© The Authors 2024

1 Introduction

Venus is the most thoroughly explored planet in our Solar System after Earth. The analysis of the data reported by spacecraft measurements (e.g., Venera, Mariner, Pioneer Venus Orbiter, and Venus Express) revealed different plasma boundaries in the Venusian environment, such as the ionosphere, magnetic barrier, magnetosheath, and bow shock (Brace & Kliore 1991; Phillips & McComas 1991). The plasma environment of Venus is completely controlled by both the solar wind and solar radiation, which is a consequence of the lacking intrinsic global magnetic field on Venus. However, Venus still has weak magnetic shielding that is formed by the draping of the interplanetary magnetic field that is carried by the solar wind and by the induction of magnetic fields through the electric currents derived from the direct interaction of the solar wind with the planetary ionosphere. A unique layer is formed between the ionospheric plasma and the hot drifting shocked solar wind found in the magnetosheath/ionosheath, which is called the mantle/transition region (Spenner et al. 1980). It extends significantly along the field lines, but it is confined in the perpendicular direction (see Fig. 1 in Salem et al. 2022). The thickness of this region depends on the solar zenith angle (SZA). For instance, it is only a few hundred kilometers in the subsolar region and becomes wider with increasing SZA. It is characterized by the intermixture of both cold planetary and hot solar wind particles. The coexistence of these two populations is unstable, and it leads to wave excitations that aid the energy and momentum transfer between the shocked solar wind and the ionospheric plasma (Szegö et al. 1995).

Space plasmas are usually collisionless media (e.g., ionospheres and magnetospheres). Plasma waves are therefore considered a vital component of space plasmas as they carry energy and momentum and deposit it in other plasma regions. Plasma waves are excited through the excess of free-energy sources (solar outflows, streaming particles, gradients in density and magnetic field, temperature anisotropy, etc.) and the deviation of a Maxwellian distribution. Space plasma is always either out of thermal equilibrium or has an abundance of free energy, and hence, waves are found to be ubiquitous in these plasmas. The ion acoustic wave (IAW) is a fundamental plasma mode. Various nonlinear structures emerge from the dispersion, dissipation, or (nonlinear) amplification of these waves, such as solitons, double-layers (DLs), periodic, cnoidal, and rogue waves. Electrostatic solitary waves (ESWs) are observed in numerous regions in the near-Earth environment, the magnetosheath (Pickett et al. 2003, 2005), magnetic reconnection sites (Liu et al. 2019), lunar wake (Rubia et al. 2017), auroral regions (Temerin et al. 1982; Bounds et al. 1999; Pickett et al. 2004), in the magnetotail, and in the bow shock (Bale et al. 1998).

The detection of ESWs is not limited to Earth; they are also observed in other planetary and astrophysical plasma environments (Pickett et al. 2015; Hadid et al. 2021; Kakad et al. 2022). In the literature, compressive (positive potential) ESWs have been referred to by several terms: “electron phase-space holes”, “Debye scale structures”, “time domain structures”, and so on (Hutchinson 2017). Usually, solitary waves are observed in space as bipolar electric pulses, that is, a half sinusoid-like cycle followed by a similar half-cycle having an opposite sign. One other interesting nonlinear structure is the DLs, which is usually observed as a monopolar electric pulse. These DLs are observed in the near-Earth environment (Temerin et al. 1982; Sun et al. 2022), in the solar wind (Mangeney et al. 1999), and in the Venusian magnetosheath (Malaspina et al. 2020). ESWs and DLs in the fluid regime can be interpreted by the Sagdeev pseudo-potential technique (Sagdeev 1966) and KdV models (Washimi & Taniuti 1966). Several wave solutions can be obtained by solving a class of partial differential equations such as Korteweg-De Vries (KdV), Kadomtsev-Petviashvili (KP), and Zakharov-Kuznetsov (ZK). These evolution equations can be solved using numerous methods, such as the G′ /G-expansion method (Wang et al. 2008), a Painleve analysis (Kudryashov 1991), or the Tanh-function (Wazwaz 2004).

IAWs are observed in the Venusian environment in different regions, for instance, in the foreshock (Crawford et al. 1993a,b, 1998), bow shock (Strangeway 1991), magnetosheath (Hadid et al. 2021), transition region (Scarf et al. 1980), and the magnetotail (Intriligator & Scarf 1984). Recently, several studies have investigated the nonlinear structures in the mantle region. Afify et al. (2021) investigated the linear and nonlinear IAWs that are excited through the interaction of the shocked solar wind with the ionospheric plasma. They investigated the (in)stability of the linear structures and derived the KdV equation to study the propagation features of the solitary waves. Prasad et al. (2021) employed the Jacobi elliptic expansion technique to determine large-amplitude IAWs. They obtained three wave solutions, which are the solitary, periodic, and superperiodic wave solutions. Salem et al. (2022) proposed that ion-acoustic solitary waves (IASWs) may contribute as an energy-transfer candidate in the transition region. They presented a theoretical study that examined the influence of the solar wind on the features of fully nonlinear solitary structures. They showed that the electric pulses associated with the solitary waves when they are Fourier transformed into the frequency domain could generate a broadband electrostatic noise in the frequency range of 0.1–4 kHz, which is in line with the spacecraft observations. Rubia et al. (2023) explored the features of the arbitrary amplitude ESWs propagating at both the noon-midnight and dawn-dusk sectors in the Venusian ionosphere and transition zone. They took into consideration the population of superthermally distributed electrons in the solar wind. Their work suggests the existence of compressive slow ion-acoustic modes. They estimated the wave width, velocity, electric field amplitude, and frequency ranges as ~(1.7–53.21) m, ~(1.48–8.33) km s⁻¹, ~(0.03–27.67) mVm⁻¹ and ~9.78–8.77 kHz, respectively.

To the best of our knowledge, no previous studies have included the influence of the magnetic field and the oblique propagation on the ion-acoustic excitations. In this work, we aim to investigate three-dimensional nonlinear IAWs in the transition region. For this purpose, we use a five-component hydrodynamic model inspired by the observational data reported by PVO and VEX (Lundin et al. 2011; Knudsen et al. 2016). Our model consists of two warm singly-charged ionospheric species, namely, hydrogen H⁺ and oxygen O⁺ ions, as well as ionospheric Maxwellian electrons and the solar wind (streaming solar wind protons and electrons; see also Afify et al. 2021, and Salem et al. 2022). First, we adopt the reductive perturbation technique (RPT) to derive the three-dimensional partial differential ZK equation. Then, the G′/G-expansion method is employed to obtain the soliton and shock-like/DL wave solutions.

The manuscript is organized in the following way: Sect. 2 describes the hydrodynamic equations of our model. Sections 3 and 4 are devoted to the linear and nonlinear analyses, respectively. In Sect. 4, the numerical results of our theoretical study are elucidated. Finally, the outcomes of this work are summarized in Sect. 5.

Fig. 1 Phase velocity λ in dependence on several plasma parameters, where (a) Phase velocity λ in dependence on the relative density α(= nO0/nH0), where σsp(= Tsp/Te) = 1.5 and usp0 = 3.2. (b) Phase velocity in dependence on the relative temperature σsp, where α = 3.65 and usp0 = 3.2. (c) Phase velocity in dependence on the normalized proton streaming velocity usp0. The other plasma parameters are Q0(= mH/mO) = 1/16, Qsp(= mH/msp) = 1, σO(= TO/Te) =σH(= TH/Te) = 0.2, σse(= Tse/Te) = 2, β(= nsp0/nH0) = χ(= nse0/nH0) = 1.4.

