Grid Spacing Research Articles

Full minimal coupling Maxwell-TDDFT: An ab initio framework for light-matter interaction beyond the dipole approximation

We report an , nonrelativistic QED method that couples light and matter self-consistently beyond the electric dipole approximation and without multipolar truncations. This method is based on an extension of the Maxwell-Pauli-Kohn-Sham approach to a full minimal coupling Hamiltonian, where the space- and time-dependent vector potential is coupled to the matter system, and its back reaction to the radiated fields is generated by the full current density. The implementation in the open-source code is designed for massively parallel multiscale simulations considering different grid spacings for the Maxwell and matter subsystems. Here, we show applications of this framework to simulate renormalized Cherenkov radiation of an electronic wave packet, magneto-optical effects with nonchiral light in nonchiral molecular systems, and renormalized plasmonic modes in a nanoplasmonic dimer. We show that in some cases, the beyond-dipole effects cannot be captured by a multipolar expansion Hamiltonian in the length gauge. Finally, we discuss further opportunities enabled by the framework in the field of twisted light and orbital angular momentum, inelastic light scattering, and strong-field physics. Published by the American Physical Society 2025

Experimental research on the two-phase turbulent mixing between rod bundle subchannels with spacer grid

Experimental research on the two-phase turbulent mixing between rod bundle subchannels with spacer grid

The Relevance of Mean-State Critical Levels for the Intensification of Downslope Winds in a Coastal Mountainous Environment

Abstract In wildfire-prone coastal Santa Barbara, California, downslope winds observed on the southern slopes of the east–west-oriented Santa Ynez Mountains are known as Sundowner winds (or Sundowners). One important feature of Sundowners is the remarkable spatial and temporal variability in lee-slope jet characteristics. Besides the intensity of the flow approaching the mountain range, the acceleration of the lee-slope jet can be influenced by reflected gravity waves associated with one or more of the following mechanisms: a self-induced critical level, an inversion close to mountaintop, and the presence of a mean-state critical level (MSCL). The relative contribution of these mechanisms to the enhancement of Sundowners is yet unknown. This study uses 32-yr simulations (hourly, 1-km grid spacing) complemented with observations collected during the Sundowner Winds Experiment (SWEX) to better quantify the relative contribution of these mechanisms and to quantify the importance of MSCLs. We show that when an MSCL is present below 5 km, less atmospheric forcing is necessary to attain similar lee-slope jet strengths compared to when MSCLs are absent or above 5 km. This was evidenced from simulations and verified with observations. Although MSCLs during Sundowners occur year-round, their relative frequency increases in summer, when temperatures are high and fuels are dry, enhancing wildfire risk. Properly identifying these processes contributes to improved understanding and predictability of Sundowners and many other hazardous downslope windstorms in coastal environments. Significance Statement Wildfires in coastal Santa Barbara, Southern California, are among the most disruptive hazards and are influenced by strong winds that accelerate down the southern slopes of the Santa Ynez Mountain range. These winds are locally known as Sundowner winds, due to their appearance around sunset. Our goal is to understand how different atmospheric situations influence the surface winds at the south-facing slopes of the mountain range. We find that when the atmosphere north of this mountain range contains a wind reversal (i.e., northerly winds turning southerly aloft), weaker winds below this reversal will induce winds on the south-facing slope that are stronger than with similar upstream winds without a reversal. Our findings are relevant to improve forecasting of Sundowner winds.

An Idealized Parameter Study of Destabilization and Convection Initiation in Coastal Regions. Part II: Onshore Synoptic-Scale Flow

Abstract While it has been established that pronounced sea-breeze fronts (SBF) develop under conditions of weak or offshore flow, with the SBF serving as a focus for convection initiation (CI), less is known regarding CI processes during large-scale onshore flow, including the potential existence and influence of embedded SBF. Challenges that have limited progress in addressing this question include the limited focus on cases under this flow regime and the coarse grid spacing used in prior model-based parameter studies, artificially decreasing the magnitude of convergence or thermodynamic gradients associated with the SBF and potentially leading to substantial error in representation of surface fluxes. In Part II of this two-part series, the authors address these prior challenges through execution of a suite of Cloud Model 1 (CM1) large-eddy simulations analyzing the impact of varying the magnitude of the onshore wind component (+2.5 and +5.0 m s−1), and radiative forcing and water surface temperature consistent for summertime in the Great Lakes region. In all simulations, a mesoscale enhancement of onshore flow and return flow were present, with a thermal gradient and convergence co-located with the leading edge of this enhanced flow providing evidence for the presence of a diffuse yet discernible SBF. The SBF progressed inland and separated areas where CI did and did not occur, highlighting the need for increased focus on SBF and CI in this less-studied flow regime. In conditions of weak onshore flow, surface-based CAPE was larger, CI was more common, and occurred closer to the coast.

