Nonlinear Burgers Type Equation for Acoustic Waves in the Ray Approximation in a Moving Atmosphere (Theory, Experiment)

Abstract

The derivation of a nonlinear Burgers-type equation for acoustic waves within the ray approximation for an inhomogeneous moving dissipative atmosphere is presented. The equation is applied to investigate the propagation of three-dimensional infrasonic waves, without using significant computational resources. By means of solving the obtained equation, the shapes of infrasonic signals recorded at distances of 295 km and 305 km from the explosion with energy of 30 kt THT were calculated. The observation point at the distance of 295 km (Tsimlyansk) was located in the western direction from the source. At this place, infrasonic signals were recorded corresponding to the sound propagation in the stratospheric and thermospheric acoustic waveguides. The presence of both two stratospheric and two thermospheric rays falling into the same observation point on the Earth's surface is a wave propagation feature result here. The recorded signals in Tsimlyansk also have a complex structure, both for stratospheric and thermospheric infrasonic arrivals. The registration point at a distance of 305 km (Saratov) was located north of the source. For this point, the calculations showed the presence of only thermospheric rays. The calculation results are compared with experimental data. A satisfactory agreement between the calculated and experimental data was obtained for both observation points in Saratov and Tsimlyansk. The calculated data in Tsimlyansk include manifestation features of the multipath structure of infrasound propagation and agree with the complex structure of infrasound signals recorded in Tsimlyansk.

Appendices

Appendix A. Burgers type equation on a ray. The problem formulation and numerical integration of the equation.

To consider the issue of numerical integration of Eq. (15), it is convenient to write this equation briefly in the form

$$\frac{\partial V}{\partial \theta }+a\left(\theta \right)V+d\left(\theta \right)V\frac{\partial V}{\partial \tau }+g\left(\theta \right)\frac{{\partial }^{2}V}{\partial {\tau }^{2}}=0,$$

(1A)

where

$$a\left(\theta \right)=\frac{\left(c\left(\theta \right)+\gamma {U}_{\theta }\right)}{c\left(0\right)}\frac{n}{2\aleph \left(\theta \right)}\frac{\partial \aleph \left(\theta \right)}{\partial \theta }+\frac{1}{2{\rho }_{0}\left(\theta \right)c\left(\theta \right)}\frac{\partial {\rho }_{0}\left(\theta \right)c\left(\theta \right)}{\partial \theta }+\frac{\left(\gamma +1\right)}{2{c}_{0}}n\frac{\partial }{\partial \theta }{U}_{\theta },d\left(\theta \right)=\frac{-\left(\gamma +1\right)n\left(\theta \right)}{2{c}^{2}\left(0\right)},g\left(\theta \right)=\frac{-\nu \left(\theta \right){n}^{2}\left(\theta \right)}{2{c}^{3}\left(0\right)}.$$

The problem for Eq. (1A) is formulated as follows. For some $\theta ={\theta }_{0}$, $V\left({\theta }_{0},\tau \right)$ is known, and ${\text{lim}}_{\tau \to \pm \infty }\left|V\left({\theta }_{0},\tau \right)\right|=0$. It is required to calculate $V\left({\theta }_{0},\tau \right)$ for $\theta >{\theta }_{0}$. Since we will solve the equation by the finite difference method, we assume that the grid ${\Omega }_{\tau }=\left\{{\tau }_{i}:{\tau }_{i+1}={\tau }_{i}+h\right\}$ is given. We also assume that the ${\Omega }_{\theta }=\left\{{\theta }_{k}:{\theta }_{k+1}={\theta }_{k}+p\right\}$ grid is given. Here $h$ and $p$ are the steps of difference grids ${\Omega }_{\tau }$ and ${\Omega }_{\theta }$. Solutions of Eq. (1a) satisfy the following integral relations

