Complex Three-Dimensional Mathematical Model of the Ignition of a Coniferous Tree via a Cloud-to-Ground Lightning Discharge: Electrophysical, Thermophysical and Physico-Chemical Processes

Abstract

Thunderstorms are the main natural source of forest fires. The ignition mechanism of trees begins with the impact of cloud-to-ground lightning discharge. A common drawback of all predicting systems is that they ignore the physical mechanism of forest fire as a result of thunderstorm activity. The purpose of this article is to develop a physically based mathematical model for the ignition of a coniferous tree via cloud-to-ground lightning discharge, taking into account thermophysical, electrophysical, and physicochemical processes. The novelty of the article is explained by the development of an improved mathematical model for the ignition of coniferous trees via cloud-to-ground lightning discharge, taking into account the processes of soot formation caused by the thermal decomposition phase of dry organic matter. Mathematically, the process of tree ignition is described by a system of non-stationary nonlinear differential equations of heat conduction and diffusion. In this research, a locally one-dimensional method is used to solve three-dimensional partial differential equations. The finite difference method is used to solve one-dimensional heat conduction and diffusion equations. Difference analogues of the equations are solved using the marching method. To resolve nonlinearity, a simple iteration method is used. Temperature distributions in a structurally inhomogeneous trunk of a coniferous tree, as well as distributions of volume fractions of phases and concentrations of gas mixture components, are obtained. The conditions for tree trunk ignition under conditions of thunderstorm activity are determined. As a result, a complex three-dimensional mathematical model is developed, which makes it possible to identify the conditions for the ignition of a coniferous tree trunk via cloud-to-ground lightning discharge.

1. Introduction

Predicting forest fires can help to reduce damage [1]. There are various methods for predicting forest fires, ranging from empirical to complex deterministic–probabilistic ones [2,3]. As a rule, a specific information-computing system is based on some specific method of predicting forest fire danger. For example, in the Russian Federation, there is an information system for remote the monitoring of forest fires known as ISDM Rosleskhoz [4]. The predictive component of this system is based on the Nesterov index [5].

It should be noted that in recent years, a deterministic–probabilistic method for predicting forest fire danger has been intensively developed [6]. This method is based on deterministic mathematical models of the drying and ignition of forest fuels and probabilistic criteria [7]. It is known [8] that forest fires from thunderstorms often occur in remote areas. Therefore, it is important to develop mathematical models for the ignition of trees in conditions of lightning activity. The ignition of a tree by lightning is a complex phenomenon that includes various electrophysical, thermophysical and physicochemical processes. These processes occur both in the tree trunk and in the near-wall region of the trunk in the gas phase [9,10,11,12,13,14,15].

In the forestry structures of various states in the world community, instrumental systems are used to obtain information on lightning discharge. First of all, forest services are interested in information on the distribution of cloud-to-ground lightning discharges in time and space. There are examples of lightning direction-finding systems that provide information online:

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(EU: www.euclid.org, accessed on 10 June 2023)

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(Germany: http://www.blitzortung.org, accessed on 10 June 2023)

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(USA: http://webflash.ess.washington.edu/, accessed on 10 June 2023).

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(Russian Federation: http://www.meteorf.ru/product/info/, accessed on 10 June 2023).

On every continent, there are regions with a higher density of lightning discharges that lead to forest fires. For example, in Austria there is the Austrian Lightning Detection and Information System (ALDIS). This system can detect lightning strikes and provide access to information on lightning current, polarity, and complexity. In addition [16], information is analyzed using the time and place of fire in the forest, the type of vegetation, and the area covered by the fire. Using this information, the following [17] were calculated: FFMC—Fine Fuel Moisture Code; BUI—Build Up Index; and FWI—Fire Weather Index. An analysis of the application of ALDIS showed that this system makes it possible to recognize more than 90% of cloud-to-ground lightning discharges over the territory of Austria [18,19]. A comparative analysis showed that this is one of the most accurate systems for the direction finding of such lightning discharges [20,21,22,23,24,25].

In turn, a lightning direction-finding system operates in Asia using SAFIR 300 sensors, which work according to the IMPACT method [26,27,28]. In addition, the LINET method [29,30,31,32,33] is used, which makes it possible to determine almost half of cloud-to-ground lightning discharges. It should be noted that similar studies were carried out in Malaysia and Colombia [27,29]. Lightning direction-finding systems can use two methods to determine the point of application of lightning discharge [16]: Magnetic Direction Finding (MDF) and Time of Arrival (TOA). These methods allow the system to determine the following discharge characteristics [21]: geographic coordinates, polarity, current, and number of strokes in a complex discharge. In addition, similar studies using lightning direction-finding systems were carried out in the territory of the Iberian Peninsula [34,35,36] and in the mountainous regions of the Alps [37]. It should be noted that there is also a European Cooperation for Lightning Detection network (EUCLID) project in Europe [38].

The assessment of the impact of thunderstorm activity on the occurrence of forest fires can be carried out using maps of the density of lightning discharges passed from cloud to ground over a controlled forested area [39]. It should be noted that precipitation is of great importance for the occurrence of forest fires under conditions of thunderstorm activity [40,41]. On the one hand, a positive relationship has been established between precipitation and thunderstorm activity [42,43,44,45,46]. This reduces the probability of forest fires occurring in rainy conditions. On the other hand, there are so-called dry thunderstorms, when the probability of ignition of forest fuels via cloud-to-ground lightning discharge is high [47]. There are also more complex approaches to modeling thunderstorm activity in order to predict forest fires [48]. For example, in Portugal a system has been developed using the CELLS atmospheric electrical model and the Meso-NH non-hydrostatic atmospheric model [49,50] to provide meteorological parameters based on the SURFEX model [51] and CELLS [52].

In addition, studies on the analysis of the characteristics of lightning discharges were carried out in the USA [53,54,55,56] and China [57]. For example, an analysis of the characteristics of cloud-to-ground lightning discharges was carried out using the YNLDN [57,58]. It was found that the maximum current is >70 kA and the minimum is <30 kA. For this analysis, a dataset of 400,000 lightning discharges was used [58].

This review provides information on lightning observation and direction-finding systems in various countries of the world community. Existing systems, such as WWLLN, can identify the duration and location of a lightning strike, as well as the strength of the discharge. There are mathematical expressions that allow researchers to obtain the value of peak lightning current. The duration and discharge current are the input data for the proposed mathematical model. Currently, a scenario-modeling approach is used when these parameters change within certain ranges. However, in future studies, when organizing interactions with GIS systems to predict forest fire danger, there will be a geographic reference to the locations of registered cloud-to-ground lightning discharges in a controlled forested area. Therefore, the review is directly related to the research topic.

