# Dynamo Action of Jupiter’s Zonal Winds

Johannes Wicht  Max Planck Institute for Solar System Research, Justus-von-
Liebig-Weg 3, 37077 Göttingen, Germany  Thomas Gastine  College of
Engineering, Mathematics and Physical Sciences, University of Exeter, Physics
building, Stocker Road, Exeter, EX4 4QL, United Kingdom  Lucia D. V. Duarte
IPGP, Institution for Higher Education and Research 1, rue Jussieu, 75238
Paris cedex 05, France  Wieland Dietrich  Max Planck Institute for Solar
System Research, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany

###### Abstract

The new data delivered by NASA’s Juno spacecraft significantly increase our
understanding of Jupiter’s internal dynamics. The gravity data constrain the
depth of the zonal flows observed at cloud level and suggest that they slow
down considerably at a depth of about $0.96\,r_{J}$, where $r_{J}$ is the mean
radius at the one bar level. Juno’s magnetometer reveals the planet’s internal
magnetic field. We combine the new zonal flow and magnetic field models with
an updated electrical conductivity profile to assess the zonal wind induced
dynamo action, concentrating on the outer part of Jupiter’s molecular hydrogen
region where the conductivity increases very rapidly with depth. Dynamo action
remains quasi-stationary and can thus reasonably be estimated where the
magnetic Reynolds number remains smaller than one, which is roughly the region
above $0.96\,r_{J}$. We calculate that the locally induced radial magnetic
field reaches rms values of about $10^{-6}\,$T in this region and may just be
detectable by the Juno mission. Very localized dynamo action and a distinct
pattern that reflects the zonal wind system increases the chance to
disentangle this locally induced field from the background field. The
estimates of the locally induced currents also allow calculating the zonal
flow related Ohmic heating and associated entropy production. The respective
quantities remain below new revised predictions for the total dissipative
heating and total entropy production in Jupiter for any of the explored model
combinations. Thus neither Ohmic heating nor entropy production offer
additional constraints on the depth of the zonal winds.

###### Acknowledgements

This work was supported by the German Research Foundation (DFG) in the
framework of the special priority programs ’PlanetMag’ (SPP 1488) and
’Exploring the Diversity of Extrasolar Planets’ (SPP 1992).

##  1 Introduction

Two of the main objectives of NASA’s Juno mission are to measure Jupiter’s
magnetic field with unprecedented resolution and to determine the depth of the
fierce zonal winds observed in the planet’s cloud layer. The first Juno-based
internal magnetic field model JRM09 (Connerney et al., 2018) already provides
the internal magnetic field up to spherical harmonic degree $10$ and shows
several interesting features that seem unique to Jupiter’s dynamo (Moore et
al., 2018). Better resolved models are expected as the mission continues.

Based on Juno gravity measurements (Iess et al., 2018), Kaspi et al. (2018)
deduce that the speed of the equatorially antisymmetric zonal flow
contributions must be significantly reduced at a depth of about $3000\,$km
below the one bar level, which corresponds to a radius if $0.96\,r_{J}$. Kong
et al. (2018) come to roughly similar conclusions with a different inversion
procedure, but they also point out that the solution is not unique. While the
gravity data only allow constraining the equatorially antisymmetric winds, the
results likely also extend to the symmetric contributions. New interior models
(Guillot et al., 2018; Debras & Chabrier, 2019) and also the width of the
dominant equatorial jet (Gastine et al., 2014; Heimpel et al., 2016) both
support the idea that the fast zonal winds are roughly confined to the outer
$4$% in radius.

The fast planetary rotation enforces geostrophic flow structures with minimal
variation along the direction of the rotation axis. Geostrophic zonal winds
are thus expected to reach right through the planet’s gaseous envelope, and it
remains unclear which mechanism limits their extend in Jupiter. The demixing
of hydrogen and helium and the subsequent precipitation of helium deeper into
the planet offers one possible explanation (Militzer et al., 2016). This
process would have established a helium gradient that suppresses convection.
In Jupiter, this stable helium-rain layer may start somewhere between $0.93$
and $0.90\,r_{J}$ and perhaps extends down to $0.80\,r_{J}$ (Debras &
Chabrier, 2019). Note, however, that ab initio simulations by Schöttler &
Redmer (2018) predict that the hydrogene/helium demixing may not even have
started. Recent analysis of gravity measurements by the Cassini spacecraft
suggest that Saturn’s zonal winds may only reach down to about $0.85\,r_{S}$
(Iess et al., 2019; Galanti et al., 2019). Since the stably stratified layer
is thought to start significantly deeper, at about $0.62\,r_{S}$ according to
(Schöttler & Redmer, 2018), it cannot be the reason for this limited depth
extend of Saturn’s zonal winds.

