# Zebra stripes with high gyro-harmonic numbers

Jan Benáček  Marian Karlický  Center for Astronomy and Astrophysics, Technical
University of Berlin, 10623 Berlin, Germany  Astronomical Institute, Czech
Academy of Sciences, 251 65 Ondřejov, Czech Republic

###### keywords:

Radio Bursts, Type IV; Radio Emission, Theory; Instabilities

##  1 Introduction

sec:intro Solar radio bursts and their fine structures are used to diagnose
solar flare plasmas. Among them, zebra patterns (zebras) belong to the most
frequently investigated bursts. An example of the zebra observed in 14
February 1999 is shown in Figure fig0. In the radio spectrum, they appear in
the form of regularly spaced emission stripes (zebra stripes). Many papers and
monographs were devoted to them, e.g., Slottje (Apr 1982); Chernov (2006,
2011); Tan et al. (2012, 2014); Chernov, Formichev, and Fainshtein .

For the interpretation of the zebra stripes, several models have been proposed
(Rosenberg, 1972; Kuijpers, 1975; Zheleznyakov and Zlotnik, 1975; Chernov,
1976, 1990; LaBelle et al., 2003; Kuznetsov and Tsap, 2007; Bárta and
Karlický, 2006; Ledenev, Yan, and Fu, 2006; Laptukhov and Chernov, 2009; Tan,
2010; Karlický, 2013). In the published literature, the most often used model
of zebras is the one based on the double plasma resonance (DPR) condition
(Kuijpers, 1975; Zheleznyakov and Zlotnik, 1975; Kuijpers, 1980; Mollwo, 1983,
1988; Winglee and Dulk, 1986; Zlotnik, 2013)

$\ilabel{eq1}\omega-\frac{s\omega_{\mathrm{ce}}}{\gamma}-\frac{k_{\parallel}u_{\parallel}}{\gamma}=0,$
(1)

where $\omega$ and ${\mathbfit k}=(k_{\parallel},k_{\perp})$ are the frequency
and wave vector of the resonant wave, $\omega_{\mathrm{ce}}$ is the electron
cyclotron frequency, $s$ is the gyro-harmonic number,
$\gamma=\sqrt{1+(u_{\parallel}^{2}+u_{\perp}^{2})/c^{2}}$ is the relativistic
Lorentz factor, $u_{\parallel}=p_{\parallel}/m_{\mathrm{e}}$,
$u_{\perp}=p_{\perp}/m_{\mathrm{e}}$, in the electron momentum ${\mathbfit
p}=(p_{\parallel},p_{\perp})$, and $m_{\mathrm{e}}$ is the electron mass. This
condition may be simplified assuming that the generated waves are upper-hybrid
(UH) waves in a non-relativistic plasma and the DPR resonance is at integer
gyro-harmonics of the electron-cyclotron frequency

$f_{\rm UH}=(f_{\rm pe}^{2}+f_{\rm ce}^{2})^{1/2}\approx sf_{\rm
ce},\ilabel{eq2}$ (2)

where $f_{\rm UH}$ and $f_{\rm pe}$ are the UH and electron-plasma
frequencies, $f_{\rm ce}$ is the electron-cyclotron frequency. In the DPR
model, superthermal electrons are considered as the driver of the zebra
emission. Those electrons are considered to be trapped in a flare loop and
have a loss-cone distribution. In the flare loop locations where such a
distribution is present and the DPR condition is fulfilled, the DPR
instability generates the UH waves. These UH waves are then converted to the
observed electromagnetic waves by merging with low-frequency waves or by
scattering on plasma particles. This model is supported by positional
observations made by the Owens Valley Solar Array (Chen et al., 2011).

Figure 1.: An example of the zebra observed at 14 February 1999 by the
Ondřejov radiospectrograph (Jiřička and Karlický, 2008).fig0

When applying the DPR model to observed zebras and to estimate plasma
parameters at the zebra source, the most important is to determine the gyro-
harmonic numbers of zebra stripes $s$. From the zebra stripe frequencies and
their gyro-harmonic numbers, the electron cyclotron frequency and magnetic
fields are consequently estimated. Several methods for their determination
were developed, as shown, e.g., in Karlický and Yasnov (2015); Yasnov and
Karlický (2020); Yasnov (2021).

These methods are based on fitting a set of the observed zebra-stripe
frequencies $f(n)$ by the set of theoretically derived zebra-stripe
frequencies $f_{\mathrm{th}}(n)$ in the form (Yasnov, 2021)

$f_{\mathrm{th}}(n)=\frac{(s_{1}-(n-1))f(s_{1})}{s_{1}}\left(\frac{(s_{1}-(n-1))^{2}-1}{s_{1}^{2}-1}\right)^{\frac{1}{R-2}},\ilabel{eq3}$
(3)

where $n$ is the ordinal number of the zebra stripes counted from the lowest
to higher frequencies, $f(s_{1})$ is the frequency of the stripe with the
lowest frequency, $s_{1}$ is the corresponding gyro-harmonic number,
$R=L_{\mathrm{bh}}$/$L_{\mathrm{nh}}$ is the ratio of spatial scales, which
exponentialy depend on the magnetic field and density, in the region where
zebra-stripes are generated, and $R=R_{1}+R_{2}(n-1)$ is taken as a linear
function of $n$, where $R_{1}$ and $R_{2}$ are constants.