Effect of numerical resolution on synthetic observables of simulated coronal loops

Abstract Increasingly realistic simulations of the corona are used to predict synthetic observables for instruments onboard both existing and upcoming heliophysics space missions. Synthetic observables play an important role in constraining coronal heating theories. Choosing the spatial resolution of numerical simulations involves a trade-off between accuracy and computational cost. Since the numerical resolution not only affects the scale of structures that can be resolved, but also thermodynamic quantities such as the average coronal density, it is important to quantify the effect on synthesized observables. Using 3D radiative MHD simulations of coronal loops at three different grid spacings, from 60 km down to 12 km, we find that changes in numerical resolution lead to differences in thermodynamic quantities and stratification as well as dynamic behaviour. Higher grid resolution results in a more complex and dynamic atmosphere. The resolution affects the emission intensity as well as the velocity distribution, thereby affecting synthetic spectra derived from the simulation. The distribution of synthetic coronal loop strand sizes changes as more fine-scale structure is resolved. A number of parameters, however, seem to start to saturate from our chosen medium grid resolution on. Our study shows that while choosing a sufficiently high resolution matters when comparing forward-modelled observables with data from current and future space missions, for most purposes not much is gained by further increasing the resolution beyond a grid spacing of 24 km, which seems to be adequate for reproducing bulk loop properties and forward-modelled emission, representing a good trade-off between accuracy and computational resource.

Closing the Circulation Budget

AbstractCirculation budgets can identify physical processes underpinning tropical cyclones, mesoscale convective vortices, and other weather systems where there are interactions across scales. It is unclear, however, how well these budgets close in practice. The present study uses the rapid intensification of Tropical Cyclone Nepartak (2016) as a case study to quantify the practical limitations of calculating circulation budgets using standard reanalyzes and numerical weather model data. First, we evaluate the circulation budget with ERA5. The budget residual can be reduced considerably by including contributions to circulation changes from subgrid‐scale momentum transports, and reduced further with 24‐hr smoothing, which dampens the discontinuous effects of data assimilation. Second, using a high‐resolution Met Office Unified Model simulation, we examine how the choice of the path used (the domain boundary) affects the budget closure. Third, the truncation errors associated with numerical differentiation in time and space are investigated. The circulation budget improves as the model data are analyzed with more frequent time output intervals, and as the output grid spacing decreases. For the tropical convective examples evaluated here, the column mean budget residuals increase by up to 50% as the output intervals increase from 5 min to 3 hr. Errors also increase if the data are regridded to a coarser horizontal grid spacing and when convection straddles the domain boundary. A key result is that the circulation budget need not close for physical inferences made about the circulation and its evolution to be meaningful, thus validating the use of the technique in prior studies.

Refinement of Finite Element Method Analysis Model of Pressurized Water Reactor Nuclear Fuel Spacer Grid Based on Experimental Data

A Finite Element Method (FEM) analysis of the nuclear fuel spacer grid was conducted to assess the strength of components for the safety of nuclear power plants. The fuel assembly consists of fuel rods, upper end-fitting, lower end-fitting, guide tubes, and spacer grids. Spacer grids play a critical role in maintaining the proper spacing between fuel rods within a fuel assembly and ensuring smooth coolant flow. This role becomes particularly crucial during unforeseen emergencies, such as seismic loads, where minimizing deformation caused by external forces is essential. Therefore, this study proposes FEM models of spacer grids, mesh refinement of models, and analysis of the stiffness of the spacer grids concerning the presence or absence of pellet and clad. The results revealed significantly lower shear stiffness compared to normal stiffness, indicating potential vulnerability of the fuel assembly to large loads such as those experienced during seismic events.