2 Fluid model equations

We aim to investigate the electrostatic ion-acoustic excitations propagating in the Venusian mantle. To do this, we employed the hydrodynamic model for a collisionless magnetized plasma consisting of two warm positive singly-charged planetary ions, namely hydrogen H⁺ and oxygen O⁺, inertialess isothermal planetary electrons (e), as well as streaming solar wind protons (sp) and inertialess solar wind electrons (se) (Afify et al. 2021; Salem et al. 2022). The hydrodynamic equations of H⁺ ionic species read $$\frac{\partial {\overline{n}}_{\text{H}}}{\partial \overline{t}}+\overline{\nabla }\cdot \left({\overline{n}}_{\text{H}}{\overline{u}}_{\text{H}}\right)=0,$$(1) $$\frac{\partial {\overline{u}}_{\text{H}}}{\partial \overline{t}}+\left({\overline{u}}_{\text{H}}\cdot \overline{\nabla }\right){\overline{u}}_{\text{H}}+\overline{\nabla }\overline{\varphi }-{\overline{u}}_{\text{H}}\times {\overline{\Omega }}_{\text{H}}\widehat{x}+\frac{5}{3}{\sigma }_{\text{H}}{\overline{n}}_{\text{H}}^{-1/3}\mathbf{\nabla }{\overline{n}}_{\text{H}}=0,$$(2)

where the σ_H = T_H/T_e is the temperature ratio of the H⁺ temperature and that of planetary electrons, while ${\overline{\Omega }}_{\text{H}}={\Omega }_{\text{cH}}/{\omega }_{\text{pH}}$, where Ω_cH is the hydrogen cyclotron frequency, and ω_pH is the hydrogen plasma frequency. The third term of the momentum equation represents the electric force, that is, E = −∇ϕ, where ϕ is the electrostatic potential. Furthermore, the fourth term represents the magnetic force, and the last term is the adiabatic thermal pressure gradient force, which is taken into account for all ions. The O⁺ ions are expressed by the following set of equations: $$\frac{\partial {\overline{n}}_{\text{O}}}{\partial \overline{t}}+\overline{\nabla }\cdot \left({\overline{n}}_{\text{O}}{\overline{u}}_{\text{O}}\right)=0,$$(3) $$\frac{\partial {\overline{u}}_{\text{O}}}{\partial \overline{t}}+\left({\overline{u}}_{\text{O}}\cdot \overline{\nabla }\right){\overline{u}}_{\text{O}}+Q_{\text{O}}\overline{\nabla }\overline{\varphi }-{\overline{u}}_{\text{O}}\times {\overline{\text{Ω}}}_{\text{O}}\widehat{x}+\frac{5}{3}{Q}_{\text{O}}{\sigma }_{\text{O}}{\overline{n}}_{\text{O}}^{-1/3}\overline{\mathbf{\nabla }}{\overline{n}}_{\text{O}}=0,$$(4)

where σ_O = T_O/T_e is the ratio of the oxygen ion temperature and that of planetary electrons, and Q_O = m_H/m_O is the ratio of the hydrogen mass m_H to the oxygen mass m_O. Here, ${\overline{\Omega }}_{\text{O}}={\Omega }_{\text{cO}}/{\omega }_{\text{pH}}$, where Ω_cO is the oxygen cyclotron frequency. The streaming solar wind protons are expressed by the following continuity and momentum equations: $$\frac{\partial {\overline{n}}_{\text{sp}}}{\partial \overline{t}}+\overline{\nabla }\cdot \left({\overline{n}}_{\text{sp}}{\overline{u}}_{\text{sp}}\right)=0,$$(5) $$\frac{\partial {\overline{u}}_{\text{sp}}}{\partial \overline{t}}+\left({\overline{u}}_{\text{sp}}\cdot \overline{\nabla }\right){\overline{u}}_{\text{sp}}+Q_{\text{sp}}\overline{\nabla }\overline{\varphi }-{\overline{u}}_{\text{sp}}\times {\overline{\text{Ω}}}_{\text{sp}}\widehat{x}+\frac{5}{3}{Q}_{\text{sp}}{\sigma }_{\text{sp}}{\overline{n}}_{\text{sp}}^{-1/3}\overline{\mathbf{\nabla }}{\overline{n}}_{\text{sp}}=0,$$(6)

where σ_sp = T_sp/T_e is the ratio of the solar wind proton temperature and that of planetary electrons, and Q_sp = m_H/m_sp = 1 is the ratio of the hydrogen ionic mass mH to the solar wind proton mass m_sp. Here, ${\overline{\Omega }}_{\text{sp}}={\Omega }_{\text{csp}}/{\omega }_{\text{pH}}$ where Ω_csp(= Ω_cH) is the proton cyclotron frequency. The massless planetary electrons and solar wind electrons are described by the following Boltzmann distribution: $${\overline{n}}_{\text{e}}=\exp (\overline{\varphi }).$$(7) $${\overline{n}}_{\text{se}}=\exp \left(\overline{\varphi }/{\sigma }_{\text{se}}\right),$$(8)

where σ_se = T_se/T_e is the ratio of the solar wind electron temperature and that of planetary electrons. The system of equations is closed by the Poisson equation, $${\overline{\nabla }}^2\overline{\varphi }=v{\overline{n}}_{\text{e}}+\chi {\overline{n}}_{\text{se}}-{\overline{n}}_{\text{H}}-\alpha {\overline{n}}_{\text{o}}-\beta {\overline{n}}_{\text{sp}},$$(9)

where $\nabla =\widehat{x}{\partial }_x+\widehat{y}{\partial }_y+\widehat{z}{\partial }_z,\widehat{x},\widehat{y}$, and $\widehat{z}$ are the unit vectors along the x-, y-, and z-axis, respectively. v = n_e0/n_H0, χ = n_se0/n_H0, α = n_O0/n_H0, and β = n_sp0/n_H0 are all the relative densities. The normalized plasma parameters (indicated by upper bars) are defined as $$\begin{array}{ll}{\overline{n}}_j=\frac{n_j}{n_{j0}},\hfill & {\overline{u}}_i=u_i{\left(\frac{k_{\text{B}}{T}_{\text{e}}}{m_{\text{H}}}\right)}^{-1/2}\text{, }\hfill \\ \overline{t}=t{\left(\frac{4\pi {\text{e}}^2{n}_{\text{H}0}}{m_{\text{H}}}\right)}^{1/2},\hfill & \overline{\varphi }=\varphi {\left(\frac{k_{\text{B}}{T}_{\text{e}}}{e}\right)}^{-1},\hfill \\ \overline{\nabla }=\nabla {\left(\frac{k_{\text{B}}{T}_{\text{e}}}{4\pi e^2{n}_{\text{H}0}}\right)}^{-1/2},\hfill & {\overline{\Omega }}_i={\Omega }_i{\left(\frac{4\pi n_{\text{H}0}{\text{e}}^2}{m_{\text{H}}}\right)}^{-1/2}.\hfill \end{array}$$

In Eqs. (1)–(9), n_j refers to the densities and n_j0 refers to the equilibrium density of the jth species, while u_i refers to the ion velocities where j = H⁺ , O⁺ e, sp, and se, and i = H⁺, O⁺ and sp.

