ross Sea shelf (ERSS) is crucial for understanding the stability of ice shelves in the Ross Sea. In this study, a 6‐year (2005–2010) simulation from the Southern Ocean State Estimate (SOSE) is employed to analyse the interannual variations of heat budget in the lower layer of the ERSS and the controlling mechanisms. The results reveal that the annual change in the heat content of the study region is dominated by the horizontal heat advection term, and only in 2015 the vertical advection and diffusion terms also play an important role. The horizontal advection term is affected by the intrusion of warm circumpolar deep water (CDW) onto the shelf across the northern boundary of the ERSS, the transport of cold dense shelf water (DSW) from the western Ross Sea shelf, and the transport of CDW across the western boundary of the ERSS. The contributions of these physical processes to the change in annual heat content vary between years. The interannual variation of the CDW intrusion is associated with the strength of eddy activity over the slope, and the interannual variation of heat transports associated with DSW is affected by both the extent of DSW on the western shelf and the coastal circulations that affect the eastward spread of DSW.

Relating the artificial chemotaxis of catalysts to a gradient descent of the free energy.

Recent experiments suggest that mesoscale catalysts are active materials that power their motion with chemical free energy from their environment and also "chemotax" with respect to substrate gradients. In the present work, we explore a thermodynamic framework for relating this chemotaxis to the evolution of a system down the gradient of its free energy. This framework builds upon recent studies that have employed the Wasserstein metric to describe diffusive processes within the Onsager formalism for irreversible thermodynamics. In this work, we modify the Onsager dissipation potential to explicitly couple the reactive flux to the diffusive flux of catalysts. The corresponding gradient flow is a modified reaction-diffusion equation with an advective term that propels the chemotaxis of catalysts with the free energy released by chemical reactions. In order to gain first insights into this framework, we numerically simulate a simplified model for spherical catalysts undergoing artificial chemotaxis in one dimension. These simulations investigate the thermodynamic forces and fluxes that drive this chemotaxis, as well as the resulting dissipation of free energy. Additionally, they demonstrate that chemotaxis can delay the relaxation to equilibrium and, equivalently, prolong the duration of nonequilibrium conditions. Although future simulations should consider a more realistic coupling between reactive and diffusive fluxes, this work may provide insight into the thermodynamics of artificial chemotaxis. More generally, we hope that this work may bring attention to the importance of the Wasserstein metric for relating nonequilibrium relaxation to the thermodynamic free energy and to large deviation principles.

Numerical Accuracy Necessary for Large-Eddy Simulation of Planetary Boundary Layer Turbulence Using the Discontinuous Galerkin Method

Abstract In large-eddy simulations (LES), it is crucial to ensure that discretization errors do not contaminate the subgrid effect of the turbulence model in a wavelength range larger than the effective resolution. Recently, we showed that a seventh- or eighth-order accuracy is required for advection terms in planetary boundary layer simulations when using conventional gridpoint methods. However, a significant amount of communication between parallel computers is necessary to achieve high-order accuracy in gridpoint methods, and this can degrade computational efficiency. The discontinuous Galerkin method (DGM) is a promising approach for overcoming these limitations. Therefore, this study focuses on the numerical criteria of the DGM at LES from the viewpoint of numerical diffusion and dispersion. We extend our earlier study to the DGM framework and clarify the necessary order of the polynomial (p). We find that p = 4 is required based on the numerical criteria at the grid spacing of O(10) m with sufficiently scale-selective modal filters. The examination of temporal accuracy suggests that the fourth-order is sufficient when a fully explicit temporal scheme is used. In addition, we investigate the effect of hyperupwinding that is usually met when the Rusanov flux is employed in the low Mach number flows. It suggests that the choice of numerical flux has little effect on simulation results when the high-order DGM is used. Furthermore, we perform a series of LES in the planetary boundary layer and confirm that the indication obtained from the criteria holds for an actual LES.

