# Generation mechanism and beaming of Jovian nKOM from 3D numerical modeling
of Juno/Waves observations

###### Abstract

The narrowband kilometric radiation (nKOM) is a Jovian low-frequency radio
component identified as a plasma emission produced in the region of the Io
plasma torus. Measurements from the Waves instrument onboard the Juno
spacecraft permitted to establish the distribution of nKOM occurrence and
intensity as a function of frequency and latitude. We have developed a 3D
geometrical model that can simulate at large scale the plasma emissions
occurrence observed by a spacecraft based on an internal Jovian magnetic field
model and a diffusive equilibrium model of the plasma density in Jupiter’s
inner magnetosphere. With this model, we propose a new method to discriminate
the generation mechanism, wave mode, beaming and radio source location of
plasma emissions. Here, this method is applied to the study of the nKOM
observed from all latitudes by the Juno/Waves experiment to identify which
conditions reasonably reproduce the observed occurrence distribution versus
frequency and latitude. The results allow us to exclude the two main nKOM
models published so far, and to show that the emission must be produced at the
local plasma frequency and beamed along its local gradient in the direction of
decreasing frequencies. We also propose that depending on its latitude, Juno
observes two distinct kinds of nKOM: the low frequency nKOM in ordinary mode
at high latitudes and high frequency nKOM on extraordinary mode at low
latitudes. Both radio source locations are found to be distributed near the
centrifugal equator from the outer edge to the inner edge of the Io plasma
torus.

Journal of Geophysical Research

LESIA, Observatoire de Paris, CNRS, PSL, Sorbonne Université, Université Paris
Cité, Meudon, France Observatoire Radioastronomique de Nançay, ORN,
Observatoire de Paris, CNRS, PSL, Université d’Orléans, Nançay, France School
of Cosmic Physics, DIAS Dunsink Observatory, Dublin Institute for Advanced
Studies, Dublin, Ireland Department of Electrical Engineering and Information
Science, National Institute of Technology (KOSEN), Niihama College, Niihama,
Japan

Adam<EMAIL_ADDRESS>
We developed a 3D modeling method to test the generation mechanism and beaming
of plasma emissions from Jupiter’s inner magnetosphere.

nKOM occurrence distribution is reproduced with plasma emissions at $f_{pe}$
beamed in the opposite direction of their frequency gradient.

nKOM is compatible with O-mode at high latitudes and X-mode at low latitudes,
from radio sources distributed near the centrifugal equator.

## Plain Language Summary

This paper investigates a specific type of Jupiter’s natural radio emissions
called “narrowband kilometric radiation” (nKOM). The nKOM is produced within
the region of Jupiter’s Io plasma torus. Using data collected by the Waves
instrument on the Juno spacecraft, we analyze when and how the nKOM occurs
across different frequencies and latitudes. To better understand these radio
emissions, we have developed a 3D model that simulates the occurrence of nKOM.
Additionally, we introduce a novel methodology for determining key
characteristics of nKOM, including its generation mechanism, propagation mode,
beaming direction, and the source location of the radio emissions. Applying
his model to Juno’s observations, we found that the previously proposed
theoretical models for nKOM generation do not match the observed data.
Instead, the study suggests that nKOM is generated at the local plasma
frequency and is beamed along its local gradient, in the direction of
decreasing frequencies. The study also revealed, two distinct types of nKOM
depending on Juno’s latitude: low-frequency nKOM in ordinary mode at high
latitudes and high-frequency nKOM in extraordinary mode at low latitudes. Both
types of nKOM are found to originate near the Jupiter’s centrifugal equator.

##  1 Introduction

A large variety of radio emissions are produced below 40 MHz in the inner
Jovian magnetosphere. Only the decameter emissions are accessible to ground-
based observations due to the Earth’s ionospheric cutoff at 10 MHz. But
observations from the two Voyager spacecraft extended Jupiter’s radio spectrum
to the hectometer and kilometer wavelength ranges [Warwick . (19792), Warwick
. (19791)]. Two components were discovered in the kilometer range: a broadband
($\sim$10 kHz to $\sim$1 MHz) sporadic emission called bKOM [Desch Kaiser
(1980)], and a narrowband smoother one restricted to frequencies around 100
kHz called nKOM [Kaiser Desch (1980)]. Both kilometer wave components display
a minimum of occurrence for an observer near Jupiter’s equator, more marked
for bKOM [Kurth . (1980), Daigne Leblanc (1986), Louis, Zarka, Dabidin .
(2021)]. This led jones1980,jones1986,jones1987 to propose for both bKOM and
nKOM a generation theory based on the linear conversion of Z-mode
electrostatic waves at the upper hybrid resonance frequency
$f_{uh}=(f_{pe}^{2}+f_{ce}^{2})^{1/2}$ (with $f_{pe}$ the electron plasma
frequency and $f_{ce}$ the electron cyclotron frequency) into O-mode waves.
But studies of Voyager data soon showed that for bKOM, polarization is
consistent with X-mode waves and occurrence is modulated at Jupiter’s rotation
period (system III, $\sim 9.925$h) [Desch Kaiser (1980), Leblanc Daigne
(1985)], while nKOM is more consistent with an O-mode emission and is
modulated at a variable period $\sim$3% to 5% longer than Jupiter’s system III
rotation period [Kaiser Desch (1980), Daigne Leblanc (1986)]. Next, radio
goniopolarimetric (i.e. direction-finding + polarization) measurements by the
Ulysses spacecraft showed unambiguously that bKOM originates from the high-
latitude auroral region [Ladreiter . (1994)] whereas nKOM originates from the
vicinity of Io’s plasma torus (IPT, extending from slightly under 6 $R_{J}$ to
$\sim$ 12 $R_{J}$, with $R_{J}~{}=~{}71492$ km the Jovian radius) [Reiner .
(1993)]. This scenario is supported by Juno measurements [Imai . (2017)].
These results identified the bKOM as an X-mode auroral component produced by
the electron cyclotron maser mechanism near the local $f_{ce}$ [Leblanc Daigne
(1985), CS. Wu (1985), Zarka (1998), Treumann (2006)], and they confirmed the
early hypothesis that nKOM was a low-latitude plasma emission (near $f_{pe}$
or $2~{}f_{pe}$) mainly produced on the O-mode near the outer edge of the IPT
[Kaiser Desch (1980), Carr . (1983), Daigne Leblanc (1986), Reiner . (1993)].

Beyond these qualitative conclusions, there is no consensus about the detailed
conditions for generation of nKOM, namely its frequency (near the local
$f_{pe}$ [Jones (1980), Jones (1986), Jones (1987)] or $f_{uh}$, or its first
harmonic [Fung Papadopoulos (1987)]), its beaming (along the local magnetic
field $\bf{B}$, the electron density gradient $\nabla n_{e}$, or at some
specific angle relative to these vectors), and its dependence on the density
gradient strength $||\nabla n_{e}||$, the magnetic field amplitude $|\bf{B}|$,
or the angle between $\nabla n_{e}$ and $\bf{B}$. Despite O-mode being
seemingly dominant, as nKOM have also been observed in X-mode [Reiner .
(1993)], its favored mode of emissions and their sources locations, remains to
be confirmed.

In the absence of wave and particle measurements inside nKOM radio sources,
that would allow us to answer the above questions as was done for auroral
radio sources [<]see e.g.¿[]zarka1998, louarn2017, louarn2018, louis2020,
collet2023, we investigate here if large-scale modeling can reproduce the
observed statistical distribution of nKOM occurrence versus frequency and
latitude [Louis, Zarka, Dabidin . (2021)], in order to provide strong
constraints on its conditions of emission and thus answer at least part of the
above questions.

In section 2, we detail key aspects of the theories proposed for nKOM
generation, and the hypotheses and parameters that will be tested via our
modeling. In section 3, we present the observational data that our modeling
intends to reproduce, for testing the relevance of each theory and hypothesis.
Section 4 describes the plasma and field models used and our modeling
framework. The results are presented in section 5, and discussed in section 6
that also gives further perspectives to this work.