3 Nonlinear analysis

3.1 Derivation of the evolution equation

We considered that our weakly nonlinear excitations propagate in three-dimensional space, and we adopted the RPT (Washimi & Taniuti 1966) to derive the ZK equation. A stretching of the independent variables is defined as $$\begin{array}{llll}X\hfill & ={ϵ}^{1/2}(x-\lambda t),\hfill & Y={ϵ}^{1/2}y,\hfill & Z={ϵ}^{1/2}z,and\hfill \\ T\hfill & ={ϵ}^{3/2}t,\hfill & \hfill & \hfill \end{array}$$(10)

where ϵ is a small parameter (0 < ϵ < 1) that determines the perturbation amplitude, and λ represents the wave linear phase velocity (parallel to the magnetic field), which is determined below. The flow variables n_j, u_i, and ϕ are expanded as a power series in terms of e around their equilibrium values as follows: $$\begin{array}{l}{n}_j=1+ϵn_j^{(1)}+{ϵ}^2{n}_j^{(2)}+{ϵ}^3{n}_j^{(3)}+\dots \dots \dots \dots \dots \dots \mathrm{..},\\ u_{ix}=u_{ix}^{(0)}+ϵu_{ix}^{(1)}+{ϵ}^2{u}_{ix}^{(2)}+{ϵ}^3{u}_{ix}^{(3)}+\dots \dots \dots \dots \dots \dots ,\\ u_{iy}={ϵ}^{3/2}{u}_{iy}^{(1)}+{ϵ}^2{u}_{iy}^{(2)}+{ϵ}^{5/2}{u}_{iy}^{(3)}+{ϵ}^3{u}_{iy}^{(4)}+\dots \dots ,\\ u_{iz}={ϵ}^{3/2}{u}_{iz}^{(1)}+{ϵ}^2{u}_{iz}^{(2)}+{ϵ}^{5/2}{u}_{iz}^{(3)}+{ϵ}^3{u}_{iz}^{(4)}+\dots \dots ,\\ \varphi =ϵ{\varphi }^{(1)}+{ϵ}^2{\varphi }^{(2)}+{ϵ}^3{\varphi }^{(3)}+\dots \dots \dots \dots \dots \dots .\end{array}$$(11)

for j = H⁺, O⁺, e⁻ sp⁺, and se⁻, and i = H⁺, O⁺, and sp⁺. We note that the perpendicular velocity components have higher-order perturbations compared to the parallel component since in the presence of a magnetic field, the plasma is anisotropic, so that the ion gyromotion becomes a higher-order effect (Elkamash & Kourakis 2021).

The charge-neutrality condition is always maintained through the relation $$v+\chi -\alpha -\beta -1=0.$$(12)

We used Eqs. (10) and (11) in the normalized set of equations and then collected the lowest order in ϵ, which gives the following relations: $$\begin{array}{ll}{n}_{\text{H}}^{(1)}\hfill & =\frac{1}{{\lambda }^2-\frac{5}{3}{\sigma }_{\text{H}}}{\varphi }^{(1)},n_{\text{O}}^{(1)}=\frac{Q_{\text{O}}}{{\lambda }^2-\frac{5}{3}{Q}_{\text{O}}{\sigma }_{\text{O}}}{\varphi }^{(1)},\hfill \\ n_{\text{sp}}^{(1)}\hfill & =\frac{Q_{\text{sp}}}{V^2-\frac{5}{3}{Q}_{\text{sp}}{\sigma }_{\text{sp}}}{\varphi }^{(1)},\hfill \end{array}$$(13)

where $V=\left(\lambda -u_{\text{sp}}^{(0)}\right)$, $$\begin{array}{l}{u}_{\text{Hx}}^{(1)}=\frac{\lambda }{{\lambda }^2-\frac{5}{3}{\sigma }_{\text{H}}}{\varphi }^{(1)},u_{\text{Ox}}^{(1)}=\frac{\lambda Q_{\text{O}}}{{\lambda }^2-\frac{5}{3}{Q}_{\text{O}}{\sigma }_{\text{O}}}{\varphi }^{(1)},\\ u_{\text{spx}}^{(1)}=\frac{V{Q}_{\text{sp}}}{V^2-\frac{5}{3}{Q}_{\text{sp}}{\sigma }_{\text{sp}}}{\varphi }^{(1)},\end{array}$$(14) $$\begin{array}{l}{u}_{\text{Oy}}^{(1)}=\frac{-3Q{\lambda }^2}{{\Omega }_{\text{O}}\left(3{\lambda }^2-5Q{\sigma }_{\text{O}}\right)}\frac{\partial {\varphi }^{(1)}}{\partial Z},\hfill \\ u_{\text{spy}}^{(1)}=\frac{-3Q_{\text{sp}}{V}^2}{{\Omega }_{\text{sp}}\left(3V^2-5Q_{\text{sp}}{\sigma }_{\text{sp}}\right)}\frac{\partial {\varphi }^{(1)}}{\partial Z},\hfill \\ u_{\text{Hz}}^{(1)}=\frac{-3{\lambda }^2}{{\Omega }_{\text{H}}\left(3{\lambda }^2-5{\sigma }_{\text{H}}\right)}\frac{\partial {\varphi }^{(1)}}{\partial Y},\hfill \end{array}$$(15) $$\begin{array}{l}{u}_{\text{Hz}}^{(1)}=\frac{3{\lambda }^2}{{\Omega }_{\text{H}}\left(3{\lambda }^2-5{\sigma }_{\text{H}}\right)}\frac{\partial {\varphi }^{(1)}}{\partial Y},\hfill \\ u_{\text{Oz}}^{(1)}=\frac{3Q_{\text{O}}{\lambda }^2}{{\Omega }_{\text{O}}\left(3{\lambda }^2-5Q_{\text{O}}{\sigma }_{\text{O}}\right)}\frac{\partial {\varphi }^{(1)}}{\partial Y},\hfill \\ u_{\text{spz}}^{(1)}=\frac{3Q_{\text{sp}}{V}^2}{{\Omega }_{\text{sp}}\left(3V^2-5Q_{\text{sp}}{\sigma }_{\text{sp}}\right)}\frac{\partial {\varphi }^{(1)}}{\partial Y},\hfill \end{array}$$(16) $$\begin{array}{cc}{n}_{\text{e}}^{(1)}={\varphi }^{(1)},& n_{\text{se}}^{(1)}={\varphi }^{(1)}/{\sigma }_{\text{se}}\text{.}\end{array}$$(17)

Combining the lowest-order contributions (Eqs. (13)–(17)) in the Poisson equation, we obtain the compatibility condition as $$\frac{3}{3{\lambda }^2-5{\sigma }_{\text{H}}}+\frac{3Q_{\text{O}}\alpha }{3{\lambda }^2-5Q_{\text{O}}{\sigma }_{\text{O}}}+\frac{3Q_{\text{sp}}\beta }{3V^2-5Q_{\text{sp}}{\sigma }_{\text{sp}}}-\frac{\chi }{{\sigma }_{\text{se}}}-v=0.$$(18)