High accuracy compact difference and multigrid methods for two-dimensional time-dependent nonlinear advection-diffusion-reaction problems

We focus on high-accuracy and stable numerical methods for the time-dependent nonlinear advection-diffusion-reaction problems in this work. A novel fourth-order fully implicit compact difference scheme is derived, in which we make use of the fourth-order compact formulas to discretize the diffusion terms, the fourth-order Padé formulas to compute the nonlinear advection terms, and the fourth-order backward differencing formula to approximate the time derivative term. Convergence and stability of the new scheme are analysed by the energy method. On the basis of the novel scheme, a multigrid algorithm is established such that a time advancement algorithm for solving these nonlinear problems are obtained. We provide various numerical experiments to verify the performances of the proposed scheme including the reliability, stability and computational efficiency. And the results obtained by our method show it is more accurate than most same kind schemes reported in the literature.

Spatiotemporal velocities, aggregation, and false dispersion: numerical considerations for the modeling of colonial and motile harmful algae

PurposeThe purpose of this study is to investigate the errors arising from the numerical treatment of model processes, paying particular attention to the impact of key system features including widely variable dispersion coefficients, spatiotemporal velocities of algal cells, and the aggregation of algae from single cells to large colonies. An advection–dispersion model has been presented to describe the vertical transport of colonial and motile harmful algae in a lake environment.Design/methodology/approachModel performance is examined for two different numerical treatments of the advective term: first-order upwind and quadratic upwind with a stability-preserving flux limiter (SMART). To determine how these schemes impact predictions, comparisons are made across a sequence of models with increasing complexity.FindingsUsing first-order upwinding for advection–dispersion calculations with a time oscillating velocity field leads to oscillatory numerical dispersion. Subjecting an initially uniform distribution of large-sized algal colonies to a spatiotemporal velocity creates a concentration pulse, which reaches a steady-state width at high-grid Peclet numbers when using the SMART scheme; the pulse exhibits contraction–expansion behavior throughout a velocity cycle at all Peclet numbers when using first-order upwinding. When aggregation dynamics are included with advection-dominated spatiotemporal transport, results indicate the SMART scheme predicts larger peak concentration values than those predicted by first-order upwind, but peak location and the time to large colony appearance remain largely unchanged between the two advective schemes.Originality/valueTo the best of the authors’ knowledge, this study is the first numerical investigation of a novel advection–dispersion model of vertical algal transport. In addition, a generalized expression for the effective dispersion coefficient of temporally variable flow fields is presented.

Winter and summer sedimentary dynamic process observations in the sea area off Qinhuangdao in the Bohai Sea, China

Sedimentary processes in marginal seas play an important role in the biology, physics, and geochemistry as well as ecology of coastal environments and contain abundant information about the material transfer from land to ocean and the regional circulation. Due to the huge sediment discharge of the Yellow River, the Bohai Sea, China is one of the areas with the highest suspended sediment concentration (SSC) in the world. Interestingly, the SSC at the west of the Bohai Sea is low all year round. Thus, it is of great significance to examine the sedimentary dynamic process in this area for better understanding the circulation structure, material exchange and regional environment of the Bohai Sea. Using seabed base observation platform measurements obtained in February 2017 and August 2019, this study examines the winter and summer hydrography and suspended sediment concentration in the sea area off Qinhuangdao located to the west of the Bohai Sea. In summer, the relatively weak residual currents flowed northeastward and showed little correlation with the wind field, especially in the middle layer of the water column. In winter, the residual currents were strengthened, flowing to the northeast during strong wind periods, and predominantly to the southwest during intermittent periods. Moreover, driven by the pressure gradient force associated with the wind-induced sea surface height variations, the winter current was closely related to the wind speed, with correlation coefficients ranging from 0.4 to 0.6 and a time lag of 10 h. The summer SSC was lower and mainly controlled by the tidal current, whereas in winter, owing to the enhanced Reynolds stress and turbulent kinetic energy, strong wind bursts triggered significant sediment resuspension and led to a higher SSC. For the suspended sediment flux (SSF), the advection terms contributed more than 80% in the winter and summer, while the vertical circulation terms contributed 13% in winter, and approximately half that much in summer. Generally, the suspended sediment is transported back and forth, with a little net northward and northeastward motions in winter and summer, respectively. This may explain the low SSC sustaining in coastal Qinhuangdao all year round. These results provide a reference for sedimentary studies conducted in other coastal waters, especially in monsoon-dominating shelf seas.