Analysis of electromagnetic wave characteristics of heterogeneous plasma sheath based on the ZT-DGTD

A hypersonic plasma sheath is an unmagnetized, weakly ionized, and inhomogeneous plasma flow that causes a blackout in radio communication. Studying the propagation of electromagnetic waves in a plasma flow is of significant importance for addressing potential communication disruptions. The discontinuous Galerkin time domain method based on the Z-transform for dispersive media is derived, and the derivation of the Z-transformation is concise and effective. According to different sizes of the plasma parameter, the non-uniform flow field was divided into two enhancement regions and two attenuation regions. By observing the changes in amplitude and phase of electro-magnetic fields in different regions of the flow field, the effects of enhancement and attenuation regions on electromagnetic waves can be clearly observed. When the plasma flow field has a significant effect on electromagnetic scattering, regions with different parameters in the flow field exhibit different or even opposite changes in the amplitude and phase of the electromagnetic field. The effects of the enhanced and attenuated regions on electromagnetic scattering can cancel each other out. Under different band conditions, two regions play a dominant role in electromagnetic scattering. The dust plasma flow field was expressed by the relative dielectric constant of weakly ionized dust plasma, and the influence of the dust plasma on electromagnetic scattering was studied. The influence of plasma flow on electromagnetic scattering is weakened by dust particles. This study provides new insights into the influence of non-uniform flow fields on incident waves.


I. INTRODUCTION
When a hypersonic vehicle reenters the atmosphere, the intense friction between the vehicle and the surrounding air creates a plasma sheath, 1,2 which can strongly interact with electromagnetic waves, significantly affecting their transmission. 3everal scholars have conducted extensive research to address this issue.Rusch and Yeh. 4 simulated the actual plasma sheath profile of a re-entering aircraft using a parabolic electron density profile.Scarabosio et al. 5 used the eikonal approximation in the large inhomogeneous plasma region, and compute radiation and scattering via the equivalence theorem, the results show that significant radio link path losses can be associated with plasma spatial variations (gradients) and collisional losses.Chen et al. analyzed the effects of the incident wave frequency, incidence angle, scattering angle, polarization angle, altitude, and Mach number on the radar cross section (RCS). 6The reflection coefficient and absorption rate of terahertz waves in the plasma flow at different electron density relaxation times, temperatures, and pressures were calculated. 7Yusuke et al. 8 elaborates on the distribution of charged particles and the complex behavior of electromagnetic waves, including attenuation and reflection.The power reflection and transmission coefficients of electromagnetic waves in the plasma sheath at four different heights were calculated. 9The interaction mechanism between the pulsed magnetic field and the hypersonic flow was physically analyzed. 10Wei et al. 11 calculated the electromagnetic scattering of a hypersonic aircraft with a plasma sheath using the discontinuous Galerkin time domain (DGTD) method with different incident angles.Yutong et al. 12 calculated the radar signal attenuation coefficients of terahertz waves under different external magnetic fields and dust plasma parameters.Yuan et al. developed a new estimation model that can predict the attenuation of terahertz signals in hypersonic flow. 13Sun et al. 14 studied the backward radar cross section of blunt cones surrounded by hypersonic flow at different speeds and heights.Guo et al. 15 used the Wentzel Kramers Brillouin (WKB) method to calculate the electromagnetic wave attenuation coefficients of layered non-uniform plasma in two relative motion modes.This study aimed to address the non-uniformity and time-varying distribution of electron density in weakly ionized dust plasmas and further discuss its impact on the propagation characteristics of electromagnetic waves.Hong et al. 16 considered three main collisions in dust plasmas and improved the Drude model to describe the dielectric properties of dust plasmas, which explains collisions involving electron molecule, electron ion, and electron dust particles.
Based on different plasma parameters, we divided the non-uniform flow field into two enhanced regions and two attenuated regions and studied the effects of these two regions on incident waves in different wavelength bands as well as the influence of dust particles in the plasma flow field on the RCS.To the best of our knowledge, this is the first analysis of plasma flow fields in terms of the amplitude and phase of electromagnetic fields.It also investigates the different effects of different regions in a non-uniform plasma flow field on the amplitude and phase of the electromagnetic field.The paper is arranged as follows: In Sec.II, the Drude model of dispersive media is introduced, and the calculation process of the discontinuous Galerkin method is described.In Sec.III, the effectiveness of the proposed algorithm is demonstrated, and an analysis of the distinct influences of different regions in the plasma flow field, with or without dust electromagnetic waves is presented.