This equation is used below to determine the wave -phase velocity λ. We note that λ is dependent on the physical parameters of Venus, such as the relative densities and temperatures. We then combined the higher-order contributions to obtain the following equation: $$\frac{\partial {\varphi }^{(1)}}{\partial T}+A{\varphi }^{(1)}\frac{\partial {\varphi }^{(1)}}{\partial X}+B\frac{{\partial }^3{\varphi }^{(1)}}{\partial X^3}+C\frac{\partial }{\partial X}\left(\frac{{\partial }^2}{\partial Y^2}+\frac{{\partial }^2}{\partial Z^2}\right){\varphi }^{(1)}=0,$$(19)

where ϕ⁽¹⁾ is the electrostatic potential (leading-order disturbance), A is the nonlinearity coefficient, and B and C are the dispersive coefficients in the longitudinal and transverse direction(s), which are given as follows: $$\begin{array}{c}A=B(\frac{3\left(27{\lambda }^2-5{\sigma }_{\text{H}}\right)}{{\left({\lambda }^2-5{\sigma }_{\text{H}}\right)}^3}+\frac{3\alpha Q^2\left(27{\lambda }^2-5Q{\sigma }_{\text{O}}\right)}{{\left({\lambda }^2-5Q{\sigma }_{\text{O}}\right)}^3}\\ +\frac{3\beta Q_{\text{sp}}^2\left(27V^2-5Q_{\text{sp}}{\sigma }_{\text{sp}}\right)}{{\left(3V^2-5Q_{\text{sp}}{\sigma }_{\text{sp}}\right)}^3}-\frac{\chi }{{\sigma }_{\text{se}}^2}-v),\end{array}$$(20) $$B=\frac{1}{2}{\left(\frac{3\lambda }{{\left(3{\lambda }^2-5{\sigma }_{\text{H}}\right)}^2}+\frac{3\alpha Q_{\text{O}}\lambda }{{\left(3{\lambda }^2-5{\sigma }_{\text{O}}{Q}_{\text{O}}\right)}^2}+\frac{3\beta Q_{\text{sp}}V}{{\left(3V^2-5{\sigma }_{\text{sp}}{Q}_{\text{sp}}\right)}^2}\right)}^{-1},$$(21) $$\begin{array}{l}C=B(1+\frac{9{\lambda }^4}{{\Omega }_{\text{H}}^2{\left(3{\lambda }^2-5{\sigma }_{\text{H}}\right)}^2}\hfill \\ +\frac{9\alpha Q_{\text{O}}{\lambda }^4}{{\Omega }_{\text{O}}^2{\left(3{\lambda }^2-5{\sigma }_{\text{O}}{Q}_{\text{O}}\right)}^2}+\frac{9\beta V^4{Q}_{\text{sp}}}{{\Omega }_{\text{sp}}^2{\left(3V^2-5{\sigma }_{\text{sp}}{Q}_{\text{sp}}\right)}^2}).\hfill \end{array}$$(22)

3.2 Nonlinear solutions of the ZK equation

3.2.1 Soliton wave solution

To obtain a stationary soliton solution for Eq. (19), we used the traveling-wave transformation, $$\eta =L_{1}X+L_{2}Y+L_{3}Z-MT,$$(23)

where η is the transformed coordinate relative to the frame that moves with the velocity M. L₁, L₂, and L₃ are the directional cosines of the wave vector k along X, Y, and Z, respectively, satisfying the equation $L_1^2+L_2^2+L_3^2=1$. For simplicity, we assumed that ϕ⁽¹⁾ = ϕ. By integrating Eq. (19) with respect to the variable η and using the vanishing boundary condition for φ and its derivatives up to second order for |η|→ ∞, the soliton solution is given by $$\phi (\eta )={\phi }_0{\mathrm{Sech}}^2\left(\frac{\eta }{w}\right),$$(24)

where φ₀ = 3M/AL₁ is the soliton amplitude and $w=2\sqrt{L_1\left(B{L}_1^2+C\left(L_2^2+L_3^2\right)\right)/M}$ is the soliton width.

3.2.2 Double-layer (shock-like) wave solution

We adopted the G′/G-expansion technique (Wang et al. 2008) to find possible solutions of Eq. (19). According to this approach, Eq. (19) is transformed into an ordinary differential equation (ODE) using the independent variable (23). Thus, Eq. (19) is written as $$\frac{{\text{d}}^3\phi }{\text{d}{\eta }^3}+H_1\phi \frac{\text{d}\phi }{\text{d}\eta }-H_2\frac{\text{d}\phi }{\text{d}\eta }=0,$$(25)

where $$\begin{array}{c}{H}_1=\frac{A}{2\left(B{L}_1^2+C\left(L_2^2+L_3^2\right)\right)},\\ H_2=\frac{M}{2\left(B{L}_1^3+C{L}_1\left(L_2^2+L_3^2\right)\right)}.\end{array}$$(26)

Equation (25) can be integrated with respect to η to obtain $$\frac{{\text{d}}^2\phi }{\text{d}{\eta }^2}+\frac{H_1}{2}{\phi }^2-H_2\phi =c,$$(27)

where c is the constant of integration. The wave solution φ can be expressed as $$\phi (\eta )={\displaystyle \sum _{l=0}^n{h}_l}{\left(\frac{G^{\prime }}{G}\right)}^l+{\displaystyle \sum _{l=1}^n{g}_l}{\left(F_l(\eta )+\frac{G^{\prime }}{G}\right)}^{-l},$$(28)

where F_l(η) are functions of η, h_l and g_l are arbitrary constants, and n = 2, which is determined by considering the homogeneous balance of the highest-order nonlinear terms and the highest-order derivatives appearing in Eq. (27). Using Eq. (28) in Eq. (27) and following the usual procedure of the G′ /G-expansion technique, we obtained the following four sets of arbitrary constants h_l and g_l: $$\begin{array}{l}{h}_0=\frac{(-5+\sqrt{5})H_1}{10H_2},h_1=\frac{-\sqrt{H_1}}{H_2{5}^{1/4}},h_2=\frac{-1}{H_2},\hfill \\ g_1=\frac{H_1^{3/2}}{2H_2{5}^{3/4}},g_2=\frac{H_1^2}{20H_2},\hfill \end{array}$$(29) $$\begin{array}{l}{h}_0=\frac{(-5-\sqrt{5})H_1}{10H_2},h_1=\frac{i\sqrt{H_1}}{H_2{5}^{1/4}},h_2=\frac{-1}{H_2},\hfill \\ g_1=\frac{-i{H}_1^{3/2}}{2H_2{5}^{3/4}},g_2=\frac{-H_1^2}{20H_2},\hfill \end{array}$$(30) $$\begin{array}{l}{h}_0=\frac{(-5-\sqrt{5})H_1}{10H_2},h_1=\frac{-i\sqrt{H_1}}{H_2{5}^{1/4}},h_2=\frac{-1}{H_2},\\ g_1=\frac{i{H}_1^{3/2}}{2H_2{5}^{3/4}},g_2=\frac{-H_1^2}{20H_2},\end{array}$$(31) $$\begin{array}{l}{h}_0=\frac{(-5+\sqrt{5})H_1}{10H_2},h_1=\frac{\sqrt{H_1}}{H_2{5}^{1/4}},h_2=\frac{-1}{H_2},\\ g_1=\frac{H_1^{3/2}}{2H_2{5}^{3/4}},g_2=\frac{-H_1^2}{20H_2}.\end{array}$$(32)

The function G(η) satisfies the Riccati equation, $$\frac{d^{2}G}{ \text{d}{\eta }^2}+{\beta }_1\frac{\text{d}G}{ \text{d}\eta }+{\beta }_{2}G=0,$$(33)

where β₁ and β₂ are constants. Equation (33) has a solution given by $$G(\eta )=\exp \left(\frac{-{\beta }_1}{2}\eta \right)\left(c_1\sinh \left[{\theta }_1\eta \right]+c_2\cosh \left[{\theta }_1\eta \right]\right),$$(34)

with $c_1>c_2,{\left({\beta }_1^2-4{\beta }_2^2\right)}^{1/2}>0$, and ${\theta }_1={\left({\beta }_1^2-4{\beta }_2^2\right)}^{1/2}$.

