Published: 02 April 2024

# Tensor analysis of tornadoes: a new analytical and numerical model

Mustamina Maulani1
Valentinus Galih Vidia Putra2
1Department of Petroleum Engineering, Universitas Trisakti, Jakarta, Indonesia
2Basic and Applied Science Research Group in Theoretical and Plasma Physics, Department of Textile Engineering, Politeknik STTT Bandung, Bandung, Indonesia
Corresponding Authors:
Mustamina Maulani, Valentinus Galih Vidia Putra
Article in Press
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#### Highlights

• A new analytical and numerical model of tornado
• Rigorous analysis
• Compare the model with experiment
• Combine tensor analysis, computational modeling, as well as 2D and 3D simulations for simulating tornadoes

## 2. Research methods

The foundation of our work is the theoretical framework of variables used to describe motion. Eq. (1) expresses the acceleration in a fixed system in terms of location, speed, and acceleration in the rotating system:

1
$a={\stackrel{¨}{r}}^{\text{'}}+\stackrel{˙}{\omega }×{r}^{\text{'}}+2\omega ×{\stackrel{˙}{r}}^{\text{'}}+\omega ×\omega ×{r}^{\text{'}},$

where $\omega$ is the angular velocity, $\stackrel{˙}{\omega }$ is the time derivative of $\omega$ and $r\text{'}$, ${\stackrel{˙}{r}}^{\text{'}}$ and ${\stackrel{¨}{r}}^{\text{'}}$ are the position in the unit (m), velocity in the unit (m/s), and acceleration in the unit (m/s2), respectively. In the case where the primed system undergoes both translation and rotation, we obtain general equations for transforming from a fixed to a moving and rotating system, as shown in Eq. (2). This comprises the generic equations for transforming a stationary system into a moving and rotating one:

2
$a={\stackrel{¨}{r}}^{\text{'}}+\stackrel{˙}{\omega }×{r}^{\text{'}}+2\omega ×{\stackrel{˙}{r}}^{\text{'}}+\omega ×\omega ×{r}^{\text{'}}+A.$

After we have the motion equation in moving coordinates, we can write it as shown in Eqs. (3) to (6):

3
$F=ma=m\left({\stackrel{¨}{r}}^{\text{'}}+\stackrel{˙}{\omega }×{r}^{\text{'}}+2\omega ×{\stackrel{˙}{r}}^{\text{'}}+\omega ×\omega ×{r}^{\text{'}}+A\right),$
4
$F-m\left(\stackrel{˙}{\omega }×{r}^{\text{'}}+2\omega ×{\stackrel{˙}{r}}^{\text{'}}+\omega ×\omega ×{r}^{\text{'}}+A\right)=m{\stackrel{¨}{r}}^{\text{'}},$
5
$F-\left(m\stackrel{˙}{\omega }×{r}^{\text{'}}+2m\omega ×{\stackrel{˙}{r}}^{\text{'}}+m\omega ×\left(\omega ×{r}^{\text{'}}\right)+mA\right)=m{\stackrel{¨}{r}}^{\text{'}},$
6
$F-\left({F}_{Eul}+{F}_{cor}+{F}_{cent}+{F}_{t}\right)=m{\stackrel{¨}{r}}^{\text{'}},$

where $F$ is the physical force, ${F}_{eul}=m\stackrel{˙}{\omega }×{r}^{\text{'}}$ is the Euler force, $\omega$ is the angular velocity, ${F}_{cor}=2m\omega ×{\stackrel{˙}{r}}^{\text{'}}$ is the Coriolis force, ${F}_{cent}=m\omega ×\left(\omega ×{r}^{\text{'}}\right)$ denotes the centrifugal force, and ${F}_{t}=mA$ denotes the force due to the translation of the coordinate system. The equation of motion in a moving system can be written as Eq. (7):

7
$\sum {F}_{all}=m{\stackrel{¨}{r}}^{\text{'}}=\rho V{\stackrel{¨}{r}}^{\text{'}},\sum f=\rho {\stackrel{¨}{r}}^{\text{'}},$

where $f$ is the total force per unit volume, $\rho$ is the density of the particle with a certain mass, and ${\stackrel{¨}{r}}^{\text{'}}$ is the acceleration of the particle with a certain mass. We can extend the model by using Cauchy's equation in Eq. (8) and the generalized equation of motion in 3D movement proposed by [22] as in Eqs. (9) and (10):

8
$\nabla \bullet \sigma +{f}_{tot}=\rho {\stackrel{¨}{r}}^{\text{'}},$
9
$\nabla \bullet \sigma +{f}_{g}-\left({f}_{Eul}+{f}_{cor}+{f}_{cent}+{f}_{t}\right)=\rho \left({\stackrel{¨}{x}}^{\text{'}}+{\stackrel{¨}{y}}^{\text{'}}+{\stackrel{¨}{z}}^{\text{'}}\right),$
10
$\nabla \bullet \sigma +{f}_{g}-{f}_{Eul}-{f}_{cor}-{f}_{cent}-{f}_{t}=\rho \left({\stackrel{¨}{x}}^{\text{'}}+{\stackrel{¨}{y}}^{\text{'}}+{\stackrel{¨}{z}}^{\text{'}}\right),$

where ${f}_{tot}$ is the total force per unit volume, ${f}_{g}$ is the gravitational attraction force, $\rho$ is the density of the particle with a certain mass, and $\sigma$ is the stress tensor. Consider the Euler force ${f}_{Eul}=0$, the force due to the translation of the coordinate system ${f}_{t}=0$, and since the centrifugal force is so small compared to the other terms, we can neglect it. The equation of motion then becomes:

11
$\left(\frac{\partial {\sigma }_{xx}}{\partial x}+\frac{\partial {\sigma }_{yx}}{\partial y}+\frac{\partial {\sigma }_{zx}}{\partial z}\right){\mathbf{i}}^{\mathbf{\text{'}}}+\left(\frac{\partial {\sigma }_{xy}}{\partial x}+\frac{\partial {\sigma }_{yy}}{\partial y}+\frac{\partial {\sigma }_{zy}}{\partial z}\right){\mathbf{j}}^{\mathbf{\text{'}}}+\left(\frac{\partial {\sigma }_{xz}}{\partial x}+\frac{\partial {\sigma }_{yz}}{\partial y}+\frac{\partial {\sigma }_{zz}}{\partial z}\right){\mathbf{k}}^{\mathbf{\text{'}}}-\rho g{\mathbf{k}}^{\mathbf{\text{'}}}-2\rho \left({\omega }_{x}{\mathbf{i}}^{\mathbf{\text{'}}}+{\omega }_{y}{\mathbf{j}}^{\mathbf{\text{'}}}+{\omega }_{z}{\mathbf{k}}^{\mathbf{\text{'}}}\right)×\left({\stackrel{˙}{x}}^{\text{'}}{\mathbf{i}}^{\mathbf{\text{'}}}+{\stackrel{˙}{y}}^{\text{'}}{\mathbf{j}}^{\mathbf{\text{'}}}+{\stackrel{˙}{z}}^{\text{'}}{\mathbf{k}}^{\mathbf{\text{'}}}\right)=\rho \left({{\stackrel{¨}{x}}^{\text{'}}+{\stackrel{¨}{y}}^{\text{'}}+{\stackrel{¨}{z}}^{\text{'}}}^{\mathbf{\text{'}}}\right),$
12
${f}_{px}{\mathbf{i}}^{\mathbf{\text{'}}}+{f}_{py}{\mathbf{j}}^{\mathbf{\text{'}}}+{f}_{pz}{\mathbf{k}}^{\mathbf{\text{'}}}-\rho g{\mathbf{k}}^{\mathbf{\text{'}}}-2\rho \left({\omega }_{x}{\mathbf{i}}^{\mathbf{\text{'}}}+{\omega }_{y}{\mathbf{j}}^{\mathbf{\text{'}}}+{\omega }_{z}{\mathbf{k}}^{\mathbf{\text{'}}}\right)×\left({\stackrel{˙}{x}}^{\text{'}}{\mathbf{i}}^{\mathbf{\text{'}}}+{\stackrel{˙}{y}}^{\text{'}}{\mathbf{j}}^{\mathbf{\text{'}}}+{\stackrel{˙}{z}}^{\text{'}}{\mathbf{k}}^{\mathbf{\text{'}}}\right)=\rho \left({{\stackrel{¨}{x}}^{\text{'}}+{\stackrel{¨}{y}}^{\text{'}}+{\stackrel{¨}{z}}^{\text{'}}}^{\mathbf{\text{'}}}\right).$

To simplify the calculation of the model, it can be assumed that:

$\left(\frac{\partial {\sigma }_{xx}}{\partial x}+\frac{\partial {\sigma }_{yx}}{\partial y}+\frac{\partial {\sigma }_{zx}}{\partial z}\right){\mathbf{i}}^{\mathbf{\text{'}}}+\left(\frac{\partial {\sigma }_{xy}}{\partial x}+\frac{\partial {\sigma }_{yy}}{\partial y}+\frac{\partial {\sigma }_{zy}}{\partial z}\right){\mathbf{j}}^{\mathbf{\text{'}}}+\left(\frac{\partial {\sigma }_{xz}}{\partial x}+\frac{\partial {\sigma }_{yz}}{\partial y}+\frac{\partial {\sigma }_{zz}}{\partial z}\right){\mathbf{k}}^{\mathbf{\text{'}}}={f}_{px}{\mathbf{i}}^{\mathbf{\text{'}}}+{f}_{py}{\mathbf{j}}^{\mathbf{\text{'}}}+{f}_{pz}{\mathbf{k}}^{\mathbf{\text{'}}},$
${\omega }_{x}{\mathbf{i}}^{\mathbf{\text{'}}}+{\omega }_{y}{\mathbf{j}}^{\mathbf{\text{'}}}+{\omega }_{z}{\mathbf{k}}^{\mathbf{\text{'}}}=\left(0{\mathbf{i}}^{\mathbf{\text{'}}}+\omega \mathrm{c}\mathrm{o}\mathrm{s}\lambda {\mathbf{j}}^{\mathbf{\text{'}}}+\omega \mathrm{s}\mathrm{i}\mathrm{n}\lambda {\mathbf{k}}^{\mathbf{\text{'}}}\right),$

yield:

13
${f}_{px}{\mathbf{i}}^{\mathbf{\text{'}}}+{f}_{py}{\mathbf{j}}^{\mathbf{\text{'}}}+{f}_{pz}{\mathbf{k}}^{\mathbf{\text{'}}}-\rho g{\mathbf{k}}^{\mathbf{\text{'}}}=\rho \left({\stackrel{¨}{\mathbf{r}}}^{\mathbf{\text{'}}}\right)+2\rho \left({\omega }_{x}{\mathbf{i}}^{\mathbf{\text{'}}}+{\omega }_{y}{\mathbf{j}}^{\mathbf{\text{'}}}+{\omega }_{z}{\mathbf{k}}^{\mathbf{\text{'}}}\right)×\left({\stackrel{˙}{x}}^{\text{'}}\mathbf{i}+{\stackrel{˙}{y}}^{\text{'}}\mathbf{j}+{\stackrel{˙}{z}}^{\text{'}}\mathbf{k}\right)$
14
${f}_{px}{\mathbf{i}}^{\mathbf{\text{'}}}+{f}_{py}{\mathbf{j}}^{\mathbf{\text{'}}}+{f}_{pz}{\mathbf{k}}^{\mathbf{\text{'}}}-\rho g{\mathbf{k}}^{\mathbf{\text{'}}}=\rho \left({\stackrel{¨}{\mathbf{r}}}^{\mathbf{\text{'}}}+2\left(0{\mathbf{i}}^{\mathbf{\text{'}}}+\omega \mathrm{c}\mathrm{o}\mathrm{s}\lambda {\mathbf{j}}^{\mathbf{\text{'}}}+\omega \mathrm{s}\mathrm{i}\mathrm{n}\lambda {\mathbf{k}}^{\mathbf{\text{'}}}\right)×\left({\stackrel{˙}{x}}^{\text{'}}\mathbf{i}+{\stackrel{˙}{y}}^{\text{'}}\mathbf{j}+{\stackrel{˙}{z}}^{\text{'}}\mathbf{k}\right)\right).$

We can write $g=\mathcal{}\mathcal{g}-{\omega }^{2}R\mathrm{c}\mathrm{o}\mathrm{s}\lambda$ because of the effect of Earth’s rotation. Where $\mathcal{g}$ denotes the actual gravitation acceleration, and ${\omega }^{2}R\mathrm{c}\mathrm{o}\mathrm{s}\lambda$ denotes the centripetal acceleration for the Earth’s radius, $R$, and geocentric latitude, $\lambda$. In this study, we choose the coordinate axis $O\text{'}x\text{'}y\text{'}z\text{'}$ such that the $z\text{'}$ is vertical, the $x\text{'}$ axis to the east, and the $y\text{'}$ axis points north. The coordinate axes for analyzing tornado motion can be shown in Fig. 1.

Fig. 1Coordinate axes for analyzing tornado motion.

We also use ${\omega }_{x}=0$, ${\omega }_{y}=\omega \mathrm{c}\mathrm{o}\mathrm{s}\lambda \text{,}$ and ${\omega }_{z}=\omega \mathrm{s}\mathrm{i}\mathrm{n}\lambda .$ Eq. (14) can be solved computationally, and we get Eqs. (15) and (16):

15
${f}_{px}{\mathbf{i}}^{\mathbf{\text{'}}}+{f}_{py}{\mathbf{j}}^{\mathbf{\text{'}}}+{f}_{pz}{\mathbf{k}}^{\mathbf{\text{'}}}-\rho g{\mathbf{k}}^{\mathbf{\text{'}}}=\rho {\stackrel{¨}{\mathbf{r}}}^{\mathbf{\text{'}}}+2\rho \left({\stackrel{˙}{z}}^{\text{'}}\omega \mathrm{c}\mathrm{o}\mathrm{s}\lambda -{\stackrel{˙}{y}}^{\text{'}}\omega \mathrm{s}\mathrm{i}\mathrm{n}\lambda \right){\mathbf{i}}^{\mathbf{\text{'}}}+2\rho \left({\stackrel{˙}{x}}^{\text{'}}\omega \mathrm{s}\mathrm{i}\mathrm{n}\lambda \right){\mathbf{j}}^{\mathbf{\text{'}}}-2\rho \left({\stackrel{˙}{x}}^{\text{'}}\omega \mathrm{c}\mathrm{o}\mathrm{s}\lambda \right){\mathbf{k}}^{\mathbf{\text{'}}},$
16
$\left(\frac{\partial {\sigma }_{xx}}{\partial x}+\frac{\partial {\sigma }_{yx}}{\partial y}+\frac{\partial {\sigma }_{zx}}{\partial z}\right){\mathbf{i}}^{\mathbf{\text{'}}}+\left(\frac{\partial {\sigma }_{xy}}{\partial x}+\frac{\partial {\sigma }_{yy}}{\partial y}+\frac{\partial {\sigma }_{zy}}{\partial z}\right){\mathbf{j}}^{\mathbf{\text{'}}}+\left(\frac{\partial {\sigma }_{xz}}{\partial x}+\frac{\partial {\sigma }_{yz}}{\partial y}+\frac{\partial {\sigma }_{zz}}{\partial z}\right){\mathbf{k}}^{\mathbf{\text{'}}}-\rho g{\mathbf{k}}^{\mathbf{\text{'}}}=\rho {\stackrel{¨}{\mathbf{r}}}^{\mathbf{\text{'}}}+2\rho \left({\stackrel{˙}{z}}^{\text{'}}\omega \mathrm{c}\mathrm{o}\mathrm{s}\lambda -{\stackrel{˙}{y}}^{\text{'}}\omega \mathrm{s}\mathrm{i}\mathrm{n}\lambda \right){\mathbf{i}}^{\mathbf{\text{'}}}+2\rho \left({\stackrel{˙}{x}}^{\text{'}}\omega \mathrm{s}\mathrm{i}\mathrm{n}\lambda \right){\mathbf{j}}^{\mathbf{\text{'}}}-2\rho \left({\stackrel{˙}{x}}^{\text{'}}\omega \mathrm{c}\mathrm{o}\mathrm{s}\lambda \right){\mathbf{k}}^{\mathbf{\text{'}}}.$

We can solve Eq. (16); hence we find Eqs. (17) to (19):