##  2 Theories

As mentioned in the introduction, observations from Voyager, Ulysses and Juno
suggest that nKOM is produced in or near the IPT. In that region $f_{pe}\geq
f_{ce}$ or $f_{pe}\sim f_{ce}$, favouring mode conversion mechanisms [Melrose
(2017)]. The literature proposes two classes of theories for the generation of
nKOM, from which we define four scenarios below.

The first theory, developed by jones1980, jones1986, jones1987, is based on
the radio window theory [Budden (1966), Budden Terry (1971), Jones (1976),
Budden (1980)] applied to the Jovian magnetosphere to explain the latitudinal
beaming of the nKOM observed by Voyager. They predict Z-mode electrostatic
waves, to produce Cerenkov radiation when $f_{pe}<f<f_{uh}$ that cannot escape
the Io plasma torus. If the Z-mode wave propagates toward the decreasing
plasma density, it is either absorbed or reflected at the upper-hybrid
resonance $f\sim f_{uh}$. If the Z-mode wave propagate toward the increasing
plasma density, it is absorbed when reaching the plasma frequency $f\sim
f_{pe}$. However, if the Z-mode and the O-mode refractive indexes are close
when $f\sim f_{pe}$, the Z-mode is able to convert into O-mode through the so-
called “radio window”. The process requires density gradients $\nabla n_{e}$
to be weak enough for the radio window to occur according to the ray theory
approximations. They also suggest Z-mode quasi-perpendicular propagation with
respect to the magnetic field $\bf{B}$ (which is equivalent to quasi-
perpendicularity between $\bf{B}$ and $\nabla n_{e}$) to full-fill upper-
hybrid resonance and maximize the mode conversion efficiency. The O-mode radio
wave, first generated along $\nabla n_{e}$, is finally beamed in two opposite
directions making angles $\beta~{}=~{}\arctan(f_{pe}/f_{ce})^{1/2}$ and
$\pi~{}-~{}\beta$ with respect to $\bf{B}$ in the plane defined by ($\nabla
n_{e}$, $\bf{B}$). This will be our scenario #1.

The theory of fung1987 conversely suggests that nKOM is produced at $2f_{uh}$
by nonlinear coupling of two electrostatic waves each at $f_{uh}$, propagating
through density inhomogeneities at the outer periphery of the IPT. Emission is
favoured at places where the density gradient $\nabla n_{e}$ is weak and makes
a specific angle ($\sim 47^{\circ}$) with the local $\bf{B}$. The theory
predicts both O-mode and X-mode emissions, beamed in a plane nearly
perpendicular to $\bf{B}$. This will be our scenario #2.

The radio window theory has been updated by budden1986, removing the ray
theory approximation, as observations have shown that larger density gradients
are likely to be present in the potential plasma emissions generation regions.
Applied to the Jovian magnetosphere it can predict the conversion of Z-mode
into O-mode and X-mode if the density gradient is strong enough. Following the
results from budden1986, the beaming angle $\beta$ from jones1980 might not be
relevant to describe the beaming of O-mode nKOM when the density gradient is
strong. Thus, we will also consider a scenario #3 where emission is produced
(through mode conversion) at the local $f_{pe}$ along its local gradient
$-\nabla f_{pe}$ and, for completeness, a similar scenario #4 with emissions
produced at the local $f_{uh}$ along its local gradient ($-\nabla f_{uh}$).
These four nKOM generation scenarios are summarized in Table 1. A more
detailed version of this table is provided in the appendix (Table
LABEL:tab_appendix:cases).

##  3 Data

Figure 1: Two 24-hour dynamic spectra of calibrated Juno/Waves data [Louis,
Zarka, Dabidin . (2021)] observed (a) from low latitudes (day-of-year
2017-122, i.e. 05/02/2017) and (b) around the first perijove PJ1 (day-of-year
2016-240, i.e. 27/08/2017). The white boxes correspond to the cataloged nKOM
components with their labels displayed in black [Louis, Zarka, Cecconi Kurth
(2021)]. The horizontal axis corresponds to the Juno observations time period,
radial distance (Rj), system III longitude (LonIII) and jovicentric latitude
(Lat). The vertical axis represent the Juno/Waves frequency channels from 10
kHz to 141 kHz, distributed on a logarithmic scale with a ratio $\times 1.12$
between consecutive channels. The saturated band at 140 kHz correspond to the
channel 61 of the HFR-low receiver that has a poor sensitivity. The two color
bars represent the calibrated flux densities measured by the Waves experiment
at different times. The intensity scales are adjusted to highlight the nKOM
occurrences.

In Voyager [Kaiser Desch (1980), Daigne Leblanc (1986)], Galileo [Louarn .
(1998)], Cassini [Zarka . (2021)] and Juno [Imai . (2017), Louis, Zarka,
Dabidin . (2021)] data, nKOM is detected as smooth patches lasting for one to
a few hours, that can reoccur every 10.3h to 10.5h during several days (cf.
Figure 1). This periodicity is consistent with the plasma motion in the outer
parts of the IPT, whose inertia causes slight subcorotation relative to
Jupiter’s magnetic field rotation at the system III period.
louarn1998,louarn2000 further proposed a scenario explaining intermittent
activity periods of 3-4 days by centrifugal ejections of plasma from the IPT,
exciting locally nKOM and also causing brightenings of auroral emissions
(bKOM, hectometer emission). An effect of the solar wind causing variable
magnetospheric compressions is not excluded [Nozawa . (2006), Louis . (2023)],
although it is expected to be stronger in the outer magnetosphere [Zarka .
(2021)] than in the IPT. This intermittency of nKOM makes difficult to
characterize it on an event by event basis. We thus focused on statistical
constraints.

Figure 2: nKOM occurrence probability versus frequency and latitude derived
from the first 3 years of Juno/Waves low-frequency observations around Jupiter
by louis2021. Latitude bins are 1° wide and frequencies correspond to
Juno/Waves frequency channels from 10 kHz to 141 kHz, distributed on a
logarithmic scale with a ratio $\times 1.12$ between consecutive channels.
This figure is based on the same data as Fig. S4a from louis2021, except that
no smoothing has been applied.

Based on 6 months of Cassini radio observations from low Jovian latitudes,
zarka2004 found that nKOM covers the range $\sim 60-160$ kHz (very different
from the more sporadic bKOM that covers the range $\sim 23-400$ kHz in that
same data set). Statistics over 3 years of Juno radio observations from all
latitudes (from 9 April 2016 to 24 June 2019) allowed louis2021 to extend this
range down to 20-30 kHz and to characterize the variations of nKOM occurence
(and intensity, but we only use the occurrence here) with the latitude of the
observer (cf. Figure 2). Juno could not precise the upper frequency limit of
nKOM because its radio instrument has a very limited sensitivity in the range
140 kHz–3 MHz. The time-averaged distribution of Figure 2 is asymmetrical
between the two Jovian hemispheres. Above 60 kHz, nKOM occurrence peaks at
$-40^{\circ}\pm 15^{\circ}$ in the South and $+15^{\circ}\pm 15^{\circ}$ in
the North, with a minimum of occurrence about $-15^{\circ}$, and a gap around
$+30^{\circ}$ below $\sim$100 kHz. At lower frequencies, one observes
extensions of nKOM occurrence down to $-70^{\circ}$ latitude around 50 kHz,
and a very strong peak up to $+75^{\circ}$ latitude at $30-50$ kHz.