The Development of a Regional, High-Resolution OSSE Framework to Study the Impact of Observations from Uncrewed Aircraft Systems on Numerical Weather Prediction

Abstract Using Uncrewed Aircraft Systems (UAS) for routine weather observations is becoming increasingly feasible, but two open questions are the impact UAS observations will have on operational NWP and the optimal configuration of a UAS observing network. Deploying hundreds to thousands of UAS across the US to answer these questions is not feasible, so we propose using an Observing System Simulation Experiment (OSSE) instead. This article describes the development and validation of an OSSE framework for examining the impact of UAS observations on regional, convection-allowing NWP. The OSSE includes two week-long Nature Runs (NRs) over CONUS with 1-km grid spacing, simulated conventional observations, a 3-km forecast system similar to the prototype Rapid-Refresh Forecast System (RRFS), and verification workflows. Comparisons between the NR and various observations show that the NR generally mirrors reality, though the NR tends to produce too many reflectivity objects that are too large and larger coverage of intense precipitation rates compared to reality. Comparing real-data and OSSE RRFS runs indicates that, for the majority of the variables examined, forecast errors are not systematically smaller in the OSSE, suggesting that there is no identical twin issue. Data-denial experiments using either real or simulated conventional observations indicate that the OSSE has a similar ordering of most to least impactful observations, though the impact of aircraft observations tend to be larger in the OSSE. Altogether, these results suggest the OSSE framework can be used to glean meaningful information about UAS observation impacts on regional NWP.

Km-scale climate simulations over Madeira and Canary Islands under present and future conditions: a model intercomparison study

This study seeks to explore two new km-scale regional climate simulations prepared through the European Climate Prediction project over the Madeira and Canary islands, which are Portuguese and Spanish archipelagos located in the North Atlantic, off the African coast. The simulations are based on two models using different modelling approaches: COSMO-CLM with a three-step nesting at 50, 25 and 3 km grid spacing using a time-slice approach driven by a global climate model, and COSMO-crCLIM with a two-step nesting at 12 and 1 km grid spacing using the pseudo-global warming approach, where the current-day simulations are driven by ERA-Interim reanalysis. Although the modelling approaches are different, several findings are highlighted: (1) the use of km-scale simulation is essential to properly represent temperature and precipitation mean and extremes over small islands that are characterized by complex topography; (2) the projected changes in temperature and precipitation mean and extremes are qualitatively similar in all seasons except autumn; (3) the differences in the autumn projections are shown to be due to the large-scale driving conditions. Small islands, such as the Canary and Madeira ones, are often neglected by large modelling initiatives, so the presented simulations contribute to filling this gap for local policy makers, stakeholders and climate services. The encouraging results highlight the need for further coordinated km-scale projections.

Two-phase flow evolution and interfacial area transport downstream of the mixing-vane spacer grid in rod bundle channels

This study investigated the axial two-phase flow evolution and interfacial area transport characteristics downstream of the mixing vane spacer grid (MVSG) in a tight lattice rod bundle channel with a subchannel hydraulic diameter of Dh = 17.27 mm. The experiment was conducted under the air-water two-phase flow at room temperature and atmospheric pressure. The phase distributions of 6 positions downstream of MVSG (Z/Dh = 9.15, 14.94, 20.73, 26.52, 32.31, and 61.26) were measured utilizing a self-developed double-layer wire-mesh sensor (WMS). 42 cases were obtained, involving the bubbly flow, cap-bubbly flow, and slug flow. Void fraction, bubble velocity, interfacial area concentration (IAC), and bubble size distribution (BSD) databases were built. Under the group-1 (G-1) flow, due to the swirling flow generated by MVSG, the G-1 bubbles were drawn from the bundle surface and the gap region into the subchannel center to form a spindle core-peak distribution. BSD results demonstrate that the swirling flow promotes bubbles’ coalescence. The one-dimensional (1-D) void fraction and IAC downstream of the MVSG decrease first and then recover, which contrasts with the non-mixing vane spacer grid (N-MVSG) effect. It was attributed to the increment of the bubbles’ velocity after MVSG due to the bubbles’ migration to the channel center and coalescence. Under the group-2 (G-2) flow, MVSG intensifies the core peak distribution of void fraction, and the void fraction profile is also spindle-shaped. The obvious bubbles’ break-up occurs at Z/Dh = 14.94 attributed to the swirling decay. The 1-D void fraction and IAC transport indicate, that with the liquid-velocity increment, the MVSG effect is different from the N-MVSG effect, which is mainly achieved by affecting the G-1 bubbles’ dynamic behaviors. The model evaluation utilizing the interfacial area transport equation (IATE) closing bubble interaction source/sink terms indicates that the advection effect dominates the IAC evolution and the contribution of the pressure effect is weak. The remarkable bubbles’ coalescence occurs under the high liquid velocity. In future studies, the additional bubble break-up and coalescence source/sink term model considering the MVSG effect should be developed based on these cases to improve the IATE prediction ability.