Shortcomings or open questions in previously published work should be noted. Despite the fact that there has been intensive development in recent decades in the field of predicting forest fire danger, almost all existing systems do not have a physical basis. The forest fire danger prediction systems used in various countries do not have software modules that would allow researchers to take into account physical and chemical processes when a lightning discharge impacts trees. Typically, forest fire statistics and lightning direction-finding data are used. Therefore, it is necessary to develop new systems for predicting forest fire danger based on the real physical and chemical processes that occur during the impact of a cloud-to-ground lightning discharge with a coniferous tree. As a consequence, this is impossible without the development of complex mathematical models of the ignition of coniferous trees by lightning discharges.

The novelty of the article is explained by the development of an improved mathematical model for the ignition of coniferous trees via cloud-to-ground lightning discharge, taking into account the processes of soot formation caused by the thermal decomposition phase of dry organic matter. The development of this mathematical model allows us to get closer to the creation of hardware and software systems for the visual observation of the presence of soot particles in the observation zone using LiDAR atmospheric sensing technologies. The formation of soot emissions can be monitored, and the occurrence of a surface forest fire can be predicted. The mathematical model can be used to develop a new information and computing system for the scenario modeling of forest fire danger in conditions of thunderstorm activity.

The purpose of this article is to develop a physically based mathematical model for the ignition of coniferous trees via cloud-to-ground lightning discharge, taking into account thermophysical, electrophysical, and physicochemical processes.

Research objectives: (1) The formulation of a physical model of the coniferous tree trunk lightning ignition, taking into account the heating of wood due to release according to the Joule–Lenz law, thermal decomposition of dry organic matter, chemical reaction in the gas phase, and the formation of soot particles. (2) The development of a spatial mathematical model, taking into account the indicated physical effects and its software implementation. (3) The scenario modeling of heat and mass transfer in a tree trunk under the influence of a cloud-to-ground lightning discharge. (4) The formulation of conclusions and proposals for the practical application of the developed mathematical model in order to predict forest fires from the observation of thunderstorms.

2. Materials and Methods

2.1. Physical Statement

System of basic assumptions:

(1)

The subcrustal zone contains moisture and is a conductor of electric current, like a resistor [11,12];

(2)

In the structure of the trunk, there is heterogeneity in the lower part of the branch in the form of reactive wood [11];

(3)

In the structure of the cortical layer, there is a heterogeneity in the form of a crack that runs along the entire vertical of the trunk (Figure 1);

(4)

Only heat transfer in that part of the branch that grows out of the trunk is taken into account;

(5)

The main gaseous combustible component of the thermal decomposition products of dry organic matter is carbon monoxide;

(6)

The oxidation reaction of carbon monoxide into carbon dioxide is the leading reaction in the gas phase;

(7)

Heating of wood occurs as a result of heat release according to the Joule–Lenz law [12];

(8)

Pyrolysis products enter the gas phase instantly;

(9)

Thermophysical characteristics do not depend on temperature.

The ignition of a coniferous tree trunk occurs when a critical temperature is reached in the zone of localization of the leading reaction and when the heat flux from the chemical reaction exceeds the heat flux from the subcrustal zone.

The solution area is shown in Figure 2a and the boundaries of the regions are indicated in Figure 2b.

2.2. Mathematical Statement

The process of tree lightning ignition is described via a system of three-dimensional non-stationary nonlinear equations of heat conduction and diffusion (1)–(19). A locally one-dimensional finite-difference method was used for numerical implementation [59]. Difference analogs of one-dimensional heat conduction equations are solved by using the marching method in combination with the simple iteration method [59].

$$\begin{array}{l}{\rho }_1{c}_1\frac{\mathit{\partial }{T}_1}{\mathit{\partial }t}=\frac{{\lambda }_1}{r}\frac{\mathit{\partial }}{\mathit{\partial }r}\left(r\frac{\mathit{\partial }{T}_1}{\mathit{\partial }r}\right)+\frac{{\lambda }_1}{r^2}\frac{{\mathit{\partial }}^2{T}_1}{\mathit{\partial }{\phi }^2}+{\lambda }_1\frac{{\mathit{\partial }}^2{T}_1}{\mathit{\partial }{z}^2}-\\ -Q_p{k}_p{\rho }_1{\phi }_{13}\exp \left(-\frac{E_1}{R{T}_1}\right)\end{array},$$

(1)

$$\begin{array}{l}{\rho }_2{c}_2\frac{\mathit{\partial }{T}_2}{\mathit{\partial }t}=\frac{{\lambda }_2}{r}\frac{\mathit{\partial }}{\mathit{\partial }r}\left(r\frac{\mathit{\partial }{T}_2}{\mathit{\partial }r}\right)+\frac{{\lambda }_2}{r^2}\frac{{\mathit{\partial }}^2{T}_2}{\mathit{\partial }{\phi }^2}+{\lambda }_2\frac{{\mathit{\partial }}^2{T}_2}{\mathit{\partial }{z}^2}+\\ +JU-Q_p{k}_p{\rho }_2{\phi }_{13}\exp \left(-\frac{E_1}{R{T}_2}\right)\end{array},$$

(2)

$$\begin{array}{l}{\rho }_3{c}_3\frac{\mathit{\partial }{T}_3}{\mathit{\partial }t}=\frac{{\lambda }_3}{r}\frac{\mathit{\partial }}{\mathit{\partial }r}\left(r\frac{\mathit{\partial }{T}_3}{\mathit{\partial }r}\right)+\frac{{\lambda }_3}{r^2}\frac{{\mathit{\partial }}^2{T}_3}{\mathit{\partial }{\phi }^2}+{\lambda }_3\frac{{\mathit{\partial }}^2{T}_3}{\mathit{\partial }{z}^2}-\\ -Q_p{k}_p{\rho }_3{\phi }_{13}\exp \left(-\frac{E_1}{R{T}_3}\right)\end{array},$$

(3)

$$\begin{array}{l}{\rho }_4{c}_4\frac{\mathit{\partial }{T}_4}{\mathit{\partial }t}=\frac{{\lambda }_4}{r}\frac{\mathit{\partial }}{\mathit{\partial }r}\left(r\frac{\mathit{\partial }{T}_4}{\mathit{\partial }r}\right)+\frac{{\lambda }_4}{r^2}\frac{{\mathit{\partial }}^2{T}_4}{\mathit{\partial }{\phi }^2}+{\lambda }_4\frac{{\mathit{\partial }}^2{T}_4}{\mathit{\partial }{z}^2}-\\ -Q_p{k}_p{\rho }_4{\phi }_{13}\exp \left(-\frac{E_1}{R{T}_4}\right)\end{array},$$