We display on Figure 1 representative examples of the Juno/Waves nKOM
observations at two geometrical configurations: (a) when Juno is far from
Jupiter and at low latitudes and (b) when Juno is close to Jupiter and at high
latitudes (near its perijove). Panel (a) of Figure 1 displays two occurrences
of nKOM events separated by $\sim$ 10h, while panel (b) displays two distinct
nKOM events observed each for a longer period of time but without periodicity.
The corresponding radio sources may be wider in longitudes than panel (a), or
the occurrence is here dominated by the variation of position of Juno along
its perijove orbit. Furthermore, when Juno is (a) at low latitudes, the nKOM
is observed for frequencies roughly $>40$ kHz, while when Juno is (b) at high
latitudes, the nKOM is observed at significantly lower frequencies, with a
different frequency range for the nKOM observed before the perijove (30-50
kHz) and the nKOM observed after the perijove (40-60 kHz).

Figures 1 and 2 suggest that nKOM is produced at specific locations in the
IPT. The variable frequency ranges observed at low and high latitudes by Juno,
should provide information on the large-scale beaming and distribution of the
radiosource inside the IPT.

##  4 Modeling

Our purpose is to derive significant constraints on nKOM generation from the
modeling of its large-scale time-averaged latitude-frequency distribution of
Figure 2. We will thus apply the conditions defined in each scenario to a grid
of modeled plasma density and magnetic field values in Jupiter’s inner
magnetosphere, synthesize the frequency-latitude occurrence distribution of
nKOM detected along the trajectory of Juno between 9 April 2016 and 24 June
2019, and compare it to the latitude versus frequency distribution of the
observations displayed on Figure 2.

###  4.1 Plasma Density & Magnetic Field Models

To model the plasma density in Jupiter’s inner magnetosphere and IPT, we use
the diffusive density model of imai2016 that computes the densities of the
main torus species (electrons, many positive ions of $S$ & $O$, $H^{+}$ and
$Na^{+}$) for radial distances between 4 and 13 $R_{J}$ and jovicentric
latitudes below $80^{\circ}$. This model is based on time-averaged plasma
(density and temperature) observations of Voyager 1 and 2 as a function of the
radial distance to Jupiter [Bagenal (1994), Moncuquet . (2002)]. The plasma
density is then predicted in a meridian plane by extrapolating the radial
profiles along Jupiter’s magnetic field lines using the VIP4 magnetic field
model plus current sheet [Connerney . (1998)]. For 3D representation of the
plasma density, cylindrical symmetry around the jovicentrifugal axis is
assumed. The imai2016 model has been preferred to the model by divine1983
because it computes the density in a more consistent way out of the region
where measurements were made.

The VIP4 magnetic field model (based on Voyager and Pioneer magnetic field
measurements plus infrared observations of the Io flux tube footprints) is
limited to order 4. More accurate models have been built from Juno magnetic
measurements. The most recent one, JRM33 [Connerney . (2022)], includes
spherical harmonics up to order 13. But to describe the magnetospheric
magnetic field between 4 and 13 $R_{J}$, the VIP4 plus current sheet model is
still adequate, because at these distances the influence of higher order terms
in the magnetic field becomes negligible, and it is fully consistent with the
imai2016 plasma model.

Figure 3: Meridian maps of (a) the plasma frequency
$f_{pe}~{}=~{}\sqrt{n_{e}e^{2}/4\pi^{2}m_{e}\epsilon_{0}}$ computed from the
electron density $n_{e}$ of the diffusive density model of imai2016, (b) the
electronic cyclotron frequency $f_{ce}~{}=~{}eB/2\pi m_{e}$ computed from the
magnetic field amplitude $B$ from the VIP4 magnetic field model [Connerney .
(1998)] and (c) the upper hybrid frequency
$f_{uh}~{}=~{}\sqrt{f_{pe}^{2}+f_{ce}^{2}}$. The meridian plane here is where
the jovicentric, jovicentrifugal and jovimagnetic equators are aligned
(defined by the system III longitude $\varphi~{}=~{}200.769^{\circ}$). The
vertical axis is the cartesian coordinate $z$. The horizontal axis is the
radial cylindrical coordinate $\rho$ in the meridian plane. The color maps
have (0.1 $R_{J}$)2 pixels. Jupiter’s disk is displayed in black, the dotted
circles around it mark radial distance by 1 $R_{J}$ steps, the dotted straight
lines mark jovicentric latitudes by $10^{\circ}$ steps, and the bold dotted
line is the jovicentric equator, aligned with the jovicentrifugal and
jovimagnetic equators in this plane.

###  4.2 Synthesizing Occurrence Probability Distributions versus Frequency
and Latitude

The above models allow us to compute the electron density $n_{e}$, its
gradient $\nabla n_{e}$ and the vector magnetic field $\bf{B}$ at any point of
Jupiter’s inner magnetosphere. For our study, we computed these quantities
over a 3D cartesian grid $-15\leq x,y\leq+15$ $R_{J}$ in the equatorial plane
and $-10\leq z\leq+10$ $R_{J}$ along the rotation axis, with a grid step
$dx~{}=~{}dy~{}=~{}dz~{}=~{}0.1$ $R_{J}$. Figure 3 displays the meridian
distributions (a) of $f_{pe}$ and $n_{e}$ predicted by Imai’s (2016) diffusive
density model, (b) of $f_{ce}$ and $B$ distributions by the VIP4 model and (c)
of $f_{uh}$ distribution inside the IPT by combining both models. Although it
counts 2.3 million points, our grid is coarse because its step is much larger
than the wavelength of the nKOM radio waves and cannot capture any detail of
the wave-particle or wave-wave interactions involved in their generation.

Following the prescriptions of each scenario of Table 1, for each Juno/Waves
frequency of Figure 2, we identify all points of the grid matching that
frequency $\pm 6$%. Juno/Waves frequency channels from 10 kHz to 141 kHz are
distributed on a logarithmic scale with a ratio $\times 1.12$ between
consecutive channels [Kurth . (2017)], thus this interval of $\pm 6$% ensures
a continuous frequency coverage. These are the potential sources at that
frequency. We compute the angle $\alpha$ between $\nabla n_{e}$ and $\bf{B}$,
as well as the density gradient strength $||\nabla n_{e}||$ for each potential
source. Figure 4 displays the meridian distributions of (a) $\alpha$ and (b)
$||\nabla n_{e}||$. We define $\epsilon$ as the percentile of the $||\nabla
n_{e}||$ distribution at the considered frequency. Figure 10 illustrates the
distribution of $||\nabla n_{e}||$ versus $f_{pe}$, $f_{uh}$ and $2f_{uh}$,
with four particular values of $\epsilon$ indicated as examples. $\alpha$ and
$\epsilon$ are the two parameters that will be explored through our modeling.
For a selected range of $\alpha$ and $\epsilon$, the corresponding potential
sources become active sources. We then launch one or several rays (depending
on the scenario, cf. Tables 1 and LABEL:tab_appendix:cases) from each active
source. As ray-tracing would require a much finer grid and be computationally
intractable, we propagate these rays in straight line to infinity, unless a
ray crosses a grid cell in which the local cutoff frequency
$f_{O}~{}=~{}f_{pe}$ in O-mode or
$f_{X}~{}=~{}\sqrt{f_{pe}^{2}+(f_{ce}/2)^{2}}+f_{ce}/2$ in X-mode, is larger
than the wave frequency. In that case, the ray is considered absorbed by the
plasma.

Figure 4: Meridian maps of (a) the angle between the magnetic-field and the
density gradients $\alpha$, and (b) the density gradient strength $||\nabla
n_{e}||$. The meridian plane, axis, color map parameters and annotations are
defined as in Figure 3.

The directions of the rays are then compared to the jovicentric coordinates of
Juno along its 2016-2019 trajectory (sampled by $1.5^{\circ}$ steps in
latitude and $2^{\circ}$ steps in longitude). Each ray, from each active point
source, beamed at an angle of Juno’s direction $\leq\delta\theta$ (we used
$\delta\theta~{}=~{}2^{\circ}$ in our study), as seen from the corresponding
point source, is considered to illuminate and thus be detected by the
spacecraft if the ray is not absorbed by the plasma before reaching Juno. We
count the number of rays thus detected by Juno along its trajectory. This
provides us with a modeled time series of nKOM occurrence at the selected
frequency. We repeat this operation for each Juno/Waves frequency of Figure 2
and obtain a modeled distribution of nKOM occurrence probability versus
frequency and latitude.