17
${\stackrel{¨}{\mathbf{x}}}^{\mathbf{\text{'}}}=\frac{1}{\rho }\left(\frac{\partial {\sigma }_{xx}}{\partial x}+\frac{\partial {\sigma }_{yx}}{\partial y}+\frac{\partial {\sigma }_{zx}}{\partial z}\right)-2\omega \left({\stackrel{˙}{z}}^{\text{'}}\mathrm{c}\mathrm{o}\mathrm{s}\lambda -{\stackrel{˙}{y}}^{\text{'}}\mathrm{s}\mathrm{i}\mathrm{n}\lambda \right),$
18
${\stackrel{¨}{\mathbf{y}}}^{\mathbf{\text{'}}}=\frac{1}{\rho }\left(\frac{\partial {\sigma }_{xy}}{\partial x}+\frac{\partial {\sigma }_{yy}}{\partial y}+\frac{\partial {\sigma }_{zy}}{\partial z}\right)-2{\stackrel{˙}{x}}^{\text{'}}\omega \mathrm{s}\mathrm{i}\mathrm{n}\lambda ,$
19
${\stackrel{¨}{\mathbf{z}}}^{\mathbf{\text{'}}}=\left(\frac{1}{\rho }\left(\frac{\partial {\sigma }_{xz}}{\partial x}+\frac{\partial {\sigma }_{yz}}{\partial y}+\frac{\partial {\sigma }_{zz}}{\partial z}\right)-g\right)+2{\stackrel{˙}{x}}^{\text{'}}\omega \mathrm{c}\mathrm{o}\mathrm{s}\lambda .$

Assuming that $\frac{\partial {\sigma }_{zz}}{\partial z}=\frac{R\rho T}{T}\frac{\partial \left(T\right)}{\partial z}+\frac{R\rho T}{\rho }\frac{\partial \left(\rho \right)}{\partial z}={\sigma }_{o}$, and ${\sigma }_{xx}={\sigma }_{xy}={\sigma }_{xz}={\sigma }_{yy}={\sigma }_{yz}=0$, hence we get ${e}_{zz}=\frac{{\sigma }_{zz}}{E}$, ${e}_{yy}=-v\frac{{\sigma }_{zz}}{E}$, and ${e}_{xx}=-v\frac{{\sigma }_{zz}}{E}$, ${e}_{xy}={e}_{xz}={e}_{yz}=0$, and we find Eqs. (20) to (22):

20
${\stackrel{¨}{\mathbf{x}}}^{\mathbf{\text{'}}}=\frac{1}{\rho }\left(\frac{\partial {\sigma }_{xx}}{\partial x}+\frac{\partial {\sigma }_{yx}}{\partial y}+\frac{\partial {\sigma }_{zx}}{\partial z}\right)-2\omega \left({\stackrel{˙}{z}}^{\text{'}}\mathrm{c}\mathrm{o}\mathrm{s}\lambda -{\stackrel{˙}{y}}^{\text{'}}\mathrm{s}\mathrm{i}\mathrm{n}\lambda \right)=-2\omega \left({\stackrel{˙}{z}}^{\text{'}}\mathrm{c}\mathrm{o}\mathrm{s}\lambda -{\stackrel{˙}{y}}^{\text{'}}\mathrm{s}\mathrm{i}\mathrm{n}\lambda \right),$
21
${\stackrel{¨}{\mathbf{y}}}^{\mathbf{\text{'}}}=\frac{1}{\rho }\left(\frac{\partial {\sigma }_{xy}}{\partial x}+\frac{\partial {\sigma }_{yy}}{\partial y}+\frac{\partial {\sigma }_{zy}}{\partial z}\right)-2{\stackrel{˙}{x}}^{\text{'}}\omega \mathrm{s}\mathrm{i}\mathrm{n}\lambda =-2{\stackrel{˙}{x}}^{\text{'}}\omega \mathrm{s}\mathrm{i}\mathrm{n}\lambda ,$
22
${\stackrel{¨}{\mathbf{z}}}^{\mathbf{\text{'}}}=\left(\frac{1}{\rho }\left(\frac{\partial {\sigma }_{xz}}{\partial x}+\frac{\partial {\sigma }_{yz}}{\partial y}+\frac{\partial {\sigma }_{zz}}{\partial z}\right)-g\right)+2{\stackrel{˙}{x}}^{\text{'}}\omega \mathrm{c}\mathrm{o}\mathrm{s}\lambda =\frac{{\sigma }_{o}}{\rho }-g+2{\stackrel{˙}{x}}^{\text{'}}\omega \mathrm{c}\mathrm{o}\mathrm{s}\lambda .$

We can integrate once concerning $t$ to get the component of velocity, and we find, as shown in Eqs. (23) to (25):

23
${\stackrel{˙}{\mathbf{x}}}^{\mathbf{\text{'}}}={\stackrel{˙}{{x}_{o}}}^{\mathbf{\text{'}}}-2\omega \left({z}^{\text{'}}\mathrm{c}\mathrm{o}\mathrm{s}\lambda -{y}^{\text{'}}\mathrm{s}\mathrm{i}\mathrm{n}\lambda \right),$
24
${\stackrel{˙}{\mathbf{y}}}^{\text{'}}={\stackrel{˙}{{y}_{o}}}^{\mathbf{\text{'}}}-2{x}^{\text{'}}\omega \mathrm{s}\mathrm{i}\mathrm{n}\lambda ,$
25
${\stackrel{˙}{\mathbf{z}}}^{\mathbf{\text{'}}}={\stackrel{˙}{{z}_{o}}}^{\mathbf{\text{'}}}+\left(\frac{{\sigma }_{o}}{\rho }-g\right)t+2{x}^{\text{'}}\omega \mathrm{c}\mathrm{o}\mathrm{s}\lambda .$