Validation of the Ensemble Generalized Analog Regression Downscaling (En-GARD) Model to Downscale Near-Surface Wind Speed for the Northeast United States

Abstract Wind, often overlooked in climate change impact assessments, is now a crucial atmospheric variable due to our imminent reliance on renewable energy sources like wind energy. Therefore, establishing a wind speed database in higher temporal and spatial resolution is of great importance to assess wind power in historical periods and future projections. This study focuses on enhancing wind data representation and reliability of statistically downscaled model outputs by refining grid spacing from coarser to a desirable finer scale. Employing the Ensemble Generalized Analog Regression Downscaling (En-GARD) technique, we downscaled near-surface wind speed across a northeast (NE) U.S. domain that combines onshore and offshore wind farm areas. We tested multiple atmospheric variable combinations and identified near-surface wind speed (wind speed at 10 m), longitudinal and latitudinal wind at 10 m, and atmospheric temperature at 2 m, as the set that effectively guides the selection of analog days for the analog regression downscaling approach. Validation involved using ERA5 atmospheric reanalysis products (31 km) and high-resolution (4 km) Weather Research and Forecasting (WRF) Model simulations. Downscaling was conducted using a leave-one-year-out approach over 12 years (2001–12). Statistical analysis of the downscaled output over this period demonstrated a significant correlation and low error metrics (less than 20% for normalized RMSE and 16% for normalized MAE), indicating the reliability of the method and paving the way for its future application to downscale climate projection outputs. Significance Statement Wind speed has become a significant atmospheric variable due to our society’s imminent reliance on renewable energy resources like wind power. Often overlooked in climate change assessments, wind speed at a local scale over wind farm locations plays a crucial role in assessing and planning wind power resources in a future climate. We are providing the successful validation of a statistical downscaling methodology for near-surface wind speed at 4-km grid spacing using reanalysis data. The validity of the methodology gives confidence to downscale climate projection models and assess changes in wind speed for future climate scenarios in a follow-up study.

Enhancing structural analysis of nonrigid mechanisms in space grids through the Finite Particle Method

Enhancing structural analysis of nonrigid mechanisms in space grids through the Finite Particle Method

Regulable crack patterns for the fabrication of high-performance transparent EMI shielding windows.

Crack pattern-based metal grid film is an ideal candidate material for transparent electromagnetic interference shielding optical windows. However, achieving crack patterns with narrow grid spacing, small wire width, and high connectivity remains challenging. Herein, an aqueous acrylic colloidal dispersion was developed as a crack precursor for preparing crack patterns. The ratio of hard monomers in the precursor, the coating thickness, and the drying mediation strategy were systematically varied to control the spacing and width of the crack patterns. The resulting dense and narrow crack patterns served as sacrificial templates for the fabrication of patterning metal grid films on transparent substrates, intended for optoelectronic applications. These films demonstrated excellent optoelectronic properties (82.7% transmission at 550nm visible light, sheet resistance 4.1Ω/sq) and strong EMI shielding effectiveness (average shielding effectiveness 33.6 dB at 1-18 GHz), showcasing their potential as a scalable and effective transparent EMI shielding solution.

