Wave excitation by energetic ring-distributed electron beams in the solar corona

TL;DR

This study uses kinetic simulations to investigate wave excitation by energetic ring-distributed electron beams in the solar corona, revealing conditions for electromagnetic wave escape and implications for solar radio burst observations.

Contribution

It provides the first detailed analysis of wave excitation by ring-beam electrons in the solar corona using self-consistent kinetic simulations, highlighting the role of plasma magnetization and electron density ratios.

Findings

Electrostatic waves dominate wave energetics in the corona.

Escaping electromagnetic waves are effectively excited in strongly magnetized plasmas.

Wave polarization varies with ring-beam electron density, explaining solar radio burst properties.

Abstract

We analyzed properties of waves excited by mildly relativistic electron beams propagating along magnetic field with a ring-shape perpendicular momentum distribution in neutral and current-free solar coronal plasmas. These plasmas are subject to both the beam and the electron cyclotron maser (ECM) instabilities driven by the positive momentum gradient of the ring-beam electron distribution in the directions parallel and perpendicular to the ambient magnetic field, respectively. To explore the related kinetic processes self-consistently, 2.5-dimensional fully kinetic particle-in-cell (PIC) simulations were carried out. To quantify excited wave properties in different coronal conditions, we investigated the dependence of their energy and polarization on the ring-beam electron density and magnetic field. In general, electrostatic waves dominate the energetics of waves and nonlinear waves are ubiquitous. In weakly magnetized plasmas, where the electron cyclotron frequency $\omega_{ce}$ is lower than the electron plasma frequency $\omega_{pe}$, it is difficult to produce escaping electromagnetic waves with frequency $\omega > \omega_{pe}$ and small refractive index $|c k / \omega| < 1$ ($k$ and $c$ are the wavenumber and the light speed, respectively). Highly polarized and anisotropic escaping electromagnetic waves can, however, be effectively excited in strongly magnetized plasmas with $\omega_{ce}/\omega_{pe} \geq 1$. The anisotropy of the energy, circular polarization degree (CPD), and spectrogram of these escaping electromagnetic waves strongly depend on the number density ratio of the ring-beam electrons to the background electrons. In particular, their CPDs can vary from left-handed to right-handed with the decrease of the ring-beam density, which may explain some observed properties of solar radio bursts (e.g., radio spikes) from the solar corona.

Equations

$$F_{rb}(u_{\parallel },u_{\perp })=F_{rb\parallel }(u_{\parallel })F_{rb\perp }(u_{\perp })$$

$$F_{rb\parallel }(u_{\parallel })=\frac{1}{\sqrt{2\pi }{u}_{th\parallel }}\exp \left[-\frac{(u_{\parallel }-u_{rb\parallel }{)}^2}{2u_{th\parallel }^2}\right]$$

$$F_{rb\perp }(u_{\perp })=\frac{1}{2\pi u_{th\perp }^2{Q}_{\perp }}\exp \left[-\frac{(u_{\perp }-u_{rb\perp }{)}^2}{2u_{th\perp }^2}\right]$$

$$Q_{\perp }=\exp \left[-\frac{u_{rb\perp }^2}{2u_{th\perp }^2}\right]+\sqrt{\frac{\pi }{2}}\frac{u_{rb\perp }}{u_{th\perp }}\mathrm{erfc}\left[-\frac{u_{rb\perp }}{\sqrt{2}{u}_{th\perp }}\right]$$

$$I_M(\vec{k},\omega )=\sum _m{\left|A_{m,M}(\vec{k},\omega )\right|}^2$$

$$A_{m,M}(\vec{k},\omega )=A_m(\vec{k},\omega )\cdot {\left\{\frac{1}{\sqrt{2\pi }\sigma }\exp \left[-\frac{{\left(\omega -{\omega }_{Mcold}(\vec{k})\right)}^2}{2{\sigma }^2}\right]\right\}}^{1/2}$$

$${\epsilon }_M(t)=\sum _m\sum _{\vec{k}}{\left|{\mathrm{IFFT}}_{\omega }\left[A_{m,M}(\vec{k},\omega )\right]\right|}^2\Delta \vec{k}$$

$${\epsilon }_M(\theta )=\sum _{\omega }\sum _{\vec{k}}{I}_M(\vec{k},\omega )\delta (k_{\parallel }/k-\cos \theta )\Delta \vec{k}\Delta \omega$$

$$\vec{e_{p}}=\left(\begin{array}[]{lr}\vec{e_{1}}=\vec{k}/|k|\\ \vec{e_{2}}=\vec{e_{3}}\times\vec{e_{1}}\\ \vec{e_{3}}=\vec{e_{z}}\end{array}\right)$$

$$\begin{array}[]{c}\vec{e_{l}}=\left\{\begin{array}[]{lr}(\vec{e_{2}}+i\vec{e_{3}})/\sqrt{2}\qquad[\omega\cdot k_{\parallel}>0\quad or\quad(k_{\parallel}=0\quad and\quad\omega\cdot k_{\perp}>0)]\\ (\vec{e_{2}}-i\vec{e_{3}})/\sqrt{2}\qquad[\omega\cdot k_{\parallel}<0\quad or\quad(k_{\parallel}=0\quad and\quad\omega\cdot k_{\perp}<0)]\end{array}\right.\\ \vec{e_{r}}=\left\{\begin{array}[]{lr}(\vec{e_{2}}-i\vec{e_{3}})/\sqrt{2}\qquad[\omega\cdot k_{\parallel}>0\quad or\quad(k_{\parallel}=0\quad and\quad\omega\cdot k_{\perp}>0)]\\ (\vec{e_{2}}+i\vec{e_{3}})/\sqrt{2}\qquad[\omega\cdot k_{\parallel}<0\quad or\quad(k_{\parallel}=0\quad and\quad\omega\cdot k_{\perp}<0)]\end{array}\right.\\ E_{l}=\vec{E}(\vec{k},\omega)\cdot\vec{e_{l}}\qquad\qquad E_{r}=\vec{E}(\vec{k},\omega)\cdot\vec{e_{r}}\end{array}$$

$$P=\frac{|E_r{|}^2-|E_l{|}^2}{|E_r{|}^2+|E_l{|}^2}$$

$$P=\frac{⟨|E_r{|}^2⟩-⟨|E_l{|}^2⟩}{⟨|E_r{|}^2⟩+⟨|E_l{|}^2⟩}$$

$$K^L(\omega ,\vec{k})=1-{\Sigma }_s\frac{q^2{n}_s}{{\epsilon }_0{\gamma }_s{m}_s{\omega }^2}\left[1+\frac{2\vec{k}\cdot \overrightarrow{v_{ds}}}{\omega -\vec{k}\cdot \overrightarrow{v_{ds}}}+\frac{1-{\omega }^2/(k^2{c}^2)}{(\omega -\vec{k}\cdot \overrightarrow{v_{ds}}{)}^2}(\vec{k}\cdot \overrightarrow{v_{ds}}{)}^2\right]=0$$

Full text

Beam instabilities — Electron cyclotron maser instability — Magnetic reconnection — Particle in cell simulations — Solar corona — Electromagnetic waves — Solar radio bursts

††journal: ApJ

1 Introduction

The solar corona consists of a very dynamic, hot and dilute magnetized plasma in which eruptive energy and mass releases take place, such as solar flares and coronal mass ejections (CMEs). In the course of these solar activity, energetic particles can be accelerated by magnetic reconnection (Zhou et al., 2015, 2016; Muñoz & Büchner, 2016; Muñoz & Büchner, 2018a), shocks (Aschwanden, 2002; Benz, 2008; Chen et al., 2015), as well as turbulence (Petrosian & Liu, 2004; Fletcher & Hudson, 2008; Vlahos & Cargill, 2009). These energetic particles can produce electromagnetic emissions from radio to $\gamma$-ray wavelengths. Of particular interests are solar radio bursts (SRBs) characterized by (a) high brightness temperatures, (b) short, eruptive time scales, (c) narrow frequency bands and (d) strong polarization. These characteristics indicate that the SRBs are likely due to coherent emissions of plasma waves caused by plasma instabilities in the solar corona plasma (Melrose, 2017). Therefore, SRBs carry rich information of plasma dynamics in the solar corona and they may be used to remotely study the related plasma processes.

SRBs can be classified into many types by their distinctive structures in the dynamical spectrum (or spectrogram, Wild et al., 1963). Among all types of the SRBs, spikes immediately attracted the attention of researchers since their first detections (Elgarøy, 1961; Droege & Riemann, 1961; De Groot, 1962) due to their particular observed properties: extremely short duration (down to and probably even less than a few milliseconds, limited by the time resolution of radio telescopes), narrow bandwidth ($<1\%$), and mostly X-mode dominated high degree of polarization (can be $\approx 100\%$, see, e.g., Fleishman & Mel’nikov, 1998 for a review and references therein). Solar radios spikes are closely related to particle acceleration and primary energy release processes in solar flares. They might provide direct information on the finest structure of these energy release processes (Benz, 1985, 1994). To deduce energy release information of solar flares from observations of solar radio spikes, one first needs a reliable generation mechanism for these solar radio spikes. Since right-handed polarized X-mode waves are predominant in most cases of solar radio spikes, electron cyclotron maser (ECM) emission (Twiss, 1958; Schneider, 1959; Gaponov, 1959) has been widely accepted as the most likely coherent emission mechanism for their generation (Dulk, 1985; Vlahos, 1987; Vlahos & Sprangle, 1987; Melrose, 1994; Fleishman & Mel’nikov, 1998).

The so called ECM mechanism, proposed by Twiss, 1958, is a consequence of a linear ECM instability, where electromagnetic waves absorb the energy of energetic electrons, i.e., negative absorption of waves by energetic electrons, via wave-particle interactions (see Melrose, 2017, Chap. 3.2). Besides solar radio spikes, the ECM mechanism was also applied to the generation of the Earth’s auroral kilometric radiation (AKR, Wu & Lee, 1979; Lee & Wu, 1980; Lee et al., 1980; Strangeway et al., 2001) and Jupiter’s decametric emission (DAM, Goldreich & Julian, 1969). For the ECM mechanism to operate on remote radio emissions, first the electron cyclotron frequency $\omega_{ce}$ needs to be greater than the plasma frequency $\omega_{pe}$ in the generation sites of radio emissions, since strong wave excitations by the ECM instability are mainly located around $\omega_{ce}$ and waves with frequencies below local $\omega_{pe}$ cannot escape from a plasma directly, i.e., the escape condition (Melrose, 2017). The condition $\omega_{ce}>\omega_{pe}$, however, implies high local Alfvén velocities $\sim 0.02c$ (Wu et al., 2014), which cannot be easily satisfied within the standard model of the solar atmosphere (Wild, 1985; Wu, 2012, 2014; Wu et al., 2014). Wu et al., 2014; Chen et al., 2017 suggested that the condition $\omega_{ce}>\omega_{pe}$ can be fulfilled if local density cavities form, e.g., due to fluctuations in the ubiquitous Alfvénic turbulence. Such density cavities are, indeed, found recently along the path of the electron beam propagating parallel to the low-density separatrices of strong-guide-field magnetic reconnection via 3D fully kinetic particle-in-cell (PIC) simulations (Drake et al., 2003; Pritchett & Coroniti, 2004; Muñoz & Büchner, 2018b). Observation by Régnier, 2015; Morosan et al., 2016 also demonstrated that the condition of $\omega_{ce}>\omega_{pe}$ can be satisfied within some areas of the solar corona, such as the core of a large active region.

Furthermore, to trigger the ECM emission (i.e., the ECM instability), a positive gradient is required in the electron momentum distribution perpendicular to the ambient magnetic field, i.e., $\partial f/\partial u_{\perp}>0$, where $f$ is the electron momentum distribution and $u_{\perp}$ is the perpendicular momentum of electrons. This property, called population inversion, drives the maser instability. Possible momentum distributions with $\partial f/\partial u_{\perp}>0$ include ring distributions (Pritchett, 1984; Vandas & Hellinger, 2015), loss-cone distributions (Wu & Lee, 1979; Tsang, 1984) or horseshoe distributions (Melrose & Wheatland, 2016; Pritchett et al., 1999).

