2.1 Delayed Radiation Volume-Source Model

After high-altitude nuclear explosions, the debris cloud spreads out and rises driven by a considerable share of energy. The cloud expands rapidly at the beginning and then engulfs the atmosphere. When the mass of the cloud is large enough, the driving process ends and the cloud is translated into diffusion with constant speed. Then, the cloud starts to gradually dilute. As mentioned above, there are some differences in the properties of the atmosphere at different altitudes, so the speed of diffusion is not same in all directions. In general, the density of the atmosphere decays exponentially as altitude increases, so the cloud diffuses rapidly upward but slowly downward in the vertical direction. In the horizontal direction, the density of the atmosphere basically remains constant, so the debris cloud spreads out symmetrically. Therefore, the debris cloud is approximately inverted pear-shaped^[12]. As the cloud spreads outward, its mass and geometric center also rise. The X-rays released at the beginning of the nuclear explosion heat the air to produce energy deposition below the explosion point. The buoyancy caused by the increase in the atmospheric temperature is greater than the gravity of the debris cloud itself, which leads to the rise of the debris cloud. The equations of motion for the diffusion and rise of the debris cloud are as follows

where R is the radius, h is the central height of the debris cloud, ρ_h is the atmospheric density of the explosion point, M is the total mass of the debris cloud, ρ(r) is the density distribution in the debris cloud, r is the radius vector of one point in the debris cloud, p_h is the atmospheric pressure, v is the diffusion velocity, ξ is the probability of atmospheric molecules being involved, dσ is the surface element of the debris cloud, and θ is the angle between the normal of the panel and the vertical direction. The first term on the right side of Equation (1) represents the pressure difference between the inside and outside of the debris cloud, and the second term represents the change in momentum caused by mass addition. The first item on the right side of Equation (2) is the pressure difference between the upper and lower surfaces of the debris cloud, and the second term is the gravity of the debris cloud. As shown in Figure 1, this model is used to calculate the horizontal diffusion radius of the debris cloud under the experimental parameters of 'teak’, and the theoretical calculation results agree well with the experimental values.

Fig. 1Full-screen View

Change of horizontal diffusion radius with time

2.2 Delayed Radiation Ionization Model

After 10 μs of the nuclear explosion, the delayed γ-rays begin to radiate. For the fission bomb, the radioactive intensity A(t)(Ci) of γ-rays changing with time is given as

where Q is the fission equivalent (kt), the value of b is between 4 and 8, and the value of n is related to t (min). When the value of t is less than 30 min, the value of n is 0.89. When the value of t is between 30 and 1440 min, the value of n is 1.11. When the value of t is greater than 30 min, the value of n is 1.26. For the same equivalent nuclear explosion, the radioactive intensity of delayed β-rays is 1.2-2.5 times larger than that of the delayed γ-rays, but the variation law of intensity with time is the same. The energy spectrum of the delayed γ-rays and β-rays can be obtained from a previous study^[13].

The γ-rays emitted from fission fragments in the radioactive cloud undergo the photoelectric effect, Compton scattering, and electron pair effect with molecules or atoms in the atmosphere, generating high-energy electrons. Because the energy of γ-rays is not very high, the electron pair effect can be ignored. Simultaneously, the debris cloud keeps on releasing high-energy β rays. These β-rays and high-energy electrons generated from reactions between γ-rays and particles in the atmosphere move spirally along the magnetic line of force in the geomagnetic field, undergoing the collision ionization effect with air particles in the zone of electron motion to consume themselves and generate secondary low-energy electrons. The geomagnetic field can be described by the magnetic dipole field as

where R_E is the radius of the Earth, and B₀ is the geomagnetic field intensity at 80 km, generally taken as 0.5 Gs. The gyration radius r_g of the high-energy electrons can be obtained from the following equation:

where $F_{\text{L}}$ is the Lorentz force, $\zeta$ is the relativity factor, and $m_{\text{e}}$ and $v_{\text{e}}$ are the mass and velocity of the electron, respectively. Then, the ionization of the atmosphere near the magnetic field line can be obtained under the action of the Lorentz force.