The observed distribution may result from the combination of plasma emission
generated at different values of $\alpha$ and $\epsilon$. Therefore, for each
scenario of Table 1 (and LABEL:tab_appendix:cases), we perform a parametric
study with $0^{\circ}\leq\alpha\leq 90^{\circ}$ by steps
$\delta\alpha~{}=~{}3^{\circ}$ and 0% $\leq\epsilon\leq$ 100% by steps
$\delta\epsilon~{}=~{}10$%. Each of the 300 resulting occurrence probability
distributions, computed on the same grid as Figure 2, are compared the to
observed distribution with a specifically built correlation-inclusion
coefficient $C^{*}_{\alpha,\epsilon}$ (see Figure 11 for a detailed example of
the calculation of $C^{*}$). Physically, $C^{*}_{\alpha,\epsilon}$ is
justified by the fact that a simulated distribution for a specific selection
of parameters $\alpha$ (i.e. angle B, $\nabla n_{e}$) and $\epsilon$ (i.e.
$\nabla n_{e}$ strength) is not expected to coincide with the observed
distribution, that may result from plasma emissions generated in a range of
values for $\alpha$ and $\epsilon$, but must be included on it. For a single
couple ($\alpha$, $\epsilon$), emissions simulated but not observed are not
acceptable, whereas emission observed but not simulated are acceptable. Then,
an overall occurrence probability distribution is computed by combining the
distributions simulated for values of $\alpha$ and $\epsilon$ corresponding to
$C^{*}_{\alpha,\epsilon}$ above a given threshold. This overall distribution
is then compared with the observed one using the Pearson linear correlation
coefficient $C$.

Table 1: Basic prescriptions of the studied scenarios. Scenario | Reference | Frequency | Beaming
---|---|---|---
#1 |  jones1980, jones1986, jones1987 | $f_{pe}$ |  $\beta~{}=~{}(\textbf{k}_{+},\textbf{B})$, $\pi~{}-~{}\beta~{}=~{}(\textbf{k}_{-},\textbf{B})$
#2 |  fung1987 | $2f_{uh}$ | $\textbf{k}\perp\textbf{B}$
#3 | This study | $f_{pe}$ | $\textbf{k}\parallel-\nabla f_{pe}$
#4 | This study | $f_{uh}$ | $\textbf{k}\parallel-\nabla f_{uh}$

##  5 Results

For each scenario described in Table 1, considering the plasma emission cutoff
either of O-mode or X-mode, we have combined from the 300 simulated
distributions, those for which the value of $C^{*}_{\alpha,\epsilon}$ is above
a threshold $p$. This threshold $p$ is defined as a fraction of
max$(C^{*}_{\alpha,\epsilon})$. The evolution of the linear correlation
coefficient $C$ with the observations, as a function of $p$, for the overall
distribution obtained by selecting ranges of $(\alpha,\epsilon)$ for which
$C^{*}_{\alpha,\epsilon}\geq p$ of max($C^{*}_{\alpha,\epsilon}$), is
displayed on the Figure 5. The blue curves correspond to the simulations where
we considered the emission cutoff in O-mode and the orange curves in X-mode.
The dashed vertical lines correspond to the threshold $p$ for which we
obtained the highest value of $C$.

Panel (a) of Figure 5, shows that the distributions simulated in the scenario
#1 are poorly correlated with the nKOM distribution in both O-mode and X-mode.
In O-mode $C$ varies slightly with $p$, with a maximum at $C~{}=~{}17\%$ for
$p=40\%$, and in X-mode $C$ remains roughly constant, with a maximum at
$C~{}=~{}11\%$ for $p=0\%$. Panel (b) of Figure 5, shows that scenario #2
distributions are also poorly correlated with the nKOM distribution in both
O-mode and X-mode. In both cases, $C$ increases with $p$ until its maximum at
$C~{}=~{}17\%$ for $p=90\%$ in O-mode, and at $C~{}=~{}18\%$ for $p=100\%$ in
X-mode. Panel (c) of Figure 5, shows that the distributions simulated in the
scenario #3 are better correlated with the nKOM distribution than for the
previous scenarios, especially in O-mode. Indeed, in O-mode, $C$ increases
with $p$, with its maximum at $C~{}=~{}37\%$ for $p=85\%$, showing a partial
correlation with the nKOM distribution. In X-mode, $C$ varies only slightly
with $p$ until its maximum at $C~{}=~{}22\%$ for $p=75\%$, before decreasing
greatly. Finaly, Figure 5 panel (d), shows that scenario #4 distributions are
also poorly correlated with the nKOM distribution in both O-mode and X-mode.
In O-mode, $C$ increases with $p$ until its maximum value at $C~{}=~{}18\%$
for $p=80\%$ and in X-mode, $C$ remains constant until its maximum at
$C~{}=~{}20\%$ for $p=25\%$, before rapidly decreasing.

Figure 5: Linear correlation coefficient $C$ between the modeled and observed
latitude versus frequency distributions of nKOM occurrence, as a function of
the threshold $p$ on the correlation-inclusion coefficient
$C^{*}_{\alpha,\epsilon}$ defined as a fraction of its maximum value
max($C^{*}_{\alpha,\epsilon}$). The blue curves correspond to the cases where
we considered the simulated emissions with a cutoff in O-mode and the orange
curves in X-mode. Panels (a), (b), (c) and (d) correspond respectively to
scenarios 1 to 4.

Figures 6 and 7 synthesize the results obtained for each scenario,
respectively in O-mode and X-mode, for the thresholds $p$ (cf. Figure 5) that
return the highest correlation coefficients $C$ of the overall simulated
distributions with the observed nKOM. The left column of each figure displays
the correlation-inclusion coefficients $C^{*}$ between the modeled and
observed frequency–latitude distributions of nKOM occurrence probability, as a
function of $\alpha$ (angle between $\nabla n_{e}$ and $\bf{B}$) and
$\epsilon$ (percentile of the density gradient strength $||\nabla n_{e}||$
distribution at each frequency). On each of these panels, the intervals of
$\alpha$ and $\epsilon$ corresponding to the distributions above $p$ are
highlighted (with black contours). The central column of each figure displays
the simulated distributions of nKOM occurrence probability versus frequency
and latitude for the highlighted selections of the left column, with the
overall linear correlation coefficient $C$ noted above each panel. The right
side columns display the jovicentrifugal distributions of the sources
corresponding to the distributions of the central columns. The black contours
correspond to values of the emitted frequencies in each scenario ($f_{pe}$,
$f_{uh}$ or $2f_{uh}$) in the meridian plane where the jovicentrifugal and
jovimagnetic equators are aligned. Below, we investigate the 4 scenarios and
comment the parameters selection, distributions and source locations obtained
in O-mode and X-mode.