$\partial f/\partial u_{\perp}>0$ was obtained by considering particle acceleration in the outflow region of magnetic reconnection, where cup-like momentum distributions are found (Büchner & Kuska, 1996). Energetic particles’ magnetic gradient drifts can also cause a redistribution of the energy of parallel flowing beam particles to the perpendicular direction (Zhou et al., 2015), forming ring and gyro-phase restricted as well as crescent-shaped momentum distributions in the perpendicular direction (Voitcu & Echim, 2012, 2018). Vlahos & Sprangle, 1987; Vlahos, 1987 mentioned that interaction of quasi-perpendicular shocks with the ambient solar coronal plasma might lead to formation of a ring momentum distribution in the direction perpendicular to the ambient magnetic field. A quasi-perpendicular shock related SRB event during a solar flare was reported by Chen et al., 2015. Additionally, by means of fully kinetic PIC simulations, it has been proved that ring momentum distribution in the direction perpendicular to the ambient magnetic field can indeed be produced during magnetic reconnection (Shuster et al., 2014; Bessho et al., 2014; Shuster et al., 2015). Moreover electron holes in the electron exhaust regions at the X-points of magnetic reconnection could also provide $\partial f/\partial u_{\perp}>0$ for the ECM emissions (Treumann et al., 2011, 2012; Treumann & Baumjohann, 2017). Note that gradients in the parallel direction $\partial f/\partial u_{\parallel}$ can also drive ECM emissions. This requires, however, extremely anisotropic electron momentum distributions, e.g., $(\Delta u_{\perp}/c)^{2}\geq\Delta u_{\parallel}/c$ for a bi-Maxwellian electron momentum distribution, where $\Delta u_{\perp}$, $\Delta u_{\parallel}$ and $c$ are the perpendicular, parallel thermal momenta of electrons and the speed of light, respectively (Melrose, 1973, 2017). There is, however, no observational evidence for the existence of such strong anisotropy in the solar corona.

On the other hand, energetic electrons always follow a beam momentum distribution in the direction along the coronal magnetic field based on some high-energy phenomena in the solar corona (Cairns et al., 2018; Chen et al., 2015, 2018), e.g., Type @slowromancapiii@ SRBs, hard X-ray bursts, solar energetic particle (SEP) events. 3D fully kinetic PIC and test particle simulations have also shown that strongly energized electron beams can be generated by guide-field magnetic reconnection (Büchner et al., 2018; Muñoz & Büchner, 2018a; Zhou et al., 2016). The beam momentum distribution is unstable to the beam instability driven by free energies from electrons with a momentum distribution $f$ containing $u_{\parallel}\cdot\partial f/\partial u_{\parallel}>0$ (see Melrose, 1986; Gary, 1993). The classical theory of plasma emission, suggested by Ginzburg & Zhelezniakov, 1958, is based on this beam instability.

The plasma emission mechanism contains nonlinear three-wave interaction processes. The theory starts with the excitation of electrostatic Langmuir waves ($L$) via the beam instability. Then backward-directed Langmuir ($L^{\prime}$) waves can be generated via the electrostatic decay or induced backscattering of forward-directed Langmuir waves by fluctuations of ions ($L\rightarrow L^{\prime}\pm S$, where $S$ represents ion-acoustic wave, see Umeda, 2010). Electromagnetic decay or coalescence of $L$ and $S$ waves will lead to the fundamental electromagnetic emission ($T_{\omega_{pe}}$) at the electron plasma frequency $\omega_{pe}$ ($L\rightarrow T_{\omega_{pe}}\pm S$). While the second harmonic electromagnetic emission ($T_{2\omega_{pe}}$) at $2\omega_{pe}$ can be produced by the coupling of $L$ and $L^{\prime}$ waves ($L+L^{\prime}\rightarrow T_{2\omega_{pe}}$) (Karlický & Bárta, 2011; Melrose, 2017; Henri et al., 2019). Generally, the classical plasma emission processes will lead to excitations of the $L$, $L^{\prime}$, $T_{\omega_{pe}}$, $T_{2\omega_{pe}}$ as well as $S$ waves due to the beam instability. Recently, Umeda, 2010 proposed an alternative mechanism for the generation of the $L^{\prime}$ waves with two symmetric counter-propagating electrons beams, where the $L$ and $L^{\prime}$ waves can be directly induced by the forward and backward-propagating electron beams, respectively (Ganse et al., 2012a, b; Thurgood & Tsiklauri, 2015). The plasma emission theory has been widely used to explain the formations of Type @slowromancapi@, @slowromancapii@ and @slowromancapiii@ SRBs (Aschwanden, 2005; Melrose, 2017). In situ spacecraft observations of the interplanetary Type @slowromancapiii@ SRB have also confirmed the plasma emission theory  (Lin et al., 1981; Ergun et al., 1998). By considering many propagation effects (e.g., wave scattering, decreasing magnetic field strength, interplanetary shocks) during the transportation of energetic electrons from the solar corona to the interplanetary medium (IPM), momentum distributions of the energetic electrons in the solar corona should be quite different from those in the IPM.

Both the beam and ECM instabilities (driven by free energies in $u_{\parallel}\cdot\partial f/\partial u_{\parallel}>0$ and $\partial f/\partial u_{\perp}>0$ distributions, respectively) have been invoked separately to explain the coherent emission mechanism of different types of SRBs (see, e.g., Aschwanden, 2005; Melrose, 2017, for reviews of SRBs). For a more general application to the microscopic emission processes in plasmas, here we will generalize these two distributions and characterize the properties of emission processes due to both instabilities. On the other hand, based on the above mentioned theoretical studies and numerical simulations, both free energies with population inversion $u_{\parallel}\cdot\partial f/\partial u_{\parallel}>0$ and $\partial f/\partial u_{\perp}>0$ in the electron momentum distribution and density cavity with $\omega_{ce}>\omega_{pe}$ can be realized simultaneously in the dynamically evolving fast solar magnetic reconnection events in the solar corona.

In this paper, via 2.5D fully kinetic PIC simulations, we investigate properties of waves excited by mildly relativistic ring-beam electrons in neutral and current-free solar coronal plasmas. In this system, the ring-beam electrons together with protons and background electrons support the global charge and current neutralities, respectively. Many theories and observations have proved that the majority of the current induced by energetic beam electrons can be rapidly compensated by the return current of drifting background electron, e.g., Brown & Bingham, 1984; van den Oord, 1990; Melrose, 1990; Khodachenko et al., 2009. Note that non-zero net current in plasmas can not only introduce a current instability (see Melrose, 1986; Matsumoto & Omura, 1993; Wu et al., 2014; Chen et al., 2017) but also generate strong magnetic field and oscillations leading to a very complex plasma system (Henri et al., 2019).

Some parametric dependence of the wave excitations resulting from the electron ring-beam momentum distribution have been investigated by Lee et al., 2011 utilizing 2.5D fully kinetic PIC simulations. In particular, these authors explored the influence of the average kinetic energy and pitch angle of the ring-beam electrons on the wave excitations, keeping the density ratio of ring-beam and background electrons fixed ($n_{rb}/n_{bg}=1:19$) as well as the frequency ratio $\omega_{ce}/\omega_{pe}=5$. In order to derive properties of waves generated by energetic ring-beam electrons at different locations along the beam trajectory in the solar corona, we utilize a 2.5D version of the fully kinetic PIC code ACRONYM to explore the dependence of the nonlinear wave generation and saturation by energetic ring-beam electrons on the frequency ratio of $\omega_{ce}/\omega_{pe}$ and the density ratio of the ring-beam $n_{rb}$ to background $n_{bg}$ electrons with the average kinetic energy and pitch angle of the ring-beam electrons fixed.

Compared with previous studies (e.g., Pritchett, 1984; Lee et al., 2009, 2011), we have developed precise diagnostics to investigate the nonlinear evolution, saturation, and anisotropy of different electromagnetic wave modes guided by the dispersion relations of a magnetized cold plasma (see Sect.2.2.1). Since only electromagnetic waves with frequency $\omega>\omega_{pe}$ and small refractive index $|ck/\omega|<1$ can escape from their generation sites and might be detected by remote detectors, polarization, spectrogram and anisotropies of these escaping electromagnetic waves are explored to compare with the ground-based observations of solar radio spikes (see Sect.2.2.2).

This paper is organized as follows: after the introduction, we present the numerical simulation model in Sect.2. Sect.3 contains the key results of this study and in Sect.4 we draw our conclusions and discuss the application of our results.

2 Numerical Simulation

2.1 Model and Setup

Since the excitation and growth of waves driven by plasma instabilities are, in general, kinetic and nonlinear processes, self-consistent kinetic simulations are required to investigate it. In these simulations, particle interactions through the electromagnetic fields and effects of particles’ motions on the electromagnetic fields as well as nonlinear wave-wave, wave-particle interactions can be correctly incorporated. Our tool of choice is a fully kinetic PIC code, which can model all those processes from a first-principle approach. In a fully kinetic PIC code, generally, the electromagnetic fields are calculated from the Maxwell’s equations with the charge and current densities by knowing the positions and velocities of all particles. And then the particles move due to these electromagnetic fields from the Newton-Lorentz equation of motion and the new positions and velocities of particles lead to a new state of the electromagnetic fields. This step is equivalent to solve the effective Vlasov equation for the (numerical) particles. These steps are repeated until the end of a simulation. Therefore, the fully kinetic PIC algorithm solves the full set of the Vlasov-Maxwell equations, (see, e.g., Birdsall & Langdon, 1991; Tskhakaya et al., 2007; Lapenta, 2012; Vay & Godfrey, 2014, for reviews of the basic theories and applications of fully kinetic PIC codes).

We performed this study with the fully kinetic PIC code — ACRONYM (http://plasma.nerd2nerd.org/, Kilian et al., 2017), a fully relativistic electromagnetic code tuned for the study of kinetic-scale plasma wave phenomena and interactions in collisionless plasmas in a wide variety of physical environments (see, e.g., Ganse et al., 2012a; Kempf et al., 2016; Muñoz & Büchner, 2016; Schreiner et al., 2017; Muñoz & Büchner, 2018b; Büchner et al., 2018). We use its version in two spatial dimensions and three dimensions (i.e., 2.5D) in momentum and components of the electromagnetic fields.

The 2D simulation box contains $1024\times 1024$ grid points in the $x-y$ plane. Periodic boundaries are applied in both directions for both fields and particles. In our simulations, to keep the global charge neutrality, three species of particles are employed, one for the mildly relativistic ring-beam electrons and other two species for the protons and the background electrons. For physically realistic results, the proton-to-electron mass ratio has been chosen as the physical $m_{p}/m_{e}=1836$. Initially, 2000 particles per cell are implemented to reduce the numerical noise (Hockney, 1971; Dawson, 1983; Birdsall & Langdon, 1991). These particles are homogeneously distributed in the whole simulation domain with a constant ambient magnetic field $\vec{B_{0}}=B_{0}\vec{x}$ along the x-axis, since typical domain sizes modelled by fully kinetic PIC simulations are much smaller than the typical length scale of the density gradient in the solar corona.