4 Numerical results

In this paper, we aim to investigate the electrostatic solitary and DL ion-acoustic waves driven by the interaction of the solar wind and the Venutian ionosphere. We expect that these nonlinear waveforms may serve as an energy transfer candidate or an energization process at the transition region (~ 1000–2000 km). We adopted the observational data of plasma parameters reported by VEX, and we compared our results, for example, along with the traveling wave solutions derived in Sect. 3, with the measured wave frequencies and DL structures via PVO and PSP, respectively. The plasma configurations representative of the transition region are α ∈ [3.33, 4], β ∈ [1.3, 1.5], χ ∈ [1.3, 1.5], Q_sp = 1, Q_O = 1/16, σ_O = 0.2, σ_H = 0.2, σ_sp ∈ [1, 2], σ_se ∈ [1, 3], u_sp0 ∈ [2.6, 3.9], and B₀ ∈ [80 × 10⁻⁵, 120 × 10⁻⁵] G (∈ [80, 120] nT) (Bertucci et al. 2003; Lundin et al. 2011; Knudsen et al. 2016). Based on these values, the nonlinear coefficient A and dispersion coefficient B (defined in Eqs. (20) and (21)) are always positive. When A is positive, only compressive (positive polarity) IAWs exist in the mantle.

In the following, we examine both types of wave structures in two cases: i) oblique propagation, and ii) quasi-parallel propagation. However, we note that we only included the results of the case of quasi-parallel DLs because the oblique DLs results were unreasonable. The electric field wave amplitudes were unreasonably high, which does not match our weakly nonlinear analysis. This suggests that oblique DLs would need a different mathematical treatment. Moreover, in our analysis, we set the value of the constant of integration at Eq. (27) to zero because it only causes a constant shift to the wave amplitude.

We defined our cases (i.e., the propagation angles) according to the directional cosines L₁, L₂, and L₃ mentioned previously in Sect. 3. From the equation $L_1^2+L_2^2+L_3^2=1$, we have $L_1=\sqrt{1-L_2^2-L_3^2}$, where L₁, L₂, and L₃ are the directional cosines of the wave vector k along the X–, Y–, and Z–axes. To obtain the quasi-parallel case, we set L₂ and L₃ to very low values (i.e., L_2,3 ≳ 0). Hence, the transverse perturbation will be weak and the wave would behave as a wave propagating in only one direction (goes back to a KdV solitary wave). On the other hand, to obtain the oblique case, we set finite values for L₂ and L₃ (i.e., 0 < L_2,3 < 1). This means that the transverse perturbation will be stronger and will influence the wave dynamics.

To obtain insights into the phase velocity of our nonlinear structures, we rearranged the compatibility condition Eq. (18) to obtain a sixth-order polynomial in λ. Then, we solved it numerically to obtain six different roots. Each root represents a possible mode with a certain phase velocity. The six roots are illustrated in Fig. 1 versus the physical parameters α, σ_sp, and u_sp0; no other parameters affect the phase velocity, and therefore, we did not include them. Figure 1a shows four forward ion-acoustic modes (λ > 0), λ_1–4. The rest of them are backward modes (λ < 0), λ_5,6. Forward modes are divided into two supersonic/fast ion-acoustic modes (λ > 1), λ_1,2, the others are subsonic/slow ion-acoustic modes (λ < 1), λ_3,4, and the backward modes are only slow modes. In our analysis, we considered only supersonic velocities because they are in line with spacecraft observations (Lundin et al. 2011). To define each mode, we considered both the initial velocity and mass of the species. Faster modes belong to lighter and faster species, and slower modes belong to the heavier and slower species. Therefore, we can characterize them as follows: λ_1,2 belong to the solar wind protons, λ_3,6 belong to the planetary hydrogen ion, and λ_4,5 to the planetary oxygen ion. In Fig. 1b, higher values of σ_sp increase the phase velocity of one of the solar wind proton branches, but reduce the other. Moreover, higher streaming velocities increase the phase velocity of both branches of the solar wind protons Fig. 1c.

4.1 Solitary waves

First, we examined the effect of the relative densities α(= n_O0/n_H0), β (= n_sp0/n_e0), and χ (= n_se0 /n_H0) on the profile of solitary pulses (Eq. (24)), as shown in Fig. 2. As clearly illustrated in Figs. 2a,b, increasing α reduces the amplitude of the solitary pulse in both cases (i.e., oblique and quasi-parallel propagation angles) and slightly reduces the width, which means that lower α values enhance the nonlinearity of the wave structures. In Figs. 2c,d, we examine the effect of β. Evidently, increasing β amplifies the amplitude of the solitary wave with a slight increase in the wave width. Figures 2e,f show that solitary waves change very little when χ varies in both cases. One clear difference between the two cases in all figures is the spatial size of the solitary pulse. It extends to ~10⁴ λ_DH in the case of oblique propagation, but is only ~30 λ_DH for parallel propagation. This can be explained by the fact that the wavelength should be greater than the plasma characteristic scale length L in the fluid limit, and L changes as we change the propagation angle. In the case of quasi-parallel propagation, the IAW behaves as it is propagating in unmagnetized plasma, where L ~ λ_D. For instance, using the mantle plasma parameters reported by VEX and PVO (Bertucci et al. 2003; Knudsen et al. 2016; Lundin et al. 2011), we found that λ_D ~ 7 m for planetary hydrogen ion in Venus. In the case of oblique propagation, the wave starts to be affected by the magnetic field, and here, L ~ r_L, where the planetary hydrogen Larmor radius is r_LH ~ 2.5 km. Therefore, we can observe that quasi-parallel pulses extend to a few dozen meters, while oblique pulses extend to much greater distances and reach a few dozen kilometers.

In Fig. 3, we examine the effect of the relative temperatures σ_sp(= T_sp/T_e) and σ_se(= T_se/T_e) on solitary structures. As illustrated in Figs. 3a,b, increasing σ_sp diminishes the solitary-pulse amplitude significantly, with a slight increase in the spatial size of the pulse in both cases. This means that greater σ_sp values would enhance the wave dispersion and reduce the wave non-linearity. In Figs. 3c,d, higher values of σ_se amplify the wave amplitude and slightly increase the wave width. We examined the effect of both σ_O and σ_H, but the waveform showed no variation. We therefore did not include any figures for them.

In Fig. 4, we investigate the influence of the streaming velocity of the solar wind proton u_sp0 solitary waves. As clearly illustrated, the pulse does not show any change when the value of u_sp0 varies. This may be because of the relatively high velocities of the solar wind compared to the planetary plasma. This is in line with results obtained by Salem et al. (2022).

The effect of the magnetic field strength on the solitary structures is illustrated in Fig. 5. Greater magnetic field strengths do not affect the pulse amplitude at all. Otherwise, the pulse only becomes narrower with stronger magnetic fields. In other words, stronger magnetic fields reduce the wave dispersion without an effect on the wave nonlinearity.