COMPUTING THE LOCAL QUALITY CHARACTERISTIC OF TETRAHEDRAL MESH ELEMENTS AS A SOLUTION EXTREME TASK

The paper discusses the problem of calculating the quality characteristic of elements of a triangulation grid (in multidimensional space). We call quality the degree of difference between a mesh element in a sense critical. For example, the difference between a triangle and a degenerate triangle. To calculate this value, the following construction is used. Mesh element vertices are treated as a numbered set of 𝑋 — points point family. In the space of such sets, the metric proposed in early works is introduced. As a result, the problem boils down to calculating the value of dist(𝑋,𝒴) — the distance between a set 𝑋 and a set 𝒴 defined by critical elements. In early works, the problem was solved through the empirical classification of families 𝑌 ∈ 𝒴 and excluding those that are not known to be closest to 𝑋. Namely, such that the offset some point in the 𝑌 family is given by the 𝑌 ′ ∈ 𝒴 family, which is closer to 𝑋. At the same time, empirical classification led to a large amount of calculations. This work gives a standardized classification of the families of the set 𝒴, allowing you to accurately describe the set of 𝒴− ⊂ 𝒴 of such families for which The specified conversion is not possible. That is, 𝒴− is approved dist(𝑋,𝒴) = dist(𝑋,𝒴−) . In addition, a diagram is given for constructing families of the set 𝒴− — analogues of the points of the assumed extremum in the extremum research problem of the function of numerical variables.

Optimal Location Method of Urban and Rural Emergency Shelter Based on Improved Ant Colony Algorithm

In order to realize the automatic location of urban and rural emergency shelters, the optimal location method of urban and rural emergency shelters based on the improved ant colony algorithm is proposed. Combined with the method of urban and rural land block grid space planning, the spatial grid planning of urban and rural distribution and the method of remote sensing image covering big data detection are adopted to extract the spatial grid geospatial detection model of urban and rural emergency shelters. The remote sensing images of urban and rural emergency shelters and the wireless sensor information sampling method of unmanned aerial vehicle (UAV) are adopted to establish the remote sensing images of urban and rural emergency shelters and the monitoring map database of UAV wireless sensor information. The improved ant colony optimization algorithm is adopted to optimize the path in the process of urban and rural emergency shelters’ site selection, and the overall geographical and geometric characteristics of urban and rural emergency shelters’ site selection are analyzed. The texture, color, shape and other characteristics related to the location of urban and rural emergency shelters are extracted, and the distributed optimization control method of dynamic ant colony is adopted to detect the distribution of geometric deformation characteristics and optimize the location of urban and rural emergency shelters. The location is realized by spatial enhancement of remote sensing dynamic information and the location of spatial feature points. The simulation results show that this method has a good positioning ability and a strong ability to optimize the location of emergency shelters in urban and rural areas.

A Numerical Study of the Impact of Topography on the July 2021 Extreme Rainfall Event in Zhengzhou, China

AbstractAn unprecedented extreme rainfall event occurred in Zhengzhou, Henan Province, China, in July 2021. To understand the impact of local topography on this extreme rainfall event, the Weather Research and Forecasting model is configured with 27 and 9 km model grid spacings (MGS), along with United States Geological Survey (USGS) topography data at 8.3 and 0.9 km resolutions, called MGS27_USGS8.3, MGS9_USGS8.3, and MGS9_USGS0.9. Results show that the 9 km MGS, permitting activities with coarse γ scales (∼20 km), successfully reproduces the generation of mesoscale cyclones. However, the finer topography enables a more accurate representation of orographic blocking and lifting effects, thereby adjusting the position of the mesoscale cyclone. It can depict the location of adiabatic processes, local circulation, and vertical pressure gradient forces more accurately, thereby adjusting the position of topography‐induced vertical motions. The turbulence diagnostics show that the topography‐induced lifting motion enhances clouds that block longwave radiation, leading to local environment warming, which in turn enhances turbulence and further amplifies the updrafts, ultimately improving the spatial distribution and temporal variation of precipitation. This study provides insights for an in‐depth understanding of the mechanisms of topography on extreme rainfall.

Dealing with steep slopes when modeling stable boundary‐layer flow in Alpine terrain

AbstractSteep slopes in mountainous terrain are challenging for commonly used numerical weather prediction models using terrain‐following coordinates since they can generate numerical errors when evaluating horizontal gradients, and numerical instabilities. The model orography is therefore generally smoothed locally so as to remove slopes exceeding a defined threshold, typically in the range 30°–40°. However, this process can lead to undesired changes to orographic features at the scale of the mountain range. So as to preserve properties of the orography, e.g., mean terrain height across the domain, a global smoothing algorithm is developed. The algorithm is then applied to a domain centred on the Alpine Grenoble valley. Exploration of the parameters of the algorithm allows to minimize the changes in the orography in the domain, especially that of the two main valleys (of width 2 and 5 km and depth about 2000 m). Results of numerical model simulations are next analyzed to evaluate the effects of the slope threshold on local‐ and valley‐scale atmospheric dynamics for a cold‐air pool episode. Model results from simulations with slope thresholds of 28° and 42° are compared, the lower threshold allowing for a reduction (albeit small in practice) of the near‐surface vertical grid spacing. Remarkably, not only are the near‐surface boundary‐layer temperature and wind fields similar for both simulations (and in very good agreement with observations), but this similarity holds in the whole inversion layer throughout the episode. The reasons for this similarity are that, for this cold‐air pool episode, the valley atmospheric dynamics is confined within the inversion layer and the orography is essentially unchanged by the smoothing algorithm in that layer for the slope threshold of 28° and 42°.