The fitting procedure is mathematically expressed as a search for the minimum
value in the relation

$\displaystyle\frac{1}{N}\sum_{n=1}^{N}(f_{\mathrm{th}}(n)-f(n))^{2},$ (4)

where $N$ is the total number of the analyzed zebra stripes. The parameters
$s_{1},R_{1},R_{2}$, and $f(s_{1})$ are the result of this fitting procedure.
For more details about this method, see Yasnov and Karlický (2020) and Yasnov
(2021).

Using this method, seven detected zebras were analyzed at selected times, and
gyro-harmonic numbers $s_{1}$ were determined for the zebra-stripe with the
lowest analyzed frequency $f(s_{1})$, see Table tab1. As can be seen there,
the gyro-harmonic number $s_{1}$ is greater than 50 in all these zebras. It is
surprising that the physical process that generates zebra stripes operates on
such high harmonics instead of the usually preferred low harmonics, where the
resonance frequency might be achieved easily. These gyro-harmonic numbers are
also significantly higher than those $s$ that have been considered in previous
theoretical studies of the DPR instability. Moreover, the theoretical
estimations made until now show that both growth rates and saturation energies
of the UH waves caused by the DPR instability decrease with increasing gyro-
harmonic number (Kuznetsov and Tsap, 2007; Benáček and Karlický, 2018).
However, the growth rates and saturation energies of the DPR instability have
been so far mainly computed for gyro-harmonic numbers $s\lesssim 20$ (Winglee
and Dulk, 1986; Benáček, Karlický, and Yasnov, 2017; Lee et al., 2018; Ni et
al., 2020; Li et al., 2021). Only Kuznetsov and Tsap (2007) calculated the
growth rates for $s\lesssim 50$ for a power-law velocity distribution with a
loss-cone angle $30^{\circ}$, showing that the growth rate maxima are well
distinct for selected parameters.

In the present paper, we propose a solution to the problem of the generation
of the zebra stripes with high gyro-harmonic numbers by considering a loss-
cone distribution with large loss-cone angles. We compute the growth rates of
the DPR instability in a broad interval of UH to cyclotron frequency ratios,
corresponding to observed gyro-harmonic numbers up to $s\sim$100\. We solve
this problem numerically, integrating the analytical expressions for these
growth rates. We consider the loss-cone $\kappa$-distribution and vary the
loss-cone angle, electron energies, background plasma temperature, and the
$\kappa$ parameter. Besides the general results covering a broad range of
gyro-harmonic numbers, we also present some results for the zebra observed on
the 14 February 1999.

The paper is organized as follows: In Section 2, we describe the method of the
growth rate calculation. Section 3 presents the results of these calculations
for a broad range of gyro-harmonic numbers. Finally, we summarize our results
and discuss how zebra stripes with high values of $s$ can be generated in
Section 4.

Date | Time | f ($s_{1}$) | Gyro-harmonic
---|---|---|---
| (UT) | (MHz) | number $s_{1}$
25 Oct 1994 | 10:08:23.1 | 144.8 | 78
17 Aug 1998 | 07:06:31.0 | 260.3 | 78
14 Feb 1999 | 12:08:57.0 | 1526 | 64
21 Apr 2002 | 01:45:49.5 | 2668 | 52
1 Aug 2010 | 08:21:16.0 | 1120 | 114
24 Feb 2011 | 07:41:08.8 | 2977 | 70
21 Jun 2011 | 03:22:27.4 | 160.7 | 152
Table 1.: Parameters of analyzed zebras according to Karlický and Yasnov
(2021).tab1

##  2 Method for the Growth Rate Calculation

We assume a collisionless plasma consisting of a cold background plasma and
hot electron species that are uniformly distributed in space and embedded in a
uniform magnetic field. The background plasma consists of cold electrons with
number density $n_{\mathrm{b}}$ and cold protons with density
$n_{\mathrm{i}}$. Hot electrons of density $n_{\mathrm{h}}$ are also
considered. The plasma satisfies the charge neutrality condition,
$n_{\mathrm{i}}=n_{\mathrm{b}}+n_{\mathrm{h}}$. Both cold particle species
have Maxwell velocity distributions with thermal velocities
$v_{\mathrm{b,i}}=\sqrt{k_{\mathrm{B}}T_{\mathrm{b,i}}/m_{\mathrm{b,i}}}$,
where $k_{\mathrm{B}}$ is the Boltzmann constant, $T_{\mathrm{b,i}}$ are the
thermodynamic temperatures, and $m_{\mathrm{b,i}}$ are the masses of cold
electrons and protons, respectively.