The initial momentum distributions of all particles (both electrons and protons) are characterized in terms of momentum per unit mass, $\vec{u}=\gamma\vec{v}$, where $\gamma=1/\sqrt{1-v^{2}/c^{2}}=\sqrt{1+u^{2}/c^{2}}$. Note that, hereafter, we will simply call ”momentum per unit mass” as ”momentum”. Correspondingly, the momentum distribution for the mildly relativistic ring-beam electrons is (Umeda et al., 2007; Lee et al., 2011; Kainer & MacDowall, 1996):

[TABLE]

where $u_{\parallel},u_{\perp}$ are the particle momenta along and perpendicular to the ambient magnetic field $\vec{B_{0}}$, respectively. ($u_{rb\parallel}$, $u_{rb\perp}$) and ($u_{th\parallel}$, $u_{th\perp}$) are their corresponding bulk drift and thermal momenta ($v_{th\parallel}=u_{th\parallel}/\gamma_{th}=\sqrt{k_{B}T_{e,\parallel}/m_{e}}$ and $v_{th\perp}=u_{th\perp}/\gamma_{th}=\sqrt{k_{B}T_{e,\perp}/m_{e}}$, where $T_{e,\parallel}$ ($T_{e,\perp}$) is the parallel (perpendicular) electron temperature, $k_{B}$ is the Boltzmann’s constant and $\gamma_{th}=\sqrt{1+(u_{th\parallel}^{2}+u_{th\perp}^{2})/c^{2}}$). According to our simulations, the parallel and perpendicular directions are along the x-axis and y-axis, respectively. Considering the typical velocity of energetic beam electrons (Wild et al., 1959; Alvarez & Haddock, 1973; Suzuki & Dulk, 1985; Reid & Ratcliffe, 2014) and typical temperature in the solar corona, initially we take $\gamma=\sqrt{1+(u_{rb\parallel}^{2}+u_{rb\perp}^{2})/c^{2}}=1.2$ ($\sim 100~{}keV$ and $\sqrt{u_{rb\parallel}^{2}+u_{rb\perp}^{2}}=0.67~{}c$) as the average initial kinetic energy of the ring-beam electrons. Their averaged pitch angle is $\phi_{0}=\tan^{-1}(u_{rb\perp}/u_{rb\parallel})=30^{\circ}$, which indicates that the ring-beam electrons have more energies in the $\vec{B_{0}}$ parallel direction. And $u_{th\parallel}=u_{th\perp}=u_{rbth}=u_{th}=0.025c$. $Q_{\perp}$ in Eq.(1) is the normalization constant

[TABLE]

For a current-free system, all the background electrons drift in the opposite direction to that of the ring-beam electrons with a momentum $u_{bg\parallel}=-u_{rb\parallel}n_{rb}/n_{bg}$, here $n_{rb}$ and $n_{bg}$ are the number density of the ring-beam and background electrons, respectively (Karlický & Bárta, 2009; Ganse et al., 2012b).

The background electrons, hence, follow a drifting Maxwellian momentum distribution with a thermal spread $u_{bgth}=0.05c$ along each dimension. The backward-drifting background electrons, hence, also contain free energies $u_{\parallel}\cdot\partial f/\partial u_{\parallel}>0$ for the beam instability. Protons are used for the global charge neutrality. They are assumed to follow an isotropic Maxwellian momentum distribution with the same temperature as the background electrons. Note that protons are quite important for the generation of the ion-acoustic waves in the plasma emission mechanism, so that we let them move freely, i.e., our simulations also solve the equations of motion for the protons, even though they respond to electromagnetic forces at much larger timescales than electrons.

Due to the free energies provided by the ring-beam and backward-drifting background electrons, both the beam and ECM instabilities can be driven due to the positive gradients $u_{\parallel}\cdot\partial f/\partial u_{\parallel}>0$ and $\partial f/\partial u_{\perp}>0$ in the electron momentum distribution, respectively. To distinguish contributions from the beam and ECM instabilities, we also carry out simulations with only either a pure-beam or a pure-ring momentum distribution for the energetic electrons ($u_{rb\perp}=0$ or $u_{rb\parallel}=0$, respectively), while other parameters are the same as these ring-beam simulations.

In this study, all quantities are solved in real (spatial-temporal) space and all simulations have the same spatial and time resolution. In particular, the grid cell size is $\Delta x=\Delta y\simeq\lambda_{De}$, where $\lambda_{De}=u_{th}/\omega_{pe}$ is the electron Debye length and $\omega_{pe}$ is the total electron plasma frequency, i.e., $\omega_{pe}=\sqrt{\omega_{prb}^{2}+\omega_{pbg}^{2}}=\sqrt{4\pi n_{t}e^{2}/m_{e}}$, where $n_{t}=n_{rb}+n_{bg}$ and $e$ are the total electron number density and charge of electrons, respectively. And $\omega_{prb}$ ($\omega_{pbg}$) is the plasma frequency of the ring-beam (background) electrons. The timestep in our simulations is determined by the inherent length and timescale requirements in a fully kinetic PIC code, i.e., the Courant-Friedrichs-Lewy (CFL) condition for the speed of light $c$. Correspondingly, our simulations can cover $|k_{x,y}/(\omega_{pe}/c)|<92.2$ and $|\omega/\omega_{pe}|<12.3$ with resolutions $\Delta k_{x,y}=0.18\omega_{pe}/c$ and $\Delta\omega=0.015\omega_{pe}$, respectively, in the wavevector-frequency ($\vec{k}-\omega$) space.

Variable parameters are $n_{rb}/n_{t}$ (being equal to $5\%$, $10\%$ , $20\%$, $30\%$, $40\%$, $50\%$ with fixed $\omega_{ce}/\omega_{pe}=5$, see Sect.3.1) and $\omega_{ce}/\omega_{pe}$ (being equal to $0.2$, $0.3$, $0.5$, $1$, $2$, $3$ with fixed $n_{rb}/n_{t}=5\%$, see Sect.3.2). Note that simulation with $n_{rb}/n_{t}=5\%$ and $\omega_{ce}/\omega_{pe}=5$ can be compared with the Case B in the study of Lee et al., 2011. The ambient magnetic field $\vec{B_{0}}$ is initialized based on the frequency ratio $\omega_{ce}/\omega_{pe}$. Note that the values of the beam to total density ratio are probably much higher than those thought to exist in the solar corona, but they could be considered appropriate for density cavities, where the background density drops considerably. In addition, fully kinetic PIC simulations of magnetic reconnection tend to generate electron beams, propagating through the low density separatrices, with similar density ratios (Muñoz & Büchner, 2016).

Normalizations used throughout this paper are as follows: $\omega_{norm}=5.0\omega_{pe}$ is the normalization of frequency. Time, momentum and distance are normalized by $1/\omega_{norm}$, $c$ and $c/\omega_{norm}$, respectively. $B_{norm}$ is the normalization of the electric and magnetic field strengths and corresponds to the ambient magnetic field $\vec{B_{0}}$ for $\omega_{ce}/\omega_{pe}=5.0$. Energy is normalized by the magnetic field energy $\varepsilon_{norm}$ corresponding to a homogeneous and uniform $B_{norm}$ in the whole simulation domain.

2.2 Diagnostic Method

2.2.1 Energy in wave modes

Since all quantities in our simulations are given in real (spatial-temporal) space, to characterize the plasma waves, one applies fast Fourier transforms (FFTs) on the electromagnetic fields over the space-time domain of the simulations. Meanwhile, different wave modes are distinguished by their own dispersion relation. To estimate the energy contained by different wave modes, we should consider their dispersion relation. As a simplification, we take the wave dispersion relations in the magnetized cold plasma as an approximation (see, e.g., Andre, 1985; Melrose, 1986; Stix, 1992), despite the criteria for the validity of the cold plasma approximation (see Melrose, 1986; Stupp, 2000) could not be always satisfied in our simulations. For a numerical simulation, different from theoretical studies, these criteria are, however, difficult to adopt since the effective electron temperature in simulated plasmas are quite inhomogeneous and dynamic. Generally, the cold plasma dispersion relation constitutes a good approximation to the full hot plasma dispersion relation in many conditions (Chen et al., 2013).

We also assume that energy spectral density of a wave mode $M$, i.e., $I_{M}(\vec{k},\omega)$ follows a Gaussian frequency distribution around its dispersion surface in the $\vec{k}-\omega$ space (Comişel et al., 2013):

[TABLE]

[TABLE]

where $\omega_{Mcold}(\vec{k})$ denote the frequency of the wave mode $M$ at the given wavevector $\vec{k}$ (dispersion relation) in the magnetized cold plasma approximation. $\sigma$ characterizes the frequency broadening for the wave mode $M$ around its corresponding cold-plasma dispersion relation surface. Here we use $\sigma=0.05\omega_{norm}$for each wave mode as a simplification. And $m$ indicates different components of electromagnetic field. We apply FFTs on the electromagnetic field components $a_{m}(x,y,t)$ over the entire space and time domain of our simulations to get their fluctuations $A_{m}(\vec{k},\omega)=\operatorname{FFT}_{x,y,t}[a_{m}(x,y,t)]$ in $\vec{k}-\omega$ space. And then Gaussian filter is applied on $A_{m}(\vec{k},\omega)$ to get the electromagnetic field component fluctuations of the wave mode $M$, i.e., $A_{m,M}(\vec{k},\omega)$. This method is hereinafter called Gaussian filter method.

For the (temporal) evolution of energy of the wave mode $M$, an inverse fast Fourier transform (IFFT) is implemented on $A_{m,M}(\vec{k},\omega)$ (Eq.4) in the frequency $\omega$ space. An integration over the wavevector $\vec{k}$ space is applied on the IFFT results to get the energy evolution of the the electromagnetic field component $m$, i.e.,

[TABLE]

For the study of the wave energy along different wave propagation directions, we integrate $I_{M}(\vec{k},\omega)$ in the $\vec{k}-\omega$ space only if $\cos\theta=k_{\parallel}/k$ is satisfied, where $\theta$ is the pitch angle between the $\vec{k}$ and $\vec{B_{0}}$ and $k_{\parallel}$ is the $\vec{B_{0}}$ parallel component of the wave vector $\vec{k}$, then the energy of the wave mode M along the direction $\theta$:

[TABLE]

where $\delta(\ast)$ is the Dirac delta function.

2.2.2 Polarization

To get the polarization of waves propagating along each direction, the polarization vector $\vec{e_{p}}$ is defined with respect to the wave propagation vector $\vec{k}$ in the $x-y$ plane (Melrose, 1986; Bittencourt, 2004; Willes & Cairns, 2000):

[TABLE]

where $\vec{e_{z}}=\vec{e_{x}}\times\vec{e_{y}}$ is the unit vector in the direction perpendicular to the $x-y$ plane. To separate the left and right-handed polarized components ($E_{l}(\vec{k},\omega),E_{r}(\vec{k},\omega)$) of the transverse electric fields, i.e., perpendicular components of $\vec{E}(\vec{k},\omega)$ respecting to $\vec{k}$, a circular basis ($\vec{e_{l}},\vec{e_{r}}$) is defined based on the polarization vector $\vec{e_{p}}$ in Eq.(7):

[TABLE]

where $\vec{E}(\vec{k},\omega)$ is the electric field in the $\vec{k}-\omega$ space and it is obtained via the FFT. With the definition of the Eq.(8), the polarization state of a wave also refers to the ambient magnetic field (Stix, 1962; Gary, 1993). Hence, the right and left-handed polarized waves rotate in the same sense as an electron and a proton, respectively, as far as they propagate along (either parallel or antiparallel to) the ambient magnetic field. Also note that wave and its polarization make no sense when $\omega=0$ and/or $|k|=0$. Hence the contribution of $\omega=0$ and/or $|k|=0$ to the polarization is not considered in our calculations.

Following the definition of the Stokes parameters (McMaster, 1954; Carozzi et al., 2001), for a wave at a given time, its circular polarization degree (CPD, $P$) can be calculated as:

[TABLE]

where the vertical bars $|*|$ indicate the amplitude of the respective quantity. In order to determine the CPD, $P$ in a plasma (with many waves) at a given time or a wave over a period, instead of taking the average value of CPDs from different waves, we use:

[TABLE]

where the angle brackets $\langle*\rangle$ indicate the average value of each corresponding quantity. Note that for different studies, averages are calculated in different spaces, i.e., (1) averages over the $\vec{k}$ space are considered for the evolution of the CPD, (2) for CPD along a wave propagation direction $\theta$ respect to $\vec{B_{0}}$, we take averages in both $\omega$ and $k_{\parallel}/k_{\perp}=\cos\theta$ spaces. Definition in Eq.(10), hence, can give us a direct idea that which polarization is energetically dominant. We thus can verify that the polarization is circular with a right- or left-hand sense according to $P>0$ or $P<0$, respectively. A value of $P=1$ $(-1)$ corresponds to fully right- (left-)hand circular polarization and $P=0$ indicates a linear polarization.