A second possibility to brake the zonal winds at depth are Lorentz forces.
Lorentz forces are tied to dynamo action and thus to the electrical
conductivity profile. Ab initio simulations for Jupiter suggest that
ionization effects lead to a super-exponentially increase of the electrical
conductivity in the outermost molecular gas envelope. We will refer to this
layer as Jupiter’s Steeply Decaying Conductivity Region (SDCR) in the
following. At about $0.9\,r_{J}$, hydrogen, the planet’s main constituent,
becomes metallic, and the conductivity increases much more smoothly with depth
(French et al., 2012) (see panel a) of fig. 1). Though dynamo action and the
potential braking of the zonal winds due to Lorentz forces are classically
attributed to the metallic region, they may already become significant where
the electrical conductivity reaches sizable levels in the SDCR.

Different dynamo-related arguments have been evoked to estimate the depth of
the zonal winds without, however, directly addressing the role of the Lorentz
forces. Liu et al. (2008) estimate that the Ohmic heating caused by zonal-wind
related induction would exceed the total heat emitted from Jupiter’s interior,
should the winds reach deeper than $0.96\,r_{J}$ with undiminished speed.
Ridley & Holme (2016) argue that the secular variation of the magnetic field
over 30 years of pre-Juno observations is rather small and thus likely
incompatible with an advection by undiminished zonal winds. They conclude that
the winds cannot reach to depths where the magnetic Reynolds number exceeds
one and more significant induction can be expected. This puts the maximum
depth somewhere between $0.96\,r_{J}$ and $0.97\,r_{J}$, as we will discuss
below. A recent analysis by Moore et al. (2019) suggests that the observations
over a $45\,$year time span including Juno data would by compatible with zonal
wind velocities of $2.4\,$m/s at $0.95\,r_{J}$, two orders of magnitude
smaller than observed in the cloud layer.

Another interesting question is how much the dynamo action in the SDCR
contributes to Jupiter’s total magnetic field. Using a simplified mean-field
approach, Cao & Stevenson (2017) predict that the radial component of the
Locally Induced Field (LIF) may reach $1\,$% of the background field and could
thus be detectable by the Juno magnetometer. Wicht et al. (2019) analyze the
dynamo action in the SDCR of fully self-consistent numerical simulations that
yield Jupiter-like magnetic fields. Because of the dominance of Ohmic
diffusion, the dynamo dynamics becomes quasi-stationary in the SDCR of their
simulations. A consequence is that the locally induced electric currents and
field can be estimated with decent precision when flow, electrical
conductivity profile, and the surface magnetic field are known. Refined
information on all three ingredients has recently become available for
Jupiter, allowing for a fresh look on the problem.

Here we use three different zonal flow models, two electrical conductivity
models, and the new Juno-based magnetic field model JRM09 to predict the
electric currents and magnetic fields produced in Jupiter’s SDCR. In addition,
we also derive new estimates for the total dissipative heating and related
entropy production and explore whether either value is exceeded by the zonal-
flow related Ohmic dissipation.

The article starts off with outlining the methods and introducing the used
data in sect. 2. Sect. 3 discusses dissipative heating and entropy production
in Jupiter. Estimates for dynamo action, Ohmic heating, and entropy production
are then presented in sect. 4. Sect. 5 closes the article with a discussion
and conclusion.

##  2 Methods and Data

###  2.1 Estimating Dynamo Action

The ratio of inductive to diffusive effects in the induction equation,

$\frac{\partial\mbox{$\boldsymbol{B}$}}{\partial
t}={\boldsymbol{\nabla}}\times\left(\mbox{$\boldsymbol{U}$}\times\mbox{$\boldsymbol{B}$}\right)-{\boldsymbol{\nabla}}\times\lambda{\boldsymbol{\nabla}}\times\mbox{$\boldsymbol{B}$}\;\;,$
(1)

can be quantified by the magnetic Reynolds number

$\mbox{Rm}=\frac{\langle\mbox{$\boldsymbol{U}$}\rangle\;D}{\lambda}\;\;,$ (2)

where $\lambda=1/(\mu\sigma)$ is the magnetic diffusivity, with $\mu$ the
magnetic permeability and $\sigma$ the electrical conductivity. Angular
brackets generally denote rms values at a given radius throughout the paper;
thus $\langle\mbox{$\boldsymbol{U}$}\rangle$ stands for

$\langle\mbox{$\boldsymbol{U}$}\rangle=\left(\frac{1}{4\pi}\int_{0}^{2\pi}\mathrm{d}\phi\;\int_{0}^{\pi}\mathrm{d}\theta\;\sin\theta\;\mbox{$\boldsymbol{U}$}^{2}\right)^{1/2}\;\;,$
(3)

$\theta$ being the colatitude and $\phi$ the longitude.

