Lagrangian Model Research Articles

Changes in moisture sources contributed to the onset and development of the 2017-2019 southeast Australian drought

In 2017-2019 southeast Australia experienced one of its most severe droughts since 1900. Rainfall over the region encompassing the Murray-Darling Basin was consistently below average for three consecutive cool seasons, an unprecedented event on record. A strong positive Indian Ocean Dipole event has been previously suggested to have intensified the conditions of the drought in 2019, however the state of the climate drivers cannot fully explain the onset and development of this drought. In this study, we adopt a different approach to investigate processes other than remote climate drivers that may have triggered the drought. Using a Lagrangian model to identify moisture sources to the Murray Darling Basin, we show that local processes were crucial in explaining the onset and development of the drought. We identify the oceanic and terrestrial sources of atmospheric moisture over the Murray Darling Basin and show for the first time a significant decline in rainfall moisture supply from the Tasman Sea in 2017 and 2018. We further show that anomalous atmospheric circulation transported the expected moisture northward toward the Maritime Continent. Our results provide an explanation for the moisture and rainfall deficit that caused the 2017-19 southeast Australian drought. Understanding the processes that led to the 2017-2019 drought is an important step towards improved predictions and planning for future multi-year droughts in Australia.

On the autonomous validation and comparison of particle models for a Newtonian laminar flow mixing model using PEPT

A methodology for the validation of a Computational Fluid Dynamics (CFD) mixing model using Positron Emission Particle Tracking (PEPT) was applied to a horizontal laminar stirred tank containing Newtonian glycerol. Particle trajectories recorded using PEPT were compared to three one-way particle-coupled CFD models (CFD-DEM, and CFD-Lagrangian, and CFD-Massless) through time-averaging and binning into a grid of 3-D voxels. A cell-by-cell comparison of particle velocities between the computational models and PEPT highlighted the accuracy of the CFD-DEM and CFD-Massless models over the Lagrangian model. Furthermore, using PEPT as a validation dataset the Pearson r-squared was used as a cost function to guide an autonomous calibration of a 5-dimensional DEM parameter space. A novel statistic, the simulation quality coefficient Qsim, was applied to assess accuracy between trajectory datasets and revealed the extent to which a model can be considered calibrated. Considering the balance between computational cost and precision, the CFD-massless model was the most effective particle model for this validation workflow. Further investigations will apply this methodology to calibrate non-Newtonian laminar CFD models.

Control of the Weld Heterogeneity by Using a Novel Counter Variable Pin Shoulder Rotation Friction Stir Welding: A Simulation and Experimental Study

Abstract Friction stir welding (FSW) produces inhomogeneous mechanical and metallurgical properties in the weld, which further require post-weld processing to control the heterogeneity. In the present study, the heterogeneity in the weld is reduced through counter-variable rotation friction stir welding (CVRFSW). The material flow and temperature distribution significantly affect the inhomogeneity of the FSWed properties which has been studied by developing a three-dimensional Lagrangian method-based viscoplastic model. The material flow, strain rate, and temperature distribution in conventional FSW (CFSW) and CVRFSW are studied quantitatively. The study revealed that CVRFSW improved joint strength and reduced the inhomogeneity of temperature, strain, and hardness. At a 10% lower shoulder speed than a pin, the weld strength improved by 16%. The simulation predicted that the temperature difference between the advancing side (AS) and the retreating side (RS) was 36 °C in CFSW, which was reduced to 8 °C in CVRFSW. Material deformation in CVRFSW occurred at a strain rate more than twice that of CFSW, and the asymmetry of strain rate between AS and RS reduced to one-fifth. Microstructures and their orientations of the welds were studied in detail. These findings contribute to the understanding of CVRFSW processes for enhanced weld quality and mechanical performance for industrial applications.