In our diagnostics, magnetic fields are used to determine the energies of electromagnetic wave modes. Adopting magnetic fields can automatically filter out electrostatic waves since an electrostatic wave does not contain magnetic fluctuations. For calculations related to polarization, however, electric fields are used. Note that we will not investigate evolutions of the anisotropy and CPD, since the whole time duration of our simulations (dozens of microseconds) are much shorter than the time resolutions of the remote detectors (more than milliseconds).

3 Simulation Results

As mentioned in Sect.2.2.1, the dispersion relation of magnetized cold plasmas will be applied to identify which wave has been excited. There are 5 different wave branches in the cold plasma limit. While each mode branch (or surface) can be differently named for different frequencies and/or wave propagation directions (Andre, 1985). For an example, when waves propagate along the ambient magnetic field, the X mode is usually called (right-handed polarized) R-mode , while the O mode is associated with the (left-handed polarized) L-mode. In this paper, however, we will simply call them as ion-cyclotron, whistler, slow extraordinary (Z), ordinary (O) as well as fast extraordinary (X) modes from the low to high frequencies, respectively. For the applied physical proton-to-electron mass ratio $m_{p}/m_{e}=1836$, the frequencies in the ion-cyclotron branch are marginally resolved in our calculations. In the following, we will ignore the ion-cyclotron branch.

3.1 $n_{rb}/n_{t}$ Dependence

In this section, dependence of excited wave properties on the number density ratio between the ring-beam and total electrons $n_{rb}/n_{t}$ are discussed. While the ratio between the electron cyclotron frequency $\omega_{ce}$ and the electron plasma frequency $\omega_{pe}$ is fixed $\omega_{ce}/\omega_{pe}=5$.

Both beam and ECM instabilities can occur with the ring-beam momentum distribution. Note that many instabilities can fit with the description of the beam instability, since their free energy sources come from the drifting beam population, e.g., the reactive beam instability, kinetic Langmuir beam (or bump-in-tail) instability, whistler heat flux instability, firehose instability, etc (Melrose, 1986; Gary, 1993). With the initial setup in this study (see Sect.2.1), these instabilities may occur at the different stages of the free energy release. We will not distinguish these instabilities and call them simply as the beam instability in this study.

3.1.1 Statistics of particles

The beam and ECM instabilities, in general, are triggered by the electron free energy in the directions along and perpendicular to the ambient magnetic field $\vec{B_{0}}$, respectively. Evolution of the electron momentum along each direction can, hence, give us insights on the growth and saturation of these instabilities. Panels (a) to (d) of Fig.1 show the evolution of the bulk (or average) drift momenta and thermal spreads in the directions along and perpendicular to $\vec{B_{0}}$ for both the ring-beam and background electrons, respectively. The bulk drift momentum and thermal spread of different electron species ($s$) along different directions ($t$) are defined as $u_{d,s,t}=\left(\sum\limits_{i}u_{s,t,i}\right)/N$ and $u_{th,s,t}=\sqrt{\left[\sum\limits_{i}(u_{s,t,i}^{2}-u_{d,s,t}^{2})\right]/N}$, where $s=rb$ or $bg$ for the ring-beam or background electrons and $t=\parallel$ or $\perp$ for the direction along or perpendicular to $\vec{B_{0}}$, respectively. While $u_{s,t,i}$ is the parallel or perpendicular momentum of a single electron $i$ and $N$ is the total electron number in species $s$. The evolution of the perpendicular bulk drift momentum of the background electrons is not shown in panel (b) of Fig.1, since it is negligible compared to that of the ring-beam population. Note that we stopped our simulations when these quantities reach quasi-steady values, i.e., there is no obvious energy exchange between electromagnetic fields and particles.

Panels (a) and (b) of Fig.1 show the evolution of the bulk drift momenta in the parallel and perpendicular directions, respectively. The first minima of these curves indicate the saturation of their corresponding instabilities. One can see that the reduction of the free energy, generally, is faster and larger in the parallel than in the perpendicular direction. The faster free energy reduction in the parallel direction implies that waves excited by the beam instability will saturate earlier than those by the ECM instability. (Note that, in this paper, the saturation of a wave corresponds to the end of the growth phase in its energy evolution profile.)

While the free energy release rates of the beam instability do not vary significantly among cases with dense ring-beam electrons ($n_{rb}/n_{t}\geq 20\%$, panel a of Fig.1), the free energy for the ECM instability decrease faster with the increase of the ring-beam electron population (panel b of Fig.1). Moreover, the greater free energy reduction in the parallel direction implies that waves induced by the beam instability should contain more energy than those due to the ECM instability. This difference becomes larger with the increase of $n_{rb}/n_{t}$.

Panel (a) of Fig.1 shows that both the ring-beam and background electrons simultaneously lose their bulk drift energies along $\vec{B_{0}}$, making contributions to the wave excitation driven by the beam instability. Generally, in both the parallel and antiparallel directions, the release of the electron drift energy increases monotonically with the increase of $n_{rb}/n_{t}$. Evolution of the bulk drift momentum of the ring-beam electrons in the direction perpendicular to $\vec{B_{0}}$ (panel b of Fig.1) is, however, more complicated than the parallel direction, i.e., the decrease of the perpendicular bulk drift momentum in the cases with $n_{rb}/n_{t}=5\%$ and $10\%$ are slower but even more significant than the cases with $n_{rb}/n_{t}\geq 20\%$. This indicates that different dynamic processes are underway between cases with $n_{rb}/n_{t}\leq 10\%$ and $n_{rb}/n_{t}\geq 20\%$, which we will clarify later.

Part of the released energies from the bulk drift motion are, however, absorbed again by electrons themselves via wave-particle interactions, leading to electron heating and acceleration. Electron thermal spread is, hence, strongly enhanced and have opposite behavior to their corresponding bulk drift motion in both parallel and perpendicular directions, see panels (c) and (d) of Fig.1. Especially in the direction along $\vec{B_{0}}$ (panel c), the final thermal spread of the ring-beam electrons already reach relativistic regime ($>0.4c$) in the cases with $n_{rb}/n_{t}>30\%$. For the case with $n_{rb}/n_{t}=50\%$, this is almost equal to its initial parallel drift momentum. In the final quasi-steady state, the thermal spread of the background electrons is, in general, smaller than that of the ring-beam electrons, and the thermal spread of all electrons is much wider in the parallel direction than in the perpendicular direction, which agrees with the distributions of the parallel and perpendicular momenta shown in Fig.2.

Fig.2 shows the evolution of the parallel, perpendicular momentum and energy distributions of all electrons. One can see that when the plasma system is close to its quasi-steady state ($t=1275\omega_{norm}^{-1}$, column f), the initial free energies for the beam ($u_{\parallel}\cdot\partial f/\partial u_{\parallel}>0$, row a) and ECM ($\partial f/\partial u_{\perp}>0$, row b) instabilities are almost totally dissipated and plateau momentum distribution forms in all directions. Meanwhile, with panels in rows (a) and (b), strong electron acceleration can also be seen along each direction, particularly, in the cases with larger $n_{rb}/n_{t}$. Note that the high momentum tail in the antiparallel direction contains reflected ring-beam electrons and reflection of the ring-beam electrons is suppressed in cases with $n_{rb}/n_{t}\leq 10\%$. The reflection of the ring-beam electrons makes the wave generation more symmetric with respect to the plane perpendicular to $\vec{B_{0}}$. Petrosian & Liu, 2004 found that acceleration of particles via resonant wave-particle interactions can be enhanced significantly if particle can resonate with multi-waves simultaneously. Hence reflection of the ring-beam electrons will increase their acceleration efficiency. This may explain the correlation between the ring-beam electron reflection and their acceleration in the perpendicular direction. Strong perpendicular acceleration also leads to the late increase of perpendicular drift momentum in the case with the maximum ring-beam electron density ($n_{rb}/n_{t}=50\%$ in panel b of Fig.1). Interestingly, in each $n_{rb}/n_{t}$ case, a double power-law distribution forms in the high energy tail with $\gamma-1>0.1\sim 50~{}keV$ when the plasma system is close to its quasi-steady state, i.e., after the release of the free energy for both the beam and ECM instabilities (see the right-bottom panel c-f). These break energies are located around the initial energy ($\gamma\sim 1.2$) of the ring-beam electrons.

3.1.2 Excited electrostatic waves

Based on the coordinates of our simulations, electric component $E_{z}$ is purely transverse, while the character of the $E_{x}$ and $E_{y}$ components change with wave propagation direction, i.e., $E_{x}$ is a purely longitudinal (transverse) component when waves propagate along (perpendicular to) the ambient magnetic field $\vec{B_{0}}$, i.e., $\vec{k}\parallel\vec{B_{0}}$ and $\theta=0^{\circ}$ or $180^{\circ}$ ($\vec{k}\perp\vec{B_{0}}$ and $\theta=90^{\circ}$ or $270^{\circ}$). But, in general, $E_{x}$ and $E_{y}$ represent a mixture of both longitudinal and transverse electric field components. Note that $\theta\leq 90^{\circ}$ together with the sign of $\pm k$ gives two supplementary wave propagation directions in the wave $\vec{k}-\omega$ (or dispersion) spectra of Figs.3, 4, 7 and 10.

Hence, in row (a) of Fig.3, one can mainly find excited electrostatic modes, i.e., Langmuir and (electron) beam modes. Hereinafter, we define a wave mode being excited if its spectral intensity is significantly higher than that of an isotropic equilibrium Maxwellian plasma, which has the same thermal spread and $\omega_{pe}$ as the background and total electrons in the ring-beam simulations (see Sect.2.1), respectively, corresponding to $n_{rb}/n_{t}=0\%$ in Figs.5 and 6. Since the ECM instability, in general, mainly excites electromagnetic modes, the excitation of the electrostatic Langmuir and beam modes should be mostly due to the beam instability.

Similar to Karlický & Bárta, 2009; Ganse et al., 2012b, antiparallel-propagating Langmuir waves ($L^{\prime}$) are also excited in all ring-beam plasmas (see left-half panels in row a of Fig.3, where $\theta=0^{\circ}$ and $k<0$). As mentioned in Sect.1, generation of the $L^{\prime}$ waves could be attributed to electrostatic decay of the parallel-propagating Langmuir waves ($L\rightarrow L^{\prime}+S$) and/or free energies for the beam instability in the antiparallel-drifting background electrons. In our simulations, we indeed found both intensity enhancement in the ion density fluctuation spectra for plasmas with dense ring-beam electrons $n_{rb}/n_{t}\geq 20\%$ (similar to Fig.4 of Thurgood & Tsiklauri, 2015) as well as reduction of the free energy in the antiparallel-drifting background electrons (see row a of Fig.2).

Furthermore, one can see the rise of these excited $L^{\prime}$ branch toward larger $\omega$ at a given wavenumber $k<0$ with the increase of $n_{rb}/n_{t}$, which agrees with a higher effective electron temperature of the antiparallel-moving electrons in plasmas with a larger $n_{rb}/n_{t}$ (panel c of Fig.1), since higher electron temperature will lead to a larger slope ($d\omega/dk$) in the dispersion relation of the Langmuir wave, i.e., $\omega^{2}=\omega_{pe}^{2}+3k^{2}v_{the}^{2}$, where $v_{the}\propto\sqrt{T_{e}}$ is the effective thermal velocity of electrons. Correspondingly, excitation of the $L^{\prime}$ waves appears at increasingly smaller wavenumbers $|k|$. That could be due to electron Landau damping (Landau, 1946; Tsurutani & Lakhina, 1997) of longitudinal electric fluctuations with large $|k|$ in hot plasmas, where Langmuir waves will get damped when their $|k|$ become larger than $1/\lambda_{De}$ ($\lambda_{De}\propto\sqrt{T_{e}}$ and $T_{e}$ is the effective electron temperature).

In the direction antiparallel to $\vec{B_{0}}$, except for the $L^{\prime}$ wave, intensity of the beam mode $\omega=kv_{b}<\omega_{pe}$ is also enhanced. The typical drift velocity $v_{b}$ of these excited antiparallel-propagating beam modes also increases with the increase of $n_{rb}/n_{t}$, since initially we have $u_{bg\parallel}=-u_{rb\parallel}n_{rb}/n_{bg}$, i.e., the initial bulk drift momentum of the background electrons $u_{bg\parallel}$ increases with the increase of $n_{rb}/n_{t}$ (see dashed lines at $t=0$ in panel a of Fig.1).