The hot electrons have a loss-cone $\kappa$-distribution that can be expressed
in cylindrical coordinates as

$f({\mathbfit
u},\theta_{\mathrm{c}},\Delta\theta_{\mathrm{c}},\kappa)=f_{\kappa}({\mathbfit
u},\kappa)\,\varphi(\theta_{\mathrm{c}},\Delta\theta_{\mathrm{c}}),$ (5)

where $\varphi$ characterizes the linear transition at loss cone angles

$\varphi(\theta_{\mathrm{c}},\Delta\theta_{\mathrm{c}})=\left\\{\begin{array}[]{ll}0,&~{}~{}~{}~{}~{}~{}\theta_{\mathrm{c}}-\Delta\theta_{\mathrm{c}}\geq\theta,\\\
\frac{\theta-(\theta_{\mathrm{c}}-\Delta\theta_{\mathrm{c}})}{2\Delta\theta_{\mathrm{c}}},&~{}~{}~{}~{}~{}~{}\theta_{\mathrm{c}}+\Delta\theta_{\mathrm{c}}>\theta>\theta_{\mathrm{c}}-\Delta\theta_{\mathrm{c}},\\\
1,&\pi-\theta_{\mathrm{c}}-\Delta\theta_{\mathrm{c}}\geq\theta\geq\theta_{\mathrm{c}}+\Delta\theta_{\mathrm{c}},\\\
\frac{(\pi-\theta_{\mathrm{c}}+\Delta\theta_{\mathrm{c}})-\theta}{2\Delta\theta_{\mathrm{c}}},&\pi-\theta_{\mathrm{c}}+\Delta\theta_{\mathrm{c}}>\theta>\pi-\theta_{\mathrm{c}}-\Delta\theta_{\mathrm{c}},\\\
0,&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\theta\geq\pi-\theta_{\mathrm{c}}+\Delta\theta_{\mathrm{c}},\\\
\end{array}\right.$ (6)

where $\theta_{\mathrm{c}}$ is the loss-cone angle,
$\Delta\theta_{\mathrm{c}}$ is the angular half-width of the linear loss-cone
transition, $\theta=\arctan(u_{\perp}/u_{\parallel})$ is the particle pitch
angle, $f_{\kappa}({\mathbfit u},\kappa)$ is the $\kappa$-distribution
(Livadiotis and McComas, 2013)

$f_{\kappa}({\mathbfit
u},\kappa)=\frac{n_{\mathrm{h}}}{n_{\mathrm{e}}}\frac{1}{(\pi\kappa
v_{\mathrm{\kappa}}^{2})^{\frac{3}{2}}\cos(\theta_{\mathrm{c}})}\frac{\Gamma(\kappa+1)}{\Gamma(\kappa-\frac{1}{2})}\left(1+\frac{u_{\perp}^{2}+u_{\parallel}^{2}}{\kappa
v_{\mathrm{\kappa}}^{2}}\right)^{-\kappa-1},$ (7)

characterized by the index $\kappa=\left(\frac{3}{2},\infty\right)$,
$v_{\mathrm{\kappa}}$ is the $\kappa$-distribution velocity, and $\Gamma$ is
the Gamma function. The distribution function is normalized to the loss-cone
angle as $1/\cos(\theta_{\mathrm{c}})$, assuming that
$\Delta\theta_{\mathrm{c}}\ll 1$. Unless otherwise mentioned, we use
$\kappa=2$ and $n_{\mathrm{e}}/n_{\mathrm{h}}=32$. The $\kappa$ index is in
the interval of the observed values during solar flares, see, e.g., Lörinčík
et al. (2020). The loss-cone angular half-width is taken as
$\Delta\theta_{\mathrm{c}}=2.5^{\circ}$.

The growth rate of the DPR instability is computed as (Melrose, 1986)

$\ilabel{eq:growth-
rate}\gamma(\omega,k_{\perp})=-\frac{\mathrm{Im}\,\epsilon_{\parallel}^{(1)}}{\left[\frac{\partial\mathrm{Re}\,\epsilon_{\parallel}^{(0)}}{\partial\omega}\right]_{\epsilon_{\parallel}^{(0)}=0}},$
(8)

where $\mathrm{Re}\,\epsilon_{\parallel}^{(0)}$ and
$\mathrm{Im}\,\epsilon_{\parallel}^{(1)}$ are the real and imaginary parts of
the plasma dispersion tensor component along the magnetic field, that are
obtained by a perturbation approach from the total dispersion tensor component

$\epsilon_{\parallel}=\epsilon_{\parallel}^{(0)}+\epsilon_{\parallel}^{(1)}+\mathcal{O}(\epsilon_{\parallel}^{(2)}).$
(9)

The imaginary part of the tensor component in Equation eq:growth-rate for the
growth rate was derived by Kuznetsov (2005)

$\displaystyle\mathrm{Im}(\epsilon_{\parallel}^{(1)})$ $\displaystyle=$
$\displaystyle-2\pi^{2}m_{\mathrm{e}}^{3}\frac{\omega_{\mathrm{pe}}^{2}}{k^{2}}\sum_{l=s+1}^{\infty}a^{3}\times$
(10)
$\displaystyle\times\int_{0}^{\pi}J_{l}\left(\frac{\gamma_{\mathrm{rel}}k_{\perp}v_{\perp}}{\omega_{\mathrm{ce}}}\right)\frac{\gamma_{\mathrm{rel}}^{2}\sin\phi}{\frac{\partial\psi}{\partial\rho}}\frac{1}{v_{\perp}}\frac{\partial
f}{\partial u_{\perp}}\,\mathrm{d}\phi,$

$\frac{\partial\psi}{\partial\rho}=\frac{1}{c^{2}}\left(v_{\parallel}^{2}+v_{\perp}^{2}\right).$
(11)