Figure 6: O-mode synthetic results derived from Figure 5: Rows (a), (b), (c)
and (d) correspond respectively to scenarios 1 to 4. (Left column)
correlation-inclusion coefficients $C^{*}_{\alpha,\epsilon}$ between the
modeled and observed frequency–latitude distributions of nKOM occurrence
probability, as a function of $\alpha$ (within $3^{\circ}$ wide bins) and
$\epsilon$ (within 10% wide bins). The regions where
$C^{*}_{\alpha,\epsilon}>p$ of max($C^{*}_{\alpha,\epsilon}$), are contoured
in black. The value of max($C^{*}_{\alpha,\epsilon}$), annotated
$C^{*}_{max}$, is indicated above each panel of this column. (Central column)
simulated distributions of nKOM overall occurrence probability versus
frequency and latitude for all values of $\alpha$ and $\epsilon$ within the
contoured black region in the left panel considered together. The value of the
Pearson correlation coefficient $C$ of this distribution with the observed one
of Figure 2 is indicated above each panel. The color scale is adjusted in each
panel. (Right column) meridian projection color maps of the sources
distribution in the IPT in each scenario (indicated above each panel), in the
jovicentrifugal coordinates. The black contours correspond the values
$f~{}=~{}[10,20,40,80,160,320]$ kHz of the emitted frequencies in each
scenario. In the $2f_{uh}$ case, the contour starts at $f~{}=~{}20$ kHz. The
color map has (0.1 $R_{J}$)2 pixels. Jupiter’s disk is displayed in black, the
dotted circles around it mark radial distance by 1 $R_{J}$ steps, the dotted
straight lines mark jovicentrifugal latitudes by $10^{\circ}$ steps, and the
bold dotted line is the jovicentrifugal equator. The inner IPT extends from
$\leq$ 6 to $\sim$ 8 $R_{J}$, the outer IPT from $\sim$ 8 to $\sim$ 10 $R_{J}$
and the inner plasma disc beyond 10 $R_{J}$. Figure 7: Same as Figure 6, but
derived from the X-mode modeling results of Figure 5.

###  5.1 Scenario #1: Jones theory

The results for scenario #1 in O-mode are displayed on row (a) of Figure 6.
There is a clear dependence of the correlation-inclusion value on $\alpha$ and
$\epsilon$, as the contoured area $C^{*}_{\alpha,\epsilon}>40\%$ of
max($C^{*}_{\alpha,\epsilon}$) (Figure 6a, left) is very localised at
moderately large angles $\alpha~{}=~{}63^{\circ}~{}-~{}72^{\circ}$ and
intermediate gradients $\epsilon~{}=~{}40\%~{}-~{}50\%$. But
$C^{*}_{\alpha,\epsilon}$ is mostly negative and the overall distribution
(Figure 6a, center), is poorly correlated with the observed nKOM distribution
($C~{}=~{}17\%$). Nevertheless, it has some interesting features: it presents
a North-South asymmetry, and an occurrence patch $>+30^{\circ}$ in the North
at 35–80 kHz. The predicted sources (Figure 6a, right) are distributed from
the middle to the inner edge of the IPT. The low frequency sources below 20
kHz are located along the inner edge of the IPT at 4–5 $R_{J}$, within $\pm
5^{\circ}$ of the jovicentrifugal equator while the high frequency sources
above 80 kHz are located at 5–6 $R_{J}$ and 8–9 $R_{J}$ for jovicentrifugal
latitudes around $\pm 10^{\circ}$. The sources responsible for the occurrence
patch $>+30^{\circ}$ in the North, are located in the inner edge of the IPT at
4–5 $R_{J}$ for jovicentrifugal latitudes around $\pm 35^{\circ}$.

The results for scenario #1 in X-mode are displayed on row (a) of Figure 7.
There seems to be a dependence of $C^{*}_{\alpha,\epsilon}$ on $\alpha$ and
$\epsilon$, as the contoured area $C^{*}_{\alpha,\epsilon}>0\%$ of
max($C^{*}_{\alpha,\epsilon}$) (Figure 7a, left), i.e. the regions where the
$C^{*}_{\alpha,\epsilon}>0$, is localised at intermediate angles
$\alpha~{}=~{}36^{\circ}~{}-~{}39^{\circ}$ and
$\alpha~{}=~{}54^{\circ}~{}-~{}57^{\circ}$, and intermediate gradients
$\epsilon~{}=~{}50\%~{}-~{}60\%$ and strong gradients
$\epsilon~{}=~{}80\%~{}-~{}90\%$. However, $C^{*}_{\alpha,\epsilon}$ is mostly
negative and the overall distribution (Figure 7a, center) is almost empty,
meaning that the majority of the emissions are absorbed by the plasma. The
right panel on Figure 6a right, shows that only 12 sources are observed (while
the order of magnitude in the other cases is $\sim 1000$). The resulting
distribution does not match with the nKOM observations.

###  5.2 Scenario #2: Fung and Papadopoulos theory

The results for scenario #2 in O-mode are displayed on row (b) of Figure 6.
There is a dependence of the correlation-inclusion value on $\alpha$, as the
contoured $C^{*}_{\alpha,\epsilon}>90\%$ of max($C^{*}_{\alpha,\epsilon}$)
(Figure 6b, left) correspond to intermediate angles
$\alpha~{}=~{}39^{\circ}~{}-~{}45^{\circ}$ and moderately large angles
$\alpha~{}=~{}60^{\circ}~{}-~{}69^{\circ}$. On the other hand, the dependence
of $C^{*}_{\alpha,\epsilon}$ on $\epsilon$ is not convex, as
$C^{*}_{\alpha,\epsilon}>90\%$ of max($C^{*}_{\alpha,\epsilon}$) correspond to
both weak gradients $\epsilon~{}=~{}10\%~{}-~{}20\%$ and strong gradients
$\epsilon~{}=~{}70\%~{}-~{}80\%$. The overall distribution (Figure 6b, center)
is poorly correlated with the observed nKOM: it does not present a North-South
asymmetry; at low latitudes it presents very strong peaks above 60 kHz and
weak peaks down to 15 kHz and it does not predict occurrences at high
latitudes below 55 kHz. The predicted sources (Figure 6b, right) are
distributed from the middle to the outer edge of the IPT for jovicentrifugal
latitudes $<15^{\circ}$. The sources below 80 kHz are mainly located at the
outer edge of the IPT at the limit of validity of Imai’s model ($13$ $R_{J}$).
The high frequency sources above 80 kHz are distributed in the middle part of
the IPT at 8–10 $R_{J}$ and 10–12 $R_{J}$ for jovicentrifugal latitudes within
5∘–13∘.

The results for scenario #2 in X-mode are displayed on row (b) of Figure 7.
The contoured
$C^{*}_{\alpha,\epsilon}~{}=~{}\mathrm{max}(C^{*}_{\alpha,\epsilon})$ (Figure
7b, left) correspond to moderately large
$\alpha~{}=~{}57^{\circ}~{}-~{}60^{\circ}$ and strong gradients
$\epsilon~{}=~{}60\%~{}-~{}70\%$. The overall distribution (Figure 7b, center)
is poorly correlated with the observed nKOM and shares similar defects with
the O-mode case (cf. Figure 6b, center), but with a sharp cutoff in the
observation below 50 kHz. The predicted sources (Figure 7b, right) are
distributed as in the O-mode case, but with significantly less low frequency
sources $>10$ $R_{J}$.

###  5.3 Scenario #3: emission at $f_{pe}$ beamed along -$\nabla f_{pe}$

The results for scenario #3 in O-mode are displayed on row (c) of Figure 6.
This case clearly stands out, as the correlation-inclusion value seems to be
strongly controlled by $\alpha$ and $\epsilon$. The contoured area
$C^{*}_{\alpha,\epsilon}>85\%$ of max($C^{*}_{\alpha,\epsilon}$) (Figure 6c,
left) corresponds to
$C^{*}_{\alpha,\epsilon}~{}=~{}\mathrm{max}(C^{*}_{\alpha,\epsilon})$
localised at intermediate angles $\alpha~{}=~{}54^{\circ}~{}-~{}57^{\circ}$
and strong gradients $\epsilon~{}=~{}80\%~{}-~{}90\%$. The overall
distribution (Figure 6c, center), shows partial correlation with the nKOM
observations as it shares most of its features. It presents a strong North-
South asymmetry, an occurrence peak in the South around $-40^{\circ}$ at
$20~{}-~{}141$ kHz, with a high latitude extension down to $<-70^{\circ}$ at
$40~{}-~{}50$ kHz, and a very strong peak in the North $>+40^{\circ}$ at
$40~{}-~{}50$ kHz. However, some significant features are missing: the
distribution does not present any occurrence within $\pm 15^{\circ}$ of the
jovicentrifugal equator above 50 kHz and the strong peak in the North does not
extend to 30 kHz. The predicted sources (Figure 6c, right) are distributed
from the inner edge to the outer edge of the IPT within $\pm 20^{\circ}$ of
the jovicentrifugal equator. They can be divided into four groups. The first
group is located at the outer edge of the IPT at $11-13$ $R_{J}$, along the
jovicentrifugal equator, and is responsible for the occurrence peak in the
South at $-40^{\circ}$ at 20-100 kHz and the weak occurrence above 50 kHz in
North near $+30^{\circ}$. The second and third groups are located in the
middle part of the IPT, respectively at $9-10$ $R_{J}$ and $6-8$ $R_{J}$, for
jovicentrifugal latitudes below $\pm 10^{\circ}$. They both contribute to the
high occurrence peak above 80 kHz. The fourth group is located at the inner
edge of the IPT at $4-6$ $R_{J}$ below $\pm 40^{\circ}$, following the contour
of the low density cavity predicted by Imai’s model (cf. Figure 3a), and is
responsible for the North and South high latitudes occurrence peaks.