We were interested in estimating the ESWs characteristics as the magnitude of the wave corresponding bipolar electric field, the wave time duration, and the wave frequency range. For this purpose, we derived the electric field profiles and used a Fourier transform to derive their frequency spectrum, as illustrated in Fig. 6. We applied this procedure to both cases, but we only included the oblique propagation case because the quasi-parallel propagation case results match the results obtained by Salem et al. (2022) and Rubia et al. (2023). The estimates of Salem et al. (2022) were a frequency range of 0.1–4 kHz, a maximum electric field amplitude E_m of 7.5 mV m⁻¹, and a time duration of τ ~ 3 ms. The estimates of Rubia et al. (2023), on the other hand, were E_m = (0.03–27.67) mV m⁻¹, τ = (0.34–22) ms, width w = (1.7–53.21)m, and ƒ = 9.8−8.7 kHz. Our results are as follows: a frequency range of ~630–3160 Hz, E_m ~ 8 mV m⁻¹, and τ ~ 1.6 ms across a spatial width of (140–200) m. These outcomes may explain the measurements of electrostatic activity in the frequency range 730 Hz and 5.4 kHz (with a bandwidth of ±15% of the central frequency) carried outby the electric dipole onboard PVO (Scarf et al. 1979; Yadav 2021). While no simultaneous electric and magnetic field observations were made with PVO and the frequency coverage of the electric field observations was noncontinuous, our theoretical results demonstrate that IAWs should arise at frequencies where PVO OEFD would have observed them in the 730 Hz and 5.4 kHz channels, so that it is valid to interpret the mantle waves observed in these channels as IAWs. Recently, solitary structures have been detected in Venus via PSP in both the magnetosheath and in the transition region (Malaspina et al. 2020; Hadid et al. 2021), but there are not enough studies about their nature so far. Therefore, our estimates suggest additional explanations for these structures in the transition region, as reported by Malaspina et al. (2020). For the oblique case, we present the influence of the magnetic field strength on the ESW and its corresponding electric field in Figs. 6a,b, respectively. To make the waveforms relevant to space observations, we replotted the solitary profile with the variation in the magnetic field strength for a non-normalized electrostatic potential Φ and spatial width w. Clearly, stronger magnetic fields amplify the electric disturbances associated with the solitary pulses, where Φ_max = 0.4 V and E_m ~ 0.024 mV m⁻¹. The pulse spatial width and temporal width are ~40–80 km and ~0.4 s. Furthermore, the fast Fourier transform (FFT) of the bipolar pulse generates a broadband electrostatic activity with a frequency spanning a range of ~ 1.6–10 Hz, as illustrated in Fig. 6c. The wave features vary in both cases. However, this is expected because IAWs are low-frequency waves with frequencies lower than the plasma characteristic frequency (i.e., ω ≤ ω_p in non-magnetized plasma and ω ≤ Ω_c in magnetized plasma). Typical parameters representative of the Venutian transition zone are Ω_pH = 4556.34 rad s⁻¹ (ƒ_pH = 725 Hz) and Ω_cH = 9.56 rad s⁻¹ (ƒ_cH = 1.52 Hz).

Fig. 2 Profiles of solitary waves in case of oblique propagation (left) and quasi-parallel propagation (right), where (a) and (b) are the solitary profiles in dependence on α(= nO0/nH0), where β (= nsp0/nH0) = χ(= nse0/nH0) = 1.4. (c) and (d) are the solitary profiles in dependence on β where α = 3.5 and χ = 1.4. (e) and (f) are the solitary profiles in dependence on χ, where α = 3.5 and β = 1.4. The other plasma parameters are QO(= mH/mO) = 1/16, Qsp (= mH/msp) = 1, σO(= TO/Te) = σH (= TH/Te) = 0.2, σsp(= Tsp/Te) = 1 σse( = Ts/Te) = 2, usp0 = 3.3, and B0 = 100 × 10−5 G (100 nT).

Fig. 3 Profiles of solitary waves in case of oblique propagation (left) and quasi-parallel propagation (right), where (a) and (b) are the solitary profiles in dependence on σsp(= Tsp/Te), where σse(= Tse/Te) = 2. (c) and (d) are the solitary profiles in dependence on σse, where σsp = 1. The other plasma parameters are QO(= mH/mO) = 1/16, Qsp (= mH/msp) = 1, α(= nO0/nH0) = 3.5, β (= nsp0/nH0) = χ (= nseo/nH0) = 1.4, σO (= TO/Te) = σH (= TH/Te) = 0.2, usp0 = 3.3, and B0 = 100 × 10−5 G (100 nT).

Fig. 4 Profiles of solitary waves in case of oblique propagation (left) and quasi-parallel propagation (right), where (a) and (b) are the solitary profiles in dependence on usp0 = 3.3, where the other plasma parameters are QO (= mH/mO) = 1/16, Qsp (= mH/msp) = 1, α (= nO0/nH0) = 3.5, β (= nsp0/nH0) = χ (= nse0/nH0) = 1.4, σO (= TO/Te) = σH(= TH/Te) = 0.2, σsp(= Tsp/Te) = 1, σse(= Tse/Te) = 2, and B0 = 100 × 10−5 G (100 nT).

Fig. 5 Profiles of solitary waves in case of oblique propagation (left) and quasi-parallel propagation (right), where (a) and (b) are the solitary profiles in dependence on B0 = 85 × 10−5 G (85 nT) (orange), B0 = 100 × 10−5 G (100 nT) (blue), B0 = 120 × 10−5 G (120 nT) (purple). The other plasma parameters are QO (= mH/mO) = 1/16, Qsp(= mH/msp) = 1, α(= nO0/nH0) = 3.5, β (= nsp0/nH0) = χ (= nse0/nH0) = 1.4, σO (= TO/Te) = σH (= TH/Te) = 0.2, σsp (= Tsp /Te) = 1, σse (= Tse/Te) = 2, and usp0 = 3.3.

Fig. 6 Profile of the oblique solitary pulse with its corresponding bipolar electric pulse and the FFT power spectra. (a) Solitary pulse electrostatic potential expressed in volts (V) in dependence on the spatial width in Km. (b) Associated electric pulse for different values of the magnetic field. (c) Corresponding FFT spectra of the electric waveforms, where the x-axis represents log10v, where v is the frequency in Hz, and the y-axis signifies the power of the electric pulse expressed in decibels $\text{dB}(mV/v\sqrt{(Hz)})$.

4.2 Double-layers

We investigated another nonlinear wave structure, which is the DL or the shock-like soliton wave. This wave structure is known to induce strong field-aligned electric fields, and therefore, they usually serve as a particle acceleration process. Ion-acoustic DLs were observed in the Venusian magnetosheath with the Parker Solar Probe (PSP; Malaspina et al. 2020). Malaspina et al. (2020) suggested that these DL structures may be excited through the mixing of hot sheath electrons with the relatively cold solar wind electrons. They also suggested that these structures might cause accelerating or heating solar wind electrons. Generally, one of the processes that causes these excitations is the interaction between hot and cold plasma (Hultqvist 1971). In the case of Venus, they may be excited through the mixing of the solar wind and the relatively cold planetary plasma in the transition zone. In the following, we present the profile of the electrostatic potential across the DLs versus other physical parameters in Figs. 7–11.