II. ITERATIVE EQUATION OF ZT-DGTD
This section derives the ZT-DGTD method, which better represents the computational relationship between the frequency of electromagnetic waves and the plasma and expansion frequencies through further transformation.It helps to study the amplitude and phase of electromagnetic waves with different frequencies in non-uniform plasma flow fields.

A. Drude model for dispersive medium
The dielectric coefficient can be used to express the characteristics of a non-magnetized medium.The dielectric coefficient related to frequency can be expressed as eðxÞ ¼ e 0 e 0 r ðxÞ À je 00 r ðxÞ where e 1 is the relative dielectric constant at infinity, e 0 is the dielectric coefficient in vacuum, and v x is the polarizability function.
The non-uniform plasma flow of hypersonic vehicles is usually treated as cold plasma, which is expressed by a Drude dispersive medium.The concentration of electrons, density of neutral particles, and temperature in non-uniform plasma flow are important parameters for calculating the dielectric coefficient.In the Drude dispersive medium, 17 v x is given by where x p , the plasma collision frequency, and c , the plasma frequency, are given by m e e 0 s ; where m e is the electron mass, n e is the number density of electrons, n s is the number density of species s, e is the magnitude of electronic charge equal to 4:802 98 Â 10 À10 esu, and j is the Boltzmann constant.The effective electron-neutral energy exchange cross section is defined by a curve fit of the form The constants for Eq. ( 3) are presented in Table I. 17

B. Formulation in computational domain
The Maxwell equations are given by Substituting Eq. (1) into Eq.( 4), we obtain Space X is divided into non-overlapping and continuous subregions.Each sub-region is a separate hexahedral element, and its weight function (also called the basis function) is / i (i is the serial number of the unit).The weighted integral in Eq. ( 5) in each element is taken and set to zero, such that The numerical fluxes are defined at the boundary between two elements as where n is the unit outward normal vector on each surface of the element; E Ã and H Ã are the numerical fluxes; E þ and E þ are in the fields of the adjacent element; E and H are in the fields of elements.The coefficients j e , j h , t e , and t h are presented in Table II, 18 ð Substituting Eq. ( 8) into Eq.( 6), we obtain C. Z-DGTD transformation Substituting Eq. ( 2) into J p , we obtain By Z-transformation, 19,20 we obtain Performing z À1 J p ðzÞ !J p nÀ1 on the above formula, we obtain Based on the above derivation, the relationship between the plasma and incident wave frequency is not significant, and we need further transformation.The time expression is as follows: where k is the wavelength of the incident wave, f is the frequency of the incident wave, and Dl is the distance traveled by the light in each time step.Substituting Eq. ( 12) into Eq.( 11), we obtain The above formula represents the Z-transform method for dispersive media.The propagation of electromagnetic waves in dispersive media can be calculated using Eqs.( 9) and ( 13).The time domain stepping steps are summarized as follows: To obtain the scattering cross section/amplitude, it is necessary to set up a closed surface within the calculation area according to the principle of equivalence, and then extrapolate the equivalent electromagnetic current on this surface to obtain scattering.The monostatic RCS can be represented using the scattering field equation 21 given by where f is the incident wave frequency, E x is the far-field, and E i is the incident wave.

III. NUMERICAL SIMULATION RESULTS
This section provides three examples of numerical analysis.The first analyzes the effectiveness of the calculation method proposed in this paper, the second analyzes the amplitude and phase of electromagnetic waves with different frequencies in a non-uniform plasma flow field, and the third studies the propagation characteristics of electromagnetic waves in dust plasma flow.

A. Numerical verification
We calculated the monostatic RCS of a plasma sphere with a radius of 0.00375 m.The plasma frequency and collision frequency were x p ¼ 2p Â 28:7 Â 10 9 rad=s; t c ¼ 2:0 Â 10 10 Hz, respectively.The computational domain was discretized into 207 292 hexahedrons, with a discrete size of 0.000 021 721 875 m.
Mie obtained a strict solution for elastic scattering by a uniform dielectric sphere by solving Maxwell's equations, which can be

Numerical flux j e j h t e t h
Centered flux 1 2 arbitrary in terms of size and material. 22As shown in Fig. 1, very good agreement is achieved between the results obtained using the DGTD method and Mie theory when the incident wave frequency is near 40.0 GHz.As the frequency increased, the RCS error also increased (as shown in Table III).This is mainly because as the frequency increases, the wavelength decreases, and the wavelength dispersion decreases.