(4)

$$\begin{array}{l}{\rho }_5{c}_5\frac{\mathit{\partial }{T}_5}{\mathit{\partial }t}=\frac{{\lambda }_5}{r}\frac{\mathit{\partial }}{\mathit{\partial }r}\left(r\frac{\mathit{\partial }{T}_5}{\mathit{\partial }r}\right)+\frac{{\lambda }_5}{r^2}\frac{{\mathit{\partial }}^2{T}_5}{\mathit{\partial }{\phi }^2}+{\lambda }_5\frac{{\mathit{\partial }}^2{T}_5}{\mathit{\partial }{z}^2}+\\ +JU-Q_p{k}_p{\rho }_5{\phi }_{13}\exp \left(-\frac{E_1}{R{T}_5}\right)\end{array},$$

(5)

$$\begin{array}{l}{\rho }_6{c}_6\frac{\mathit{\partial }{T}_6}{\mathit{\partial }t}=\frac{{\lambda }_6}{r}\frac{\mathit{\partial }}{\mathit{\partial }r}\left(r\frac{\mathit{\partial }{T}_6}{\mathit{\partial }r}\right)+\frac{{\lambda }_6}{r^2}\frac{{\mathit{\partial }}^2{T}_6}{\mathit{\partial }{\phi }^2}+{\lambda }_6\frac{{\mathit{\partial }}^2{T}_6}{\mathit{\partial }{z}^2}-\\ -Q_p{k}_p{\rho }_6{\phi }_{13}\exp \left(-\frac{E_1}{R{T}_6}\right)\end{array},$$

(6)

$$\begin{array}{l}{\rho }_7{c}_7\frac{\mathit{\partial }{T}_7}{\mathit{\partial }t}=\frac{{\lambda }_7}{r}\frac{\mathit{\partial }}{\mathit{\partial }r}\left(r\frac{\mathit{\partial }{T}_7}{\mathit{\partial }r}\right)+\frac{{\lambda }_7}{r^2}\frac{{\mathit{\partial }}^2{T}_7}{\mathit{\partial }{\phi }^2}+{\lambda }_7\frac{{\mathit{\partial }}^2{T}_7}{\mathit{\partial }{z}^2}-\\ -Q_p{k}_p{\rho }_7{\phi }_{13}\exp \left(-\frac{E_1}{R{T}_7}\right)\end{array},$$

(7)

$$\begin{array}{l}{\rho }_8{c}_8\frac{\mathit{\partial }{T}_8}{\mathit{\partial }t}=\frac{{\lambda }_8}{r}\frac{\mathit{\partial }}{\mathit{\partial }r}\left(r\frac{\mathit{\partial }{T}_8}{\mathit{\partial }r}\right)+\frac{{\lambda }_8}{r^2}\frac{{\mathit{\partial }}^2{T}_8}{\mathit{\partial }{\phi }^2}+{\lambda }_8\frac{{\mathit{\partial }}^2{T}_8}{\mathit{\partial }{z}^2}+JU-\\ -Q_p{k}_p{\rho }_8{\phi }_{13}\exp \left(-\frac{E_1}{R{T}_8}\right)\end{array},$$

(8)

$$\begin{array}{l}{\rho }_9{c}_9\frac{\mathit{\partial }{T}_9}{\mathit{\partial }t}=\frac{{\lambda }_9}{r}\frac{\mathit{\partial }}{\mathit{\partial }r}\left(r\frac{\mathit{\partial }{T}_9}{\mathit{\partial }r}\right)+\frac{{\lambda }_9}{r^2}\frac{{\mathit{\partial }}^2{T}_9}{\mathit{\partial }{\phi }^2}+{\lambda }_9\frac{{\mathit{\partial }}^2{T}_9}{\mathit{\partial }{z}^2}-\\ -Q_p{k}_p{\rho }_9{\phi }_{13}\exp \left(-\frac{E_1}{R{T}_9}\right)\end{array},$$

(9)

$$\begin{array}{l}{\rho }_g{c}_g\frac{\mathit{\partial }{T}_g}{\mathit{\partial }t}=\frac{{\lambda }_g}{r}\frac{\mathit{\partial }}{\mathit{\partial }r}\left(r\frac{\mathit{\partial }{T}_g}{\mathit{\partial }r}\right)+\frac{{\lambda }_g}{r^2}\frac{{\mathit{\partial }}^2{T}_g}{\mathit{\partial }{\phi }^2}+{\lambda }_g\frac{{\mathit{\partial }}^2{T}_g}{\mathit{\partial }{z}^2}+\\ +Q_5(1-{\nu }_5)R_5\end{array},$$

(10)

$$\frac{\mathit{\partial }{C}_{10}}{\mathit{\partial }t}=\frac{D}{r}\frac{\mathit{\partial }}{\mathit{\partial }r}\left(r\frac{\mathit{\partial }{C}_{10}}{\mathit{\partial }r}\right)+\frac{D}{r^2}\frac{{\mathit{\partial }}^2{C}_{10}}{\mathit{\partial }{\phi }^2}+D\frac{{\mathit{\partial }}^2{C}_{10}}{\mathit{\partial }{z}^2}-R_5\frac{M_4}{2M_5},$$

(11)

$$\frac{\mathit{\partial }{C}_{11}}{\mathit{\partial }t}=\frac{D}{r}\frac{\mathit{\partial }}{\mathit{\partial }r}\left(r\frac{\mathit{\partial }{C}_{11}}{\mathit{\partial }r}\right)+\frac{D}{r^2}\frac{{\mathit{\partial }}^2{C}_{11}}{\mathit{\partial }{\phi }^2}+D\frac{{\mathit{\partial }}^2{C}_{11}}{\mathit{\partial }{z}^2}-R_5,$$

(12)

$${\displaystyle \sum _{i=10}^{12}{C}_i}=1,$$

(13)

$${\displaystyle \sum _{i=13}^{14}{\phi }_i}=1,$$

(14)

$${\rho }_i\frac{\mathit{\partial }{\phi }_{13}}{\mathit{\partial }t}=-k_p{\rho }_i{\phi }_{13}\exp \left(-\frac{E_1}{R{T}_i}\right),i=1, \dots , 9$$

(15)

$$R_5=k_5{M}_{11}{T}^{-2.25}\exp \left(-\frac{E_5}{R{T}_g}\right)\cdot \left\{\begin{array}{l}{x}_{10}^{0.25}{x}_{11},x_{10}>0.05\\ x_{10}{x}_{11},x_{10}\le 0.05\end{array}\right.,$$

(16)

$$x_i=\frac{C_i}{{\displaystyle \sum _{k=10}^{12}\frac{C_k}{M_k}{M}_i}},$$

(17)

$$P=\frac{\rho RT}{M},\frac{1}{M}=\frac{C_{10}}{M_{10}}+\frac{C_{11}}{M_{11}}+\frac{C_{12}}{M_{12}}.$$

(18)

$${\rho }_s\frac{\mathit{\partial }{\phi }_{15}}{\mathit{\partial }t}={\alpha }_s{k}_p{\rho }_s{\phi }_{13}\exp \left(-\frac{E_1}{R{T}_i}\right),i=1, \dots , 9$$

(19)

Initial and boundary conditions for the system of Equations (1)–(19).