$$\begin{array}{c}{\int }_{-\infty }^{\infty }V\left(\theta ,\tau \right)d\tau ={\int }_{-\infty }^{\infty }V\left({\theta }_{0},\tau \right)d\tau {\text{exp}}\left\{{\int }_{{\theta }_{0}}^{\theta }a\left(\theta {^\prime}\right)d\theta {^\prime}\right\}\\ {\int }_{-\infty }^{\infty }{V}^{2}\left(\theta ,\tau \right)d\tau ={\int }_{{\theta }_{0}}^{\theta }a\left({\theta }{^\prime}\right)\left({\int }_{-\infty }^{\infty }{V}^{2}\left({\theta }{^\prime},\tau \right)d\tau \right)d{\theta }{^\prime}\\ -{\int }_{{\theta }_{0}}^{\theta }g\left({\theta }{^\prime}\right)\left({\int }_{-\infty }^{\infty }{\left(\frac{\partial V}{\partial \tau }\left({\theta }{^\prime},\tau \right)\right)}^{2}d\tau \right)d{\theta }{^\prime}\end{array}$$

(2A)

Integral relations (2A) generalize the integral relations known for the Burgers equation to the case of an equation with variable coefficients (1A). Obviously, the finite-difference method for solving the Burgers-type Eq. (1A) must support grid analogues of the integral relations (2A). The limiting case, when the term $g\left(\theta \right)\frac{{\partial }^{2}V}{\partial {\tau }^{2}}$ is small, deserves special consideration.

When we solve Eq. (1A) numerically, due to the grid step finiteness, the finite-difference approximation of the term $g\left(\theta \right)\frac{{\partial }^{2}V}{\partial {\tau }^{2}}$ can be vanishingly small. That is, for sufficiently small $g\left(\theta \right)$, the dissipative term actually disappears due to the specifics of numerical methods, and we should keep in mind this obstacle. In this case, the simplified equation

$$\frac{\partial \widetilde{V}}{\partial \theta }+a\left(\theta \right)\widetilde{V}+d\left(\theta \right)\widetilde{V}\frac{\partial \widetilde{V}}{\partial \tau }=0,$$

(3A)

is of the quasilinear type. We have changed the notation of the sought function, and we write $\widetilde{V}$ in (3A) instead of $V$, since Eq. (3A) does not coincide with (1A), and, therefore, the solutions are different. Equations of this type are well studied mathematically. It is known that, with smooth initial data, smooth solutions of equations of this type exist only up to some ${\theta }_{break}$ that depends on the “initial condition”. The ${\theta }_{break}$ value is called the breaking point. For $\theta >{\theta }_{break}$, only a generalized (weak) solution can be constructed, which is defined in a special way. Moreover, mathematical studies have shown that the generalized solution can be defined in various ways that lead to different mathematically rigorous answers [Lax, 1954; Lax, 1957]. In this specific case, we search for the generalized solution, which has a clear physical meaning. Such a physically justified solution can naturally be defined in terms of the following limit

$$\widetilde{V}\left(\theta \text{,}\tau \right)=\underset{\varepsilon \to 0}{\text{lim}}\widetilde{\widetilde{V}}\left(\theta \text{,}\tau ,\varepsilon \right),$$

where $\widetilde{\widetilde{V}}\left(\theta \text{,}\tau \text{,}\varepsilon \right)$ is the solution of the regularized equation

$$\frac{\partial \widetilde{\widetilde{V}}}{\partial \theta }+a\left(\theta \right)\widetilde{\widetilde{V}}+d\left(\theta \right)\widetilde{\widetilde{V}}\frac{\partial \widetilde{\widetilde{V}}}{\partial \tau }+\varepsilon \frac{{\partial }^{2}\widetilde{\widetilde{V}}}{\partial {\tau }^{2}}=0,\varepsilon >0$$

(4A)