Then substitute ${\stackrel{˙}{\mathbf{z}}}^{\mathbf{\text{'}}}$ and $\stackrel{˙}{\mathbf{y}}\text{'}$ into Eq. (20), we find Eqs. (26) and (27):

26
${\stackrel{¨}{\mathbf{x}}}^{\mathbf{\text{'}}}=-2\omega \left({\stackrel{˙}{z}}^{\text{'}}\mathrm{c}\mathrm{o}\mathrm{s}\lambda -{\stackrel{˙}{y}}^{\text{'}}\mathrm{s}\mathrm{i}\mathrm{n}\lambda \right)=-2\omega \left(\left[{\stackrel{˙}{{z}_{o}}}^{\mathbf{\text{'}}}+\left(\frac{{\sigma }_{o}}{\rho }-g\right)t+2{x}^{\text{'}}\omega \mathrm{c}\mathrm{o}\mathrm{s}\lambda \right]\mathrm{c}\mathrm{o}\mathrm{s}\lambda -\left[{\stackrel{˙}{{y}_{o}}}^{\mathbf{\text{'}}}-2{x}^{\text{'}}\omega \mathrm{s}\mathrm{i}\mathrm{n}\lambda \right]\mathrm{s}\mathrm{i}\mathrm{n}\lambda \right),$
27
${\stackrel{¨}{\mathbf{x}}}^{\mathbf{\text{'}}}=-2\omega \left(\left[{\stackrel{˙}{{z}_{o}}}^{\mathbf{\text{'}}}+\left(\frac{{\sigma }_{o}}{\rho }-g\right)+2{x}^{\text{'}}\omega \mathrm{c}\mathrm{o}\mathrm{s}\lambda \right]\mathrm{c}\mathrm{o}\mathrm{s}\lambda -\left[{\stackrel{˙}{{y}_{o}}}^{\mathbf{\text{'}}}-2{x}^{\text{'}}\omega \mathrm{s}\mathrm{i}\mathrm{n}\lambda \right]\mathrm{}\mathrm{s}\mathrm{i}\mathrm{n}\lambda \right)\cong -2\omega {\stackrel{˙}{{z}_{o}}}^{\mathbf{\text{'}}}\mathrm{c}\mathrm{o}\mathrm{s}\lambda -2\mathrm{\omega }\left(\frac{{\sigma }_{o}}{\rho }-g\right)t\mathrm{c}\mathrm{o}\mathrm{s}\lambda +2\omega {\stackrel{˙}{{y}_{o}}}^{\mathbf{\text{'}}}\mathrm{s}\mathrm{i}\mathrm{n}\lambda =2\omega \left(g-\frac{{\sigma }_{o}}{\rho }\right)t\mathrm{c}\mathrm{o}\mathrm{s}\lambda -2\mathrm{\omega }\left({\stackrel{˙}{{z}_{o}}}^{\mathbf{\text{'}}}\mathrm{c}\mathrm{o}\mathrm{s}\lambda -\stackrel{˙}{{y}_{o}}\mathbf{\text{'}}\mathrm{s}\mathrm{i}\mathrm{n}\lambda \right).$

We integrate Eq. (27) again to get $\stackrel{˙}{x}\mathbf{\text{'}}$, as shown in Eq. (28) and Eq. (29):

28
$\int d\stackrel{˙}{x}\text{'}=\int \left[2\omega \left(g-\frac{{\sigma }_{o}}{\rho }\right)t\mathrm{c}\mathrm{o}\mathrm{s}\lambda -2\omega \left({\stackrel{˙}{{z}_{o}}}^{\mathbf{\text{'}}}\mathrm{c}\mathrm{o}\mathrm{s}\lambda -\stackrel{˙}{{y}_{o}}\mathbf{\text{'}}\mathrm{s}\mathrm{i}\mathrm{n}\lambda \right)\right]dt,$
29
${\stackrel{˙}{x}}^{\mathbf{\text{'}}}=\omega \left(g-\frac{{\sigma }_{o}}{\rho }\right){t}^{2}\mathrm{c}\mathrm{o}\mathrm{s}\lambda -2\omega t\left({\stackrel{˙}{{z}_{o}}}^{\mathbf{\text{'}}}cos\lambda -\stackrel{˙}{{y}_{o}}\mathbf{\text{'}}\mathrm{s}\mathrm{i}\mathrm{n}\lambda \right)+{\stackrel{˙}{{x}_{o}}}^{\text{'}},$

and finally, we find ${x}^{\text{'}}$ by integrating Eq. (29):

30
${x}^{\text{'}}=\frac{\omega \left(g-\frac{{\sigma }_{o}}{\rho }\right){t}^{3}}{3}\mathrm{c}\mathrm{o}\mathrm{s}\lambda -\mathrm{\omega }{t}^{2}\left({\stackrel{˙}{{z}_{o}}}^{\mathbf{\text{'}}}\mathrm{c}\mathrm{o}\mathrm{s}\lambda -\stackrel{˙}{{y}_{o}}\mathbf{\text{'}}\mathrm{s}\mathrm{i}\mathrm{n}\lambda \right)+{\stackrel{˙}{{x}_{o}}}^{\text{'}}t+{x}_{o}^{\text{'}}.$

Then substitute Eq.(30) into Eqs. (24) and (25), we find Eqs. (31) and (32):