B. Analysis of vehicle with non-uniform plasma flow
In this example, the monostatic and bistatic RCS of the blunted cone surrounded by a non-uniform plasma flow is illuminated.A Gaussian pulse incident from the front of the blunted cone ðh ¼ 90; / ¼ 0Þ was considered.The blunted cone has a blunted nose of radius 0.01 m, a conical cross section with half-angle of 9 , and a whole length of 0.1 m.The plasma flow distribution of the RAM-II aircraft was simulated using the CFD-Fastran software, and its numerical algorithm was based on the finite volume method.
The flow field parameters were located at the center of each structured grid in the flow field calculation domain, whereas the electromagnetic parameters were located at the nodes of the grid in the electromagnetic calculation domain.The CFD (Computational Fluid Dynamics) grid was very dense and the CEM (computational electromagnetics) grid was relatively sparse compared to the CFD grid.The meshes for CFD and CEM are not consistent.When coupling data between the flow field grid and the electromagnetic field grid, the plasma frequency and collision frequency on the flow field grid nodes are first calculated, and then the average scheme is used to calculate the plasma frequency and collision frequency in the CEM grid, using Eq. ( 15) to calculate the CFD elements data.Subsequently, we obtain J p using Eq. ( 13) and put it into the DGTD.We have In Fig. 3, the influence of flight altitude and Mach number on the backward RCS of blunt cones at altitudes of 50 km and speeds of 20 Mach are analyzed.When the frequency is below 1.8 GHz, incident waves with frequencies much lower than the plasma frequency cannot penetrate the plasma region.The plasma flow significantly enhanced the backward RCS of the vehicle.When the frequency ranged from 1.8 to 13 GHz, the plasma flow reduced the backward RCS of the blunt cone.When the frequency was above 13 GHz, the backward RCS of the blunt cone with plasma was consistent with that without plasma, indicating that the plasma flow had little influence on the electromagnetic scattering of the reentry target in the high frequency band.These results are in agreement with plasma theory. 2,23 Incident wave frequency is 1.0 GHz As shown in Fig. 4, when the incident wave frequency was 1.0 GHz, the plasma flow enhanced electromagnetic scattering.In Fig. 5, the magnetic field amplitude changes less than the electric field amplitude.By comparing Figs.5(a) and 5(b), we can observe changes in the amplitude of the electric field.By comparing Figs.5(c) and 5(d), we can observe changes in the amplitude of the magnetic field.In regions 1 and 3, x p is greater than the incident wave frequency f and t c , the electric field amplitude decreases, whereas the magnetic field amplitude only increases in region 1.In regions 2 and 4, x p and t c are less than f, and x p is less than t c , the electric field amplitude increases sharply, while the magnetic field amplitude decreases.At this point, the amplitudes of the electric and magnetic fields exhibited opposite changes.This shows that when the plasma frequency is greater than the collision frequency and incident wave frequency, the plasma has the properties of metals and enhances electromagnetic scattering.The plasma flow in regions 1 and 3 plays a key role in electromagnetic scattering, and regions 1 and 3 belong to the enhancement region.
In Fig. 6, the phase change is relatively smaller than the amplitude change.By comparing Figs.6(a) and 6(b), we can observe changes in the phase of the electric field.By comparing Figs.6(c) and 6(d), we can observe changes in the phase of the magnetic field.In the plasma flow field, the electric field phase underwent significant deformation, moving backward from the head of the vehicle.However the magnetic field phase, changes less, moving forward from the vehicle's tail, and compared to the changes in the electric field, the changes in the magnetic field are relatively small.

Incident wave frequency is 4 GHz
As shown in Fig. 7, when the incident wave frequency was 4 GHz, the plasma flow field causes attenuation of electromagnetic scattering.As shown in the diagram below, the RCS decreases in the direction surrounded by the plasma flow field at the head and side of the spacecraft, whereas it increases in the direction of the tail of the vehicle.(c) and 8(d), we can observe changes in the amplitude of the magnetic field.In most areas of region 1, x p is greater than f, and the amplitude changes in the electric and magnetic fields are the same as those in Fig. 5.In most areas of region 3, x p is less than f, and the amplitude changes of the electric and magnetic fields are relatively small compared with those in Fig. 5.In region 2, the electric field amplitude increased sharply, and there was almost no change in the magnetic field amplitude.In region 4, the amplitude changes of the electric and magnetic fields are relatively small compared with those in Fig. 5.This is because the plasma frequency and collision frequency in region 4 are smaller than those in region 2. When t c is greater than x p , the plasma leads to attenuation of electromagnetic scattering.The plasma flow in regions 2 and 4 plays a major role in electromagnetic scattering, and regions 1 and 3 belong to the attenuation region.When the scattering angle was 180 , the cancellation effect of the attenuation region on the enhancement region was relatively small.Comparing the electric field amplitudes of regions 2 and 4 in Figs. 5 and 8, x p and t c have the greatest impact on electromagnetic attenuation within a certain range.
By comparing Figs.9(a) and 9(b), we can observe changes in the phase of the electric field.By comparing Figs.9(c) and 9(d), we can observe changes in the phase of the magnetic field.In regions 1, 2, and 3, the electric field phase experienced a larger backward shift, whereas the magnetic field phase showed minimal variation.In the first half of the plasma flow field, the changes in the electric and magnetic field phase are the most significant.When plasma attenuates electromagnetic scattering and only considers the effect of plasma on the magnetic field phase, the influence of the plasma is minimized.