The convergence is understood in the norm of ${L}_{1}$. The described method for constructing a generalized (weak) solution is called the regularization method. It was proposed by Neumann (1944) and then developed further (Neumann & Richtmayer, 1950). The fact that described regularization method gives a weak solution of Eq. (3A) was proved for the case of an equation with constant coefficients by Lax (1954, 1957). This proof can be easily generalized to the case of the Eq. (3A) with variable coefficients. The proved theorem (Lax, 1954, 1957) allows, when solving Eq. (3A]), to replace this equation with the regularized Eq. (4A) with sufficiently small, but still finite $\varepsilon$. The above allows using the following numerical scheme to solve Eq. (1A):

$${V}_{i}^{j+\frac{1}{2}}={V}_{i}^{j}-\left({a}_{j+\frac{1}{2}}{V}_{i}^{j+\frac{1}{2}}\right.+$$

$$\frac{+{d}_{j+\frac{1}{2}}}{3}\left(\frac{{V}_{i+1}^{j+\frac{1}{2}}-{V}_{i-1}^{j+\frac{1}{2}}}{2h}{V}_{i}^{j+\frac{1}{2}}+\frac{{\left({V}_{i+1}^{j+\frac{1}{2}}\right)}^{2}-{\left({V}_{i-1}^{j+\frac{1}{2}}\right)}^{2}}{2h}\right)+$$

$$+\left.{q}_{j+\frac{1}{2}}\frac{{V}_{i+1}^{j+\frac{1}{2}}-2{V}_{i}^{j+\frac{1}{2}}+{V}_{i-1}^{j+\frac{1}{2}}}{{h}^{2}}\right)\frac{p}{2}$$

(5A)

$$\begin{array}{c}{V}_{i}^{j+1}=2{V}_{i}^{j+\frac{1}{2}}-{V}_{i}^{j},\end{array}$$

where.

${V}_{i}^{j}=V\left({\tau }_{i},{\theta }_{j}\right),{V}_{i}^{j+\frac{1}{2}}=V\left({\tau }_{i},{\theta }_{j}+\frac{p}{2}\right),{a}_{j+\frac{1}{2}}=a\left({\theta }_{j}+\frac{p}{2}\right),{d}_{j+\frac{1}{2}}=d\left({\theta }_{j}+\frac{p}{2}\right)$.

The formula for ${q}_{j+\frac{1}{2}}$ is more convoluted:

$${q}_{j+\frac{1}{2}}=\left\{\begin{array}{ccc}g\left({\theta }_{j}+\frac{p}{2}\right),& {\text{for}}& {d}_{j+\frac{1}{2}}{A}^{j}h\le g\left({\theta }_{j}+\frac{p}{2}\right),\\ {d}_{j+\frac{1}{2}}{A}^{j}h,& {\text{for}}& {d}_{j+\frac{1}{2}}{A}^{j}h>g\left({\theta }_{j}+\frac{p}{2}\right).\end{array}\right.$$

(6A)

The regularization of Eq. (1A) is carried out in formula (6A) and its meaning is as follows: for waves with the scale of order of the grid step $h$, the contribution of the dissipative term should always be no less than the contribution of the nonlinear term. If the dissipative coefficient is too small and cannot met this requirement, then we artificially increase it to satisfy the imposed condition. In (6A) ${A}^{j}=ma{x}_{i}\left(\left|{\theta }_{i}^{j}\right|\right)$. If the numerical grid ${\Omega }_{\tau }$ is fine enough, then the condition in the top row (6A) is automatically satisfied, and equation regularization is not needed and is not automatically applied.

Equations (5A) are solved using a convergent iterative procedure. The numerical method (5A), (6A) ensures the implementation of grid analogues to the integral relations (2A). The Burgers-type Eq. (15) uses the variable $=t-\frac{\theta }{c\left(0\right)}$. After solving the equations, a transition to the natural variable $t$ is performed.

As for the point where $\aleph =0$ (caustics), the variable $\aleph$ never vanishes under the algorithm for calculating $\aleph$ described below, although it may be small. We do not use the transition to a new variable in the vicinity of the point where $\aleph =0$, as it was done by Sabatini et al. (2016), since we did not find the need for such a transformation within the framework of our calculation algorithm. Equation (3A) does not take into account some physical phenomena important in the vicinity of the point where $\aleph =0$. Therefore, it is simply important that computational stability is preserved in the vicinity of the point where $\aleph =0$; and this requirement is met, as practical calculations have shown.