Lagrangian modeling of a nonhomogeneous turbulent shear flow: Molding homogeneous and isotropic trajectories into a jet

Lagrangian modeling of a nonhomogeneous turbulent shear flow: Molding homogeneous and isotropic trajectories into a jet

A Lagrangian meshfree model for solidification of liquid thin-films

In this paper, a new method to model solidification of thin liquid films is proposed. This method is targeted at applications like aircraft icing and tablet coating where the formation of liquid films from impinging droplets on a surface form a critical part of the physics of the process. The proposed model takes into account the (i) unsteadiness in temperature distribution, (ii) heat transfer at the interface between the solid and the surface, (iii) volumetric expansion/contraction and (iv) the liquid thin-film behaviour, each of which are either partly or fully ignored in existing models. The liquid thin-film, modelled using the Discrete Droplet Method (DDM), is represented as a collection of discrete droplets that are tracked in a Lagrangian sense. The height of the liquid film is estimated as a summation of Gaussian kernel functions associated with each droplet. At each droplet location, a solid height is also computed. The evolution of the solid height is governed by the Stefan problem. The flow of the liquid thin-film is solved just as in the case of DDM, while also taking into consideration the shape of the solidified region lying beneath the droplet. The results presented in this work show the reliability of the proposed model in simulating solidification of thin-films and its applicability to complex problems such as ice-formation on aircraft wings. The model has been verified for canonical problems that have analytical solutions. For the more complex problems of icing, the results of the model are compared with data from literature, without considering a background air flow. The comparison can be improved by coupling this model with suitable air flow solvers, as shown in the final test case.

A closed-form Timoshenko beam element with multiple localized singularities for nonlinear material behavior using the strong discontinuity approach

The development of a novel closed-form beam element for nonlinear material behavior in structures is described, which is formulated according to Timoshenko beam theory and the embedded strong discontinuity approach. Multiple strong transverse and rotation discontinuities are considered to model several dissipative mechanisms due to localized singularities in the shear strain and curvature fields. Issues of dependence on numerical techniques, as occurs in displacement- and forced-based beam elements with the finite element method, are overcome since the formulated element is derived from a pure mathematical treatment. Theoretical principles are stemming from a Lagrangian variational model, from which singularities are naturally derived. The proposed formulation is solved by the conventional methods of calculus of variations, resulting in the boundary value problem. Then, closed-form expressions for any boundary conditions are obtained. By particularizing closed-form expressions for a basic system, a symmetric closed-form flexibility matrix is obtained. This matrix naturally leads to a symmetric closed-form stiffness matrix for any type of loading and unloading process, in which the nodeless degrees of freedom of the multiple embedded strong discontinuities are naturally condensed. Since Gauss integration methods are not used, the multiple localized singularities are not influenced by predefined positions of the integration points. However, the accuracy depends on the chosen number of singularities and their positions. Then, the proposed element is mesh independent when sufficient discontinuities are considered, which does not imply an increase in the degrees of freedom. Softening discontinuities are modeled at arbitrary positions in a single element, which overcomes the drawbacks of other embedded beam theories where more than one element is necessary to identify failure since strong discontinuities are generally located at the midpoint of the beam element. The obtained closed-form stiffness matrix is not always positive definite for strain-softening problems. Nevertheless, there are no major convergence issues as it is a symmetric matrix that does not require a constraint equation, so even the complex snap-back behavior in structures is adequately modeled. The capability of the formulated beam element for nonlinear material behavior in structures is validated with representative examples.

Lagrangian Stochastic Modeling of Stratified Atmospheric Boundary Layer

A single-column turbulence model for stratified atmospheric boundary layer (ABL), which solves the transport equations of turbulence probability density function (PDF) using a Lagrangian stochastic modeling (LSM) approach, is proposed in this study. This study adopts previously developed stochastic differential equations (SDEs) for particle velocity and temperature and extends the LSM to simulate inhomogeneous turbulence. The proposed LSM is tested for its ability to fully simulate statistics of inhomogeneous stratified turbulence. In the model, particles evolve by SDEs, and turbulence statistics are calculated by averaging the properties of particles. The model provides a full representation of turbulence PDF and simulates turbulent transport without any modeling assumption. The model performance is evaluated against large-eddy simulation (LES) results in the simulations of convective and stable ABL cases. For the convective ABL, LSM realistically simulates the entrainment process with the temperature and heat flux profiles that closely match with LES. The joint PDF simulated by LSM reproduces a curved and highly skewed shape, and some distinct features, like the asymmetric distribution of vertical velocity and the separation of the PDF in the entrainment zone, are simulated. LSM also reproduces the entrainment enhancement by wind shear in the simulation of sheared convective ABL. The LSM simulation of stable ABL predicts realistic turbulence intensity and mean field profiles, where Gaussian-like PDFs are simulated both in LSM and LES.