Incident wave frequency is 15 GHz
As shown in Fig. 10, when the incident wave frequency was 15 GHz, the RCS of the plasma flow surrounding the blunt cone tended to be similar to that of the blunt cone.At this point, the plasma flow has little effect on electromagnetic scattering.As shown in Fig. 11, in most areas of regions 1 and 3, x p is less than f.In most areas of regions 2 and 4, x p and t c are much less than f.The enhancement and attenuation effects of the plasma on electromagnetic scattering were not significant.By comparing Figs.11(a) and 11(b), the electric field amplitude experienced a certain decrease, whereas, by comparing Figs.11(c) and 11(d), the magnetic field amplitude was found to experience a certain increase.However, both changes were negligible.When the scattering angle is between 45 and 315 , there is a small attenuation of electromagnetic scattering.When the scattering angle is 180 , the cancellation effect of the attenuation region on the enhancement region was relatively small.
As shown in Figs.12(a) and 12(b), in most areas of the plasma flow field, the phases of the electric fields remain almost unchanged.As shown in Figs.12(c) and 12(d), the phases of the magnetic fields remain almost unchanged.

C. Analysis of vehicle with non-uniform dust plasma flow
Based on the analysis of weakly ionized dust plasma, 3,7,8 we substituted the dispersion expression of weakly ionized dust plasma into SO-DGTD to study the weakly ionized dust plasma flow field.The relative dielectric constant of the weakly ionized dust plasma can be obtained as follows: where t is the average collision frequency of the electrons, V eff is the effective collision frequency, r d is the dust particle radius, N d is the concentration of dust particles, N e is the electron density, g ed is the charging response factor.We calculated the monostatic RCS of a plasma sphere with a radius of 1 m.The plasma frequency is x p ¼ 6:75 Â 10 8 rad=s, average collision frequency of electrons is t ¼ 7:5 Â 10 7 Hz, dust particle radius is r d ¼ 1 Â 10 À5 m, concentration is N d ¼ 1 Â 10 9 m À3 , effective collision frequency is V eff ¼ 1 Â 10 9 Hz.
As shown in Fig. 13, very good agreement was achieved between the results obtained using the SO-DGTD method and the Dust-Mie theory when the incident wave frequency was near 0.15 GHz.As the frequency increases, the RCS error increases.This is mainly because as  the frequency increases, the wavelength decreases, and the wavelength dispersion decreases.
The influence of flight altitude and Mach number at altitudes of 50 km and speeds of 20 mach with dust on the backward RCS of blunt cones was analyzed.The average collision frequency of electrons is t ¼ 7 Â 10 11 Hz, the effective collision frequency is V eff ¼ 1 Â 10 11 Hz, and the radius and concentration of dust particles are shown in Table IV.
As shown in Figs.14(c) and 14(d), the higher the concentration of dust particles, the greater the RCS attenuation.As shown in Figs.14(a) and 14(b), the smaller the radius of the dust particles, the greater the RCS attenuation.The larger the radius and the lower the concentration of dust particles, the easier it is for them to weaken the influence of the plasma flow field on the RCS.When the frequency is below 1.8 GHz or above 13 GHz, the dust plasma flow field has almost no effect on the RCS.When the frequency is between 1.8 and 13 GHz, there is a certain decrease in the RCS.