$v_{\parallel}=-a\cos(\phi),\quad v_{\perp}=a\sin(\phi),\quad
a^{2}=\frac{c^{2}(l^{2}\omega_{\mathrm{ce}}^{2}-\omega^{2})}{l^{2}\omega_{\mathrm{ce}}^{2}}$
(12)
$\ilabel{eq:lambda}\omega_{\mathrm{pe}}^{2}=\omega_{\mathrm{pb}}^{2}+\omega_{\mathrm{ph}}^{2}=\frac{(n_{\mathrm{b}}+n_{\mathrm{h}})e^{2}}{m_{\mathrm{e}}\epsilon_{\mathrm{0}}},\qquad\lambda=\frac{k_{\perp}^{2}v_{\mathrm{tb}}^{2}}{\omega_{\mathrm{ce}}^{2}},$
(13)

where $\omega_{\mathrm{pe}}$ is the plasma frequency of both electron species,
$\omega_{\mathrm{ce}}=eB/m_{\mathrm{e}}$ is the electron cyclotron frequency,
$\mathbf{k}=(k_{\perp},k_{\parallel})$ is the wave vector of the electrostatic
waves and we assume $k_{\parallel}=0$, $\omega$ is the wave frequency,
$I_{l}(\lambda)$ is the modified Bessel function of the first kind of the
$l$-th order, $\lambda$ is the dimensionless parameter of the Bessel function,
$e$ is the electron charge, $\epsilon_{0}$ is the permittivity of vacuum, and
$s$ is the gyro-harmonic number. The integration is over the resonance ellipse
of Equation eq1.

The denominator in Equation eq:growth-rate can be expressed as

$\ilabel{eq:partial_{e}psilon_{o}mega}\frac{\partial\epsilon_{\parallel}^{(0)}}{\partial\omega}=4\omega\omega_{\mathrm{pb}}^{2}\frac{e^{-\lambda}}{\lambda}\sum_{l=1}^{\infty}\frac{l^{2}I_{l}(\lambda)}{(\omega^{2}-l^{2}\omega_{\mathrm{ce}}^{2})^{2}}.$
(14)

Equation eq:growth-rate can be mathematically evaluated in the $\omega-
k_{\perp}$ domain; however, only the waves that are solutions of the plasma
dispersion relation are actually present in the plasma and can grow. To
simplify the calculation of growth rates, we choose the dispersion relation of
the UH waves as

$\ilabel{eq:UH-
waves}\omega^{2}=\omega_{\mathrm{pb}}^{2}+\omega_{\mathrm{ce}}^{2}+3v_{\mathrm{tb}}^{2}k_{\perp}^{2},$
(15)

where we assume that the dispersion relation is given only by the background
plasma component, $\epsilon_{\parallel}^{(0)}$.

The numerical procedure for the computation of a growth rate value for a given
set of plasma parameters ($\omega_{\mathrm{pe}}/\omega_{\mathrm{ce}}$,
$v_{\mathrm{tb}}$, $v_{\mathrm{\kappa}}$, $\kappa$,
$\Delta\theta_{\mathrm{c}}$, and $\theta_{\mathrm{c}}$) is the following. We
equidistantly divide the wavenumber space of the UH branch into $N=1500$ grid
points in the $k_{\perp}c/\omega_{\mathrm{pe}}\in[1,40]$ interval. According
to our tests, this interval is wide enough for the whole range of studied
background temperatures. Approximately, the typical wavenumber $k_{\perp}$
increases with decreasing background thermal velocity $[v_{\mathrm{tb}}]$ as
follows from Equation eq:lambda when the parameter $\lambda$ is constant. The
unstable waves are typically located in an interval
$k_{\perp}c/\omega_{\mathrm{pe}}\approx[4,15]$ for a temperature of 3 MK and
in an interval $k_{\perp}c/\omega_{\mathrm{pe}}\approx[10,37]$ for a
temperature of 0.5 MK. Moreover, these intervals slightly vary with other
parameters. The grid cell size is $\Delta
k_{\perp}c/\omega_{\mathrm{pe}}=0.026$. For each grid cell $k_{\perp i}$,
where $i=1,\ldots,1500$ is the grid cell index, we compute the frequency of
the UH wave $\omega_{i}$ from Equation eq:UH-waves. For the frequency
[$\omega_{i}$], we evaluate the growth rate $\gamma_{i}$ in each grid cell
[$k_{\perp i}$]. Then, we compute the mean value of the growth rate over all
positive $\gamma_{i}$ as

$\ilabel{eq:integrated}\Gamma=\frac{1}{\Gamma_{0}}\sum_{i}\gamma_{i}(\omega_{i},k_{\perp
i})\sigma_{i}(\omega_{i},k_{\perp i})\Delta k_{\perp},$ (16)

where $\sigma_{i}$ is the characteristic width of the dispersion branch for
each $i$-th element (Benáček and Karlický, 2019; Manthei et al., 2021). The
characteristic width $\sigma_{i}$ is estimated as

$\sigma_{i}=\left.\left(\frac{\partial\epsilon_{\parallel}^{(0)}}{\partial\omega}\right)^{-1}\right|_{\omega_{i},k_{\perp
i}},$ (17)

$\Gamma_{0}$ is the normalization parameter

$\Gamma_{0}=\sum_{i}\sigma_{i}(\omega,k_{\perp})\Delta k_{\perp},$ (18)

We denote the mean growth rate value $\Gamma$ as integrated growth rate.