The results for scenario #3 in X-mode are displayed on row (c) of Figure 7.
There is a weak dependence of $C^{*}_{\alpha,\epsilon}$ on $\alpha$ and
$\epsilon$, as the contoured area $C^{*}_{\alpha,\epsilon}>75\%$ of
max($C^{*}_{\alpha,\epsilon}$) (Figure 7c, left) corresponds to intermediate
angles $\alpha~{}=~{}48^{\circ}~{}-~{}57^{\circ}$ and large to quasi-
perpendicular angles $\alpha~{}=~{}72^{\circ}~{}-~{}90^{\circ}$, and strong
gradients $\epsilon~{}=~{}60\%~{}-~{}80\%$ and
$\epsilon~{}=~{}90\%~{}-~{}100\%$. The overall distribution (Figure 7c,
center), is poorly correlated with the nKOM observations as it does not
presents any occurrences at high Northern $>40^{\circ}$ and Southern
$<-50^{\circ}$ latitudes. However, some interesting features are present: a
peak in the South around $-40^{\circ}$ and in the North around $+15^{\circ}$
above 50 kHz. Furthermore, it presents low but non-zero occurrences near the
equator, that were missing in the O-mode case. The predicted sources (Figure
7c, right) are distributed from the middle to the outer edge of the IPT for
jovicentrifugal latitudes below $\pm 5^{\circ}$. The sources below 80 kHz are
located at the outer edge of the IPT at 11–13 $R_{J}$ close to the
jovicentrifugal equator. The high frequency sources above 80 kHz are located
in the middle part the IPT at 9–10 $R_{J}$.

###  5.4 Scenario #4: emission at $f_{uh}$ beamed along -$\nabla f_{uh}$

The results for scenario #4 in O-mode are displayed on row (d) of Figure 6.
The contoured $C^{*}_{\alpha,\epsilon}>80\%$ of max($C^{*}_{\alpha,\epsilon}$)
(Figure 6d, left) correspond to quasi-parallel angles
$\alpha~{}=~{}3^{\circ}~{}-~{}15^{\circ}$ and intermediate angles
$\alpha~{}=~{}33^{\circ}~{}-~{}39^{\circ}$, and weak gradients
$\epsilon~{}=~{}0\%~{}-~{}10\%$ and $\epsilon~{}=~{}20\%~{}-~{}30\%$. The
overall distribution (Figure 6d, center) is poorly correlated with the
observed nKOM: it does not present an inverted North-South asymmetry, it
presents a very strong peak around 140 kHz near the equator and it does not
predict occurrences at high latitudes below 55 kHz. The sources (Figure 6d,
right) are scattered from the inner edge to the outer edge of the IPT at very
different jovicentrifugal latitudes.

The results for scenario #4 in X-mode are displayed on row (d) of Figure 7.
The dependence of $C^{*}_{\alpha,\epsilon}$ on both $\alpha$ and $\epsilon$ is
noisy, as the contoured area $C^{*}_{\alpha,\epsilon}>75\%$ of
max($C^{*}_{\alpha,\epsilon}$) (Figure 7d, left) corresponds to values of
$\alpha$ and $\epsilon$ spread all over the parameter space. The overall
distribution (Figure 7d, center), is poorly correlated with the nKOM
observations as it resembles to a smoother version the distribution obtained
for the scenario #3 in X-mode (Figure 7c, center). The predicted sources
(Figure 7d, right) are distributed from the inner edge to the outer edge of
the IPT at very different jovicentrifugal latitudes in a similar way as for
the scenario #3 in X-mode (Figure 7c, right). The sources below 90 kHz are
located at the outer edge of the IPT at 11–13 $R_{J}$ close to the
jovicentrifugal equator. The high frequency sources above 90 kHz are located
in the middle part the IPT at 9–10 $R_{J}$. In addition, some sources in
60–140 kHz are distributed from the middle to the inner part of the IPT in 5–8
$R_{J}$ at jovicentrifugal latitudes around $\pm 25^{\circ}$.

##  6 Discussion and Perspectives

We have developed a 3D numerical large-scale model to test the nKOM generation
mechanism and beaming, based on the plasma density model of imai2016 and the
VIP4 internal Jovian magnetic field model [Connerney . (1998)] (cf. Figure 3).
We have tested the model predictions against the observed frequency–latitude
nKOM occurrence probability measured by Juno/Waves over the first 3 years of
its Jovian tour (cf. Figure 2). We have explored 4 emission scenarios based on
assumed emission frequencies and beaming patterns (cf. Table 1), for each of
them we have either considered the emission cutoff in O-mode or X-mode, and
performed a parametric study versus the angle $\alpha$ between $\bf{B}$ and
$\nabla n_{e}$, and the (large-scale) density gradient strength percentile
$\epsilon$ at each frequency (cf. Figure 10).

Our results allow us to exclude scenario #1, based on the Jones theory of
linear mode conversion of Z-mode electrostatic waves at the upper-hybrid
resonance, to O-mode electromagnetic waves via the “radio window” [Jones
(1980), Jones (1986), Jones (1987)]. This theory postulates very specific
beaming angles ($\textbf{k},\textbf{B}$) that, in O-mode, are not supported by
the comparison of all modeled occurrence probability distributions with the
observed one, especially for the emissions observed at low latitudes. It
predicts a high Northern latitude occurrence peak that resembles the nKOM
observations with slightly higher frequency, but it requires large but not
perpendicular angles $\alpha$ and relatively strong gradient $\epsilon$, that
are incompatible with this theory. In X-mode, all the distributions modeled
are either empty or anti-correlated with the nKOM, confirming that this theory
cannot be used to describe the beaming for conversion from Z-mode to X-mode.

We can also confidently exclude scenario #2, based on fung1987 theory that
predicts emission at $2f_{uh}$ (resulting from nonlinear coupling of two
electrostatic waves at $f_{uh}$) beamed in a plane nearly perpendicular to the
local $\bf{B}$. The modeled occurrence probability distributions, in both
O-mode and X-mode, do not match the observed one, primarily because emissions
at $2f_{uh}$ cover a too broad frequency range compared to nKOM emissions, and
secondly because its beaming does not predict nKOM low-frequency extensions at
high latitudes. We can extrapolate this result to exclude emissions at
$2f_{pe}$ or any higher harmonic of $f_{pe}$ or $f_{uh}$, to be nKOM below 141
kHz.

Finally, we exclude scenario #4, i.e. plasma emissions at $f_{uh}$ beamed
along their local negative frequency gradient (i.e. toward decreasing
frequencies). The modeled distribution in O-mode is not compatible with the
nKOM observations at high and low latitudes. The modeled distributions in
X-mode, on the other hand, bear some resemblance with the nKOM distribution at
mid and low latitudes from $-50^{\circ}$ to $<+30^{\circ}$. However, the
distribution of $\alpha$ and $\epsilon$ leading to the best simulated
distribution are scatter, which is difficult to justify physically.