Boundary conditions outside the branch area:

$${\left.T_i\right|}_{t=0}=T_{i0},$$

(20)

$$r=0, {\lambda }_i\frac{\mathit{\partial }{T}_i}{\mathit{\partial }r}=0,$$

(21)

$$r=R_2, {\lambda }_1\frac{\mathit{\partial }{T}_1}{\mathit{\partial }r}={\lambda }_2\frac{\mathit{\partial }{T}_2}{\mathit{\partial }r}, T_1=T_2$$

(22)

$$r=R_1, {\lambda }_2\frac{\mathit{\partial }{T}_2}{\mathit{\partial }r}={\lambda }_3\frac{\mathit{\partial }{T}_3}{\mathit{\partial }r}, T_2=T_3$$

(23)

$$r=R_S, {\lambda }_3\frac{\mathit{\partial }{T}_3}{\mathit{\partial }r}={\lambda }_g\frac{\mathit{\partial }{T}_g}{\mathit{\partial }r}, T_3=T_g$$

(24)

$$r=R_e, T_g=T_e,$$

(25)

$$\phi =0, \frac{\mathit{\partial }{T}_i}{\mathit{\partial }\phi }=0,$$

(26)

$$\phi =\pi , \frac{\mathit{\partial }{T}_i}{\mathit{\partial }\phi }=0,$$

(27)

$$z=z_b, {\lambda }_i\frac{\mathit{\partial }{T}_i}{\mathit{\partial }z}=0,$$

(28)

$$z=z_t, {\lambda }_i\frac{\mathit{\partial }{T}_i}{\mathit{\partial }z}=0,$$

(29)

$$z=z_t, {\lambda }_i\frac{\mathit{\partial }{T}_i}{\mathit{\partial }z}=0,$$

(30)

Boundary conditions on the inner side of the branch:

$${\Gamma }_0, {\lambda }_4\frac{\mathit{\partial }{T}_4}{\mathit{\partial }r}={\lambda }_1\frac{\mathit{\partial }{T}_1}{\mathit{\partial }r},T_4=T_1$$

(31)

$${\Gamma }_1, {\lambda }_7\frac{\mathit{\partial }{T}_7}{\mathit{\partial }r}={\lambda }_1\frac{\mathit{\partial }{T}_1}{\mathit{\partial }r},T_7=T_1$$

(32)

Conditions on the boundary of the right side of the branch and the crack:

$${\Gamma }_2, {\lambda }_4\frac{\mathit{\partial }{T}_4}{\mathit{\partial }\phi }={\lambda }_1\frac{\mathit{\partial }{T}_1}{\mathit{\partial }\phi }, T_4=T_1$$

(33)

$${\Gamma }_3, {\lambda }_5\frac{\mathit{\partial }{T}_5}{\mathit{\partial }\phi }={\lambda }_2\frac{\mathit{\partial }{T}_2}{\mathit{\partial }\phi }, T_3=T_2$$

(34)

$${\Gamma }_4, {\lambda }_6\frac{\mathit{\partial }{T}_6}{\mathit{\partial }\phi }={\lambda }_g\frac{\mathit{\partial }{T}_g}{\mathit{\partial }\phi }, T_6=T_g$$

(35)

$${\Gamma }_5, {\lambda }_7\frac{\mathit{\partial }{T}_7}{\mathit{\partial }\phi }={\lambda }_1\frac{\mathit{\partial }{T}_1}{\mathit{\partial }\phi }, T_7=T_1$$

(36)

$${\Gamma }_6, {\lambda }_8\frac{\mathit{\partial }{T}_8}{\mathit{\partial }\phi }={\lambda }_2\frac{\mathit{\partial }{T}_2}{\mathit{\partial }\phi }, T_8=T_2$$

(37)

$${\Gamma }_7, {\lambda }_9\frac{\mathit{\partial }{T}_9}{\mathit{\partial }\phi }={\lambda }_g\frac{\mathit{\partial }{T}_g}{\mathit{\partial }\phi }, T_9=T_g$$

(38)

Boundary conditions on the outer cut of the branch:

$${\Gamma }_8, {\lambda }_6\frac{\mathit{\partial }{T}_6}{\mathit{\partial }r}={\lambda }_g\frac{\mathit{\partial }{T}_g}{\mathit{\partial }r}, T_6=T_g$$

(39)

$${\Gamma }_9, {\lambda }_9\frac{\mathit{\partial }{T}_9}{\mathit{\partial }r}={\lambda }_g\frac{\mathit{\partial }{T}_g}{\mathit{\partial }r}, T_9=T_g$$

(40)

Boundary conditions on the lower edge of the branch:

$${\Gamma }_{10}, {\lambda }_1\frac{\mathit{\partial }{T}_1}{\mathit{\partial }z}={\lambda }_7\frac{\mathit{\partial }{T}_7}{\mathit{\partial }z}, T_1=T_7$$

(41)

$${\Gamma }_{11}, {\lambda }_2\frac{\mathit{\partial }{T}_2}{\mathit{\partial }z}={\lambda }_8\frac{\mathit{\partial }{T}_8}{\mathit{\partial }z}, T_2=T_8$$

(42)

$${\Gamma }_{12}, {\lambda }_3\frac{\mathit{\partial }{T}_3}{\mathit{\partial }z}={\lambda }_9\frac{\mathit{\partial }{T}_9}{\mathit{\partial }z}, T_3=T_9$$

(43)

Boundary conditions on the left side of the branch:

$${\Gamma }_{13}, {\lambda }_1\frac{\mathit{\partial }{T}_1}{\mathit{\partial }\phi }={\lambda }_7\frac{\mathit{\partial }{T}_7}{\mathit{\partial }\phi }, T_1=T_7$$

(44)

$${\Gamma }_{14}, {\lambda }_2\frac{\mathit{\partial }{T}_2}{\mathit{\partial }\phi }={\lambda }_8\frac{\mathit{\partial }{T}_8}{\mathit{\partial }\phi }, T_2=T_8$$

(45)

$${\Gamma }_{15}, {\lambda }_3\frac{\mathit{\partial }{T}_3}{\mathit{\partial }\phi }={\lambda }_9\frac{\mathit{\partial }{T}_9}{\mathit{\partial }\phi }, T_3=T_9$$

(46)

$${\Gamma }_{16}, {\lambda }_1\frac{\mathit{\partial }{T}_1}{\mathit{\partial }\phi }={\lambda }_4\frac{\mathit{\partial }{T}_4}{\mathit{\partial }\phi }, T_1=T_4$$

(47)

$${\Gamma }_{17}, {\lambda }_2\frac{\mathit{\partial }{T}_2}{\mathit{\partial }\phi }={\lambda }_5\frac{\mathit{\partial }{T}_4}{\mathit{\partial }\phi }, T_2=T_4$$