The typical length scale $D$ is hard to estimate, and the planetary radius is
often used for simplicity. Where $\sigma$ decreases steeply in the SDCR,
however, the length scale is determined by the conductivity or magnetic
diffusivity scale height

$D_{\lambda}=\frac{\lambda}{\partial\lambda/\partial
r}=-\frac{\sigma}{\partial\sigma/\partial r}\;\;,$ (4)

and the modified magnetic Reynolds number

$\mbox{Rm${}^{(1)}$}=\frac{\langle\mbox{$\boldsymbol{U}$}\rangle\;D_{\lambda}}{\lambda}$
(5)

should be used. Since $D_{\lambda}$ is small and $\lambda$ decreases steeply
with radius, most of the SDCR is characterized by a small magnetic Reynolds
number $\mbox{Rm${}^{(1)}$}<1$, and the magnetic field dynamics becomes quasi-
stationary (Liu et al., 2008), obeying the simplified induction equation

${\boldsymbol{\nabla}}\times\frac{\mbox{$\boldsymbol{j}$}}{\sigma}\approx{\boldsymbol{\nabla}}\times\left(\mbox{$\boldsymbol{U}$}\times\tilde{\mbox{$\boldsymbol{B}$}}\right)\;\;.$
(6)

Here, $\boldsymbol{j}$ is the current density and
$\tilde{\mbox{$\boldsymbol{B}$}}$ the strong background field produced by the
dynamo acting deeper in the planet. The locally induced field
$\hat{\mbox{$\boldsymbol{B}$}}$ is given by Ampere’s law:

$\mbox{$\boldsymbol{j}$}={\boldsymbol{\nabla}}\times\hat{\mbox{$\boldsymbol{B}$}}\,\big{/}\,\mu\;\;.$
(7)

The steep $\sigma$ profile dominates the radial dependence of $\boldsymbol{j}$
and $\hat{\mbox{$\boldsymbol{B}$}}$ in the SDCR. The current density is thus
dominated by the horizontal components, where radial gradients in
$\hat{\mbox{$\boldsymbol{B}$}}$ contribute (Liu et al., 2008; Wicht et al.,
2018):

$\mbox{$\boldsymbol{j}$}\approx\mbox{$\boldsymbol{j}$}_{H}\approx\hat{\mathbf{r}}\times\frac{\partial}{\partial
r}\,\hat{\mbox{$\boldsymbol{B}$}}_{H}\;\;.$ (8)

Index $H$ denotes the horizontal components; the radial current can be
neglected in comparison.

Along the same lines, the horizontal components of eqn. (6) can be
approximated by

$\frac{1}{r}\frac{\partial}{\partial
r}\frac{r}{\sigma}\;\mbox{$\boldsymbol{j}$}_{H}\approx-\hat{\mathbf{r}}\times\left[{\boldsymbol{\nabla}}\times\left(\mbox{$\boldsymbol{U}$}\times\tilde{\mbox{$\boldsymbol{B}$}}\right)\right]_{H}\;\;,$
(9)

where $\hat{\mathbf{r}}$ is the radial unit vector. Integration in radius
yields the integral current density estimate introduced by Liu et al. (2008),
which we identify with an upper index $(I)$:

$\mbox{$\boldsymbol{j}$}^{(I)}_{H}=\frac{\sigma}{r}\left[\frac{r}{\sigma}\;\mbox{$\boldsymbol{j}$}_{H}\right]_{r_{J}}+\hat{\mathbf{r}}\times\frac{\sigma}{r}\;\int_{r}^{r_{J}}\,\mathrm{d}r^{\prime}\;r^{\prime}\;\left[{\boldsymbol{\nabla}}\times\left(\mbox{$\boldsymbol{U}$}\times\tilde{\mbox{$\boldsymbol{B}$}}\right)\right]_{H}\;\;.$
(10)

The square brackets with a lower index $r_{J}$ indicate that the expression
should be evaluated at the outer boundary.

For a predominantly zonal flow, we can use the approximation

$\displaystyle\left[{\boldsymbol{\nabla}}\times\left(\mbox{$\boldsymbol{U}$}\times\tilde{\mbox{$\boldsymbol{B}$}}\right)\right]_{H}$
$\displaystyle\approx$
$\displaystyle-\frac{\mbox{$\overline{U}_{\phi}$}}{r\sin\theta}\left(\frac{\partial}{\partial\phi}\,\tilde{B}_{\theta}\right)\,\hat{\mathbf{\theta}}$
$\displaystyle+\frac{1}{r}\left[\frac{\partial}{\partial
r}\,\left(r\mbox{$\overline{U}_{\phi}$}\tilde{B}_{r}\right)+\frac{\partial}{\partial\theta}\left(\mbox{$\overline{U}_{\phi}$}\tilde{B}_{\theta}\right)\right]\,\hat{\mathbf{\phi}}\;\;.$

where $\overline{U}_{\phi}$ is the zonal flow component and
$\hat{\mathbf{\theta}}$ and $\hat{\mathbf{\phi}}$ are unit vectors in
latitudinal and azimuthal direction, respectively.