Lagrangian and Eulerian perspectives of turbulent transport mechanisms in a lateral cavity

The dynamics of turbulent flows past lateral cavities is relevant for multiple environmental applications. In rivers and coastal environments, these lateral recirculating regions constitute surface storage zones, where large-scale turbulent coherent structures control the transport and fate of contaminants. Mass transport in these flows is typically represented by one-dimensional first-order equations that predict the evolution of the spatially integrated concentration between the cavity and the main channel. These models, however, cannot represent the long-term evolution of the concentration or incorporate memory effects induced by turbulence. In this investigation, we carry out large-eddy simulations (LES) of the open-channel flow with a lateral square cavity of Mignot et al. (Phys. Fluids, vol. 28, issue 4, 2016, 045104). The model is coupled with an advection–diffusion equation and a Lagrangian particle model to investigate the transport mechanisms in the cavity and across the interface. From the simulations we provide quantitative comparisons of the physical processes from both perspectives, and investigate the effects of turbulent coherent structures on residence times and trajectories from finite-time Lyapunov exponents. From the Lagrangian results, we identify general spatial distributions of time scales in the cavity associated with the dynamics of coherent structures, providing new insights into the mechanisms that drive the global transport. We also show that an upscaled model informed by LES and based on a fractional derivative captures the evolution of concentration, and the exchange between the cavity and the main channel, providing accurate predictions of mass transport and reproducing the temporal dependence observed at larger scales.

A case study analysis of the impact of a new free tropospheric turbulence scheme on the dispersion of an atmospheric tracer

AbstractMost Lagrangian dispersion models represent free tropospheric turbulence as a homogeneous steady‐state process. However, intermittent turbulent mixing in the free troposphere may be a significant source of mixing. We test a new parametrization scheme that represents spatial‐ and temporal‐varying turbulence in the free troposphere in the Met Office's Numerical Atmospheric‐dispersion Modelling Environment. We use semi‐idealized emissions of radon‐222 (Rn) from rocks and soil in the United Kingdom to evaluate the impact of using a variable free tropospheric turbulence parameterization on the dispersion of Rn. We performed two experiments, the first using the existing steady‐state scheme and the second using the newly implemented spatio‐temporal‐varying scheme, for two case periods July 2018 and April 2021. We find that the turbulence in the varying scheme (represented by the vertical velocity variance) can range by two to three orders of magnitude (10 to 10 m s) when compared with the steady‐state scheme (10 m s). In particular, low‐altitude turbulence is enhanced when synoptic conditions are conducive to forming low‐level jets. This leads to a greater dispersion in the free troposphere, reducing the mean monthly Rn concentration above the boundary layer by 20–40% relative to the steady‐state scheme. We conclude that without a space–time‐varying free tropospheric turbulence scheme atmospheric dispersion may be significantly underestimated under synoptic conditions that are favourable for low‐level jet formation. This underestimation of dispersion may potentially result in inaccurate estimations of local emissions in top‐down greenhouse gas inventory studies.

Linearization Method for Large-Scale Hydro-Thermal Security-Constrained Unit Commitment

Security-constrained unit commitment (SCUC) is one of the most fundamental optimization problems in power systems. The objective of SCUC is to minimize the operating cost while respecting both system-wide and generator-specific constraints. It leads to a large-scale and mixed-integer programming (MIP) model with a large number of binary decision variables which is difficult to solve. This paper, based on the convex hull theory of single-unit, proposes a linearization method for the hydro-thermal SCUC problem with decoupled thermal units and variable-head hydro units. Then, the strategy of embedding two types of convex hulls in a multi-unit commitment and the heuristic method of constructing a feasible solution are designed, by which the multi-UC is approximated from large-scale mixed-integer programming to linear programming that can be solved in polynomial time. Finally, we theoretically prove that the optimal solution of the proposed LP model is always better than that of the Lagrangian Relaxation model. Numerical experiments on several large-scale test systems demonstrate the effectiveness and efficiency of the proposed method. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This paper proposes a linear programming model for the SCUC problem by lifting up to a higher-dimensional space. It realizes an important innovation in reducing the computational complexity of SCUC from the perspective of linearization. The proposed method can be well applied to large-scale long-term unit commitment problems. To better use this method, the following two properties should be highlighted: 1) the error of the proposed method is less than the Lagrangian relaxation method and decreases with the increasing system scales and 2) the computational efficiency of the proposed method is 10-100 times faster than that of the MIP model. We have tested many practical power systems and find that the error of the proposed LP model is usually very small compared with the precise MIP while the computational performance is significantly improved. In some practical cases, the decision makers usually do not want to find the precise optimal solution while only an approximation under a fast speed, because the boundary condition is imprecise. The proposed method is useful. Besides, for the cases that need the precise optimal solution, the proposed method can provide a high-quality initial solution for the MIP model to accelerate the convergence.