IV. CONCLUSION
This study establishes plasma flow field models of hypersonic vehicle using the CFD-Fastran software.Our derivation of Z-transformation is concise and effective.Based on the different plasma parameters in different regions of the non-uniform flow field, the influence of different regions of the plasma flow field on the electromagnetic field was analyzed.According to the different magnitudes of plasma frequency and collision frequency, non-uniform flow fields can be divided into enhancement and attenuation regions.Under lowfrequency conditions, regions 1 and 3 near the inner edge of the flow field play a dominant role in enhancing the electromagnetic scattering.As the frequency of the incident wave increased, regions 2 and 4 near the outer edge of the flow field played a dominant role in attenuating electromagnetic scattering.A comparison of the electric field amplitudes of regions 2 and 4 shows that the plasma frequency and collision frequency have the greatest effect on electromagnetic attenuation within a certain range.When electromagnetic scattering is enhanced or attenuated under the action of the plasma flow field, the influence of the flow field on the amplitude and phase of the electromagnetic field is different: The plasma flow field has the greatest influence on the amplitude of the electric field, followed by the amplitude of the magnetic field, and the smallest influence on the phase of the magnetic field.In particular, when electromagnetic scattering is attenuated, the phase of the magnetic field undergoes only small changes and the phase of the magnetic field at the inner edge of the plasma flow field remains almost unchanged.The effects of the enhanced and attenuated regions on electromagnetic scattering can cancel each other.Under different band conditions, the two regions play a dominant role in electromagnetic scattering.The dust particles in the dust plasma weaken the influence of the plasma flow field on RCS.The larger the radius and the lower the concentration of dust particles, the easier it is for them to weaken the influence of the plasma flow field on RCS.By studying the magnitudes of plasma frequency and collision frequency in the enhanced and attenuated regions of non-uniform flow fields, the influence of plasma flow fields on electromagnetic scattering in different bands was further explained.Considering the different effects of the plasma flow field on the amplitude and phase of the electromagnetic field, when we focus more on the magnetic field in the study of the plasma flow field, the impact of the plasma flow field on the aircraft becomes smaller.The influence of plasma flow composed of parameters with different magnitudes on electromagnetic scattering will be studied in future work.
) where V n is the point of nth grid of the CEM and M is the number of grids near V n where the CFD data are recorded.As shown in Fig.2(a), the 3D geometric model is divided into hexahedral grids consisting of 1067631 hexahedrons.As shown in Fig.2(b), the plasma flow field is divided into four regions: (1) region 1, near the head of the vehicle in the plasma flow (x p ranges from 2:8 Â 10 9 to 5:9 Â 10 11 ; t c ranges from 7:5 Â 10 7 to 2:0 Â 10 10 ); (2) region 2, at the inner edge of the plasma flow (x p ranges from 2:0 Â 10 9 to 4:2 Â 10 10 ; t c ranges from 2:5 Â 10 7 to 5:6 Â 10 8 ); (3) region 3, near the front half of the outer edge of the plasma flow (x p ranges from 2:2 Â 10 5 to 7:8 Â 10 6 ; t c ranges from 1:5 Â 10 7 to 5:8 Â 10 7 ); and (4) region 4, near the latter half of the outer edge of the plasma flow (x p ranges from 1:6 Â 10 5 to 3:1 Â 10 5 ; t c ranges from 7:4 Â 10 6 to 7:8 Â 10 6 ).To truncate the open boundary of the simulation, a uniaxial perfectly matched layer (UPML) with a thickness of 0.0032 m is placed at least 0.006 m from the plasma sheath.

FIG. 9 .
FIG. 9.The frequency of the incident wave was 4 GHz: (a) electric field phase of the vehicle, (b) electric field phase of the vehicle with plasma flow, (c) magnetic field phase of the vehicle, and (d) magnetic field phase of the vehicle with plasma flow.

FIG. 11 .
FIG. 11.The frequency of the incident wave was 15 GHz: (a) electric field amplitude of the vehicle, (b) electric field amplitude of the vehicle with plasma flow, (c) magnetic field amplitude of the vehicle, and (d) magnetic field amplitude of the vehicle with plasma flow.

FIG. 12 .
FIG. 12.The frequency of the incident wave was 15 GHz: (a) electric field phase of the vehicle, (b) electric field phase of the vehicle with plasma flow, (c) magnetic field phase of the vehicle, and (d) magnetic field phase of the vehicle with plasma flow.

FIG. 14 . 8 V
FIG. 14. Monostatic RCS of the vehicle with dust plasma flow: (a) vehicle with dust plasma 2 and 3, (b) vehicle with dust plasma 1 and 4, (c) vehicle with dust plasma 2 and 4, and (d) vehicle with dust plasma 1 and 3.

TABLE I .
Constants for curve fits of electron-neutral energy exchange cross section, r es .

TABLE II .
Numerical flux coefficient.

TABLE III .
RCS error.