The integral estimates for the two horizontal current components are then
given by

$\displaystyle j^{(I)}_{\theta}$ $\displaystyle=$
$\displaystyle\frac{\sigma}{r}\left[\frac{r}{\sigma}\;j_{\theta}\right]_{r_{J}}-\frac{\sigma}{r}\;\left(\left[r\,\mbox{$\overline{U}_{\phi}$}\tilde{B}_{r}\right]_{r_{J}}-\left[r\,\mbox{$\overline{U}_{\phi}$}\tilde{B}_{r}\right]_{r}\right)$
$\displaystyle-\;\frac{\sigma}{r}\int_{r}^{r_{J}}\;\mathrm{d}r^{\prime}\;\frac{\partial}{\partial\theta}\,\left(\mbox{$\overline{U}_{\phi}$}\tilde{B}_{\theta}\right)$

and

$j_{\phi}^{(I)}=\frac{\sigma}{r}\left[\frac{r}{\sigma}\;j_{\phi}\right]_{r_{J}}-\frac{\sigma}{r}\;\int_{r}^{r_{J}}\;\mathrm{d}r^{\prime}\;\frac{\mbox{$\overline{U}_{\phi}$}}{\sin\theta}\frac{\partial}{\partial\phi}\,\tilde{B}_{\theta}\;\;.$
(13)

Since the latitudinal length scale of the zonal winds is smaller than the
azimuthal length scale of the magnetic field, we expect that the latitudinal
component dominates.

The integral estimate requires the knowledge of the surface currents. While
the surface currents are certainly very small, the scaled version
$\sigma(r)/\sigma(r_{J})\,\mbox{$\boldsymbol{j}$}$ may remain significant. Liu
et al. (2008) argue that neglecting the surface contribution at least provides
a lower bound for the rms current density.

Wicht et al. (2019) confirm that the dynamics indeed becomes quasi-stationary
in the SDCR of Jupiter-like dynamo simulations where $\mbox{Rm${}^{(1)}$}<1$
and show that $j_{\theta}$ is indeed the dominant current component in the
SDCR of their Jupiter-like dynamo simulations. They also report that the
simplified Ohm’s law for a fast moving conductor,

$\mbox{$\boldsymbol{j}$}^{(O)}=\sigma\left(\mbox{$\boldsymbol{U}$}\times\tilde{\mbox{$\boldsymbol{B}$}}\right)\;\;,$
(14)

provides a significantly better estimate than $\mbox{$\boldsymbol{j}$}^{(I)}$.
We identify the respective current estimate with an upper index $(O)$. The
general Ohm law,

$\mbox{$\boldsymbol{j}$}=\sigma\left(\mbox{$\boldsymbol{U}$}\times\mbox{$\boldsymbol{B}$}\,+\,\mbox{$\mathbf{E}$}\right)\;\;,$
(15)

also contains currents driven by the electric field, which reduces to
$\mbox{$\mathbf{E}$}=-\nabla\Phi$ in the quasi-stationary case, where $\Phi$
is the electric potential. In the SDCR, this contribution likely proves
secondary because the potential differences remain small compared to the
induction by fast zonal winds (Wicht et al., 2019).

As the electrical conductivity decreases in the SDCR, the magnetic field
approaches a potential field with its characteristic radial dependence. We use
this dependence to approximate the background field with

$\tilde{\mbox{$\boldsymbol{B}$}}_{\ell}(r,\theta,\phi)\approx\left(\frac{r_{J}}{r}\right)^{\ell+2}\;\mbox{$\boldsymbol{B}$}_{\ell}(r_{J},\theta,\phi)\;\;,$
(16)

where the index $\ell$ denotes the magnetic field contribution at spherical
harmonic degree $\ell$. This provides a decent approximation as long as the
LIF remains a small contribution of the total field (Wicht et al., 2019).

Given a surface field model and an electrical conductivity profile, Ohm’s law
for a fast moving conductor and a predominantly zonal flow suggests

$\mbox{$\boldsymbol{j}$}\approx
j^{(O)}_{\theta}=\sigma\;\mbox{$\overline{U}_{\phi}$}\;\tilde{B}_{r}\;\;.$
(17)

When using this result to constrain the outer-boundary currents, the
alternative integral estimates, eqn. (2.1) and eqn. (13), yield

$j^{(I)}_{\theta}=\sigma\;\mbox{$\overline{U}_{\phi}$}\;\tilde{B}_{r}\;-\;\frac{\sigma}{r}\;\int_{r}^{r_{J}}\,\mathrm{d}r^{\prime}\;\frac{\partial}{\partial\theta}\,\left(\mbox{$\overline{U}_{\phi}$}\tilde{B}_{\theta}\right)$
(18)

and

$j^{(I)}_{\phi}=-\frac{\sigma}{r}\;\int_{r}^{r_{J}}\,\mathrm{d}r^{\prime}\;\frac{\mbox{$\overline{U}_{\phi}$}}{\sin\theta}\frac{\partial}{\partial\phi}\,\tilde{B}_{\theta}\;\;,$
(19)

respectively. A comparison of the estimates shows that
$\mbox{$\boldsymbol{j}$}^{(I)}$ and $\mbox{$\boldsymbol{j}$}^{(O)}$ will
remain very similar at shallow depths. When the flow decays very deeply with
depth, however, the integral contributions in eqn. (18) and eqn. (19) will
dominate below some radius and cause larger deviations, as we will see below.