In Fig. 7, we present the effect of the relative densities α, β, and χ on the DL profile. Figure 7a makes clear that the amplitude of the DL fluctuates with the variation of α, where the amplitude rises somewhat with a greater value of α, and then drops significantly. Conversely, increasing β slightly reduces the DL potential and then amplifies it again (see Fig. 7b). In the case of χ, the DL amplitude shows only a slight rise with a greater value of χ (see Fig. 7c).

Figure 8 illustrates the influence of the temperature ratios σ_sp(= T_sp/T_e) and σ_se(= T_se/T_e) on the nature of DL structures. As shown in Fig. 8a, a greater σ_sp value enhances the amplitude significantly. Therefore, we expect that hotter solar wind protons may enhance the field-aligned electric field across the DL structure, leading to a stronger acceleration of particles. Conversely, the shock-like profile is not affected by the variation of σ_se (see Fig. 8b). We examined the influence of both σ_O and σ_H, but we did not include them because they do not affect the DL profile.

Figure 9 shows the influence of the proton streaming velocity in the electrostatic potential across the DL structure. It is not affected by the variation of u_sp0. This is the same behavior as in the case of solitary waves.

The effect of the magnetic field strength on the DL structure is illustrated in Fig. 10. As is clearly shown, lower magnetic field strengths enhance the potential drop of the DL. This enhancement means that stronger electric fields are generated across the DL, and hence, a stronger acceleration of particles may occur.

In Fig. 11, we investigate the DL profiles, their associated electric field, and the FFT power spectrum. As is clearly seen in Fig. 11a, the potential drop of the DL is Φ_Dl ~ (6.5– 13) V, with a spatial width of w ~ (100–120) m. Malaspina et al. (2020) estimated the potential drop and the spatial scales of the DLs reported by PSP, where Φ_Dl lies in the range of (9–86) V, with spatial scales of w = (3–155)/λ_D. Moreover, they mentioned that DLs with scales of w ≫ 500λ_D are not accepted since the charge separation cannot manifest itself over large distances. We therefore expect that highly oblique DL cannot exist in this plasma environment. Our model is restricted to the transition zone, which is the boundary at which the hot solar wind and cold planetary plasma interact. The observational data we used are reported via VEX in the noon-midnight (NM) sector, which covers the day- and nightside parts of the mantle, namely, the magnetic barrier and the magnetotail, respectively (Phillips & McComas 1991). In the literature, some authors considered this region as part of the magnetosheath, and some do not. The PSP encounter detected the fluctuations through the bow shock and the magnetosheath. Therefore, our study coincides with PSP measurements at the lower altitudes, and since our model is generic, it can be used to study structures detected at higher altitudes. The electric field profile illustrated in Fig. 11b shows that the electric field amplitude is ~(0.16–0.35) mVm⁻¹, with a temporal width of ~1 ms. The output of the FFT of this electric pulse (Fig. 11c) is a broadband electrostatic noise in the range ~630–3980 Hz.

Fig. 7 Profiles of quasi-parallel DL, where (a) the DL profiles in dependence on α(= nO0/nH0), where β(= nsp0/nH0) = χ(= nse0 / nH0) = 1.4. (b) The DL profiles in dependence on β, where α = 3.5 and χ = 1.4. (c) the DL profiles in dependence on χ, where α = 3.5 and β = 1.4. The other plasma parameters are QO (= mH/mO) = 1/16, Qsp(= mH/msp) = 1, σO (= TO/Te) = σH (= TH/Te) = 0.15, σsp (= Tsp/Te) = 1.5, σse (= Tse/Te) = 2, usp0 = 3.9, and B0 = 120 × 10−5 G (120 nT).

Fig. 8 Profiles of quasi-parallel DL, where (a) the DL profiles in dependence on σsp(= Tsp/Te), where σse(= Tse/Te) = 2. (b) The DL profiles in dependence on σse, where σsp = 1.5. The other plasma parameters are Qo(= mH/mO) = 1/16, Qsp(= mH/msp) = 1, a(= no0/nH0) = 3.5,/β(= nsp0/nH0) = χ(= nse0/nH0) = 1.4, σo(= To/Te) = σH(= TH/Te) = 0.15, usp0 = 2.6, and B0 = 120 × 10−5 G (120 nT).

Fig. 9 Profile of quasi-parallel DL in dependence on usp0 = 3.3, where other plasma parameters are QO(= mH/mo) = 1/16, Qsp(= mH/msp) = 1, α(= nO0/nH0) = 3.5, β(= nsp0/nH0) = χ(= nse0/nH0) = 1.4, σO (= TO /Te) = σH(= TH/Te) = 0.15, σsp(= Tsp/Te) = 1, σse(= Tse/Te) = 2, and B0 = 120 × 10−5 G (120 nT).

Fig. 10 Profile of quasi-parallel DL in dependence on B0, where B0 = 85 × 10−5 G (85 nT) (orange), B0 = 100 × 10−5 G (100 nT) (blue), B0 = 120 × 10−5 G (120 nT) (purple). The other plasma parameters are QO(= mH/mO) = 1/16, Qsp(= mH/msp) = 1, α(= no0/nH0) = 3.5,β(= nsp0/nH0) = χ(= nse0/nH0) = 1.4, σO(= TO/Te) = σH(= TH/Te) = 0.2, σsp(= Tsp/Te) = 1.5, σse(= Tse/Te) = 2, and usp0 = 3.

Fig. 11 Profile of the quasi-parallel DL with its corresponding electric pulse and the FFT power spectra. (a) DL electrostatic potential expressed in volts (V) in dependence on the spatial width in m. (b) Associated electric pulse for different values of the magnetic field. (c) Corresponding FFT spectra of the electric waveforms, where the x-axis represents log10v, where v is the frequency in Hz, and the y-axis signifies the power of the electric pulse, expressed in decibels dB$(mV/v\sqrt{(Hz)})$.

5 Summary and conclusions

To summarize, we examined the low-frequency nonlinear electrostatic wave activity in the Venusian mantle. With the aid of a hydrodynamic description, the RPT, and the G′ /G-expansion method, we derived the ZK equation, which describes the propagation of nonlinear IAWs. We found two analytical solutions expressing the solitary and DL waveforms. Our study complements previous works that investigated one-dimensional IAWs, where we examined the influence of obliqueness, the ambient magnetic field, and the solar wind physical parameters. Based on the observational data reported by PVO, VEX, and PSP, our study suggests the existence of supersonic compressive ionacoustic solitary and DL fluctuations. We estimated the structural characteristics of both structures, where solitary waves are predicted to have a maximum electric field of ~0.024 mVm⁻¹, with spatial and temporal widths of ~40–80 km and 0.4 s, and a frequency range of ~1.6–10 Hz. Obliqueness is found to affect the spatial size of solitary waveforms significantly, but on the other hand, it does not affect the pulse amplitude. Moreover, the pulse amplitude is enhanced by higher values of α and σ_se, but is surpassed by higher values of β and σ_se. Quasi-parallel DLs, on the other hand, are found to have a potential drop in the range ~(6.5–13) V across spatial and temporal widths of ~(100–120)m and ~1 ms, and a frequency range of ~630–3980 Hz. Furthermore, DLs are found to become steeper with the enhancement of σ_se or with weaker magnetic fields. The outcomes of this work agree well with the available wave measurements of PSP and PVO.