31
${\stackrel{˙}{\mathbf{y}}}^{\text{'}}={\stackrel{˙}{{y}_{o}}}^{\mathbf{\text{'}}}-2{x}^{\text{'}}\omega \mathrm{s}\mathrm{i}\mathrm{n}\lambda ={\stackrel{˙}{{y}_{o}}}^{\mathbf{\text{'}}}-2\left(\frac{\omega g{t}^{3}}{3}\mathrm{c}\mathrm{o}\mathrm{s}\lambda -\omega {t}^{2}\left({\stackrel{˙}{{z}_{o}}}^{\mathbf{\text{'}}}\mathrm{c}\mathrm{o}\mathrm{s}\lambda -\stackrel{˙}{{y}_{o}}\mathbf{\text{'}}\mathrm{s}\mathrm{i}\mathrm{n}\lambda \right)+{\stackrel{˙}{{x}_{o}}}^{\text{'}}t+{x}_{o}\text{'}\right)\omega \mathrm{s}\mathrm{i}\mathrm{n}\lambda \cong {\stackrel{˙}{{y}_{o}}}^{\mathbf{\text{'}}}-2\left({\stackrel{˙}{{x}_{o}}}^{\text{'}}t\omega \mathrm{s}\mathrm{i}\mathrm{n}\lambda +{x}_{o}\mathrm{\text{'}}\omega \mathrm{s}\mathrm{i}\mathrm{n}\lambda \right)$
32
${\stackrel{˙}{z}}^{\text{'}}={\stackrel{˙}{{z}_{o}}}^{\text{'}}+\left(\frac{{\sigma }_{o}}{\rho }-g\right)t+2{x}^{\text{'}}\omega \mathrm{c}\mathrm{o}\mathrm{s}\lambda ={\stackrel{˙}{{z}_{o}}}^{\mathrm{\text{'}}}+\left(\frac{{\sigma }_{o}}{\rho }-g\right)t$
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}+2\left(\frac{\omega \left(g-\frac{{\sigma }_{o}}{\rho }\right){t}^{3}}{3}\mathrm{c}\mathrm{o}\mathrm{s}\lambda -\mathrm{\omega }{t}^{2}\left({\stackrel{˙}{{z}_{o}}}^{\mathrm{\text{'}}}\mathrm{c}\mathrm{o}\mathrm{s}\lambda -\stackrel{˙}{{y}_{o}}\mathrm{\text{'}}\mathrm{s}\mathrm{i}\mathrm{n}\lambda \right)+{\stackrel{˙}{{x}_{o}}}^{\text{'}}t\right)\omega \mathrm{c}\mathrm{o}\mathrm{s}\lambda .$

As a result of integrating Eqs. (31) and (32), the positions, ${y}^{\text{'}}$ and ${z}^{\text{'}}$, are given by:

33
${y}^{\text{'}}={\stackrel{˙}{{y}_{o}}}^{\mathbf{\text{'}}}t-2\left(\frac{{\stackrel{˙}{{x}_{o}}}^{\text{'}}{t}^{2}}{2}\omega \mathrm{s}\mathrm{i}\mathrm{n}\lambda +{x}_{o}\mathrm{\text{'}}t\omega \mathrm{s}\mathrm{i}\mathrm{n}\lambda \right)+{y}_{o}^{\text{'}}={y}_{o}^{\text{'}}+{\stackrel{˙}{{y}_{o}}}^{\mathbf{\text{'}}}t-{\stackrel{˙}{{x}_{o}}}^{\text{'}}{t}^{2}\omega \mathrm{s}\mathrm{i}\mathrm{n}\lambda -2{x}_{o}^{\text{'}}t\omega \mathrm{s}\mathrm{i}\mathrm{n}\lambda \approx {y}_{o}^{\text{'}}+{\stackrel{˙}{{y}_{o}}}^{\mathbf{\text{'}}}t-2{x}_{o}^{\text{'}}t\omega \mathrm{s}\mathrm{i}\mathrm{n}\lambda ,$
34
${z}^{\text{'}}={\stackrel{˙}{{z}_{o}}}^{\mathbf{\text{'}}}t+{z}_{o}+\omega \mathrm{c}\mathrm{o}\mathrm{s}\lambda {\stackrel{˙}{{x}_{o}}}^{\text{'}}{t}^{2}+\frac{1}{2}\left(\frac{{\sigma }_{o}}{\rho }-g\right){t}^{2}={\stackrel{˙}{{z}_{o}}}^{\mathbf{\text{'}}}t+\left({\omega }^{2}{r}_{o}^{\text{'}}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{\lambda }+\frac{1}{2}\left(\frac{{\sigma }_{o}}{\rho }-g\right)\right){t}^{2}+{z}_{o}.$

According to some scientists [19, 20], a tornado is a dangerous natural event that occurs on an insignificant scale and persists for only a few minutes. Assuming that ${y}_{o}^{\text{'}}+{\stackrel{˙}{{y}_{o}}}^{\mathbf{\text{'}}}t={r}_{o}^{\text{'}}$, ${x}_{o}^{\text{'}}=\stackrel{˙}{{x}_{o}}t={r}_{o}^{\text{'}}\omega t$, and $\stackrel{˙}{{x}_{o}}=\stackrel{˙}{{y}_{o}}=\stackrel{˙}{{z}_{o}}={r}_{o}^{\text{'}}\omega$, $\stackrel{˙}{{r}_{o}}$is so small at a very short time, we get Eqs. (35) and (36):