Calculating the LIF requires to uncurl Ampere’s law, which reduces to
integrating eqn. (8) in the SDCR. When using $\mbox{$\boldsymbol{j}$}^{(O)}$,
this yields

$\hat{\mbox{$\boldsymbol{B}$}}_{H}\approx\;\int_{r}^{r_{J}}\,\mathrm{d}r^{\prime}\;\;\frac{\hat{\mathbf{r}}\times\left(\mbox{$\boldsymbol{U}$}\times\tilde{\mbox{$\boldsymbol{B}$}}\right)}{\lambda}\;\;.$
(20)

Since the electrical conductivity profile rules the radial dependence, the
integral can be approximated by

$\hat{\mbox{$\boldsymbol{B}$}}_{H}\approx\frac{D_{\lambda}}{\lambda}\;\hat{\mathbf{r}}\times\left(\mbox{$\boldsymbol{U}$}\times\tilde{\mbox{$\boldsymbol{B}$}}\right)\;\;.$
(21)

We have assumed here that the LIF vanishes at the outer boundary. For a
dominantly azimuthal flow, the primary LIF component is also azimuthal:

$\hat{\mbox{B}}_{\phi}\approx\frac{D_{\lambda}}{\lambda}\;\mbox{$\overline{U}_{\phi}$}\tilde{B}_{r}\;\;.$
(22)

This suggests that the rms value scales with Rm(1),

$\langle\hat{\mbox{$\boldsymbol{B}$}}_{H}\rangle\approx\mbox{Rm${}^{(1)}$}\langle\tilde{\mbox{$\boldsymbol{B}$}}\rangle\;\;,$
(23)

assuming that the correlation between $\overline{U}_{\phi}$ and
$\tilde{B}_{r}$ is of little relevance.

The radial LIF can be estimated based on the radial component on the quasi-
stationary induction equation (1):

$\lambda\nabla^{2}\hat{B}_{r}\approx-{\boldsymbol{\nabla}}\times\left(\mbox{$\boldsymbol{U}$}\times\tilde{\mbox{$\boldsymbol{B}$}}\right)\;\;.$
(24)

When approximating Ohmic dissipation by $\lambda\hat{B}_{r}/D_{\lambda}^{2}$,
this yields:

$\hat{B}_{r}\approx-\frac{D_{\lambda}^{2}}{\lambda}\;\hat{\mathbf{r}}\cdot{\boldsymbol{\nabla}}\times\left(\mbox{$\boldsymbol{U}$}\times\tilde{\mbox{$\boldsymbol{B}$}}\right)\;\;,$
(25)

which reduces to

$\hat{B}_{r}\approx-\frac{D_{\lambda}^{2}}{\lambda}\;\frac{\mbox{$\overline{U}_{\phi}$}}{r\sin{\theta}}\frac{\partial}{\partial\phi}\tilde{B}_{r}$
(26)

for a predominantly zonal flow. This suggest that the rms radial LIF should
roughly scale with the second modified magnetic Reynolds number

$\mbox{Rm${}^{(2)}$}=\frac{\langle\mbox{$\boldsymbol{U}$}\rangle\;D_{\lambda}^{2}}{\lambda\;D}$
(27)

like

$\langle\hat{B}_{r}\rangle\approx\mbox{Rm${}^{(2)}$}\frac{D}{D_{\phi}}\;\langle\tilde{B}\rangle\;\;.$
(28)

Here $D_{\phi}$ is the azimuthal length scale of the background field. Since
$D_{\lambda}\ll D_{\phi}$, the radial LIF is much smaller than its horizontal
counterpart (Wicht et al., 2019).

###  2.2 Data

Figure 1: (a) Electrical conductivity profiles in the outer 20% of Jupiter’s
radius. The black line shows the parametrization $\sigma_{F}(r)$ of the ab
initio simulation data points (black circles) by French et al. (2012). The
dotted red line shows the profile published in Zaghoo & Collins (2018), while
the solid red line shows the extension $\sigma_{Z}(r)$ used here. The profiles
suggested by Liu et al. (2008) (green) and Nellis et al. (1999) (blue) are
shown for comparison. (b) Magnetic diffusivity scale height
$D_{\lambda}/r_{J}$ for $\sigma_{F}$ and $\sigma_{Z}$.