Figure 2.: Growth rates as a function of the frequency and wavenumber for four
loss-cone angles $\theta_{\mathrm{c}}$, the gyro-harmonic number is $s=50$,
the temperature of the background plasma 3 MK, and the velocity of
$\kappa$-distribution $v_{\mathrm{\kappa}}=0.3\,c$. Note that the color scale
in (a) differs from those in (b-d). Green curves: the dispersion branch of UH
waves. All these UH branches are the same in all panels because they are
independent of the loss-cone distribution.fig1

##  3 Results

sec:results In the following, our results on the growth rate of the
electrostatic waves are shown as a function of the gyro-harmonic number
$s=\sqrt{\omega_{\mathrm{pe}}^{2}/\omega_{\mathrm{ce}}^{2}+1}$, as follows
from the radio emission process. Because the frequency ratio changes with the
height in the corona, zebra stripes are produced only when the profile of
integrated growth rate forms peaks at specific frequency ratios. The peaks are
not always located at integers of $s$ (harmonics of the cyclotron frequency)
because there are other effects that can shift them, such as the effect due to
the Lorentz factor in Equation eq1, as was shown by Benáček, Karlický, and
Yasnov (2017).

Figure fig1 shows growth rates in the $\omega-k_{\perp}$ space for four
selected loss-cone angles $\theta_{\mathrm{c}}$ = 30∘, 50∘, 65∘, and 80∘, the
background temperature 3 MK ($\approx$130 eV), and the velocity of
$\kappa$-distribution $v_{\kappa}=0.3$ c ($\approx$23 keV). The gyro-harmonic
number is fixed to $s=50$. The growth rates are smallest for the small loss-
cone angle $30^{\circ}$. By increasing the loss-cone angles to $50^{\circ}$
and $65^{\circ}$, the growth rate values increase and the positive growth rate
regions shift to smaller wavenumbers. For the loss-cone angle $80^{\circ}$,
the growth rates values and areas of positive growth rate regions decrease.

The growth rates are overlaid by the dispersion branch of the UH waves
(Equation eq:UH-waves). If the branch crosses a positive growth rate region,
the waves located in the intersection can grow. During the growth phase, the
kinetic energy of resonant hot electrons, whose velocities fulfill Equation
eq1, is converted into electrostatic wave energy and the particle distribution
changes. Finally, the growing waves saturate when the electron velocity
distribution becomes stable.

Each region in the $\omega-k_{\perp}$ space with a positive growth rate is
associated with a specific gyro-harmonic number $s$. These regions have a
distance in frequency dimension approximately equal to the electron cyclotron
frequency. When the frequency ratio varies, the positive regions shift in
frequency so that the crossings of positive growth rate regions with the UH
branch change. This way we expect the formation of growth rate peaks (which
may be responsible for the formation of zebra stripes). However, the peaks are
not so significant if the UH branch crosses several positive regions
simultaneously (e.g., Figure fig1(b)).

Figure 3.: Integrated growth rates in four selected intervals of the gyro-
harmonic numbers and for four loss-cone angles. The temperature of the
background plasma is 3 MK, and the velocity of the $\kappa$-distribution
$v_{\mathrm{\kappa}}=0.3\,c$.fig2

In Figure fig2, the integrated growth rates are presented as a function of the
gyro-harmonic number for four loss-cone angles, background temperature 3 MK,
and the velocity of the $\kappa$-distribution $v_{\kappa}=0.3\,c$. The
integrated growth rates are computed following Equation eq:integrated. We
selected four intervals of gyro-harmonic numbers from small values $s=7$, for
which the radio zebras are not observed, up to the most extreme case of the
gyro-harmonic number exceeding 100. In all the cases the growth rates are
positive. For low gyro-harmonic numbers $s=7-10$, the growth rates form
distinguishable peaks for all loss-cone angles. By increasing $s$ the growth
rate profiles smooth out; however, the smoothing varies between different
loss-cone angles. For the angle 30∘, the growth rate profile smooths out
already for $s\gtrsim 20$, and their values decrease for higher gyro-harmonic
numbers. For the angles 50∘ and 65∘, the growth rate profiles smooth out in
the gyro-harmonic number interval $s=50-53$. Their average values slightly
increase and decrease for $\theta_{\mathrm{c}}=50^{\circ}$ and for
$\theta_{\mathrm{c}}=65^{\circ}$, respectively. For the loss-cone angle 80∘,
the growth rates monotonically increase within the whole studied interval of
gyro-harmonic numbers. At large gyro-harmonic numbers $s=100-103$, the growth
rate peaks are still formed.

Figure 4.: Integrated growth rates as a functions of the $\kappa$-distribution
velocity, for four selected intervals of the gyro-harmonic number, the loss-
cone angle 80 ∘ and the temperature of the background plasma 3 MK. Note that
the scaling of the growth rates varies between panels.fig3 Figure 5.: Same as
Figure fig3, but for the temperature of the background plasma 0.5 MK.fig4

Because the highest growth rates for the large gyro-harmonic numbers (derived
from zebra observations) occur for the angle $80^{\circ}$, we studied the
dependence of integrated growth rate profiles on two plasma temperatures (0.5
and 3 MK) and selected values for the velocities of the $\kappa$-distribution
in the interval $v_{\kappa}\in[0.15,0.3]c$, see Figures fig3 and fig4. Though
the temperature 0.5 MK may be smaller than those estimated during flares, the
accelerated electrons may reach and emit in the colder part of the corona.
Moreover, we selected this background temperature range to clearly identify
the temperature impact as the background temperature is known to have a small
impact on the instability (Benáček, Karlický, and Yasnov, 2017). The
integrated growth rates are computed in the same gyro-harmonic number
intervals as in Figure fig2.