Коррекция и усвоение данных гидрологических наблюдений температуры водных объектов

The paper proposes an algorithm for correcting the data of hydrological observations of water temperature and an algorithm for assimilating corrected data for calculations of water body surface temperatures in Russia. The temperatures are used in initial data for computing global medium-range weather forecasts with the new version of the SLAV10 model with a grid spacing of about 10 km. The correction algorithm for hydrological observations of water temperature makes it possible to eliminate ambiguity in temperature coding at most gauging stations. The assimilation of corrected hydrological observations allows reducing water body surface temperature errors in the SLAV10 model initial data as compared to specifying water temperature from the data of the nearest land points.

The actuator farm model for large eddy simulation (LES) of wind-farm-induced atmospheric gravity waves and farm–farm interaction

Abstract. This study introduces the actuator farm model (AFM), a novel parameterization for simulating wind turbines within large eddy simulations (LESs) of wind farms. Unlike conventional models like the actuator disk (AD) or actuator line (AL), the AFM utilizes a single actuator point at the rotor center and only requires two to three mesh cells across the rotor diameter. Turbine force is distributed to the surrounding cells using a new projection function characterized by an axisymmetric spatial support in the rotor plane and Gaussian decay in the streamwise direction. The spatial support's size is controlled by three parameters: the half-decay radius r1/2, smoothness s, and streamwise standard deviation σ. Numerical experiments on an isolated National Renewable Energy Laboratory (NREL) 5MW wind turbine demonstrate that selecting r1/2=R (where R is the turbine radius), s between 6 and 10, and σ≈Δx/1.6 (where Δx is the grid size in the streamwise direction) yields wake deficit profiles, turbine thrust, and power predictions similar to those obtained using the actuator disk model (ADM), irrespective of horizontal grid spacing down to the order of the rotor radius. Using these parameters, LESs of a small cluster of 25 turbines in both staggered and aligned layouts are conducted at different horizontal grid resolutions using the AFM. Results are compared against ADM simulations employing a spatial resolution that places at least 10 grid points across the rotor diameter. The wind farm is placed in a neutral atmospheric boundary layer (ABL) with turbulent inflow conditions interpolated from a previous simulation without turbines. At horizontal resolutions finer than or equal to R/2, the AFM yields similar velocity, shear stress, turbine thrust, and power as the ADM. Coarser resolutions reveal the AFM's ability to accurately capture power at the non-waked wind farm rows, although it underestimates the power of waked turbines. However, the far wake of the cluster can be predicted well even when the cell size is of the order of the turbine radius. Finally, combining the AFM with a domain nesting method allows us to conduct simulations of two aligned wind farms in a fully neutral ABL and of wind-farm-induced atmospheric gravity waves under a conventionally neutral ABL, obtaining excellent agreement with ADM simulations but with much lower computational cost. The simulations highlight the AFM's ability to investigate the mutual interactions between large turbine arrays and the thermally stratified atmosphere.

CALCULATION OF PLASMA HEATING BY CHARGED PRODUCTS OF THERMONUCLEAR REACTIONS BASED ON A SIMPLIFIED FOKKER–PLANCK EQUATION

A two-time-layer scheme has been developed for solving the simplified kinetic Fokker–Planck equation related to the transport of charged products of thermonuclear reactions, which includes an interpolation procedure in four-dimensional grid space. Instabilities in the scheme were detected at low particle velocities and for a specific choice of particle deceleration in the ion field, which enters the kinetic equation as a parameter. It was shown that the thermalization condition, which prohibits solving the kinetic equation for particles with energy lower than the average ion energy, significantly limits the number of thermonuclear reactions where instability can manifest. The scheme was tested on the problem of relaxation to a stationary state and on a problem with a prescribed time-dependent thermonuclear reaction rate, for which an exact solution to the kinetic equation can be found.