Only the scenario #3, i.e. plasma emissions at $f_{pe}$ beamed along their
local negative frequency gradient, clearly stands out, being the only case to
show a significant positive correlation with the nKOM observations (with
$C~{}=~{}37\%$ in O-mode). It favours O-mode emission to describe the nKOM
occurrences peak observed at high latitudes, from regions of relatively strong
electron density gradients ($\epsilon~{}=~{}80\%~{}-~{}90\%$) at an angle
$\alpha\sim 55^{\circ}$ from the local magnetic field. The angle $\alpha$ is
lower than the $90^{\circ}$ value assumed in the Jones theory, as critical
coupling at $f~{}=~{}f_{pe}$ requires that $\sin(\alpha)\gg\cos(\alpha)$.
These nKOM sources are predicted to be distributed at the inner edge of the
IPT ($<~{}6~{}R_{J}$), within $\pm 40^{\circ}$ of the jovicentrifgual equator.
The X-mode modeled distribution, although poorly correlated with the observed
one, displays nKOM at mid-to-low latitudes from $-50^{\circ}$ to
$+30^{\circ}$.

In the scenario #3, the overall O-mode and X-mode distributions seems
complementary: the O-mode distribution (cf. Figure 6c, center) is missing the
low latitudes emissions, and the X-mode distribution (cf. Figure 7c, center)
is missing the low frequency high latitudes emissions. The combined
distributions, displayed on Figure 8 (with the nKOM distribution for
comparison), reaches a higher correlation with $C~{}=~{}39\%$, and possesses
all the features observed on the nKOM distribution. This simulated
distribution is not attainable with O-mode emissions alone, as generations
parameters that permits the observations of high frequency O-mode at low
latitudes, also activate sources that are observed at high latitudes but
inconsistent with the nKOM observations (i.e. the simulated distributions have
negative $C^{*}_{\alpha,\epsilon}$ values). However, as these high latitudes
emissions are beamed from regions where $f_{X}>f_{pe}$, their observation are
restricted by the X-mode cutoff frequency. This suggests that nKOM is produced
at $f_{pe}$, in both O-mode and X-mode, with the O-mode sources distributed
from the outer edge to the inner edge of the IPT, and the X-modes sources
located at the outer edge of the IPT near the jovicentrifugal equator. These
results are compatible with the updated budden1986 radio window theory,
suggesting Z-mode conversion into both O-mode and X-mode, if the density
gradients are large. It also confirms and precises the source locations
qualitatively proposed by kaiser1980 and daigne1986, and also assumed by
jones1980, in the middle part of the IPT at 8–9 $R_{J}$ for O-mode nKOM near
100 kHz, and the goniopolarimetric results of reiner1993 within the middle to
the outer part of the IPT 7–13 $R_{J}$ for X-mode nKOM.

Figure 8: (a) Scenario #3 combining O-mode and X-mode occurrence probability
distribution versus frequency and latitude, respectively Figures 6c and 6d
center panels. The nKOM occurrence probability distribution versus frequency
and latitude from Figure 2, is displayed for comparison (b).

By construction, our model cannot capture the entire physics in the sources.
Moreover, we assumed straight line radio propagation, which is validated a
posteriori by the fact that the waves are mostly emitted away from the
centrifugal equator, towards regions of lower density. In spite of these
limitations, our large-scale 3D modeling of the conditions of emission of nKOM
allowed us to discriminate between different models, exclude the two main
models proposed so far for nKOM generation, and to reach a reasonable match of
the modeled and observed distributions of nKOM occurrence probability with
simple assumptions. The fact that we obtain modeled distributions compatible
with the observed one under the assumption of radio propagation along $-\nabla
f_{pe}$ at the source can be due, in the context of our large (0.1 $R_{J}$)
grid cells, to a refraction effect close to the source. Snell-Descartes law
shows that waves beaming tends to align with the normal to the iso-surfaces of
the refractive index $N$ when $\nabla N$ is large. When applied to a wave in
perpendicular propagation in a cold collisionless magnetized plasma, this
condition is met at the conversion layer when $f_{pe}\geq f_{ce}$ for O-mode
and X-mode, if $f\sim f_{X}$ for X-mode. The refractive index $N$ varies in
the same direction as $f_{pe}$ for O-mode
($N^{2}_{O}~{}=~{}1-f_{pe}^{2}/f^{2}$), and for X-mode but only if $f\gg
f_{ce}$ (in which case $N^{2}_{O}\sim N^{2}_{X}$), which is the case for
emissions at $f_{pe}$ from the outer edge to the middle part of the IPT, near
the jovicentrifugal equator (see Figure 12). If the plasma density strongly
decreases locally to the conversion layer, the refractive index reaches
$N~{}=~{}1$ quickly, and from there, because the emission is considered
propagating in vacuum medium, the emission will be observed exiting the plasma
with a beaming close to $-\nabla f_{pe}$. As there is no obvious dependence of
$N_{O}$ and $N_{X}$ with $f_{uh}$, this would explain why the results of
scenario #4 do not fit the nKOM observations. Finally, this explanation is not
valid for waves in parallel propagation unless $f_{ce}\ll f_{pe}$, suggesting
that the O-mode nKOM in produced at the inner edge of the IPT necessarily has
a perpendicular propagation with the magnetic field locally at the radio
sources. This remains to be tested in future studies via small-scale wave
propagation close to the sources, for different initial beaming angles, with a
ray-tracing code taking into account the refraction of the waves, like
ARTEMIS-P [Gautier (2013)] or with measurement inside the sources.

Finally, we supposed nKOM to be either O-mode or X-mode emissions, as Z mode
waves are trapped in IPT and thus cannot be observed from outside. However,
Juno crossed the IPT multiple times when close to Jupiter. In these regions,
Juno might have observed Z mode emissions when crossing their sources. Based
on VIP4 and Imai’s density model, we can estimate the characteristic
frequencies of the Z-mode radio sources crossed by Juno in the IPT. In the
considered time range (from 9 April 2016 to 24 June 2019), Figure 9 displays
on panel (a) the Juno trajectory when crossing Z-mode source, and on panel (b)
the Z-mode occurrence probability versus frequency and latitude when Juno is
crossing the IPT. Figure 9 panel (b), shows that Z-mode emission should mainly
be observed in the North around $+20^{\circ}$ in 10-141 kHz. However, despite
considering the Z-mode radio sources permanent, the resulting distribution
have low probability of occurrences that does not exceed 1%, which is low in
comparison the nKOM observed at the same latitudes (cf. Figure 2). Then, it is
a valid assumption to consider the nKOM observed by Juno being either O-mode
or X-mode.

Figure 9: (a) Meridian projection of the Juno trajectory in green, crossing
the Z-mode sources of interest at $f~{}=~{}31.7$ kHz in orange, and at
$f=56.2$ kHz in blue, with (b) occurrence probability versus frequency and
latitude of the Z-mode emissions when Juno is crossing the IPT. The panel (a)
meridian plane here is where the jovicentric, jovicentrifugal and jovimagnetic
equator are aligned (defined by the system III longitude
$\varphi~{}=~{}200.769^{\circ}$). The Z-mode sources are predicted using the
$f_{pe}$ and $f_{ce}$ distributions predicted by imai2016 and VIP4 [Connerney
. (1998)] models (cf. panels (a) and (b) on Figure 3). The Z-mode frequencies
are defined by the Juno/Waves channels (cf. Sec. 4.2), considering
$f_{Z}<f<f^{res}_{Z}$ with
$f_{Z}~{}=~{}\frac{f_{ce}}{2}[(1+\frac{4f^{2}_{pe}}{f^{2}_{ce}})^{1/2}~{}-~{}1]$
and
$f^{res}_{Z}~{}=~{}\frac{f_{ce}}{\sqrt{2}}\\{(\frac{f^{2}_{uh}}{f^{2}_{ce}}+[(1~{}-~{}\frac{f^{2}_{pe}}{f^{2}_{ce}})^{2}~{}-~{}\frac{4f^{2}_{pe}}{f^{2}_{ce}}\sin^{2}\theta]\\}$.
In panel (b) each occurrence correspond to a point of the Juno trajectory
inside the plasma which we considered the Z-mode in $f_{Z}<f<f^{res}_{Z}$.