The electric current and LIF estimates discussed above require a conductivity
profile, a zonal flow model, and a surface magnetic field model. For the
heating and entropy estimates that we will derived in sect. 3, we also need
density, temperature, and thermal conductivity profiles. We adopt the interior
model calculated by Nettelmann et al. (2012) and French et al. (2012), which
is the only one proving all the required information. Note, however, that
recent Juno gravity data suggest that Jupiter’s interior may be more complex
than anticipated in this model (Debras & Chabrier, 2019).

Ab initio simulations of the electrical conductivity by French et al. (2012)
provide 12 data points at different depths. Fig. 1 shows the values in the
outer $20$% of Jupiter’s radius and the parametrization $\sigma_{F}(r)$
developed for our analysis. A linear branch,

$\sigma_{F}(r)=\sigma_{r}+(\sigma_{m}-\sigma_{r})\frac{r-r_{r}}{r_{m}-r_{r}}\;\;,$
(29)

covers the smoother inner part $r<r_{m}$. An exponential branch,

$\sigma_{F}(r)=\sigma_{m}\exp\left(\left[\frac{r-r_{m}}{r_{m}-r_{r}}+b\,\left(\frac{r-r_{m}}{r_{m}-r_{r}}\right)^{2}\right]\frac{\sigma_{m}-\sigma_{r}}{\sigma_{m}}\right)\;\;,$
(30)

describes the steeper decay for $r_{m}<r<r_{e}$ with $b=7.2$. Matching radius
$r_{m}=0.89\,r_{J}$ and reference radius $r_{r}=0.77\,r_{J}$ are chosen where
ab initio data points have been provided.

A double-exponential branch,

$\sigma_{F}(r)=\sigma_{e}\exp\left(d\,\left[\exp\left(c\frac{r-r_{e}}{r_{m}-r_{r}}\right)-1\right]\right)\;\;,$
(31)

is required to capture the super-exponential decrease for $r\geq
r_{e}=0.972\,r_{J}$. The additional free parameter is $c=10$, while
$\sigma_{e}=\sigma(r_{e})$ and

$d=\frac{1}{c}\,\left(1+2\,b\;\frac{r_{e}-r_{m}}{r_{m}-r_{i}}\right)\;\frac{\sigma_{m}-\sigma_{r}}{\sigma_{m}}\;\;.$
(32)

The dotted red line in fig. 1 shows the conductivity model used to study
dynamo action in Jupiter and Jupiter-like exoplanets by Zaghoo & Collins
(2018). This is based on measurements which suggest a higher electrical
conductivity in the metallic hydrogen phase than previous data. Unfortunately,
Zaghoo & Collins (2018) do not discuss how the results were extrapolated to
Jovian conditions. The solid red line in fig. 1 shows the respective
parametrization $\sigma_{Z}(r)$ used for our analysis, which retraces the
published curve and connects to previously published parametrizations (green
and blue) at lower densities (Nellis et al., 1996; Liu et al., 2008). Note,
however, that these parametrizations are based on data (Weir et al., 1996)
which may have been attributed to too low temperatures according to a recent
analysis by Knudson et al. (2018).

Though model $\sigma_{Z}(r)$ is somewhat arbitrary, it serves to illustrate
the impact of conductivity uncertainties in our study. Close to $r_{J}$ where
conductivities remain insignificant, $\sigma_{F}$ is many orders of magnitude
larger than $\sigma_{Z}$. The ratio $\sigma_{F}/\sigma_{Z}$ decreases with
depth, reaching $10^{2}$ around $0.97\,r_{J}$ and $10$ around $0.96\,r_{J}$.
The two models finally cross at about $0.95\,r_{J}$. At about $0.925\,r_{J}$,
the ratio reaches a minimum of $0.05$ and then slowly increases with depth to
$0.35$ at $0.8\,r_{J}$. Tab. 1 list values of both conductivity models for
selected radii.