(48)

$${\Gamma }_{18}, {\lambda }_3\frac{\mathit{\partial }{T}_3}{\mathit{\partial }\phi }={\lambda }_6\frac{\mathit{\partial }{T}_6}{\mathit{\partial }\phi }, T_3=T_6$$

(49)

Boundary conditions on the upper face of the branch:

$${\Gamma }_{19}, {\lambda }_3\frac{\mathit{\partial }{T}_3}{\mathit{\partial }z}={\lambda }_6\frac{\mathit{\partial }{T}_6}{\mathit{\partial }z}, T_3=T_6$$

(50)

$${\Gamma }_{20}, {\lambda }_2\frac{\mathit{\partial }{T}_2}{\mathit{\partial }z}={\lambda }_5\frac{\mathit{\partial }{T}_5}{\mathit{\partial }z}, T_2=T_5$$

(51)

$${\Gamma }_{21}, {\lambda }_1\frac{\mathit{\partial }{T}_1}{\mathit{\partial }z}={\lambda }_4\frac{\mathit{\partial }{T}_4}{\mathit{\partial }z}, T_1=T_4$$

(52)

Conditions at the crack boundary, excluding the right side of the branch:

$${\Gamma }_{22}, {\lambda }_3\frac{\mathit{\partial }{T}_3}{\mathit{\partial }\phi }={\lambda }_g\frac{\mathit{\partial }{T}_g}{\mathit{\partial }\phi }, T_3=T_g$$

(53)

$$\rho D\frac{\mathit{\partial }{C}_{10}}{\mathit{\partial }\phi }=0,$$

(54)

$$\rho D\frac{\mathit{\partial }{C}_{11}}{\mathit{\partial }\phi }=0,$$

(55)

$${\Gamma }_{23}, {\lambda }_g\frac{\mathit{\partial }{T}_g}{\mathit{\partial }\phi }={\lambda }_3\frac{\mathit{\partial }{T}_3}{\mathit{\partial }\phi }, T_g=T_3$$

(56)

$$\rho D\frac{\mathit{\partial }{C}_{10}}{\mathit{\partial }\phi }=0,$$

(57)

$$\rho D\frac{\mathit{\partial }{C}_{11}}{\mathit{\partial }\phi }=0,$$

(58)

$${\Gamma }_{24}, {\lambda }_2\frac{\mathit{\partial }{T}_2}{\mathit{\partial }r}={\lambda }_g\frac{\mathit{\partial }{T}_g}{\mathit{\partial }r}, T_2=T_g$$

(59)

$$\rho D\frac{\mathit{\partial }{C}_{10}}{\mathit{\partial }r}=0,$$

(60)

$$\rho D\frac{\mathit{\partial }{C}_{11}}{\mathit{\partial }r}=Y_5,$$

(61)

$${\Gamma }_{25}, T_g=T_e,$$

(62)

$$\rho D\frac{\mathit{\partial }{C}_{10}}{\mathit{\partial }r}=0,$$

(63)

$$\rho D\frac{\mathit{\partial }{C}_{11}}{\mathit{\partial }r}=0,$$

(64)

$${\Gamma }_{26}, {\lambda }_g\frac{\mathit{\partial }{T}_g}{\mathit{\partial }z}=0,$$

(65)

$$\rho D\frac{\mathit{\partial }{C}_{10}}{\mathit{\partial }z}=0,$$

(66)

$$\rho D\frac{\mathit{\partial }{C}_{11}}{\mathit{\partial }z}=0,$$

(67)

$${\Gamma }_{27}, {\lambda }_g\frac{\mathit{\partial }{T}_g}{\mathit{\partial }z}=0,$$

(68)

$$\rho D\frac{\mathit{\partial }{C}_{10}}{\mathit{\partial }z}=0,$$

(69)

$$\rho D\frac{\mathit{\partial }{C}_{11}}{\mathit{\partial }z}=0,$$

(70)

Initial conditions:

$${\left.T_i\right|}_{t=0}=T_{i0}, {\left.C_i\right|}_{t=0}=C_{i0}, {\left.{\phi }_i\right|}_{t=0}={\phi }_{i0}$$

(71)

Equation (1) describes the process of heat transfer in space and time in the core zone of a coniferous tree trunk, taking into account the heat drain due to the reaction of thermal decomposition of dry organic matter of wood.

Equation (2) describes heat transfer in time and space in the subcrustal zone of a coniferous tree trunk, taking into account the heat release according to the Joule–Lenz law and thermal decomposition of the dry organic matter of wood.

Equation (3) describes heat transfer in time and space in the region of the bark layer of the trunk, taking into account the thermal decomposition of dry organic matter of wood.

Equation (4) describes heat transfer in time and space in the zone of the upper part of the branch (grown into the core), taking into account the thermal decomposition of the dry organic matter of the branch.

Equation (5) describes heat transfer in space and time in the area of the upper part of the branch (grown into the subcrustal zone of the trunk), taking into account heat release according to the Joule–Lenz law and thermal decomposition of the dry organic matter of the branch.

Equation (6) describes heat transfer in time and space in the upper part of the branch (into the bark layer), taking into account the thermal decomposition of the dry organic matter of the branch wood.

Equation (7) describes heat transfer in time and space in the lower part of the branch (grown into the core), taking into account the thermal decomposition of the dry organic matter of the branch.

Equation (8) describes heat transfer in time and space in the lower part of the branch (grown into the subcrustal zone), taking into account the heat release according to the Joule–Lenz law and the thermal decomposition of the dry organic matter of the branch.

Equation (9) describes heat transfer in time and space in the lower part of the branch (grown into the bark), taking into account the thermal decomposition of the dry organic matter of the branch.

Equation (10) describes heat transfer in time and space in the zone of the gas mixture in a crack in the bark layer, taking into account the heat release due to the oxidation reaction of carbon monoxide to carbon dioxide.

Equation (11) describes the diffusion transfer of a substance (air oxygen), taking into account the chemical reaction in the gas phase of carbon monoxide with air oxygen.

Equation (12) describes the diffusion transfer of a substance (carbon monoxide), taking into account the chemical reaction in the gas phase of carbon monoxide with atmospheric oxygen.

Relation (13) allows us to determine the concentration of inert components based on the balance of concentrations of the gas mixture.

Relation (14) allows us to determine the volume fraction of the gas phase based on the balance of the volume fractions of the phases.

Equation (15) allows us to determine the volume fraction of dry organic matter and describes the kinetics of thermal decomposition of this phase.

Equation (16) allows us to determine the rate of the chemical reaction of the oxidation of carbon monoxide into carbon dioxide using atmospheric oxygen.

Relationship (17) is an auxiliary expression to simplify the expression for the rate of the chemical reaction of the oxidation of carbon monoxide into carbon dioxide using atmospheric oxygen.

Equation (18) is the equation of state for a gas mixture.