It will also be interesting to check the robustness of our results by
reproducing the present study with alternate plasma density models (such as
garrett2016,jun2019 or huscher2021), and magnetic field models (such as JRM09
[Connerney . (2018)] or JRM33 [Connerney . (2022)]). Alternate plasma models
might allow us to model the electron distribution over a radial range broader
than 4–13 $R_{J}$, including the plasmasphere and cool IPT below $\sim 5.5$
$R_{J}$, and a larger part of the inner plasma disc (e.g., up to 20 $R_{J}$).
Such an extension could be relevant for addressing the conditions of emission
of the narrowband low-frequency emissions discovered by Juno, refereed to as
ECR by imai2017 and nLF by louis2021. These narrowband emissions have a
frequency-latitude distribution of their occurrence probability close to that
of nKOM [Louis, Zarka, Dabidin . (2021)], but have lower frequencies (kHz to
tens of kHz). We suspect that they are also plasma emissions, and a study
similar to the present one will be dedicated to nLF generation. Extending the
observed distribution of Figure 2 (and of nLF) with four more years of
observations (i.e. over the interval 2016-2023) will also improve the
observational constraints. Furthermore, before the end of its extended orbital
tour (in 2025), Juno should cross the regions where we expect to find the
sources of nKOM emission. If nKOM sources are crossed, it will be possible to
study in situ their local characteristics (frequency, beaming, electrostatic
waves, ambient plasma) and precise their emission mechanism. Finally, a
similar study could be performed at Saturn, using a plasma model such as
persoon2020, for constraining the emission conditions of its narrowband low-
frequency radio components [S. Wu . (2021)].

##  Appendix A Detailed Table of the Scenario Beaming Characteristics

Table 2: Detailed version of Table 1, expliciting the beaming characteristics for each studied scenario. Scenario #1 is based on the theory of jones1987 that predicts Jovian plasma emissions produced at $f_{pe}$ and beamed in two opposite directions by making an angle $\beta~{}=~{}\arctan(f_{pe}/f_{ce})^{1/2}$ and $\pi~{}-~{}\beta$ with respect to $\bf{B}$ into the plane characterized by ($\nabla n_{e}$,B). Scenario #2 is based on the theory of fung1987 that predicts Jovian plasma emissions produced at $2f_{uh}$ and beamed in a plane perpendicularly to $\bf{B}$; for our simulations we assumed that the emissions are beamed in 8 directions in the plane perpendicular to $\bf{B}$. Scenarios #3 and #4 predict plasma emissions produced respectively at $f_{pe}$ and $f_{uh}$, and beamed along their local frequency gradient independently from $\bf{B}$ ($\bf{B}$ is represented on the sketches #3 and #4 for an illustrative purpose only, but its orientation can be arbitrary). Scenario : Reference | Beams/Source | Beaming | Scheme
---|---|---|---
#1 : jones1980, jones1986, jones1987 | 2 |  $\beta~{}=~{}(\textbf{k}_{+},\textbf{B})$ $\pi~{}-~{}\beta~{}=~{}(\textbf{k}_{-},\textbf{B})$ |
#2 : fung1987 | 8 | $\textbf{k}\perp\textbf{B}$ |
#3 : This study | 1 | $\textbf{k}\parallel~{}-~{}\nabla f_{pe}$ |
#4 : This study | 1 | $\textbf{k}\parallel~{}-~{}\nabla f_{uh}$ |

##  Appendix B Density Gradient Strength Distributions

Figure 10: Distributions of the number of grid cells versus the density
gradient strength $||\nabla n_{e}||$ and (a) the local plasma frequency
$f_{pe}$, (b) the local upper-hybrid frequency $f_{uh}$ and (c) the second
harmonic of the upper-hybrid frequency $2f_{uh}$, inside our simulation
domain. The horizontal axis correspond to $||\nabla n_{e}||$ values
distributed on a logarithmic scale with a ratio $\times 1.16$ between
consecutive bins. The vertical axis correspond to Juno/Waves frequencies from
10 kHz to 141 kHz, distributed on a logarithmic scale with a ratio $\times
1.12$ between consecutive channels. The color scale indicates the number of
grid cells $N$ at each frequency, for each value of $||\nabla n_{e}||$. The
dashed lines annotated $\epsilon~{}=~{}0\%$ in blue, $\epsilon~{}=~{}10\%$ in
orange, $\epsilon~{}=~{}90\%$ in green and $\epsilon~{}=~{}100\%$ in red, show
respectively as examples, the $0^{th}$, $10^{th}$, $90^{th}$ and $100^{th}$
percentiles of the $||\nabla n_{e}||$ distribution at each frequency.

##  Appendix C Correlation-Inclusion Coefficient

Figure 11: Illustration of the principle of the correlation-inclusion
coefficient $C^{*}$ between a modeled $D_{S}$ and observed distribution
$D_{O}$. $AC$ represents the region where the observed distribution
$D_{O}~{}=~{}1$ (equal to 0 elsewhere). $CD$ represents the region where the
simulated distribution $D_{S}~{}=~{}1$ (equal to 0 elsewhere). $C^{*}$
corresponds to the Pearson correlation coefficient $C$ computed after
weighting negatively the regions where $D_{S}\neq 0$ and $D_{O}~{}=~{}0$ (i.e.
emissions predicted by the simulation but not observed, here corresponding to
the region $D$), meaning that $D_{O}$ is set negative in this region.
Following empirical tests, we have set this values to $-\sigma_{O}$. In the
displayed example, $C^{*}~{}=~{}25\%$ while $C\simeq 43\%$.

##  Appendix D Plasma and Cyclotron Frequency Ratio and X-mode Frequency
Cutoff Meridian Maps

Figure 12: Meridian maps of (a) the ratio between the electron plasma
frequency $f_{pe}$ and the electron cyclotron frequency $f_{ce}$, (b) the
X-mode cutoff frequency
$f_{X}~{}=~{}\sqrt{f_{pe}^{2}+(f_{ce}/2)^{2}}+f_{ce}/2$. The meridian plane
here is where the jovicentric, jovicentrifugal and jovimagnetic equators are
aligned (defined by the system III longitude $\varphi~{}=~{}200.769^{\circ}$).
The vertical axis is the cartesian coordinate $z$. The horizontal axis is the
radial cylindrical coordinate $\rho$ in the meridian plane. The color maps
have (0.1 $R_{J}$)2 pixels. Jupiter’s disk is displayed in black, the dotted
circles around it mark radial distance by 1 $R_{J}$ steps, the dotted straight
lines mark jovicentric latitudes by $10^{\circ}$ steps, and the bold dotted
line is the jovicentric equator, aligned with the jovicentrifugal and
jovimagnetic equators in this plane.

## Data Availability Statement

The Juno/Waves calibrated data used in the manuscript is available at
https://doi.org/10.25935/6jg4-mk86 [Louis, Zarka Cecconi (2021)]. The
Juno/Waves catalog used in the manuscript is available at
https://doi.org/10.25935/6jg4-mk86 [Louis, Zarka, Cecconi Kurth (2021)].

###### Acknowledgements.

We thank L. Lamy, R. Prangé, C. Jackman and her team at DIAS, and the DIAS-
LESIA-INPE “Radioclub” for fruitful and friendly discussions at various steps
of this study. We also thank D. Santos Costa and H. Garrett for their insights
on Jovian plasma density models. C.K. Louis’s work at the Dublin Institute for
Advanced Studies was funded by Science Foundation Ireland Grant 18/FRL/6199.
The work of M. Imai was supported by the JSPS KAKENHI Grant Number JP23K13164.

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