$r/r_{J}$ | $0.98$ | $0.97$ | $0.96$ | $0.95$ | $0.94$ | $0.92$ | $0.90$
---|---|---|---|---|---|---|---
$\langle U_{G}\rangle$ [m/s] | $3.0\mbox{$\times 10^{1}$}$ | $3.2\mbox{$\times 10^{1}$}$ | $3.5\mbox{$\times 10^{1}$}$ | $3.8\mbox{$\times 10^{1}$}$ | $4.1\mbox{$\times 10^{1}$}$ | $2.5\mbox{$\times 10^{1}$}$ | $1.5\mbox{$\times 10^{1}$}$
$\langle U_{K}\rangle$ [m/s] | $2.0\mbox{$\times 10^{1}$}$ | $1.4\mbox{$\times 10^{1}$}$ | $8.0\mbox{$\times 10^{0}$}$ | $4.0\mbox{$\times 10^{0}$}$ | $2.0\mbox{$\times 10^{0}$}$ | $2.5\mbox{$\times 10^{-1}$}$ | $4.0\mbox{$\times 10^{-2}$}$
$\langle U_{Z}\rangle$ [m/s] | $2.3\mbox{$\times 10^{1}$}$ | $1.9\mbox{$\times 10^{1}$}$ | $1.5\mbox{$\times 10^{1}$}$ | $1.1\mbox{$\times 10^{1}$}$ | $6.8\mbox{$\times 10^{0}$}$ | $1.7\mbox{$\times 10^{0}$}$ | $2.4\mbox{$\times 10^{-1}$}$
$\sigma_{F}\,$[S/m] | $3.2\mbox{$\times 10^{-4}$}$ | $3.7\mbox{$\times 10^{-2}$}$ | $9.7\mbox{$\times 10^{-1}$}$ | $1.7\mbox{$\times 10^{1}$}$ | $2.1\mbox{$\times 10^{2}$}$ | $9.3\mbox{$\times 10^{3}$}$ | $8.6\mbox{$\times 10^{4}$}$
$\sigma_{Z}\,$[S/m] | $1.5\mbox{$\times 10^{-8}$}$ | $1.9\mbox{$\times 10^{-4}$}$ | $1.6\mbox{$\times 10^{-1}$}$ | $3.4\mbox{$\times 10^{1}$}$ | $2.0\mbox{$\times 10^{3}$}$ | $1.5\mbox{$\times 10^{5}$}$ | $3.9\mbox{$\times 10^{5}$}$
$\lambda_{F}\,$[m2/s] | $2.5\mbox{$\times 10^{9}$}$ | $2.2\mbox{$\times 10^{7}$}$ | $8.2\mbox{$\times 10^{5}$}$ | $4.6\mbox{$\times 10^{4}$}$ | $3.8\mbox{$\times 10^{3}$}$ | $8.5\mbox{$\times 10^{1}$}$ | $9.2\mbox{$\times 10^{0}$}$
$\lambda_{Z}\,$[m2/s] | $5.2\mbox{$\times 10^{13}$}$ | $4.1\mbox{$\times 10^{9}$}$ | $5.1\mbox{$\times 10^{6}$}$ | $2.3\mbox{$\times 10^{4}$}$ | $3.9\mbox{$\times 10^{2}$}$ | $5.5\mbox{$\times 10^{0}$}$ | $2.0\mbox{$\times 10^{0}$}$
$D_{\lambda}(\sigma_{F})/r_{J}$ | $1.4\mbox{$\times 10^{-3}$}$ | $2.9\mbox{$\times 10^{-3}$}$ | $3.2\mbox{$\times 10^{-3}$}$ | $3.7\mbox{$\times 10^{-3}$}$ | $4.4\mbox{$\times 10^{-3}$}$ | $6.6\mbox{$\times 10^{-3}$}$ | $1.4\mbox{$\times 10^{-2}$}$
$D_{\lambda}(\sigma_{Z})/r_{J}$ | $7.9\mbox{$\times 10^{-4}$}$ | $1.4\mbox{$\times 10^{-3}$}$ | $1.7\mbox{$\times 10^{-3}$}$ | $2.1\mbox{$\times 10^{-3}$}$ | $2.9\mbox{$\times 10^{-3}$}$ | $1.2\mbox{$\times 10^{-2}$}$ | $3.3\mbox{$\times 10^{-1}$}$
$\mbox{Rm${}^{(1)}$}(U_{G},\sigma_{F})$ | $1.2\mbox{$\times 10^{-3}$}$ | $2.9\mbox{$\times 10^{-1}$}$ | $9.6\mbox{$\times 10^{0}$}$ | $2.1\mbox{$\times 10^{2}$}$ | $3.2\mbox{$\times 10^{3}$}$ | $1.4\mbox{$\times 10^{5}$}$ | $1.5\mbox{$\times 10^{6}$}$
$\mbox{Rm${}^{(1)}$}(U_{K},\sigma_{F})$ | $8.2\mbox{$\times 10^{-4}$}$ | $1.3\mbox{$\times 10^{-1}$}$ | $2.2\mbox{$\times 10^{0}$}$ | $2.3\mbox{$\times 10^{1}$}$ | $1.6\mbox{$\times 10^{2}$}$ | $1.4\mbox{$\times 10^{3}$}$ | $4.2\mbox{$\times 10^{3}$}$
$\mbox{Rm${}^{(1)}$}(U_{Z},\sigma_{F})$ | $9.1\mbox{$\times 10^{-4}$}$ | $1.8\mbox{$\times 10^{-1}$}$ | $4.2\mbox{$\times 10^{0}$}$ | $6.2\mbox{$\times 10^{1}$}$ | $5.4\mbox{$\times 10^{2}$}$ | $9.3\mbox{$\times 10^{3}$}$ | $2.6\mbox{$\times 10^{4}$}$
$\mbox{Rm${}^{(1)}$}(U_{G},\sigma_{F})$ | $3.2\mbox{$\times 10^{-8}$}$ | $7.4\mbox{$\times 10^{-4}$}$ | $7.9\mbox{$\times 10^{-1}$}$ | $2.4\mbox{$\times 10^{2}$}$ | $2.1\mbox{$\times 10^{4}$}$ | $3.8\mbox{$\times 10^{6}$}$ | $1.6\mbox{$\times 10^{7}$}$
$\mbox{Rm${}^{(1)}$}(U_{K},\sigma_{F})$ | $2.2\mbox{$\times 10^{-8}$}$ | $3.2\mbox{$\times 10^{-4}$}$ | $1.8\mbox{$\times 10^{-1}$}$ | $2.5\mbox{$\times 10^{1}$}$ | $1.0\mbox{$\times 10^{3}$}$ | $3.9\mbox{$\times 10^{4}$}$ | $4.3\mbox{$\times 10^{4}$}$
$\mbox{Rm${}^{(1)}$}(U_{Z},\sigma_{F})$ | $2.4\mbox{$\times 10^{-8}$}$ | $4.5\mbox{$\times 10^{-4}$}$ | $3.4\mbox{$\times 10^{-1}$}$ | $6.9\mbox{$\times 10^{1}$}$ | $3.5\mbox{$\times 10^{3}$}$ | $2.7\mbox{$\times 10^{5}$}$ | $2.7\mbox{$\times 10^{5}$}$
Table 1: Rms flow velocities, electrical conductivities $\sigma$, magnetic
diffusivities $\lambda$, diffusivity scale heights $D_{\lambda}$, and magnetic
Reynolds numbers Rm(1) at selected radii.