For the plasma temperature 3 MK (Figures fig3), the average values (over few
$s$) of the integrated growth rates increase with the gyro-harmonic number
independently on the velocity of the $\kappa$-distribution. All growth rate
profiles smooth out for $s\sim 50$. The situation is reversed for large gyro-
harmonic numbers where the most distinct peaks are formed for the smallest
velocity of the $\kappa$-distribution $v_{\mathrm{\kappa}}=0.15$. In addition,
the integrated growth rates for this background velocity and gyro-harmonic
numbers form a double peak profile.

The behavior of the integrated growth rates as a function of the gyro-harmonic
number is different for the background temperature $0.5$ MK in a few aspects,
see Figure fig4. For velocities of the $\kappa$-distribution
$v_{\mathrm{\kappa}}\geq 0.2\,c$, the growth rate peaks are well distinct for
the whole range of the gyro-harmonic numbers. The distinction occurs because
the distribution functions of the hot and cold components are more separated
in velocity space than for the temperature 3 MK. Because of this separation,
the velocity gradient of the loss-cone distribution increases, and the growth
rate reaches higher values. For all velocities of the $\kappa$-distribution,
the growth rate values increase from $s=7-10$ to $s=20-23$. By further
increasing the gyro-harmonic numbers, the growth rate increases the most for
the smallest velocities of the $\kappa$-distribution. For the case of
$v_{\mathrm{\kappa}}=0.3\,c$, the growth rates remain almost the same as for
$s\gtrsim 20$.

Figure 6.: Comparison of the integrated growth rates for $\kappa$-indices 2
and 100, temperature 3 MK, and the velocity of the $\kappa$-distribution
$0.3\,c$.fig5

Figure fig5 shows a change in the integrated growth rates with the $\kappa$
index. We compare integrated growth rates for $\kappa=2$ and $\kappa=100$
(closer to the Maxwellian velocity distribution). The integrated growth rate
profiles for $\kappa=100$ are higher and with more distinct peaks than those
for $\kappa=2$.

###  3.1 The 14 February 1999 Zebra Case

Besides the above presented general results covering a broad range of the
gyro-harmonic number, let us summarize the results for the specific zebra
observed on the 14th February 1999 and shown in Figure fig0, as an example. In
this case the gyro-harmonic number was estimated as $s_{1}$ = 64 (Table tab1).
For this zebra, we calculated the integrated growth rates in the range
$s=64-67$, the loss-cone angles $55-80\,^{\circ}$, the temperatures 0.5 MK and
3 MK, and $v_{\kappa}\in[0.1,0.5]c$. From the analysis follows that the
emission is most likely produced by a loss-cone angle $\sim(70\pm 5)^{\circ}$.
The growth rate peaks are formed for a wider range of plasma temperatures and
velocities of the $\kappa$-distribution. For the plasma temperature of 3 MK,
the velocity of the $\kappa$-distribution must be $\geq$ 0.15 $c$. For the
plasma temperature of 0.5 MK, the velocity of the $\kappa$-distribution can be
in the whole analyzed range of $v_{\kappa}\in[0.1,0.5]c$. Our conclusion for
this zebra is that though the plasma temperature and the velocity of the
$\kappa$-distribution can be in a broad range of values to produce the
emission, the loss-cone angle should be restricted into a narrow range of
about $\sim(70\pm 5)^{\circ}$.

##  4 Discussion and Conclusions

sec:discussion The solar radio zebras are used for diagnostics of plasma
parameters during solar flares. One of the crucial parameters is the gyro-
harmonic number in the emission region, which allows to estimate the upper-
hybrid to cyclotron frequency ratio. Though the gyro-harmonic numbers
determined from zebra observations are greater than about 50 and may exceed
one hundred, most of the studies have so far focused on significantly smaller
gyro-harmonic numbers, rarely higher than 20. We addressed the question of how
the solar radio zebras can be produced with such high gyro-harmonic numbers by
analyzing the DPR instability in a broad range of gyro-harmonic numbers
$s=7-103$.

We calculated linear growth rates and studied the effects of the loss-cone
angle, velocity of the $\kappa$-distribution, the background plasma
temperature, and the $\kappa$ index on the instability. We found that the
zebras with high gyro-harmonic numbers are produced by loss-cone distributions
with large loss-cone angles. Moreover, the higher the gyro-harmonic number is,
the larger the loss-cone angle needs to be to produce the instability. For the
gyro-harmonic number $\sim 100$, a loss-cone angle $\sim 80^{\circ}$ is
sufficient. However, for even larger gyro-harmonic numbers, we expect that the
necessary high angles of the loss-cone will be hard to produce and therefore
unlikely to be observed.

###  4.1 Growth Rate Properties

Our calculations show that for increasing gyro-harmonic number the integrated
growth rates of the DPR instability increase for large values of the loss-cone
angle ($\sim 80^{\circ}$). Moreover, the growth rate peaks are distinct for
gyro-harmonic numbers $s\sim 100$ and smooth for smaller loss-cone angles
$<80^{\circ}$. The integrated growth rates are large enough for all loss-cone
angles $\gtrsim 50^{\circ}$ to produce zebra type of radio emission.