In the direction parallel to $\vec{B_{0}}$ ($\theta=0^{\circ}$ with $k>0$, right-half panels in row a of Fig.3), similar to the conditions in the antiparallel direction, enhanced intensity of both the parallel-propagating Langmuir ($L$) and beam modes can be found in each $n_{rb}/n_{t}$ case. And also, due to the Landau damping in hot plasmas, these excited $L$ waves are confined to smaller $|k|$ with the increase of $n_{rb}/n_{t}$. Landau damping of small scale longitudinal electric fluctuations will lead to electron heating discussed in Sect.3.1.1. While different from those antiparallel-propagating beam modes, the typical drift velocity $v_{b}$ of the excited parallel-propagating beam modes, however, decrease with the increase of $n_{rb}/n_{t}$ due to the stronger reduction of the parallel bulk drift energy of the ring-beam electrons in cases with larger $n_{rb}/n_{t}$ (see solid lines in panel a of Fig.1).

By comparing these electrostatic waves in the $\theta=0^{\circ}$ and $180^{\circ}$ directions (row a of Fig.3), one can see that intensity of the parallel-propagating Langmuir and beam modes are, generally, stronger than the antiparallel-propagating ones for each $n_{rb}/n_{t}$ case. This difference is, however, reduced with the increase of $n_{rb}/n_{t}$, since the free energy for the beam instability from the antiparallel-moving background electrons becomes comparable to that from the parallel-moving ring-beam electrons with the increase of $n_{rb}/n_{t}$. Moreover, for high ring-beam density cases with $n_{rb}/n_{t}\geq 20\%$, a significant fraction of the ring-beam electrons can be reflected, which also makes the wave excitation more symmetric with respect to the perpendicular plane. Additionally, besides linear waves as indicated by the cold plasma dispersion relations, diffusive nonlinear electrostatic waves are also excited.

Panel (b) of Fig.3 shows energy evolutions of the total electric fields ($\sum\limits_{x,y}\left[\vec{E}(x,y)\right]^{2}\Delta x\Delta y$) and total longitudinal electric fields ($\sum\limits_{\vec{k}}\left[(\vec{E}(\vec{k})\cdot\vec{k})/|k|\right]^{2}\Delta\vec{k}$) of all waves in the simulation domain, where $\vec{E}(\vec{k})$ is the electric field vector of waves with wavevector $\vec{k}$. Note that the total longitudinal electric fields contain not only electric fields of electrostatic waves but also the longitudinal electric component of the electromagnetic waves. One can see that, for each $n_{rb}/n_{t}$ case, the total longitudinal electric component occupies most of the total electric field energy (their energy evolution profiles are almost overlapped with each other): it is over one order of magnitude larger than the transverse electric field energy, which is shown in panel (e) of Fig.7. Similar result had also been found by Lee et al., 2009.

Energy evolutions for the electric component $E_{x}(x,y)$ ($\sum\limits_{x,y}\left[E_{x}(x,y)\right]^{2}\Delta x\Delta y$) and the longitudinal electric fields of waves propagating along $\vec{B_{0}}$ ($\sum\limits_{k_{\perp}=0}\left[\left(\vec{E}(\vec{k})\cdot\vec{k}\right)/|k|\right]^{2}$, i.e., waves shown in row a of Fig.3) are presented in panel (c) of Fig.3. One can see that the energy of $E_{x}(x,y)$ is comparable to the total longitudinal electric fields. And the energy of the $\vec{B_{0}}$ aligned longitudinal electric fields is a factor of a few lower except in the early phase of these simulations, when the energy of the total longitudinal electric fields appears to be isotropic (especially at $t=0$) and dominated by nonparallel-propagating waves. To validate our simulations, insert of panel (c), we compare the growth rate of these $\vec{B_{0}}$ aligned longitudinal electric fields to the maximum growth rate of electrostatic waves by the beam instability in the nonresonant fluid or reactive regime of unmagnetized plasmas (i.e., in cold unmagnetized plasmas). These two growth rates should agree with each other, since the $\vec{B_{0}}$ aligned longitudinal electric fields are dominated by electrostatic waves (see row a of Fig.3) and these electrostatic waves are mainly excited by the beam instability. Moreover in magnetized plasmas, excitation of the electrostatic waves propagating along the ambient magnetic field is exactly the same as that in unmagnetized plasmas (see Gary, 1993, Chap. 3.3). Additionally, the setup of our simulations are also located in the reactive regime with $(n_{rb}/n_{t})^{1/3}(u_{rb\parallel}/u_{th\parallel})\gg 1$, where $(n_{rb}/n_{t})^{1/3}(u_{rb\parallel}/u_{th\parallel})$ is a measure of the reactive ($\geq 1$) and kinetic ($<1$) nature of the beam instability (Melrose, 1986; Gary, 1993; Melrose, 2017).

The maximum growth rate of the electrostatic waves due to the reactive beam instability was obtained from the dispersion equation for unmagnetized plasmas by setting the longitudinal dielectric element $K^{L}(\omega,\vec{k})$ to be zero. In the unmagnetized cold plasma limit ($u_{rb\parallel}\gg u_{th\parallel}$, Eq.2.16 in Melrose, 1986), that is :

[TABLE]

where $\Sigma_{s}$ is for a summing over all particle species ($s$) in plasma and $v_{ds}$, $\gamma_{s}=(1-v_{ds}^{2}/c^{2})^{-1/2}$ are the bulk drift velocity and its corresponding gamma factor of particle species $s$, respectively. When $n_{rb}\ll n_{t}$, one can get the growth rate of the classical weak-beam instability from Eq.(11): $\Gamma/\omega_{pe}=\sqrt{3}(n_{rb}/n_{bg})^{1/3}/2^{4/3}$ (see Eq.3.2.9 in Gary, 1993). Growth rate of the electrostatic waves propagating along $\vec{B_{0}}$ in our simulations is evaluated via a linear fit in the range indicated by ”o” and ”x” points in panel (e). One can see that the growth rate of the electrostatic waves propagating along $\vec{B_{0}}$ is generally slightly smaller than the theoretical maximum ones by the reactive beam instability. Similar results were also found in the study of Karlický & Bárta, 2009. On the one hand, this could be due to the free energy reduction of the energetic ring-beam electrons, i.e., effective $n_{rb}$ and $u_{rb\parallel}$ for the reactive beam instability will decrease with the wave excitation. Small effective $n_{rb}$ and $u_{rb\parallel}$ leads to smaller growth rate for the reactive beam instability (Gary, 1993). On the other hand, particles can simultaneously absorb some waves during the wave excitations (i.e., plasma heating by Landau damping, see panel c of Fig.1), and increased electron momentum spread (or temperature) can also reduce the growth rate of the reactive beam instability (see Sect.3.4 in Melrose, 1986). Additionally, not all electrostatic waves propagating along $\vec{B_{0}}$ grow with the theoretical maximum rate of the reactive beam instability. Generally, values of the growth rate of the electrostatic waves propagating along $\vec{B_{0}}$ are quite similar to those of the whistler mode (panel e of Fig.5), which is also consistent with the study of Lee et al., 2011.

3.1.3 Excited electromagnetic waves

Fig.4 shows the electric field dispersion spectra of electromagnetic waves along different wave propagation directions $\theta$ ($=0^{\circ}$ or $20^{\circ}$ or $90^{\circ}$). Due to the rotation symmetry in the direction perpendicular to the ambient magnetic field $\vec{B_{0}}$, dispersion spectra of $E_{y}$ and $E_{z}$ are very similar for waves with $k_{\perp}=0$ (row a) and the dispersion spectra of waves with $k_{\parallel}=0$ are symmetric with respect to $k_{\perp}=0$ (rows c to e).

Excitation of all electromagnetic whistler, Z, O and X modes can be found in the purely transverse electric component $E_{z}$ spectra along $\vec{B_{0}}$ (row a of Fig.4). Similar to the electrostatic component $E_{x}$ along $\vec{B_{0}}$ (row a of Fig.3), there is also an asymmetry on the transverse electric intensity of waves oppositely propagating along $\vec{B_{0}}$, especially for plasmas with tenuous ring-beam electrons. It is interesting to note that the X (Z) mode dominates the transverse electric field spectra in the direction parallel (antiparallel) to $\vec{B_{0}}$ with tenuous ring-beam electrons. Between the excited X and Z modes, there are also diffusive nonlinear waves that don’t follow the dispersion relations of the linear waves in the cold plasma limit. With the increase of $n_{rb}/n_{t}$, excitations of the X and Z modes as well as the whistler and O modes become more and more symmetric with respect to the $\vec{B_{0}}$ perpendicular plane. The enhanced excitation of the antiparallel-propagating X mode in plasmas with dense ring-beam electrons may be caused by reflected ring-beam electrons (see row a of Fig.2), while the intensity of the parallel-propagating X-mode waves appears to be saturated. Along $\vec{B_{0}}$, the excitation of the whistler, Z and O modes are inefficient for plasmas with tenuous ring-beam electrons in contrast to the dense ring-beam cases.

For obliquely propagating electromagnetic waves with $\theta=20^{\circ}$ and $160^{\circ}$ (rows b of Fig.4), their transverse electric component $E_{z}$ spectra have similar properties to those of the parallel and antiparallel-propagating electromagnetic waves (in row a). $E_{z}$ intensity of these excited obliquely propagating electromagnetic waves are, however, enhanced comparing to those of the $\vec{B_{0}}$ aligned electromagnetic waves, especially in the whistler and Z modes.

Rows (c) to (e) of Fig.4 show the dispersion spectra of the electric components $E_{x}$, $E_{y}$ and $E_{z}$, respectively, for perpendicular propagating electromagnetic waves. The whistler branch is absent in these panels, since its resonance or maximum frequency $\omega_{W}^{res}\rightarrow 0$ at $\theta=90^{\circ}$ and $270^{\circ}$ with the physical proton-to-electron mass ratio $m_{p}/m_{e}=1836$ in the magnetized cold plasma limit (Melrose, 1986; Stix, 1992). It is well known that the electric field of the O (Z and X) mode is parallel (perpendicular) to $\vec{B_{0}}$, when they propagate in the direction perpendicular to $\vec{B_{0}}$, i.e., $\vec{k}\perp\vec{B_{0}}$. In row (c), hence, one can find a strong O-mode excitation. The strong Z and X-mode excitations, on the other hand, appear in the $E_{y}$ and $E_{z}$ spectra (rows d and e). The transverse electric component $E_{z}$ of the Z and X modes contains more energies than their longitudinal electric component $E_{y}$. Generally, similar to the electrostatic modes (row a of Fig.3), intensity of the O, Z and X modes also increase with the increase of $n_{rb}/n_{t}$ in the plane perpendicular to $\vec{B_{0}}$.

In rows (d) and (e) of Fig.4, the X mode appears to be enhanced just below the second harmonic of $\omega_{ce}$ and there is, additionally, an excited horizontal band located around $\omega_{ce}$ and below the cutoff frequency of the X mode $\omega_{X}^{cut}=(\omega_{ce}+\sqrt{\omega_{ce}^{2}+4\omega_{pe}^{2}})/2\approx 1.04\omega_{norm}$. Following Pritchett, 1984, we call this horizontal band as (electromagnetic) relativistic Bernstein mode. This mode result from the relativistic corrections to the classical dispersion of the magnetized cold plasma approximation, see Pritchett, 1984 for more details. This relativistic Bernstein mode is, however, evident only in the cases with $n_{rb}/n_{t}=5\%-30\%$ and the central frequency of this excited relativistic Bernstein mode increases with the increase of $n_{rb}/n_{t}$. The absence of this horizontal mode in the cases with $n_{rb}/n_{t}>30\%$ could be due to merging of the X and Bernstein modes. Based on Fig.1 in Pritchett, 1984, one can see that the cutoff frequencies of the X and Bernstein modes can be the same and above $\omega_{ce}$ in plasmas with electron temperature above $0.1~{}c$. In other words, the merging of the X and Bernstein modes indicates that the effective electron temperature could be higher than $0.1~{}c$ (i.e., efficient heating occurs) in the cases with $n_{rb}/n_{t}>30\%$.