Eulerian- lagrangian dense discrete phase model (DDPM) of stenotic LAD coronary arteries in comparison with single phase modeling

In computational fluid dynamic studies related to blood flow, investigating the behavior of blood particles is crucial, especially red blood cells as they constitute a significant proportion of blood particles. Additionally, studying red blood cell movements is necessary, especially in stenotic artery geometries. A new multiphase scheme was utilized to demonstrate the effect of red blood cells on hemodynamics in complex coronary arteries and investigate the consequence of their motion. To investigate the effect of red blood cell movement on flow, the dense discrete phase model (DDPM) was used. This simulation was performed in 3D coronary arteries with different degrees of stenosis, utilizing blood pressure as inlet and outlet boundary conditions while assuming the arterial wall to be rigid. The model prediction shows good agreement with experimental data. Velocity values were comparable in both single-phase and two-phase flow simulations, but the shear stress in two-phase modeling had higher values. In the two-phase DDPM modeling, the recirculation areas indicated a higher probability of atherosclerosis plaque re-formation in the pre-stenosis area compared to the stenosis and post-stenosis areas. The DDPM model was found to be more effective in obtaining shear stress values in the artery. Additionally, this model provides good results compared to the single-phase model in investigating the movement of particles along the artery as well as recirculation areas that lead to the deposition of particles.

Investigation of the effect of ultrasonic vibration on the performance of the friction drilling by FEM simulation

The friction drilling process is very important in various industries such as automotive, aerospace, and oil and gas industries. Therefore, they have always sought to find methods to improve this process. One of these methods is to optimize the application of ultrasonic vibrations in the process, and limited research has been done in this regard. The current research aims to investigate the effect of ultrasonic vibrations simultaneously with the parameters of drilling on the friction drilling process to optimize this process with the help of finite element simulation. Due to the nature of the complex transformations of the process, a Coupled Eulerian–Lagrangian model of the process was first created in the Abaqus software. Then, the process simulation was carried out in the presence and absence of ultrasonic vibrations. Rotational speed, feed rate, and amplitude of vibrations are the influencing parameters, and sheet temperature and axial force, and torque applied to the tool are the parameters investigated in this research. The results show that higher rotation speed and lower feed rate lead to a reduction of axial force and torque. Also, the application of ultrasonic vibration reduces up to 50.48% the axial force and up to 46.67% torque, and this reduction is improved by increasing the range of vibrations. The optimal mode was selected After analyzing 12 different modes based on the lowest amount of axial force and torque on the pin. It was found that the ideal mode occurs at a feed rate of 4.23 m/s and a rotation speed of 3000 rpm, with a 20-µm range. These findings and optimizing drilling parameters bring new horizons in material bonding techniques in various industries.

Tracking Carbon Dioxide with Lagrangian Transport Simulations: Case Study of Canadian Forest Fires in May 2021

The large amounts of greenhouse gases, such as carbon dioxide, produced by severe forest fires not only seriously affect the ecosystems in the area where the fires occur but also cause a greenhouse effect that has a profound impact on the natural environment in other parts of the world. Numerical simulations of greenhouse gas transport processes are often affected by uncertainties in the location and timing of the emission sources and local meteorological conditions, and it is difficult to obtain accurate and credible predictions by combining remote sensing satellite data with given meteorological forecasts or reanalyses. To study the regional transport processes and impacts of greenhouse gases produced by sudden large-scale forest fires, this study applies the Lagrangian particle dispersion model Massive-Parallel Trajectory Calculations (MPTRAC) to conduct forward simulations of the CO2 transport process of greenhouse gases emitted from forest fires in the central region of Saskatchewan, Canada, during the period of 17 May to 25 May 2021. The simulation results are validated with the Orbiting Carbon Observatory-2 Goddard Earth Observing System (OCO-2 GEOS) Level 3 daily gridded CO2 product over the study area. In order to leverage the high computational costs of the numerical simulations of the model, we implement the forward simulations on the Tianhe-2 supercomputer platform and the JUWELS HPC system, which greatly improves the computational efficiency through parallel computation and makes near-real-time predictions of atmospheric transport processes feasible.