In some cases, the growth rates show only small variations depending on $s$.
However, we note that these variations may produce a significant difference in
the saturation energy levels, as was shown using particle-in-cell simulations
by Benáček and Karlický (2018).

The growth rate average values (over a broader range of gyro-harmonic numbers)
monotonously decrease only for a loss-cone angle 30∘. In contrast, for the
loss-cone angle 80∘, the average values monotonously increase.

Moreover, how distinct the integrated growth rate maxima are for high gyro-
harmonic numbers $s\sim 50-100$ and for large loss-cone angles depends on the
temperature of the background plasma and the velocity of the
$\kappa$-distribution. Assuming that the instability occurs in a heated plasma
during a flare with temperature 3 MK, the growth rate profiles are smoothed
out around $s\sim 50$ but form emission peaks for $s\sim 100$. Moreover, for
the gyro-harmonic number $s\sim 100$, they produce double-peaked profiles for
small velocities of the $\kappa$-distribution $v_{\kappa}=0.15c$. If the
emission is formed in a region with a colder plasma of temperature 0.5 MK,
e.g., further from the flare region, the growth rates are distinct for the
whole range of gyro-harmonic numbers. Nonetheless, for both temperatures and
large gyro-harmonic numbers, higher growth rates are favored for smaller
$\kappa$-distribution velocities in the range $v_{\kappa}\in[0.15,0.20]c$ than
for larger values in the range $v_{\kappa}\in[0.25,0.3]c$.

###  4.2 Proposed Zebra Emission Regions

The fact that high gyro-harmonic numbers are detected in zebras means that the
zebra-stripe sources are in regions where the ratio of
$\omega_{\mathrm{pe}}/\omega_{\mathrm{ce}}$ is high. In combination with the
result that higher growth rates are favored for large loss-cone angles, two
possible explanations can be proposed:

  1. 1.

We may assume that the loop cross-section is only slightly wider at their
midpoints than at their footpoints, in agreement with Klimchuk (2000). In such
a loop, the change of the magnetic field along the loop is small, and only
superthermal electrons with high pitch angles can be trapped (Krucker, Masuda,
and White, 2020). This way, a loss-cone distribution with a high loss-cone
angle is formed. Thus, the instability can generate the UH waves and the
corresponding zebra stripes with high gyro-harmonic numbers. Moreover, for the
zebra stripes with high $s$, the magnetic field in this loop needs to be
relatively small ($\omega_{\mathrm{pe}}/\omega_{\mathrm{ce}}\gg$1). We also
note that the small change of the magnetic field strength along the loop
agrees with the small difference in magnetic field strength derived from the
neighboring zebra-stripe frequencies for high values of the gyro-harmonic
numbers. Typical intensity of the magnetic field in these regions can be
derived as $B$ = $f_{s}$/(2.8$\cdot s$)), where $B$ is the magnetic field
strength in a unit of gauss, and fs is the zebra-stripe frequency in a unit of
MHz with the gyro-harmonic number $s$ (Karlický and Yasnov, 2018). For low
values of $s$, these differences are much higher. Therefore, in a loop with a
small magnetic field gradient, spatial distances of zebra-stripe sources with
low values of $s$ will be much longer than those for high values of $s$.

  2. 2.

Considering the standard flare model (Aschwanden, 2004), a region with high
values of $\omega_{\mathrm{pe}}/\omega_{\mathrm{ce}}$ can be found below the
X-point of the flare. The plasma density here is relatively high (comparable
to the current sheet density), and the magnetic field in the reconnection
outflow is relatively low. The electrons from reconnection move downward and
penetrate into newly formed magnetic loops where they may be trapped (Švestka
et al., 1987). If the electrons initially had perpendicular velocities to the
magnetic field larger than their parallel ones, they can form loss-cone
distributions with large loss-cone angles. Electrons are accelerated not only
at the X-point of magnetic reconnection but also during the shrinking of
magnetic loops in the reconnection outflow Karlický and Kosugi (2004).

###  4.3 Double-Peaked Pattern of Growth Rates

We searched for the double-peaked pattern from Figure fig3(d), whether it also
occurs for smaller gyro-harmonic numbers, various temperatures, and
$\kappa$-distribution velocities. But it did not appear anywhere in our
calculations. Also, we performed tests to discard the numerical uncertainty at
the origin of the double-peaked profiles. We varied the ranges of calculations
and the precision of the numerical integrals and tested individual parts of
calculations for these parameters, but we found no discrepancies or errors.
Nonetheless, due to the complexity of the calculations, we could not say
whether there are hidden numerical artifacts produced by an approximation made
in the analytical growth rate derivation.

The double-peaked growth rate profile is generally formed when the growth rate
region in the $\omega-k_{\perp}$ space is not as the one shown in Figure fig1,
i.e., when it is not forming only one tadpole region for each gyro-harmonic
number, but it forms two (or more) separated regions. Therefore, the UH branch
could cross two regions and create two growth rate maxima. For this happen, it
would however require a specific kind of velocity distribution function,
probably with two distinct regions in the $\omega-k$ space of positive
velocity gradients.