3.1.4 Electromagnetic wave energy

Since solar radio emissions are electromagnetic waves, we will mainly concentrate on properties of the electromagnetic whistler, Z, O and X modes in the following. Sect.3.1.3 shows that these four mode branches dominate the excited electromagnetic waves and they roughly follow the dispersion relations of a magnetized cold plasma. Following the Gaussian filter method described in Sect.2.2.1, we extract the magnetic energy carried by each branch of these four electromagnetic modes.

Panels (a) to (d) of Fig.5 show the evolution of the magnetic energy of the whistler, X, Z and O modes, respectively, where the case with $n_{rb}/n_{t}=0\%$, an equivalent isotropic thermal plasma, shows how much these whistler, Z, O and X-mode waves are enhanced with respect to their corresponding thermal levels. Note that, in numerical simulations, the whistler, Z, O and X-mode waves can also be seen in isotropic thermal plasmas without source of free energy, which is due to the thermal noise numerically enhanced by the finite number of macroparticles (Kilian et al., 2017). Moreover, in Fig.5, the solid (dashed) lines represent the plasmas with a ring-beam (pure-beam) momentum distribution of energetic electrons. Differences in the magnetic energy evolution between the ring-beam and the associated pure-beam momentum distributions are used to assess the effects of the ring feature in the ring-beam momentum distribution.

By comparing the magnetic energy evolution between the ring-beam and the equivalent isotropic thermal plasmas, one can see that the saturation of each mode is over three orders of magnitude larger than their corresponding thermal levels. However, these magnetic energies are about one order of magnitude lower than the electric energy of the electrostatic waves propagating along $\vec{B_{0}}$ shown in panel (c) of Fig.3. All excited waves in the simulation domain are therefore dominated by electrostatic waves. Comparing the results of the ring-beam and pure-beam simulations, one can see that while the whistler mode appears to be mostly driven by the beam instability, the growth of the Z, O and X modes have two components, especially in plasmas with tenuous ring-beam electrons. Moreover, in ring-beam plasmas, the onset of the Z, O, X-mode growth appear to be dominated by the ECM instability. The beam instability has delayed contributions to the excitations of the Z, O, X modes. For each wave mode, this delay decreases with the increase of the ring-beam electron density. In a plasma, this delay appears to increase with the increase of wave frequency. The beam instability, hence, tends to excite low frequency waves first.

The saturation of the Z mode is, however, dominated by the beam instability in ring-beam plasmas. The same is true for O modes with dense ring-beam electrons ($n_{rb}/n_{t}\geq 20\%$). For $n_{rb}/n_{t}\leq 10\%$, the O-mode saturation is governed by the ECM instability and similar to the saturation time of the X mode, which corresponds well to the slow dissipation rate of the free energies in the direction perpendicular to the ambient magnetic field $\vec{B_{0}}$ (see panel b of Fig.1) in the cases with $n_{rb}/n_{t}\leq 10\%$. For the X mode, as predicted by the classical plasma emission theory, pure-beam distribution alone cannot lead to an efficient excitation of the X mode in plasmas with tenuous pure-beam electrons. On the other hand, with dense pure-beam electrons (e.g., $n_{pb}/n_{t}=50\%$, here $n_{pb}$ is the number density of the pure-beam electrons), generation of the X mode can also saturate at a quite high energy.

Note that the magnetic energy envelope of the X mode in the case with ring-beam electrons $n_{rb}/n_{t}=5\%$ and $\omega_{ce}/\omega_{pe}=5$ (solid blue line in panel b) is almost the same as those (panel i of Fig.4) in the study of Lee et al., 2011.

Generally, the magnetic energy saturation of the whistler, Z and O modes are enhanced with increasing ring-beam electron population (i.e., larger $n_{rb}/n_{t}$). But for the X mode with the ring-beam momentum distribution, its magnetic energy saturation in the cases with $n_{rb}/n_{t}=5\%$ and $10\%$ are not the smallest ones, since the free energies released from the perpendicular bulk drift momenta are, correspondingly, not the least in these two cases (see panel b of Fig.1). Furthermore, in the ring-beam plasmas with the same $n_{rb}/n_{t}$, the whistler mode has larger magnetic energy saturation than the other three (Z, O and X) modes. Saturation of the Z, O and X-mode waves decrease in that order when $n_{rb}/n_{t}\geq 20\%$, which implies that wave excitation is more efficient at lower frequencies in plasmas with dense ring-beam electrons. For $n_{rb}/n_{t}\leq 10\%$, the X mode can, however, has a larger saturation than the Z and O modes and the saturation of the Z mode becomes the least among the Z, O and X modes due to contributions from the ECM instability.

In general, Fig.1 and Fig.5 are well corrected and can be used to study energy exchange between waves and electrons. Due to the presence of waves and dynamic energy exchanges between particles and waves, the magnetic energy saturation of these electromagnetic wave modes are not exactly the same as the saturation time of their dominating instabilities (indicated by the formation of a plateau in their corresponding momentum distribution functions). In particular, a small positive gradient still remains in the perpendicular momentum distribution close to the end of simulations for plasmas with tenuous ring-beam electrons (panel b-f of Fig.2). Since the growth rates of the beam and ECM instabilities are proportional to the positive gradients of their corresponding distributions, their growth rates will become smaller when those gradients (free energy sources) are reduced (dissipated). When the gain of wave energy (due to instabilities) is equal to its losses (to heat plasma or accelerate particles), the wave energy will stop increasing and its energy saturation will be reached, no matter whether free energies for the instabilities remain or not. Moreover, the energy exchange between particles and waves can also lead to electron acceleration and heating: increase of the electron perpendicular momentum in plasmas with dense ring-beam electrons (row b of Fig.2) is likely caused by cyclotron resonances, while the spread of the electron distribution in the parallel direction is dominated by Landau damping (row a of Fig.2). The nonlinear dynamic coupling between waves and particles are simplified or ignored in most of theoretical models. But they can be self-consistently recovered in fully kinetic PIC simulations.

Panel (e) of Fig.5 shows the fitted growth rates of all four electromagnetic (whistler, X, Z and O) modes in the ring-beam plasmas with different $n_{rb}/n_{t}$, although these magnetic energy does not increase exactly exponentially with the time.The fitted ranges for these growth rates are shown in their corresponding panels of Fig.5. As one can see, the growth rate of each wave mode monotonously increases with the increase of $n_{rb}/n_{t}$ in agreement with the theoretical predictions for the growth rate of the O and X modes, e.g., Freund et al., 1983; Wu & Freund, 1984. In addition, with the fitted ranges we used, the growth rates between the whistler and Z (as well as O and X) modes are quite similar probably due to the same dominant beam (ECM) instability during their growth phases. And the whistler mode always has a larger growth rate than the X mode, which is consistent with the study of Lee et al., 2011.

Additionally, we also study anisotropy of the whistler, Z, O and X-mode magnetic energies, as shown in Fig.6. In each panel, the total magnetic energy of an electromagnetic mode, covering the whole simulation domain and time series, is divided among the different wave propagation directions $\theta$ (Eq.6). Considering the axis-symmetry of the system, we only need to investigate dependence of the energy on $\theta$ from $0^{\circ}$ to $180^{\circ}$. Note that the magnetic energy of the whistler mode at $\theta=90^{\circ}$ is not included in panel (a) due to its resonance frequency $\omega_{W}^{res}\rightarrow 0$ at $\theta=90^{\circ}$ with the physical proton-to-electron mass ratio $m_{p}/m_{e}=1836$ in the magnetized cold plasma limit.

Contrary to the electrostatic waves (Fig.3), magnetic energies are dominated by non-parallel electromagnetic waves. In plasmas with tenuous ring-beam electrons, the energy dominated waves of all mode branches propagate in the same side as the ring-beam electron propagating (i.e., $\theta<90^{\circ}$). The anisotropy of the whistler, Z and O modes decrease with the increase of the ring-beam electron density, while the anisotropy of the X mode is always high. In plasmas with dense ring-beam electrons, the X-mode magnetic energy has the strongest anisotropy than the other three (whistler, Z and O) modes. The X mode is the strongest around $\theta=60^{\circ}$, indicating influences from the beam instability on the excitation of the X mode, since the strongest energy of the X mode is exactly located at $\theta=90^{\circ}$ when the energetic electrons initially follow a pure-ring momentum distribution (Pritchett, 1984). In isotropic thermal plasmas (i.e., with $n_{rb}/n_{t}=0\%$), the magnetic energy of the whistler, Z, O, and X-mode waves are much smaller than those excited ones and more or less isotropic except of the whistler mode, which has (about one order of magnitude) less magnetic energy than the other three wave modes but its anisotropy is the strongest.

Consistent with row (a) of Fig.4, in the direction along $\theta=0^{\circ}$ and $180^{\circ}$, wave excitation is dominated by the X and Z modes, respectively, in plasmas with tenuous ring-beam electrons. And energies of the obliquely propagating whistler and Z modes with $\theta=20^{\circ}$ and $160^{\circ}$ are larger than those of the $\vec{B_{0}}$ aligned ones (rows a and b of Fig.4, respectively). As well as with the increase of the ring-beam electron density, magnetic energy of each electromagnetic mode increases in directions both along (row a of Fig.4) and perpendicular (rows c and e of Fig.4) to the ambient magnetic field $\vec{B_{0}}$, except for the quasi-parallel-propagating X-mode waves, which appears to be saturated as discussed for rows (a) and (b) of Fig.4 in Sect.3.1.3.

We note that, with the Gaussian filter method, magnetic energy for each wave mode in Fig.5 and 6 might contaminate each other when the dispersion relations of two wave modes are close to each other. For example, row (a) of Fig.4 shows that, in plasmas with tenuous ring-beam electrons $n_{rb}/n_{t}\leq 10\%$, the energy of the O mode propagating in the parallel (antiparallel) direction can have contributions from the X (Z) mode. The growth of the O mode in plasmas with tenuous ring-beam electrons (panel d of Fig.5), therefore, may be influenced by these effects. This can be clearly seen in the following section when we consider the polarization of these waves.

3.1.5 Polarization properties

The polarization of a wave depends on its propagation direction $\theta$ (Melrose, 1986). When propagating parallel to $\vec{B_{0}}$ (i.e., $\theta=0^{\circ}$), the O (X)-mode waves are fully left (right)-handed circularly polarized and the Z-mode waves are fully left (right)-handed circularly polarized when their frequencies $\omega<(>)~{}\omega_{pe}$ (see panels a and b of Fig.7). When $\theta=90^{\circ}$, both O and X-mode waves are linearly polarized (see panels c and d of Fig.7 as well as rows c to e of Fig.4), since the electric field of the O (X)-mode waves are parallel (perpendicular) to $\vec{B_{0}}$.

Following the method described in Sect.2.2.2, we separate the energy contained by the left and right-handed polarized transverse electric fields (LPTE and RPTE) in electromagnetic waves. Panel (e) of Fig.7 shows the energy evolution of the LPTE and RPTE for all electromagnetic waves in the simulation domain. In the $n_{rb}/n_{t}\leq 10\%$ cases, energy evolution profiles of the LPTE and RPTE contain two growth phases, indicating that both the beam and ECM instabilities play roles on the excitation of the electromagnetic waves in the simulation domain. These two growth phases correspond well to the obviously different dissipation rates of the free energies along the parallel and perpendicular directions for the beam and ECM instabilities, respectively (see panels a and b of Fig.1 and Fig.5). In general, in each $n_{rb}/n_{t}$ case, the RPTE dominates the transverse electric field energy during most of the simulation time, due to the energy dominance of the right-handed polarized whistler and X-mode waves, see Fig.5.