As mentioned above, the process of photoelectric effect and Compton scattering between the γ-rays and air particles only produces high-energy electrons, and these high-energy electrons and β-rays undergo collision ionization with air particles to consume high-energy electrons. Therefore, the total generation rate of high-energy electrons could be written as

where n_eH is the number density of high-energy electrons, f_com denotes the Compton scattering effect, f_pho denotes the photoelectric effect, and f_col denotes the consumption of high-energy electrons during secondary collision ionization processes. The Compton scattering process, the photoelectric effect process, and the secondary collision ionization process can all be considered two-body reactions^[14]. The contribution to the generation of high-energy electrons of the three processes and parameters can be obtained from previous studies^[11,15-17].

The atmosphere ionized by delayed γ-rays and β-rays will produce a large number of low-energy electrons and ions. There are two processes for the loss of low-energy electrons. In one process, low-energy electrons complex with positive ions in the air directly, while in the other process, low-energy electrons attach to atoms or molecules to form negative ions, leading to a complex reaction with positive ions. The atmosphere ionized by delayed γ-rays produces numerous low-energy electrons and ions. Considering 36 components and 229 reaction equations, the chemical reaction kinetics processes are very complex, including neutral particles (O, O₂, O₃, N, N₂, NO, NO₂, CO₂, H₂O, H, H₂, OH, and He), negative ions (e, O^-, O₂^-, O₃^-, O₄^-, CO₃^-, CO₄^-, O₂^-·H₂O, CO₃^-·H₂O, and CO₄^-·H₂O), and positive ions (O⁺, O₂⁺, N₂⁺, N₄⁺, NO⁺, O₄⁺, H₂O⁺, H₃O⁺, H⁺, He⁺, O₂⁺·H₂O, H₃O⁺·OH, and H₃O⁺·H₂O). Therefore, the total change rate of low-energy electrons could be written as

where f_chemL is the term of chemical reaction kinetics, ε_jm is the generation consumption factor, k_jm is the reaction coefficient, and n_j and n_m are the number density of particles involved in chemical reactions. Because one high-energy electron can generate two low-energy electrons after collision ionization, the second term on the right of Equation (13) is twice as much as consumption of high-energy electrons.

2.3 Attenuation of Radio Waves Model

When electromagnetic waves are propagated in plasma, they will drive electrons to vibrate. The electrons collide with other particles to transmit energy, so that the electromagnetic waves generate energy attenuation. For non-magnetized cold collision plasma, the relative dielectric constant ${\epsilon }_r$ and propagation constant k are respectively

where n is the refractive index of the medium, ω is the incident angular frequency of the electromagnetic waves, υ is the collision frequency of the electrons with other particles, ω_p is the angular frequency of plasma collision, k₀ is the wave number in vacuum, k_r is the attenuation constant, and k_i is the phase constant. When the electromagnetic wave frequency is less than the plasma frequency, the electromagnetic wave is cut off. When it is greater than the plasma frequency, under the WKB approximation^[19], the one-way energy attenuation Att (dB) of electromagnetic waves is given by

where P_h and P_h0 are the energy of electromagnetic waves at a height of h km and h₀ km, respectively.

3.1 Distribution of Electron Density Generated by Delayed Radiation Ionization

As shown in Figure 2, the distributions of electron number density at different moments after nuclear explosion are given. The solid lines represent the ionization caused by the delayed γ-rays, the dotted lines represent the ionization caused by the delayed β-rays, and the dots of different shapes represent the electron densities at different horizontal distances (100, 200, 500, and 1000 km) from the burst point following Earth’s surface curvature.

Fig. 2Full-screen View

(Color online) Number density of electrons generated by delayed γ-rays (solid line) and β-rays (dotted line) from a nuclear explosion at a 120 km explosion height and 1 Mt explosion equivalent at (a) 1 min, (b) 5 min, (c) 10 min, and (d) 30 min.