A question arises about how we could observationally recognize that the
adjacent zebra stripes are formed by double-peaked growth rates. Assuming that
there are two distinct regions in the $\omega-k$ space, they probably do not
have the same shape. Therefore, both generated zebra stripes could be created
at different emission intensities, with different frequency ratios between odd
and even stripes, and odd and even stripes might have different emission
profiles (e.g., odd stripes more narrow and even stripes broader).
Nonetheless, if the regions are similar and regularly distributed in the
$\omega-k$ space, also the emission profiles can be similar. Moreover, we may
speculate that a change or evolution of the velocity distribution during the
emission might allow transition from a double-peaked to a single peaked
profile, or vice versa, as in the observed zebras with merging zebra stripes
(Chernov et al., 2005).

###  4.4 Comparison with other Procedures of the Growth Rates Calculation

Figure 7.: Comparison of the growth rate calculations: (a) using the maximal
method of Kuznetsov and Tsap (2007) with (b) the maximal and (c) averaged
method used in this paper and based on an integration over the whole resonance
ellipse (Kuznetsov, 2005). The background temperature is 0.5 MK, and the
$\kappa$-distribution velocity is $v_{\mathrm{\kappa}}=0.3\,c$. Compare the
growth rates in panel (c) with the integrated growth rates for higher
background temperature in Figure fig2c.fig7

Kuznetsov and Tsap (2007) (TS07) found that growth rates decrease with
increasing gyro-harmonic number. They performed calculations for a power-law
distribution (power-law index $\delta=10$) for the loss-cone angle
$30^{\circ}$. Their approach was to estimate the growth rates only at a
specific part of the resonance ellipse (Equation eq1). In contrast, in our
approach we integrate over the whole ellipse in Equation eq-growth for each
specific point in the $\omega-k_{\perp}$ space (Kuznetsov, 2005) (K05). To
obtain the integrated growth rate, we integrate over the UH dispersion branch,
which produces more reliable results according to the tests by
electromagnetic, relativistic particle-in-cell simulations (Benáček and
Karlický, 2019; Manthei et al., 2021) (BK19). Though TS07 used a different
kind of loss-cone distribution, the decrease of the growth rates with
increasing gyro-harmonic number for the loss-cone angle 30∘ is in agreement
with our results. Nonetheless, the exact growth rate values differ.

We compare these three approaches of growth rate calculations in Figure fig7
for the gyro-harmonic number $s\sim 50$ as a function of the loss-cone angle.
In all three cases, the growth rates are selected on the UH dispersion branch
for 0.5 MK and $v_{\mathrm{\kappa}}=0.3\,c$. The maximal growth rates of the
K05 method are smaller than those of the KS07 method for loss-cone angles
$\leq$65∘. However, growth rates for the angle 80∘, which are smaller in the
KT07 method, are the largest in the K05 method. Moreover, all growth rate
profiles are relatively smoother for the KT07 method than for the K05 method.
The integrated growth rates of the BK19 method are systematically lower than
the maximal growth rates (from the definition of the average along the
dispersion branch in Equation eq:integrated). In agreement with BK19, who
studied the integrated growth rates only for small gyro-harmonic numbers
$\omega_{\mathrm{pe}}/\omega_{\mathrm{ce}}\approx 4-5.2$, the peaks of the
integrated growth rate are more distinct than (normal) growth rates also for
high gyro-harmonic numbers.

Yasnov, Benáček, and Karlický (2019) among others computed growth rates for
the loss-cone angle 80∘ using the same method as KS07. They found negative
growth rates for large loss-cone angles. However, they applied a loss-cone
$\kappa$-distribution with an additional cut-off at $v_{\mathrm{m}}=0.3\,c$,
where $v_{\mathrm{m}}$ is the cut-off velocity. For distributions without the
cut-off, they found that the growth rates are very broad for
$\theta_{\mathrm{c}}\sim 50^{\circ}$. As we show in Figure fig7, the
integrated growth rate peaks may still be distinct.

Increasing the $\kappa$ index of the $\kappa$-distribution from two to one
hundred, the integrated growth rates increase, and their maxima become more
distinct. Increasing $\kappa$ index, there are more particles with smaller
$\kappa$-distribution velocities. Therefore, these particles increase the
distribution gradients and contribute more in the integral of Equation eq-
growth over the resonance ellipse.

We conclude that the DPR instability model may produce solar radio zebras with
high gyro-harmonic numbers for large loss-cone angles $\sim 80^{\circ}$. The
growth rate maxima are distinct, and they are high enough to generate zebra
patterns.

#### Acknowledgements

We acknowledge the valuable comments by an anonymous reviewer and the proof
reading by Dr. Patricio A. Muñoz.

## Funding

J. B. acknowledges support from the German Science Foundation (DFG) project BU
777-17-1. M. K. acknowledges support from the project RVO-67985815 and GAČR
grants 20-09922J, 20-07908S, 21-16508J, and 22-34841S. The authors gratefully
acknowledge the Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) for
partially funding this project by providing computing time on the GCS
Supercomputer SuperMUC-NG at Leibniz Supercomputing Centre (www.lrz.de),
project pn73ne. This work was also supported by the Ministry of Education,
Youth and Sports of the Czech Republic through the e-INFRA CZ (ID:90140).

## Data Availability

The datasets generated during and/or analyzed during the current study are
available from the corresponding author on reasonable request.

## Conflict of interest

The authors declare that there are no conflicts of interest.

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