Panel (f) of Fig.7 shows the evolutions of the CPD (Eq.10) for all electromagnetic waves in the simulation domain. Due to the dominance of the RPTE in each $n_{rb}/n_{t}$ case, all CPDs are positive at the beginning and increase during the wave growth phase. They, however, start to decrease after the energy saturation of the transverse electric fields. For larger $n_{rb}/n_{t}$ cases, the CPDs can be close to 0 and become negative at the end of the simulations. The decreased CPD indicates more reduction of the right-handed polarized waves than the left-handed polarized ones by electrons via wave-electron cyclotron resonance interactions.

Over the whole time-frequency domain and for all electromagnetic waves in the simulation domain, energy anisotropy of their LPTE and RPTE (panel g) as well as the anisotropy of their CPDs (panel h) are also presented in Fig.7. Consistent with panels (e) and (f), RPTE predominates the total energy of transverse electric fields along most of the wave propagation directions in each $n_{rb}/n_{t}$ case. Correspondingly, the CPDs at different wave propagation angles are, hence, mostly positive (right-handed polarized). Small negative (or left-handed polarized) CPDs, however, also exist for the $n_{rb}/n_{t}=30\%,40\%$ cases, e.g., along $\theta\geq 160^{\circ}$. Furthermore, consistent with the classical definition of the perpendicular propagating electromagnetic waves, their CPDs are always around 0 (linearly polarized) at $\theta=90^{\circ}$ for all cases. For parallel-propagating electromagnetic waves ($\theta=0^{\circ}$) in $n_{rb}/n_{t}=5\%,10\%$ cases, the resulting CPDs can reach 1.0, i.e, fully right-handed circularly polarized (see panels a and b of Fig.7 for $n_{rb}/n_{t}=5\%$). There the left-handed polarized O mode has negligible contributions to the total energies of the parallel-propagating electromagnetic waves.

All electromagnetic waves in the simulation domain are included in the above discussions. From observational point of view, however, not all excited waves can be detected remotely. It is known that an electromagnetic wave can escape from an astrophysical plasma only if its refractive index is less than unity, i.e., $|ck/\omega|<1$ and its frequency is larger than the local plasma frequency, i.e., $\omega>\omega_{pe}$ (Melrose, 1986; Budden, 1988; Stix, 1992; Benz, 2002; Bellan, 2006). Escaping electromagnetic waves in plasmas, hence, are only the O and X modes. Properties of the polarization and spectrogram are, hence, investigated for these escaping electromagnetic waves with $\omega>\omega_{pe}$ and $|ck/\omega|<1$ (the escape condition), shown in Fig.8 and Fig.9, respectively. Other waves, with larger refractive indices and low frequencies, are trapped and can be absorbed or reflected (depending on their cutoff or/and resonance frequencies) during wave propagations in the interplanetary medium (IPM) or interestellar plasmas. The only way that those waves can be remotely detected is by means of conversion to escaping electromagnetic waves via mechanisms such as wave-wave coupling, coalescence or decay, antenna mechanisms or mode conversation in inhomogeneous plasmas (see, e.g., Graham et al., 2017, 2018, and references therein), which is, however, beyond the scope of this study.

Due to the removal of electromagnetic waves with $\omega\leq\omega_{pe}$ or $|ck/\omega|\geq 1$ (mostly the whistler and Z modes), energy of both the escaping RPTE and LPTE are reduced (panel e) in comparison with those of all electromagnetic waves in the simulation domain (panel e of Fig.7). Another obvious difference between the escaping RPTE, LPTE and the RPTE, LPTE of all electromagnetic waves is that, when $n_{rb}/n_{t}\leq 10\%$, the first growth phase (during $\omega_{norm}t<325$, which associated with the beam instability) in the energy evolution profile of the RPTE and LPTE of all electromagnetic waves do not exist anymore in Fig.8 for the escaping electromagnetic waves. That indicates that the excitation of the high-frequency escaping electromagnetic waves are mainly due to the the ECM instability in plasmas with tenuous ring-beam electrons ($n_{rb}/n_{t}\leq 10\%$) and the significant growth of the O mode associated with the beam instability in Fig.5 is likely caused by contamination from the Z mode in these tenuous ring-beam cases.

Panel (f) of Fig.8 shows the evolution of the CPD for the escaping electromagnetic waves. In the cases with larger $n_{rb}/n_{t}\geq 20\%$, one can see that the CPDs of the escaping electromagnetic waves are always smaller than those of all electromagnetic waves and can flip sign and be close to $-0.4$ at the end of simulations. In contrast, the CPDs of the escaping electromagnetic waves in $n_{rb}/n_{t}\leq 10\%$ cases are larger than those of all electromagnetic waves, especially around the saturation of the first growth phase in the energy evolution profile for all waves ($\sim\omega_{norm}t<325$ in panel e of Fig.7).

Panel (g) of Fig.8 shows the energy anisotropy of the LPTE and RPTE for the escaping electromagnetic waves. Different from those of all the electromagnetic waves in the simulation domain, dominance of the RPTE or LPTE changes with the wave propagation direction $\theta$ in plasmas with tenuous ring-beam electrons $n_{rb}/n_{t}\geq 10\%$. For plasmas with dense ring-beam electrons (especially $n_{rb}/n_{t}\geq 30\%$), LPTE dominates the escaping transverse electric field energy along most of the wave propagation directions. Correspondingly, the anisotropy of the CPD for the escaping electromagnetic waves (panel h of Fig.8) is also quite different from the one for all electromagnetic waves in the simulation domain (panel h of Fig.7). For the escaping electromagnetic waves, a left-handed CPD can be found at some directions for each $n_{rb}/n_{t}$ case. Furthermore, the left-handed CPD dominates over all wave propagation directions in plasmas with dense ring-beam electrons $n_{rb}/n_{t}\geq 40\%$. At $\theta=90^{\circ}$, escaping electromagnetic waves are still linearly polarized. Additionally, with the increase of $n_{rb}/n_{t}$, the CPDs of the escaping electromagnetic waves become increasingly symmetric around $\theta=90^{\circ}$, corresponding to their symmetric energies of the RPTE and LPTE (panel g of Fig.8).

Fig.9 shows dependence of the spectrogram of the RPTE and LPTE in escaping electromagnetic waves on the wave propagation direction. These spectrograms have similar anisotropy and symmetry properties (i.e., corresponds well) as their corresponding CPDs in panel h of Fig.8. With the exception of waves propagating near $\theta=90^{\circ}$, the spectrogram of the RPTE and LPTE are quite different along any other propagation directions. These differences increase with increasing value of $|\theta-90^{\circ}|$. And with Fig.9, one can also find that the CPD of these escaping electromagnetic waves depend on not only the wave propagation direction, population of the ring-beam electrons but the wave frequency and time. Furthermore, intense emission in these spectrograms are generally located around the frequencies $\omega_{pe}$ and/or $\omega_{ce}$. Bandwidth and intensity as well as pattern of these intense emissions vary a lot among different frequencies, wave propagation directions and population of the ring-beam electrons. Emission around $2\omega_{ce}$ can also be found in these spectrograms, especially for the perpendicular propagation $\theta=90^{\circ}$ and for plasmas with a dense ring-beam electron population.

3.2 $\omega_{ce}/\omega_{pe}$ dependence

As mentioned in Sect.1, $\omega_{ce}>\omega_{pe}$ is required for an efficient escaping ECM emission. Many previous numerical studies for the ECM emission (e.g., Pritchett, 1984; Lee et al., 2009, 2011), hence, considered situations with $\omega_{ce}>\omega_{pe}$. Although $\omega_{ce}>\omega_{pe}$ can exist in some density cavities due to, e.g., turbulent magnetic field fluctuations (Wu et al., 2014; Chen et al., 2017; Melrose, 2017), based on the standard solar atmosphere model (Wild, 1985), however, $\omega_{ce}<\omega_{pe}$ is typical for the solar coronal conditions. In this section, we will focus on the wave excitation dependence on the $\omega_{ce}/\omega_{pe}$ for ring-beam energetic electrons with fixed number density ratio $n_{rb}/n_{t}$ to $5\%$ and fixed total electron plasma frequency $\omega_{pe}$. This is justified considering that the typical gradient length of the particle number density is usually larger than that of the magnetic field strength in the solar corona (see Eqs.1.6.1 and 1.4.2 in Aschwanden, 2005). The ratio between the electron cyclotron frequency $\omega_{ce}$ and $\omega_{pe}$ takes $0.2$, $0.3$, $0.5$, $1$, $2$, $3$, while the case $\omega_{ce}/\omega_{pe}=5$ has been analyzed in Sect.3.1. Similar to Fig.4, the dispersion spectra dependence on the $\omega_{ce}/\omega_{pe}$ and wave propagation direction $\theta$ are presented in Fig.10.

As the cases with $\omega_{ce}/\omega_{pe}=5$, excitation of the beam, Langmuir (row a), whistler (row b), O (row c), Z and X (row d and e) modes still exist in each $\omega_{ce}/\omega_{pe}<5$ cases. Intensity of the escaping electromagnetic waves from the weakly magnetized plasmas $\omega_{ce}/\omega_{pe}<1$ are, however, significantly suppressed and negligible comparing with those from plasmas with $\omega_{ce}/\omega_{pe}>1$, which is consistent with the statement of Vlahos, 1987. With the decrease of the $\omega_{ce}/\omega_{pe}$ the dispersion relation surface of the O and X modes tend to overlap with each other in the cold plasma approximation the intensity differences between the O and X modes can not be well resolved for small $\omega_{ce}/\omega_{pe}$ cases with the limited resolution in the $\omega$ space in our simulations. Quantitative investigation on the energy and polarization property dependence on the $\omega_{ce}/\omega_{pe}$ ratio will be presented in following papers with a higher $\omega$ resolution. Here, we will concentrate on discussing the harmonic excitation of the $\omega_{pe}$ and $\omega_{ce}$.

In Fig.10, one can find that excitation at higher and higher harmonics $s_{h}$ of both $\omega_{pe}$ (rows a and b) and $\omega_{ce}$ (rows c to e) appear with the decrease of the $\omega_{ce}/\omega_{pe}$. However, excitation of harmonic $\omega_{pe}$ can only be found in cases with $\omega_{ce}/\omega_{pe}<1$, i.e., weakly magnetized plasmas. In row (b), although frequencies of the excited harmonics of $\omega_{pe}$ is evidently higher than $\omega_{pe}$, the refractive index in these harmonics of $\omega_{pe}$ are, however, much larger than 1. These waves are likely reflected at the boundary of plasmas with distinct properties and they therefore cannot be observed remotely. Meanwhile, these non-escaping harmonics of $\omega_{pe}$ are mainly located in the direction quasi-parallel to the ambient magnetic field $\vec{B_{0}}$, implying a beam instability origin. Note that these excited non-escaping harmonics of $\omega_{pe}$ are not centered exactly at the integer multiples of $\omega_{pe}$. Instead, their frequencies increase with $k$, i.e., with a small positive slope in each non-escaping harmonic $\omega_{pe}$ band. Similar results can also be found in the study of Thurgood & Tsiklauri, 2015 for the plasma emission theory. This frequency shifts in the fundamental $\omega_{pe}$ mode have been attributed to deviations from the prediction of the cold plasma theory in the case of dense beams (Fuselier et al., 1985; Cairns, 1989), where the beam-mode waves might affect the generation of the fundamental $\omega_{pe}$ mode. And the frequency shifts at higher non-escaping harmonics of $\omega_{pe}$ are, perhaps, due to the frequency shift of the fundamental $\omega_{pe}$ mode, since the fundamental mode is responsible for the excitations of other higher harmonics. Enhanced harmonics of $\omega_{ce}$ can be found in each panel of rows (c) to (e) of Fig.10. In other words, excitation of harmonic $\omega_{ce}$ does not depend on the magnetized condition of plasma $\omega_{ce}/\omega_{pe}$. Additionally, these excited harmonics of $\omega_{ce}$ are likely excited by the ECM instability, since excitation of them are mainly located in the direction perpendicular to the ambient magnetic field $\vec{B_{0}}$.