35
${x}^{\text{'}}=\frac{\omega g{t}^{3}}{3}\mathrm{c}\mathrm{o}\mathrm{s}\lambda -\omega {t}^{2}\left({\stackrel{˙}{{z}_{o}}}^{\mathbf{\text{'}}}\mathrm{c}\mathrm{o}\mathrm{s}\lambda -\stackrel{˙}{{y}_{o}}\mathbf{\text{'}}\mathrm{s}\mathrm{i}\mathrm{n}\lambda \right)+{\stackrel{˙}{{x}_{o}}}^{\text{'}}t=\frac{\omega g{t}^{3}}{3}\mathrm{c}\mathrm{o}\mathrm{s}\lambda -\omega {t}^{2}\left({\stackrel{˙}{{z}_{o}}}^{\mathbf{\text{'}}}\mathrm{c}\mathrm{o}\mathrm{s}\lambda -\stackrel{˙}{{y}_{o}}\mathbf{\text{'}}\mathrm{s}\mathrm{i}\mathrm{n}\lambda \right)+{r}_{o}^{\text{'}}\omega t\approx {r}_{o}^{\text{'}}\omega t,$
36
${y}^{\text{'}}={r}_{o}^{\text{'}}-2{r}_{o}^{\text{'}}{\omega }^{2}{t}^{2}\mathrm{}\mathrm{s}\mathrm{i}\mathrm{n}\lambda ={r}_{o}^{\text{'}}\left(1-2\mathrm{s}\mathrm{i}\mathrm{n}\lambda {\omega }^{2}{t}^{2}\right)=r\left(1-\frac{{\left(\omega t\right)}^{2}}{2}\right)={r}_{o}^{\text{'}}\mathrm{cos}\left(\omega t\right).$

Which requires that $2\mathrm{s}\mathrm{i}\mathrm{n}\lambda =\frac{1}{2}$ or $\mathrm{s}\mathrm{i}\mathrm{n}\lambda =\frac{1}{4}$ or $\lambda \approx$15° (that is near the Equator), we find:

37
${x}^{\text{'}}={r}_{o}^{\text{'}}\omega t={r}_{o}^{\text{'}}\mathrm{sin}\left(\omega t\right),$
38
${y}^{\text{'}}={r}_{o}^{\text{'}}\mathrm{cos}\left(\omega t\right),$
39
${z}^{\text{'}}={\stackrel{˙}{{z}_{o}}}^{\mathbf{\text{'}}}t+\left({\omega }^{2}+\frac{1}{2\mathrm{c}\mathrm{o}\mathrm{s}\lambda }\left(\frac{{\sigma }_{o}}{\rho }-g\right)\right){t}^{2}\mathrm{}\mathrm{c}\mathrm{o}\mathrm{s}\lambda +{z}_{o}={r}_{o}^{\text{'}}\omega t+\left({\omega }^{2}+\frac{1}{2\mathrm{c}\mathrm{o}\mathrm{s}\lambda }\left(\frac{{\sigma }_{o}}{\rho }-g\right)\right){t}^{2}\mathrm{}\mathrm{c}\mathrm{o}\mathrm{s}\lambda .$

From Eqs. (37) to (39), we get Eqs. (40) to (42):

40
${x}^{\text{'}}={r}_{o}^{\text{'}}\mathrm{sin}\left(\omega t\right)={r}_{o}^{\text{'}}\mathrm{cos}\left(9{0}^{o}-\omega t\right),$
41
${y}^{\text{'}}={r}_{o}^{\text{'}}\mathrm{cos}\left(\omega t\right)={r}_{o}^{\text{'}}\mathrm{sin}\left(9{0}^{o}-\omega t\right),$
42
${z}^{\text{'}}={r}_{o}^{\text{'}}\omega t+\left({\omega }^{2}+\frac{1}{2\mathrm{c}\mathrm{o}\mathrm{s}\lambda }\left(\frac{{\sigma }_{o}}{\rho }-g\right)\right){t}^{2}\mathrm{}\mathrm{c}\mathrm{o}\mathrm{s}\lambda .$

Assuming that ${r}_{o}^{\text{'}}={\stackrel{˙}{{r}_{o}}}^{\mathbf{\text{'}}}t$, and ignoring the effects of gravitational force and humidity, and using the equation of state, we obtain Eqs. (43) to (45):

43
${z}^{\text{'}}={r}_{o}^{\text{'}}\omega t+\left({\omega }^{2}+\frac{1}{2\mathrm{c}\mathrm{o}\mathrm{s}\lambda }\left(\frac{{\sigma }_{o}}{\rho }\right)\right){t}^{2}\mathrm{}\mathrm{c}\mathrm{o}\mathrm{s}\lambda ,$
44
${z}^{\text{'}}={r}_{o}^{\text{'}}\omega t+\left({\omega }^{2}+\frac{1}{2\mathrm{c}\mathrm{o}\mathrm{s}\lambda }\left(\frac{1}{\rho }\frac{\partial {\sigma }_{zz}}{\partial z}\right)\right){t}^{2}\mathrm{}\mathrm{c}\mathrm{o}\mathrm{s}\lambda ,$
45
${z}^{\text{'}}={\stackrel{˙}{{r}_{o}}}^{\mathbf{\text{'}}}{t}^{2}+\left({\omega }^{2}+\frac{1}{2\rho \mathrm{c}\mathrm{o}\mathrm{s}\lambda }\left(\frac{\partial {\sigma }_{zz}}{\partial z}\right)\right){t}^{2}\mathrm{}\mathrm{c}\mathrm{o}\mathrm{s}\lambda .$

According to [21], in the condition of a static atmosphere, we can write $\frac{\partial {\sigma }_{zz}}{\partial z}=\frac{\partial \left(R\rho T\right)}{\partial z}$. The density $\rho$ and pressure of the air ${\sigma }_{zz}$ in Eq (44) vary with height $z$. These changes can be calculated from the equation of state; we obtain:

46
$\frac{d\left({\sigma }_{zz}\right)}{dz}=R\rho \frac{\partial \left(T\right)}{\partial z}+RT\frac{\partial \left(\rho \right)}{\partial z},$
47
$\frac{d\left({\sigma }_{zz}\right)}{dz}=\frac{R\rho T}{T}\frac{\partial \left(T\right)}{\partial z}+R\rho T\frac{1}{\rho }\frac{\partial \left(\rho \right)}{\partial z},$
48
$\frac{1}{{\sigma }_{zz}}\frac{d\left({\sigma }_{zz}\right)}{dz}=\frac{1}{T}\frac{\partial \left(T\right)}{\partial z}+\frac{1}{\rho }\frac{\partial \left(\rho \right)}{\partial z},$

where $T$ is an absolute temperature, and $R$ is the specific gas constant of dry air. In Eq. (48), the density $\rho$ and $T$ vary with altitude, and assuming that $\frac{1}{{\sigma }_{zz}}\frac{d\left({\sigma }_{zz}\right)}{dz}=\frac{1}{2\rho }\left(\frac{\partial {\sigma }_{zz}}{\partial z}\right)$, then we obtain the position ${z}^{\text{'}}$, which indicates the height of tornadoes as shown in Eq. (49):

49
${z}^{\text{'}}={\stackrel{˙}{{z}_{o}}}^{\mathbf{\text{'}}}t+\left({\omega }^{2}\mathrm{c}\mathrm{o}\mathrm{s}\lambda +\frac{1}{2\rho }\left(\frac{\partial {\sigma }_{zz}}{\partial z}\right)\right){t}^{2}={{r}_{o}}^{\mathbf{\text{'}}}\omega t+\left({\omega }^{2}\mathrm{c}\mathrm{o}\mathrm{s}\lambda +\left(\frac{1}{T}\frac{\partial \left(T\right)}{\partial z}+\frac{1}{\rho }\frac{\partial \left(\rho \right)}{\partial z}\right)\right){t}^{2}.$

where $2\rho ={\sigma }_{zz}$ is related to tornado pressure.

### 2.2. Tornado simulation using numerical modeling

In the present research, we generated a model utilizing computational modeling that numerous researchers have performed on tornado characteristics [20, 21, 10, 11]. In this research, Eqs. (40), (41), and (49) address the difficulty of mathematically expressing positions in three dimensions when simulating tornado formation. Eqs. (29), (31), and (32) provide solutions to the difficulty of mathematically describing the velocity of the wind in growing tornadoes. In our research, MATLAB code was created to model tornado motion in two-dimensional and three-dimensional positions to explain tornado formation utilizing Eqs. (40), (41), and (49).

## 3. Results and discussions

Fig. 2 shows a two-dimensional model of a tornado with the velocity of the wind varying to the west and north throughout the same period. Modeling results show that the greater the wind velocity to the north and west, the larger the region of the tornado movement. The simulation findings reveal that the tornado’s area of rotation is affected by the velocity of the wind, tornado time, and earth rotational speed. The tornado may rotate and require Coriolis force to move.

Fig. 2Two-dimensional simulation of a tornado with wind speed variations

Fig. 3Three-dimensional simulation of a tornado with wind speed and geocentric latitude variations

Fig. 4Tornado in Gunung kencana, Banten, Indonesia [1]

The problem of mathematically expressing places in three-dimensional space when simulating tornado formation is addressed by Eqs (40), (41), and (49). Eqs. (29), (31), and (32) provide solutions to the difficulty of mathematically describing wind velocity in forming tornadoes. The modeling results demonstrate that the higher the geocentric latitude angle, the more likely a tornado will form. This research suggests that huge tornadoes can form in places with high geocentric latitudes. In this study, we discovered the equation for the motion of a tornado in three-dimensional coordinates, as shown in Eqs. (17) through (19).

## 4. Conclusions

This research reported a theoretical formulation of tornadoes in a non-inertial mechanics framework, utilizing fluid mechanics and numerical simulation. This model depicted the spiraling upward motion of air in a tornado while ignoring vertical convection. Several conditions were required for a tornado to occur, including geocentric latitude, the Coriolis effect, increased airspeed in the upper atmosphere, and increased air pressure. We calculated the airflow characteristics of a tornado and solved the three-dimensional position of the tornado or hurricane in three-dimensional (3D) space, as well as the differential equations of airflow velocity and the Earth’s rotation. To demonstrate tornado patterns, motion dynamics modeling, and numerical computations were performed using computer software. The study concluded that this model could explain tornado patterns. Using the modeling and simulation data from this work, practitioners and scientists can gain a better understanding of hurricanes. To obtain more precise models, we proposed that additional studies be performed utilizing various methodologies, such as quantum neural networks / QNNs and artificial neural networks in future research.

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01 March 2024
Accepted
22 March 2024
Published
02 April 2024
Keywords
coriolis effect
numerical model
computational dynamics
Acknowledgements

The authors have not disclosed any funding.

The authors are grateful to the Republic of Indonesia’s Ministry of Industry and Universitas Trisakti for providing adequate facilities. We also thank our colleagues who helped us with the research and analysis.

Data Availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Author Contributions

Valentinus Galih Vidia Putra and Mustamina Maulani conducted the simulations and the calculations. Valentinus Galih Vidia Putra and Mustamina Maulani wrote and revised the manuscript. All authors agreed to the final version of this manuscript.

Conflict of interest

The authors declare that they have no conflict of interest.