Equation (19) allows us to determine the volume fraction of soot particles and describes the kinetics of soot formation, which depends on the thermal decomposition of dry organic matter.

Designations of variables and indexes are presented in Appendix A. Roadmap of mathematical model design presented in Figure 3.

The workflow diagram of the presented mathematical model is shown in Figure 4.

The initial stage of the computational procedure involves establishing initial data for use in this program. Then, it is necessary to construct a three-dimensional mesh in a cylindrical system of coordinates r, φ, and z. Three-dimensional arrays of floating variables were also constructed for temperature, concentrations, and volume fractions within this stage. At the moment of time equal to zero, the initial data are arranged into arrays of variables. The next multistage step is to circle until the time is equal to the final time of tree exposure to cloud-to-ground lightning discharge.

It is necessary to compute temperature, concentrations and volume fractions for the next temporal layer. First, a set of heat conduction equations was solved using the finite difference method and locally one-dimensional method. Difference analogues of mathematical equations were solved using the marching method and the simple iteration method. Equations for core, subcrustal zone, bark, and air gap subsequently were solved at this stage. Second, a set of diffusion equations was solved using the finite difference method and locally one-dimensional method. As at the previous stage, difference analogues of mathematical equations were solved by the marching method in conjunction with the simple iteration method. The equations for carbon monoxide and oxygen were subsequently solved. Then, the concentration of inert components was calculated using the balance expression for concentration. Third, a set of ordinary differential equations were solved in order to compute the volume fractions of gas phase, dry organic matter and soot particles. Then, in the case of time being equal to exposure time, the resultant arrays were saved to files. Otherwise, the next iteration of circling was started.

Development environment: Embarcadero RAD Studio 10 Architect Edition (Delphi package) (Embarcadero, Austin, TX, USA) with Object Pascal programming language. Compilation was performed using standard Delphi tools. The visualization of the results was performed in the Origin Pro 2022 package (OriginLab Corporation, Northampton, MA, USA). Description of workstation is presented in Appendix B.

3. Results and Discussion

The distribution of temperature along the radius and height of the tree trunk at different points in time is shown in Figure 5: (a) t = 0.01 s; (b) 0.1 s; (c) 0.3 s; and (d) 0.5 s. The top figure corresponds to the section of the trunk of a coniferous tree in which the branch is missing. The lower figure corresponds to the section of the trunk of a coniferous tree in which a branch is present. An analysis of temperature distributions shows that, the subcrustal zone of a tree trunk warms up over time due to heat release according to the Joule–Lenz law. An exposure interval of lightning discharge up to 500 ms was considered. Not all cloud-to-ground lightning discharges are characterized by such an exposure time, nor do all such discharges lead to the ignition of a tree and a subsequent forest fire. The figures show that when the exposure time of lightning discharge is about 0.01 s, there is insignificant heating of the subcrustal zone of a coniferous tree trunk and practically no difference between areas with and without branches. At the time instant of 0.1 s, the temperature field at its maximum corresponds to approximately 400 K. Differences in the temperature of the regions with and without branches are already noticeable. In addition, nearby areas of the core and bark layer begin to warm up. At a time of 0.3 s, there is a noticeable difference in the temperatures of the regions with and without branches. This is due to the fact that the reactive wood of the branch has different thermophysical characteristics and this leads to a change. Namely, there is a decrease in temperature in the reactive wood zone of the coniferous branch. The maximum temperatures in the subcrustal zone and the maximum differences in branch temperatures are observed at an exposure time of 0.5 s.

Also, temperature distributions along the horizontal cut of a tree trunk at different points in time are obtained. There is also an increase in temperature in the subcrustal zone over time. The maximum differences in the temperature of the areas with and without branches are typical for an exposure time of lightning discharge of 0.5 s. Moreover, temperature distributions along the radial cut of a tree trunk at different points in time were obtained. This section was isolated in the subcrustal zone of a coniferous tree trunk. It can be seen that the reactive wood of the branch has a lower temperature. That is, a full spatial analysis of the temperature field shows that the part of the branch with reactive wood is characterized by a field of lower temperature. All areas around it have a higher temperature due to the difference in the thermophysical characteristics of various fragments of the wood of the trunk of a coniferous tree, especially the reactive wood of the branch. Based on this fact, it can be assumed that more branches there are, the less likely that the trunk of a coniferous tree will ignite when exposed to lightning discharge. In principle, this fact can be used in the analysis of descriptions of forest areas and to identify the most fire-prone trees in the controlled forested area. This can be used as the basis for a geoinformation system for assessing the fire-related danger to forests under conditions of thunderstorm activity. In addition, in relation to the most fire dangerous areas, certain measures may be introduced to care for forest areas by forest fire services. It may make sense to forcibly plant species of coniferous trees with spreading crowns and thus a large number of branches during reforestation or around industrial infrastructure. This could be a deterrent to the passage of a thunderstorm front in a controlled forested area.

The radial distribution of temperature at the moment of ignition presented in Figure 6: 1—section outside the crack in the bark; 2—section passing through a crack in the bark.

An analysis of the temperature curves in Figure 6 shows that, in the area of the subcrustal zone in the absence of a crack in the bark, an inert field of elevated temperature is formed. Under these conditions, the ignition of the tree trunk does not occur since there is no access for the oxidizer to the heated subcrustal zone. In the case of a crack in the bark layer, the situation is diametrically opposite. Gaseous pyrolysis products diffuse into the crack area and, mixing with the oxidizing agent, enter a chemical reaction at an elevated temperature. It is in the crack that the place of localization of ignition is located.

The fields of volume fractions of dry organic matter and the gas phase in the region of the subcrustal zone of a coniferous tree trunk and a crack in the bark layer are presented in Figure 7. The most intense thermal decomposition processes occur in the subcrustal zone of a coniferous tree trunk, releasing gaseous combustible pyrolysis products.

It should be noted here that there are different approaches to modeling the composition of dry organic matter [60]. For example, the following components can be suggested: lignin, cellulose and hemicellulose. Each of these components is characterized by a certain set of products that are formed during the thermal decomposition of a certain component.

In this work, we used the gross substance approach, during which the thermal decomposition of this substance produces gaseous and condensed pyrolysis products. It should be noted that the formation of liquid pyrolysis products is not taken into account in this work. It is believed that the gaseous product of pyrolysis is carbon monoxide, which reacts with oxygen and oxidizes into carbon dioxide. Other gaseous combustible components are neglected since their proportion is noticeably smaller than the formation of carbon monoxide during the thermal decomposition of dry organic matter. Accordingly, the condensed phase is represented by residues of dry organic matter and dispersed soot particles. Moreover, the part of soot particles that is formed during the period of intense thermal decomposition is considered. The soot formation process is considered to follow the thermal decomposition process.