Although the excitation mechanism for the harmonics of $\omega_{pe}$ and $\omega_{ce}$ are totally different, they still have some common characteristics. For instance, all these harmonic waves contain both longitudinal (row a for $s_{h}\omega_{pe}$, row d for $s_{h}\omega_{ce}$) and transverse (row b for $s_{h}\omega_{pe}$, rows c and e for $s_{h}\omega_{ce}$) components, but the longitudinal component is stronger than the transverse one, which is opposite to that of the Z and X modes in the perpendicular direction. Harmonic excitation of $\omega_{pe}$ with a preferential longitudinal component has been found previously by Klimas, 1983; Nishikawa & Cairns, 1991; Yoon et al., 2003; Yi et al., 2007; Rhee et al., 2009. Additionally, intensity in these excited harmonics of $\omega_{pe}$ and $\omega_{ce}$ decrease with the the increase of the harmonic number. Moreover, all excited harmonics of $\omega_{pe}$ and $\omega_{ce}$ are non-escaping modes in weakly magnetized plasmas with $\omega_{ce}/\omega_{pe}<1$.

4 Conclusions and discussion

Using 2.5D fully kinetic PIC simulations, we investigated the energy and polarization properties of electromagnetic waves excited by mildly relativistic ring-beam electrons in neutral and current-free solar coronal plasmas. These energetic ring-beam electrons could be produced by magnetic reconnection, quasi-perpendicular shocks and/or electron beams propagating in inhomogeneous magnetic fields in the solar corona. These ring-beam electrons together with the background electrons and protons support the global current and charge neutralities in these plasmas, where all background electrons drift oppositely to the ring-beam electrons (i.e., return current) to fully compensate the current induced by the ring-beam electrons, i.e., a neutral ring-beam-return current system. To apply the simulation results to solar radio observations and considering variations of the electron ring-beam density and magnetic field strength along the path of electron propagation, we explore the dependence of the electromagnetic wave excitations on the number density ratio of the ring-beam electrons over the total electrons ($n_{rb}/n_{t}$) and the ratio of the electron cyclotron frequency ($\omega_{ce}$) to the electron plasma frequency ($\omega_{pe}$).

We found that the beam and electron cyclotron maser (ECM) instabilities together can efficiently excite the whistler, Z, O, and X-mode electromagnetic waves as well as harmonics of $\omega_{pe}$ (only when $\omega_{ce}/\omega_{pe}<1$) and $\omega_{ce}$. We also found the excitations of electrostatic waves, relativistic Bernstein waves and some diffusive nonlinear waves that do not follow well defined dispersion relations. Electrostatic waves always dominate the energetics of all excited waves. These electrostatic waves can lead to a significant heating on the ring-beam and background electrons due to Landau damping.

Properties of the electromagnetic whistler, Z, O and X-mode waves were studied in detail. In order to obtain the energy evolution of these electromagnetic waves, we adopted a Gaussian filter centered on the wave dispersion surfaces of the magnetized cold plasmas in the wavevector-frequency ($\vec{k}-\omega$) space and assumed a frequency broadening of all excited electromagnetic waves $\sigma=0.05\omega_{norm}$, which is frequently seen in the spectrograms of escaping electromagnetic waves (Fig.9). For a convergence test, we also carried calculations with $\sigma=0.03\omega_{norm}$ and $0.2\omega_{norm}$, while the frequency resolution in our PIC simulations is $3\times 10^{-3}\omega_{norm}$. We found that results with $\sigma=0.03\omega_{norm}$ and $0.05\omega_{norm}$ are almost the same. We admit that this Gaussian filter method for energy of a wave mode might contain contamination from other waves particularly when the dispersion relations of two wave modes are close to each other. Additionally we might also underestimate the energy of diffusive waves with a fixed $\sigma$ for all wave modes. However, in general, this method gives more accurate information about the mode energy compared to estimates given in the previous studies,  Pritchett, 1984; Lee et al., 2009, 2011, see, e.g.,.

Based on the Gaussian filter method, we carried out detailed studies of the dependence of the excited electromagnetic whistler, Z, O, and X-mode properties on the ring-beam electron density for $\omega_{ce}/\omega_{pe}=5$ and found:

• •

Both the beam and ECM instabilities contribute to the excitation of these electromagnetic waves. The beam instability dominates the saturation of the whistler, Z modes as well as O mode in plasmas with dense ring-beam electrons. But the X-mode waves cannot be efficiently excited by the beam instability only, especially with tenuous ring-beam electrons.

• •

In the growth phase of waves, the free energy dissipation rate and the wave growth rate, in general, increase with the increase of the ring-beam density. The growth rates of the whistler and Z modes are comparable but higher than that of the O and X modes.

• •

The saturation level of different electromagnetic wave modes also increase with the increase of $n_{rb}/n_{t}$, except for the X-mode waves produced by low density ring-beam electrons. The X-mode saturation level for $n_{rb}/n_{t}=5\%$ is actually higher than that for $n_{rb}/n_{t}=10\%$.

• •

The energy of each electromagnetic wave mode is strongly anisotropic. This anisotropy is suppressed in plasmas with dense ring-beam electrons, where the X mode has the strongest anisotropy.

Although only the O and X modes are remotely detectable and related more to the remote observations of the SRBs, the individual investigation on the energy property of all the whistler, Z, O and X modes obtained here will complement their linear and quasi-linear theoretical studies. In theoretical studies, nonlinear processes (e.g., wave-wave, wave-particle cyclotron resonance interactions) and evolution of the plasma system itself (e.g., population of the energetic and background electrons, plasma temperature) cannot be usually treated self-consistently. In this study, we found that all these processes affect the energy saturation and growth rate of those wave modes.

Harmonic excitation of $\omega_{pe}$ and $\omega_{ce}$ are studied for differently magnetized plasmas with $0.2\leq\omega_{ce}/\omega_{pe}\leq 5$. Over all simulated cases, one can find that harmonics of $\omega_{ce}$ can be always excited, while there is no obvious excitation for the escaping harmonics of $\omega_{pe}$. Such results were also found by Ganse et al., 2012b. As mentioned by Thurgood & Tsiklauri, 2015 as well as according to Eq.(6.80) in Melrose, 1986 for the probability of the $L+L^{\prime}\rightarrow T_{2\omega_{pe}}$ process, the key reason for the absence of the escaping harmonics of $\omega_{pe}$ in our simulated plasmas could be the weak intensity (in plasmas with tenuous ring-beam electrons $n_{rb}/n_{t}\leq 20\%$) and/or the predominant wave intensity located at small wavenumber $|k|$ (due to Landau damping in plasmas with dense ring-beam electrons) of both the parallel and antiparallel-propagating electrostatic Langmuir waves. Based on the study of Thurgood & Tsiklauri, 2015, escaping harmonics of $\omega_{pe}$ are more likely present in plasmas with very tenuous beam electrons $n_{rb}/n_{t}<0.6\%$ and generation of the harmonics of $\omega_{pe}$ are very sensitive to the chosen parameters, like the populations of the beam and oppositely drifting electrons, the drifting velocity of the beam electrons, the magnetized condition, etc. (Rhee et al., 2009; Umeda, 2010; Ganse et al., 2012b; Thurgood & Tsiklauri, 2015; Henri et al., 2019). The setup parameters used here, however, favour more the excitation of the beam mode instead of Langmuir waves.

Non-escaping harmonics of $\omega_{pe}$ can be excited only when $\omega_{ce}/\omega_{pe}<1$. Higher and higher non-escaping harmonics of $\omega_{pe}$ are driven with the decrease of $\omega_{ce}/\omega_{pe}$. No matter whether escaping harmonics of $\omega_{pe}$ are excited or not, the beam instability can always lead to the excitation of the Langmuir, whistler, Z and O-mode waves. ECM excitation of the harmonics of $\omega_{ce}$, however, does not depend on the ratio of $\omega_{ce}/\omega_{pe}$. In other words, $s_{h}\omega_{ce}$ (as well as X mode) will be excited as long as the free energy $\partial f/\partial u_{\perp}>0$ exists for the ECM instability. The requirement of $\omega_{ce}/\omega_{pe}>1$ in the ECM emission theory is for an efficient generation of observable escaping emissions by remote detections, i.e., the escape condition (see Sect.1).

In addition, to compare with solar radio observations, we also obtained the polarization properties (circular polarization degree — CPD, spectrogram) of the electromagnetic waves, in particular of the escaping electromagnetic waves with $\omega>\omega_{pe}$ and $|ck/\omega|<1$. In summary, escaping emission decreases rapidly with the decrease of $\omega_{ce}/\omega_{pe}$. In weakly magnetized plasmas with $\omega_{ce}/\omega_{pe}<1$, most of the excited escaping electromagnetic waves are located close to the plasma frequency $\omega_{pe}$ and their energies are significantly weaker and negligible compared to those of plasmas with $\omega_{ce}/\omega_{pe}>1$, i.e., strongly magnetized plasmas. Energy and polarization properties of the escaping electromagnetic waves in strongly magnetized plasmas depend on the density ratio $n_{rb}/n_{t}$:

• •

When $n_{rb}/n_{t}\leq 10\%$, the ECM instability dominates the excitation of the escaping electromagnetic waves. Right-handed polarized electric fields (RPTEs) dominate the transverse electric field energies of the escaping electromagnetic waves. And right-handed polarized CPDs can be expected along many wave propagation directions. Moreover the strongest escaping emission is in the same side of the ring-beam electron propagation direction (i.e., $\theta<90^{\circ}$). These properties may explain observed properties of solar radio spikes.

• •

For plasmas with dense ring-beam electrons $n_{rb}/n_{t}\geq 20\%$, the escaping emissions are dominated by the ECM instability at the beginning. The beam instability plays a more important role later on, giving rise to more isotropic and left-handed polarized electric field (LPTE) dominated emissions, which may be applied to observations of Type @slowromancapiii@ bursts.

Considering the population reduction of the ring-beam electrons during their propagation in the solar corona, these results might explain the increased time delay of the Type @slowromancapiii@ bursts ($<1s$) and solar radio spikes ($2-5s$) to the hard X-ray bursts (see Fleishman & Mel’nikov, 1998 for a review of the solar radio spikes). Moreover diversities in the SRBs’ CPD and spectrogram observations may already originate from their generation sites.

Our results above deal with properties of waves driven by energetic ring-beam electrons at the site of wave generation, where the global charge and current neutralities are maintained via protons and drifting background electrons, respectively. Note that the remotely observed energy and polarization properties of the SRBs might deviate from those in their source regions, due to some propagation effects of the electromagnetic waves (e.g., reflection, refraction, Faraday rotation, energy absorption via wave-particle interaction) along the wave path in the IPM. For an accurate prediction of the remote SRB observations, one, hence, still needs to combine our simulations with a proper model describing the wave propagation effects in the IPM (Li et al., 2008a, b, 2009). This study is, however, still meaningful to gain insight into the generation mechanisms of the original coherent emission by energetic ring-beam electrons in the neutral and current-free solar coronal plasmas, where all background electrons drift oppositely to the ring-beam electrons to fully compensate the current induced by the ring-beam electrons.

Finally, we note that dynamic processes in plasmas with energetic ring-beam electrons contain not only the excitations of waves but also plasma heating and electron acceleration. Significant plasma heating and electron acceleration can be expected, particularly, in plasmas with dense ring-beam electrons, where a significant fraction of the ring-beam electrons can be reflected, making the system more or less symmetric with respect to the plane perpendicular to the ambient magnetic field $\vec{B_{0}}$. Due to the acceleration of electrons, a double power-law distribution is formed in the high energy tail ($\gamma-1>0.1\sim 50~{}keV$) of the electron energy distribution when the wave-particle plasma system reaches an equilibrium.

We gratefully acknowledge the developers of the ACRONYM code, the Verein zur Förderung kinetischer Plasmasimulationen e.V. In particular, we thank Patrick Kilian for helpful discussions and valuable suggestions. X. Zhou thanks the Chinese Academy of Sciences as well as the International Cooperation and Exchange Project of National Natural Science Foundation of China, 11761131007 for support. P. A. Muñoz acknowledges his financial support by the German Science Foundation DFG, project MU-4255/1-1. We also acknowledge the computing resources in the Max Planck Computing and Data Facility (MPCDF, formerly known as RZG) at Garching, Germany and the Max Planck Institute for Solar System Research, Germany.