After 1 min of nuclear explosion, the peak value of electron density generated by delayed ionization of γ-rays is located at a height of 100 km and the peak electron number density reaches 1×10⁷ cm^-3 at a radius of 100 km, which is higher by about three orders of magnitude than the background electron number density. As the horizontal distance increases, the electron number density gradually decreases. The peak electron number density at a horizontal distance of 1000 km is 1×10⁶ cm^-3. However, the delayed β-rays only act on the atmospheric ionization within 200 km on the north side of the explosion point after 1 min of the nuclear explosion. The electron density of ionized delayed β-rays 100 km away from the north side of the explosion point is about 2×10⁶ cm^-3, which is higher by one order of magnitude than the background electron density but lower by one order of magnitude than the electron density generated by delayed γ-rays.

After 5 min, the ionization range of the delayed β-rays extends to a horizontal distance of more than 1000 km, and the height of peak electron number density appears between 80 and 90 km. The main ionization affected height of ​​delayed β-rays is located from 70 km to 120 km. After 10 min, the peak electron number density generated by delayed γ-rays located at a horizontal distance of 100 km and at 100 km height is about 2×10⁵ cm^-3, which is an order higher than the background atmospheric electron density. After 30 min of explosion, the ionization effect of delayed γ-ray starts to gradually weaken. At this time, the ionization effect of delayed β-rays becomes dominant, and the height of peak electron density is about 90 km. The electron number density is nearly well distributed with the horizontal distance below a height of 100 km.

Figure 3 shows the spatial distribution of the electron number density generated by the total atmospheric ionization of delayed radiation on the north side of the explosion point at different moments after high-altitude nuclear explosions. The abscissa is the horizontal radius, and the ordinate is the vertical height. Because the density of the atmosphere near the explosion point is low, the spatial distribution center of the electron density caused by delayed radiation is located at a height of 90 km. As time progresses, the electron number density decreases. After 100 min, the electronic number density is 5×10⁴ cm^-3 and no more than 1×10⁵ cm^-3.

Fig. 3Full-screen View

(Color online) Distribution of electron number density on the north side of the 120-km high and 1 Mt equivalent nuclear explosion point

3.2 Attenuation of Radio Waves by Delayed Radiation

When the electromagnetic wave frequency is greater than the plasma frequency, the electromagnetic wave can propagate in the plasma, and energy attenuation is noted in the process of propagation. When the electron number density is high enough, the incident electromagnetic wave frequency is lower than the plasma frequency. At this moment, the electromagnetic wave is in the off state and cannot propagate in plasma to produce reflection. The energy attenuation of radio wave decreases as the frequency increases. When the frequency of radio wave is low, radio communication is completely interrupted.

Figure 4 shows the effect of additional ionization generated by the delayed radiation on the high-frequency (HF) and very high-frequency (VHF) radio propagation after 1 min of the 120-km high nuclear explosion with a 1 Mt explosion equivalent. In the range of 1000 km, the communication in the medium frequency (MF) band is completely interrupted, and the radio between 10 MHz and 30 MHz is affected. The one-way energy attenuation of the radio wave with 25 MHz reaches 100 decibels at a horizontal distance of 100 km. The range of the effect of delayed radiation on the VHF radio communication is mainly within 200 km, and the attenuation of 120 MHz radio is about 10 decibels at a horizontal distance of 200 km.

Fig. 4Full-screen View

(Color online) Energy attenuation of radio caused by delayed radiation from nuclear explosion

Figure 5 shows the energy attenuation of radio in some bands at some special moments. The radio communication energy attenuation in the VHF band reduces after 5 min, and a part of frequency communication is basically restored. The radio communication in the MF and HF band is still seriously affected, and the energy attenuation in the range of 500 km is higher than 10 decibels. Ten min later, the attenuation in the VHF band appears to be largely unaffected. The radio communication in the HF band is restored after 100 min.

Fig. 5.Full-screen View

(Color online) Energy attenuation of radio caused by delayed radiation from nuclear explosion