Insights into atmospheric trace gases, aerosols, and transport processes at a high-altitude station (2623 m a.s.l.) in Northeast Asia

As part of the efforts to understand long-range air pollution transport in Northeast Asia, a comprehensive field measurement campaign was conducted at the top of Changbai Mountain (2623 m a.s.l.) from 25th July to 21st August in 2021. The ambient concentrations of trace gases and chemical characteristics of fine particulate matter (PM2.5) were comparable to those at background or free tropospheric locations in China, but exceeded levels at corresponding sites in Europe and the United States. Well-defined diurnal variations were evident in most air pollutants, including carbon monoxide (CO), sulfur dioxide (SO2), and nitrogen oxide (NOx), reflecting the influence of boundary layer development and local anthropogenic activities. Ozone (O3) exhibited a unique diurnal distribution, with lower concentrations during the daytime and higher levels at nighttime, indicating a pronounced impact of downward transport of O3-rich air from the upper troposphere during the nighttime. During the campaign, organic carbon (OC) constituted the predominant component of PM2.5, comprising 43% of the total, while sulfate (SO42−) was the most abundant secondary inorganic component in PM2.5, accounting for 26%. Combining the analysis of air mass backward trajectories, we found that Changbai Mountain is primarily influenced by air masses originating from North Korea and the Sea of Japan, constituting 62% of the total, and local vegetation biological emissions and anthropogenic emissions in North Korea notably affect the observation site. Conversely, the secondary aerosol components were most abundant in marine air mass originating from the marginal seas of China and the strait between the Korean peninsula and southern Japan. Utilizing the Lagrangian photochemical trajectory model (LPTM), a significant increase in particulate SO42− concentrations was observed when air mass passed through the marginal seas of China, primarily due to SO2 oxidation reaction. This study contributes to an improved understanding of the atmospheric chemistry in the high-altitude regions of Northeast Asia and underscores the substantial role of secondary inorganic aerosol formation during the transport of marine air masses.

An Investigation on the use of air quality models in ship emission forecasts

Today's world trade operates on the basis of maritime transportation. Combating the environmental impacts of maritime transportation has become a global concern. MARPOL Annex VI contains rules for the prevention of air pollution from ships. Air pollution represents a more abstract pollution than other annexes of MARPOL. In this context, the first and most critical step is to measure air pollution. Nowadays it is possible to calculate ship emissions with individual ship activity data instead of the traditional, fuel-based approach. One of the most ideal data sources for this calculation is AIS data. The critical question to be answered in the literature is how to obtain the highest resolution output using AIS data. Here air quality modelling gains strategic importance. In this study, air quality models used in ship emission calculations were examined with the PRISMA method and the most commonly used Eulerian and Lagrangian models were discussed. For this purpose, studies using AIS data in ship emission calculations were separated through the Scopus database and air quality modelling studies were filtered. Thus, the profile of air quality models used in ship-borne air pollution studies in the literature was obtained and a methodological reference source was created for future studies.

Measurement of two phase flow concentration with a novel pressure difference method. I: Principle

Traditionally, the measurement of the concentration of a two-phase flow by gauging differential pressure is subject to pressure loss due to the flow friction along the pipe wall. A novel pressure difference method against this issue was proposed in this paper. A theoretic model is developed for the method, from the principle of hydrostatics of particle gravitational settlement. The method was essentially validated with an experiment of particle settling in a stand-alone pipe section by sealing the two terminals to form a static gas-solid two phase medium and supported with a numerical simulation based on a Lagrangian model as well. Inherently, the method is expected to measure the concentration of a two phase flow in a pneumatic conveying pipe, correspondingly by setting a flow branch which contains an erect settlement tube with two dynamically shutting valves to form a transient static particle deposition segment.

Extreme dry advection dominates the record-breaking Yangtze River heatwave in midsummer of 2022