Acknowledgments

The authors acknowledge sponsorship provided by the Alexander von Humboldt Stiftung (Bonn, Germany) in the framework of the Research Group Linkage Programme funded by the respective Federal Ministry, and the support from the Ruhr-University Bochum and the Katholieke Universiteit Leuven, and Mansoura University. These results were also obtained in the framework of the project SIDC Data Exploitation (ESA Prodex-12).

Appendix A Nomenclature

References

All Figures

Fig. 1 Phase velocity λ in dependence on several plasma parameters, where (a) Phase velocity λ in dependence on the relative density α(= nO0/nH0), where σsp(= Tsp/Te) = 1.5 and usp0 = 3.2. (b) Phase velocity in dependence on the relative temperature σsp, where α = 3.65 and usp0 = 3.2. (c) Phase velocity in dependence on the normalized proton streaming velocity usp0. The other plasma parameters are Q0(= mH/mO) = 1/16, Qsp(= mH/msp) = 1, σO(= TO/Te) =σH(= TH/Te) = 0.2, σse(= Tse/Te) = 2, β(= nsp0/nH0) = χ(= nse0/nH0) = 1.4.

In the text

Fig. 2 Profiles of solitary waves in case of oblique propagation (left) and quasi-parallel propagation (right), where (a) and (b) are the solitary profiles in dependence on α(= nO0/nH0), where β (= nsp0/nH0) = χ(= nse0/nH0) = 1.4. (c) and (d) are the solitary profiles in dependence on β where α = 3.5 and χ = 1.4. (e) and (f) are the solitary profiles in dependence on χ, where α = 3.5 and β = 1.4. The other plasma parameters are QO(= mH/mO) = 1/16, Qsp (= mH/msp) = 1, σO(= TO/Te) = σH (= TH/Te) = 0.2, σsp(= Tsp/Te) = 1 σse( = Ts/Te) = 2, usp0 = 3.3, and B0 = 100 × 10−5 G (100 nT).

In the text

Fig. 3 Profiles of solitary waves in case of oblique propagation (left) and quasi-parallel propagation (right), where (a) and (b) are the solitary profiles in dependence on σsp(= Tsp/Te), where σse(= Tse/Te) = 2. (c) and (d) are the solitary profiles in dependence on σse, where σsp = 1. The other plasma parameters are QO(= mH/mO) = 1/16, Qsp (= mH/msp) = 1, α(= nO0/nH0) = 3.5, β (= nsp0/nH0) = χ (= nseo/nH0) = 1.4, σO (= TO/Te) = σH (= TH/Te) = 0.2, usp0 = 3.3, and B0 = 100 × 10−5 G (100 nT).

In the text

Fig. 4 Profiles of solitary waves in case of oblique propagation (left) and quasi-parallel propagation (right), where (a) and (b) are the solitary profiles in dependence on usp0 = 3.3, where the other plasma parameters are QO (= mH/mO) = 1/16, Qsp (= mH/msp) = 1, α (= nO0/nH0) = 3.5, β (= nsp0/nH0) = χ (= nse0/nH0) = 1.4, σO (= TO/Te) = σH(= TH/Te) = 0.2, σsp(= Tsp/Te) = 1, σse(= Tse/Te) = 2, and B0 = 100 × 10−5 G (100 nT).

In the text

Fig. 5 Profiles of solitary waves in case of oblique propagation (left) and quasi-parallel propagation (right), where (a) and (b) are the solitary profiles in dependence on B0 = 85 × 10−5 G (85 nT) (orange), B0 = 100 × 10−5 G (100 nT) (blue), B0 = 120 × 10−5 G (120 nT) (purple). The other plasma parameters are QO (= mH/mO) = 1/16, Qsp(= mH/msp) = 1, α(= nO0/nH0) = 3.5, β (= nsp0/nH0) = χ (= nse0/nH0) = 1.4, σO (= TO/Te) = σH (= TH/Te) = 0.2, σsp (= Tsp /Te) = 1, σse (= Tse/Te) = 2, and usp0 = 3.3.

In the text

Fig. 6 Profile of the oblique solitary pulse with its corresponding bipolar electric pulse and the FFT power spectra. (a) Solitary pulse electrostatic potential expressed in volts (V) in dependence on the spatial width in Km. (b) Associated electric pulse for different values of the magnetic field. (c) Corresponding FFT spectra of the electric waveforms, where the x-axis represents log10v, where v is the frequency in Hz, and the y-axis signifies the power of the electric pulse expressed in decibels $\text{dB}(mV/v\sqrt{(Hz)})$.

In the text

Fig. 7 Profiles of quasi-parallel DL, where (a) the DL profiles in dependence on α(= nO0/nH0), where β(= nsp0/nH0) = χ(= nse0 / nH0) = 1.4. (b) The DL profiles in dependence on β, where α = 3.5 and χ = 1.4. (c) the DL profiles in dependence on χ, where α = 3.5 and β = 1.4. The other plasma parameters are QO (= mH/mO) = 1/16, Qsp(= mH/msp) = 1, σO (= TO/Te) = σH (= TH/Te) = 0.15, σsp (= Tsp/Te) = 1.5, σse (= Tse/Te) = 2, usp0 = 3.9, and B0 = 120 × 10−5 G (120 nT).

In the text

	Fig. 8 Profiles of quasi-parallel DL, where (a) the DL profiles in dependence on σsp(= Tsp/Te), where σse(= Tse/Te) = 2. (b) The DL profiles in dependence on σse, where σsp = 1.5. The other plasma parameters are Qo(= mH/mO) = 1/16, Qsp(= mH/msp) = 1, a(= no0/nH0) = 3.5,/β(= nsp0/nH0) = χ(= nse0/nH0) = 1.4, σo(= To/Te) = σH(= TH/Te) = 0.15, usp0 = 2.6, and B0 = 120 × 10−5 G (120 nT).
In the text

	Fig. 9 Profile of quasi-parallel DL in dependence on usp0 = 3.3, where other plasma parameters are QO(= mH/mo) = 1/16, Qsp(= mH/msp) = 1, α(= nO0/nH0) = 3.5, β(= nsp0/nH0) = χ(= nse0/nH0) = 1.4, σO (= TO /Te) = σH(= TH/Te) = 0.15, σsp(= Tsp/Te) = 1, σse(= Tse/Te) = 2, and B0 = 120 × 10−5 G (120 nT).
In the text

	Fig. 10 Profile of quasi-parallel DL in dependence on B0, where B0 = 85 × 10−5 G (85 nT) (orange), B0 = 100 × 10−5 G (100 nT) (blue), B0 = 120 × 10−5 G (120 nT) (purple). The other plasma parameters are QO(= mH/mO) = 1/16, Qsp(= mH/msp) = 1, α(= no0/nH0) = 3.5,β(= nsp0/nH0) = χ(= nse0/nH0) = 1.4, σO(= TO/Te) = σH(= TH/Te) = 0.2, σsp(= Tsp/Te) = 1.5, σse(= Tse/Te) = 2, and usp0 = 3.
In the text

Fig. 11 Profile of the quasi-parallel DL with its corresponding electric pulse and the FFT power spectra. (a) DL electrostatic potential expressed in volts (V) in dependence on the spatial width in m. (b) Associated electric pulse for different values of the magnetic field. (c) Corresponding FFT spectra of the electric waveforms, where the x-axis represents log10v, where v is the frequency in Hz, and the y-axis signifies the power of the electric pulse, expressed in decibels dB$(mV/v\sqrt{(Hz)})$.

In the text