It is known that there are video surveillance systems for smoke plumes in forest areas near wildland–urban interface areas. Perhaps the development of software based on the proposed mathematical model could improve the technical characteristics of such video surveillance systems for forest areas. In this article, a console application has been developed using the high-level programming language Object Pascal in RAD Studio Delphi 10 Architect. There are no obstacles to making a software module for such a system based on the software implementation of the developed mathematical model for the ignition of coniferous trees via cloud-to-ground lightning discharge.

The dependences of the volume fractions of dry organic matter and the gas phase in the subcrustal zone of a coniferous tree trunk at the location of a crack in the bark is presented in Figure 8.

It can be seen that the most significant processes of thermal decomposition of dry organic matter occur at exposure times above 0.3 s. That is, short-term discharge will not lead to the ignition of a coniferous tree trunk. This is due not only to insufficient heating of the wood in the zone of the electric current flow of a cloud-to-ground lightning discharge, but also to an insufficient concentration of combustible gaseous pyrolysis products of dry organic matter.

However, even prolonged exposure to lightning discharge does not lead to complete thermal decomposition of dry organic matter in the subcrustal zone of a coniferous tree trunk. In fact, an unstable porous structure of the material is formed in this zone, which can be subject to destruction. Previously, field observations of thunderstorm activity in the Timiryazevsky forestry of the Tomsk region were conducted [61]. As a result of numerous observations, a strike by such discharge into the trunk of a coniferous tree was recorded. A subsequent inspection of the adjacent territory showed the presence of charred wood fragments in the ground cover, which could potentially lead to a forest fire in dry thunderstorm conditions. This confirms the possibility of the destruction of the tree trunk in the subcrustal zone. Inspection of the tree trunk also confirmed the breakdown of the bark layer and the presence of traces of wood destruction in the subcrustal zone.

The dependences of the concentrations of the gas phase components at the moment of ignition presented in Figure 9: 1—section in the crack; 2—section outside the crack. An analysis of the results of numerical simulation showed that, over time, the formation of combustible gaseous pyrolysis products occurs in the subcrustal zone. It is believed that such products instantly appear in the near-wall region in the bark layer. When considering a section of the bark with no cracks, it is found that the combustible gaseous products of pyrolysis cannot reach the required critical ignition temperature. In addition, there is no oxidizing agent in this zone, which also does not contribute to ignition. In the area with a crack in the bark layer, a certain concentration of reacting components is also achieved, but this occurs when already in the presence of an oxidizing agent and when the critical temperature of the gas mixture is reached. It should be noted that the option of having a bark barrier is considered when the crack is partially filled with bark. It is found that, after reaching a thickness of the bark barrier of more than 1.5 mm, the gas mixture may not be heated to a critical temperature. As a result, the process of ignition of a coniferous tree trunk does not occur.

The dependence of the volume fractions of phases are presented in Figure 10, taking into account the formation of soot particles in the subcrustal zone of a coniferous tree trunk. These are typical simulation results. The analysis shows that the greatest soot formation processes begin when the exposure time of lightning discharge is more than 0.3 s. In addition, scenario calculations are carried out to take into account the intensity of the soot formation process. This factor is taken into account by varying the dispersion coefficient, which leads to a change in the volume fraction of soot particles formed when a coniferous tree trunk is ignited via cloud-to-ground lightning discharge.

Table 1. Wood ignition delays depending on discharge voltage at current J = 23.5 kA.

Voltage, U, kV	Ignition Delay, t*, s	Surface Temperature, K	Heat Flux to Ignition Surface, kW/m2	Fitting to Experiment [62]
1–85	No Ignition	<867	<210	No
90	0.516	>867	242	Yes
95	0.486	>867	246	Yes
100	0.463	>867	249	Yes
105	0.441	>867	252	Yes
110	0.423	>867	255	Yes

and

Table 2. Ignition delays of a tree trunk depending on the current at a voltage of U = 100 kV.

Current, J, kA	Ignition Delay, t*, s	Surface Temperature, K	Heat Flux to Ignition Surface, kW/m2	Fitting to Experiment [62]
1–20	No Ignition	<867	<210	No
23.5	0.463	>867	249	Yes
30	0.366	>867	264	Yes
35	0.317	>867	274	Yes

show the ignition delays of a coniferous tree depending on the parameters of lightning discharge. The conditions for the ignition of a coniferous tree in the context of the current–voltage characteristics of a cloud-to-ground lightning discharge are established.

It is known [62] that the ignition conditions for large wood fragments can be established by comparing the ignition surface temperature and the heat flux. Submodules are included in the computer program to calculate these parameters along with the calculation of the ignition delay time. Analysis of the data presented in Table 1 and Table 2 shows good agreement between both the experimental conditions for the ignition of wood and the conditions theoretically determined as a result of solving the presented mathematical model. Therefore, the developed mathematical model and its software implementation can be used to calculate the conditions and parameters of the ignition of a tree by an electric current of a lightning discharge passing through the tree trunk. The reliability of the numerical results is shown both when varying the electric current and the voltage of the lightning discharge.

The following indicators for assessing the results obtained have been introduced:

-

Possibility of practical use;

-

Reliability,

-

Completeness of information.

First of all, such an indicator as the completeness of information obtained using the proposed mathematical model must be discussed.

Numerical implementation of the mathematical model and scenario modeling make it possible to determine the following parameters as a result of modeling:

-

Ignition delay. This parameter can be used to parametrically assess the probability of a forest fire.

-

Temperature field in the tree trunk structure. This parameter allows us to determine the location of elevated temperatures, which can also be used to determine the probability of a forest fire. However, it must be noted that, for scenarios of modeling the formation of heated wood particles that fall on the ground layer of forest fuel material, it is also necessary to model the stress–strain state of the tree trunk, and such work will be carried out in future studies. It will then be possible to simulate the ignition of the ground cover, taking into account the size and temperature of the heated particles formed as a result of the destruction of the trunk wood. Future research will determine thermal stress in the trunk of a coniferous tree.

-

Volume fraction of soot particles. Information about this parameter can be used when setting up information and computing systems within the framework of hardware and software systems for the detection of soot and smoke particles in a controlled area. That is, information about the location of the emission of soot particles can be used to detect potential forest fires.

It is also necessary to discuss such a parameter as the possibility of the practical use of the proposed mathematical model of ignition of coniferous trees via cloud-to-ground lightning discharge. Below comparative information is presented on the execution time of program code developed on the basis of one-dimensional, two-dimensional and three-dimensional mathematical models. The following modeling scenario was used, reflecting typical situations under conditions of thunderstorm activity:

-

Exposure time 500 ms;

-

Current 23.5 kA;

-

Voltage 100 kV;

-

Summer season.

Table 3. Execution time of various software implementations.

N	Setting-Up	RAM Enough	Execution Time	r	φ	z	dt
1	1D	Yes	2 s	1000	-	-	10−4
2	2D	Yes	1 m 20 s	1000	100	-	10−4
3	3D	Yes	1 h 18 m 10 s	1000	100	200	10−3

presents the results of a comparative analysis of the execution time of various software implementations.