The Yangtze River Valley (YRV) experienced an unprecedented heatwave in midsummer of 2022. Still, the detailed physical processes involved in the influence of abnormal large-scale atmospheric circulation on the heatwave remain unexplored. Here, we show that the positive meridional gradient of anomalous atmospheric moisture at the middle-lower troposphere and associated extreme dry air advection over the YRV are vital prerequisites for forming the 2022 YRV heatwave. The 2022 YRV heatwave is dominated by interannual variability, contributing 72.7% to the total temperature anomalies. Diagnosis of the surface heat budget equation indicates that the surface cloud radiative forcing is the most critical process in driving the 2022 YRV heatwave, which is dominated by the positive surface short-wave cloud radiative forcing associated with the suppressed precipitation and the middle-low clouds. The suppressed precipitation is induced by the vertical dynamical processes of anomalous moisture advection caused by the abnormal descending flows over the YRV, which are driven by the negative advection of anomalous latent heat energy by climatological meridional wind (anomalous dry air advection) according to the atmospheric moist static energy equation. Simulations from the Lagrangian model FLEXPART further indicate that the moisture anomaly over the north of YRV mainly originated from the surface evaporation in the YRV, implying that there is a positive land-air feedback during the life cycle of the YRV heatwave. Our study enriches the mechanism understanding of the 2022 YRV heatwave from the perspective of surface energy budget and land-air feedback.

Lagrangian and Eulerian modelling of 106Ru atmospheric transport in 2017 over northern hemisphere

In September 2017, numerous measurement stations recorded large surface concentrations of Ru106 in Europe. This event was well recorded by various monitoring stations worldwide and offer a valuable framework to compare the modelling strategies deployed to quickly evaluate where the plume goes and with what concentrations. In general, the source and its intensity are not known and hypotheses have to be done. Models have to be fast and accurate: Lagrangian and Eulerian are often used but rarely compared. In this study, the FLEXPART Lagrangian model and the WRF-CHIMERE Eulerian models are used to simulate the emissions, transport and deposition of this source of Ru106. First, it is shown that the hypothesis of location, timing and intensity of the source is realistic, by comparison to surface measurements. Second, sensitivity analysis performed with the Eulerian model and several transport scheme showed that this model may provide better results than the Lagrangian one. It opens the door to further development, including chemistry and mixing with other pollutants during these specific events.

Effectiveness of portable air cleaners in mitigating respiratory virus transmission risk

Portable air cleaners (PACs) have shown promising potential in reducing the risk of SARS-CoV-2 infection by effectively removing pollutant particles and optimizing airflow patterns. This study focused on a simulated scenario where an infected source and a susceptible person engage in conversation within a naturally ventilated room. By combining the Eulerian fluid method with the Lagrangian particle tracking model, a comprehensive insight into indoor airflow patterns and the dispersion of virus-laden droplets was gained. As deposited droplets may be resuspended or in contact thereby increasing the potential risk of infection, the deposition of droplets of different sizes in different susceptible areas was also specifically analyzed. The impacts of three variables, namely the configuration of the PAC’s opening, air flow rate, and positioning, on the transmission of virus-laden droplets were investigated. The results highlighted the significant role of PAC utilization in effectively capturing droplets emitted by the infected source and reducing virus concentration in the vicinity of the susceptible person, thereby mitigating the risk of transmission. Notably, the design and orientation of the suction opening emerged as crucial factors. Among the various cases studied, the optimal control and prevention performance against the virus was achieved with a virus concentration reduction rate of 97.4% when the PAC had an opening configuration with a larger single-sided suction opening facing the infected source, an airflow rate of 200 m3 h−1, and was positioned at the center of the tabletop between the infected source and the susceptible person. This research underscored the importance of employing PACs with appropriate settings to enhance indoor air quality and minimize the potential for SARS-CoV-2 transmission in similar scenarios.

Lagrangian modelling of reactive contaminant transport in the multi-component marine medium

The particle-tracking method for transporting radionuclides in multicomponent ocean medium (water and multifractional suspended and deposited sediments) is considered using a probabilistic approach for simulating interaction processes between several states of radioactivity. The state transformations as a result of reactions of the first order were described using the master equation for the probability of the particle being in the given state. Transition probabilities between all possible states can be obtained from the numerical solution of the matrix master equation that is derived in the paper. In the first approximation, the Euler method was used to obtain a solution for the next time step. This approach can be applied to any linear system of equations describing phase transitions with any number of states, but it requires small values of the transition probabilities to ensure only a single-phase change during one time step. The paper also focuses on deriving the Lagrangian interface conditions between the water column and bottom deposition. To apply the probabilistic approach, the boundary conditions were considered as the reaction terms in a thin near-bottom interface layer in which boundary conditions were converted into source terms. For this layer, the corresponding master equation was derived to obtain transitional probabilities for particle states. The developed approaches were tested on numerical and analytical solutions of two test cases. It was found that the optimal thickness of the interface layer must be larger than the maximum vertical displacement of the particle during the one-time step, but it must be small enough to approximate the condition of uniform distribution of concentration in this layer.