Appendix B. Calculation the differential cross-section of ray tube

Equations (15), (16) contain the differential cross section $\aleph ={\aleph }_{\eta }{\aleph }_{\xi }$ of the ray tube, where ${\aleph }_{\eta }$, ${\aleph }_{\xi }$ are the differential transverse dimensions of the ray tube, defined by relations (7). Consider the question of the numerical calculation of $\aleph$. Figure 

Figure 7

Three beams coming out of the same point at close angles. The caustic is at the top, where the rays intersect, and it means ${\aleph }_{\eta }=0$

7 shows for illustration three beams starting from the horizontal plane at different angles to the horizontal: $\eta -\Delta \eta$, $\eta$, $\eta +\Delta \eta$. Let $\Delta l$ be the distance between the top ray and the bottom ray in Fig. 7 measured perpendicular to the middle beam. Then, by definition (Babich et al. 1988b), differential transverse dimension of the ray tube

$${\aleph }_{\eta }=\underset{\Delta \eta \to 0}{\text{lim}}\frac{\Delta l}{2\left(\Delta \eta \right)}.$$

Let us note that in Fig. 7 at the top, in the region of the caustic, the rays intersect, and therefore ${\aleph }_{\eta }=0$ there. ${\aleph }_{\eta }$ is approximately calculated as follows. Let, for some value of the parameter $t$ along the ray, the coordinates of the corresponding points of the rays are as follows: $\left({x}_{up}\left(t\right),{y}_{up}\left(t\right),{z}_{up}\left(t\right)\right)$, $\left({x}_{mid}\left(t\right),{y}_{mid}\left(t\right),{z}_{mid}\left(t\right)\right)$, $\left({x}_{bot}\left(t\right),{y}_{bot}\left(t\right),{z}_{bot}\left(t\right)\right)$, where "up", "mid", "bot" denote the upper, middle, and lower rays. Let the unit direction vector of the middle ray be equal to ${\overrightarrow{n}}_{mid}\left({t}_{0}\right)=\frac{\overrightarrow{b}\left(t\right)}{\left|\overrightarrow{b}\left(t\right)\right|}$, where $\overrightarrow{b}\left(t\right)=\overrightarrow{R}\left(t\right)-\overrightarrow{R}\left(t-\Delta t\right)$, $\overrightarrow{R}\left(t\right)=\left({x}_{mid}\left(t\right),{y}_{mid}\left(t\right),{z}_{mid}\left(t\right)\right)$. Let us introduce vectors

$${\overrightarrow{l}}_{up}\left(t\right)=\left({x}_{up}\left(t\right)-{x}_{mid}\left(t\right),{y}_{up}\left(t\right)-{y}_{mid}\left(t\right),{z}_{up}\left(t\right)-{z}_{mid}\left(t\right)\right),$$

$${\overrightarrow{l}}_{bot}\left(t\right)=\left({x}_{bot}\left(t\right)-{x}_{mid}\left(t\right),{y}_{bot}\left(t\right)-{y}_{mid}\left(t\right),{z}_{bot}\left(t\right)-{z}_{mid}\left(t\right)\right).$$

Then,

$${\aleph }_{\eta }\left(t\right)\approx \left(\frac{\left|{\overrightarrow{l}}_{up}\left(t\right)-\left({\overrightarrow{l}}_{up}\left(t\right){\overrightarrow{n}}_{mid}\left(t\right)\right){\overrightarrow{n}}_{mid}\left(t\right)\right|+\left|{\overrightarrow{l}}_{bot}\left(t\right)-\left({\overrightarrow{l}}_{bot}\left(t\right){\overrightarrow{n}}_{mid}\left(t\right)\right){\overrightarrow{n}}_{mid}\left(t\right)\right|}{2\left|\Delta \eta \right|}\right),$$

(1B)

where ∆η is the deviation of the upper and lower beams from the average when leaving the Earth's surface, in radians. Note that ${\aleph }_{\eta }$ calculated by formula (1B) can never be equal to zero, although it can be of a small positive value. As for ${\aleph }_{\xi }$, the wind affects much less this transverse dimension of the ray tube, and, for simplicity, we take ${\aleph }_{\xi }\left(t\right)=\sqrt{{x}_{mid}^{2}\left(t\right)+{y}_{mid}^{2}\left(t\right)}$.