Thus, the use of a three-dimensional mathematical model is possible only with parallel implementation on supercomputers with multiprocessor and multicore computer system architecture. It should be noted that parallel implementation of the proposed mathematical model can be performed using massive parallelism. In this case, iterations of cyclic constructions are spread across the different computing cores of the computer system. Since the three-dimensional equations of thermal conductivity and diffusion are solved using a locally one-dimensional method, three blocks are distinguished for which parallel calculations can be carried out: these blocks are divided along the coordinates r, φ, and z. Taking into account the given data on the spatial grid covering the modeling area, it is necessary to use at least 10 computing nodes of a multiprocessor system. If necessary, you can make a flexible algorithm that will use 100 processor cores when modeling a block at the r coordinate, 10 processor cores when modeling at the φ coordinate, and 20 processor cores when modeling at the z coordinate. However, even in the worst-case scenario, the acceleration will approach the theoretical value of 10 times. The difference from theoretical estimates is due to the fact that part of the time will be spent on sending data at the beginning of the parallel program, collecting data at the end of calculations, and exchanging data between the blocks of the parallel program.

Now, as for the reliability of the results obtained. As has already been shown in the tables, there is qualitative agreement with the results of experimental data published in [62]. The volume fraction of soot particles also qualitatively corresponds to the scenarios for modeling soot formation under the influence of a forest fire sustained by forest fuels [13] and the removal of heated particles from the forest fire front [14]. It is clear that there is a difference in the time coordinate, since the process of the impact of a lightning discharge on a tree trunk is significantly shorter than the processes indicated above. However, qualitative agreement between the results can be established. This indicates the physical adequacy and satisfactory reliability of the proposed mathematical model. However, in the future it will be necessary to conduct another series of field observations of cloud-to-ground lightning discharges in a controlled forested area.

A study of the convergence of the obtained results on a sequence of condensing grids was carried out.

Table 4. Grid parameters, obtained temperature values at the control point (exposure time of CG lightning discharge 0.3 s).

N	R	φ	z	dt	Tcontrol
1	1000	100	200	10−3	828.697
2	1000	100	200	10−1	821.004
3	1000	100	200	10−2	827.747
4	500	100	200	10−3	840.926
5	200	100	200	10−3	842.264
6	1000	50	200	10−3	758.498
7	1000	30	200	10−3	785.063
8	1000	100	100	10−3	827.485
9	1000	100	50	10−3	826.912

shows the parameters characterizing the various conditions of the performed computational experiments.

Analysis of the results shows that it is necessary to use a time step that is no more than 10⁻³ s. Among the spatial coordinates, the angular coordinate and the radius of the trunk have the maximum impact. The minimum recommended grid parameters were (1000, 100, 200).

This paper presents a generalized mathematical model for the ignition of a coniferous tree via an electric current of a lightning discharge. This model can be used for boreal forest zones since most of the trees in such forests are coniferous. In order to take into account a specific region, data on the thermophysical and thermokinetic characteristics of coniferous wood that grow in a specific controlled forest area are needed. Taking the zone of temperate forests as an example, there is a noticeable proportion of deciduous trees in such an area. The results obtained can be applied to temperate forests, but it is necessary to modernize, first of all, the geometric model of a deciduous tree. In a deciduous tree, unlike a coniferous tree, moisture is transported in vessels that are located in the core of the deciduous tree. It is necessary to develop a geometric model that takes into account the location of the network of vessels that transport moisture in the core of the tree trunk. There are no other special obstacles for the widespread application of the proposed mathematical model. In principle, a computer program can be developed that allows the scenario modeling of the ignition of both coniferous and deciduous trees. In this case, the integral level of forest fire danger can be estimated by taking into account the proportion of coniferous and deciduous trees in the controlled forest area.

For practical purposes, it is necessary to conduct a study on the time required for numerical calculation using computer technology. Two variants of software implementations were compared: (a) full three-dimensional setting; and (b) a set of two-dimensional and one-dimensional settings. The comparison showed that the calculation for two-dimensional settings is faster. The results are obtained through a cumulative analysis of 10 calculation options. The grid parameters varied over space and in relation to the current–voltage characteristics of the discharge. There was a time step of 1 ms. Such a resolution in the time coordinate is justified by the technical characteristics of systems for recording cloud-to-ground lightning discharges [63].

The limitations of this study should be pointed out. First, within the framework of this article, a fragment of a tree trunk with a single branch is considered. It is possible that, if several branches were taken into account, some cumulative effect would be observed, which may lead to some differences in the assessment of the maximum temperature in some zones of the subcrustal layer. Secondly, this study does not take into account changes in the moisture content of the subcrustal layer, which can also lead to some quantitative differences in the assessment of the maximum temperature in the subcrustal zone of a coniferous tree trunk. Thirdly, the proportion of soot formed during the combustion of dry organic matter is not considered, although a slight increase in the total amount of soot particles can be expected.

In addition, future research should be discussed. This paper presents a fairly complex mathematical model for the ignition of a coniferous tree trunk in conditions of lightning activity, taking into account electrophysical, thermophysical, and physicochemical processes. However, even this mathematical model can be improved. This can be performed, first of all, by taking into account the more plausible geometry of the study area, that is, the structure of the trunk of a coniferous tree. Secondly, for practical purposes, it is necessary to develop an application with a graphical user interface capable of organizing interactions with a lightning discharge direction-finding system, such as WWLLN [64].

4. Conclusions

In the present work, a generalized three-dimensional mathematical model for the ignition of coniferous trees via cloud-to-ground lightning discharge was developed, taking into account a complex of electrophysical, thermophysical, and physicochemical processes. It was found that the time spent on calculations is significant and the use of single-processor computing technology does not have great prospects in the context of predicting forest fire danger. It would be more logical in the future to develop a parallel software implementation of the proposed mathematical model.

Key findings and conclusions:

(1)

The presence in the branch structure of heterogeneities like reactive wood causes the formation of a low-temperature field. It is possible that trees with spreading crowns are less likely to be ignited by lightning discharge.

(2)

The electric current passes in the subcrustal zone and leads to the formation of a zone of elevated temperature and the thermal decomposition of dry organic matter, with the formation of gaseous combustible pyrolysis products.

(3)

Gaseous products of thermal decomposition enter the gas phase and ignite, which leads to the ignition of the coniferous tree trunk in the crack.

(4)

The presence of a crustal barrier in a crack makes it impossible to heat the reactants of the gas mixture in the crack and to ignite them in the gas phase.

(5)

During the thermal decomposition of dry organic matter, a certain proportion of soot particles is formed. It is possible that the remote sensing of the surface layer of the atmosphere for the presence of soot particles can be used to monitor the occurrence of forest fires.