Panel b) of fig. 1 and selected values listed in tab. 1 demonstrate that the
magnetic diffusivity scale heights $D_{\lambda}$ differ much less than the
conductivities themselves. Electric currents, locally induced fields, and
Ohmic heating depend linearly on $\sigma$ but on different powers of
$D_{\lambda}$. The differences between the results for the two conductivity
models is thus predominantly determined by $\sigma$ and can easily be scaled
from one to the other.

Figure 2: Zonal flow models used in this study. Prograde flows are shown in
red and yellow, while blue indicates retrograde directions.

The three different zonal flow models explored here are illustrated in fig. 2.
Tab. 1 lists rms values $\langle\mbox{$\overline{U}_{\phi}$}\rangle$ at
selected radii. All reproduce the observed zonal winds at $r=r_{J}$ (Porco et
al., 2003; Vasavada & Showman, 2005). We use running averages of the surface
profiles with a window width of one degree and represent the result with 256
(nearly) evenly spaced latitudinal grid points for our calculations.

The three flow models differ at depth. The most simple one, $U_{G}$, assumes
geostrophy in each hemisphere, i.e. the flow depends only on the distance
$s=r\sin\theta$ to the rotation axis. Kaspi et al. (2018) describe the depth
decay of the equatorially antisymmetric zonal flow with profiles constrained
by the Juno gravity measurements. We apply their ’latitude independent’ model
version to the total zonal flow and refer to this model as $U_{K}$. The rms
amplitude of $U_{K}$ has decreased by one order of magnitude at about
$0.95\,r_{J}$ and by two orders of magnitude around $0.925\,r_{J}$.

We also consider the ’deep’ model suggested by Kong et al. (2018), who assume
an exponential depth decay and an additional linear dependence on the distance
$z=\cos\theta$ to the equatorial plane. Like for $U_{K}$, our respective model
$U_{Z}$ assumes that the depth and $z$ dependencies, which were originally
derived for the equatorially antisymmetric contributions, apply to the whole
flow. The rms velocity in $U_{Z}$ decays smoother with depth than in $U_{K}$,
having decreased by one order of magnitude at about $0.935\,r_{J}$ and by two
orders of magnitude at about $0.905\,r_{J}$.

Fig. 2 shows that $U_{G}$ and $U_{K}$ have discontinuities at the equatorial
plane. These pose a problem when calculating the latitudinal zonal flow
derivatives required for the integral estimate $j^{(I)}_{\theta}$ (see eqn.
(18)). Formally, the derivative becomes infinite at the equator. Practically,
however, the impact of the discontinuity depends on the model setup and on the
methods used for calculating the derivatives. We tested the impact on rms
current density estimates by comparing calculations covering all latitudes
with counterparts where the derivatives were explicitly set to zero in a six-
degree belt around the equator. Simple first order finite differences with 256
grid points at each radial level are generally used for calculating the
derivative. For flow $U_{Z}$, which has been constructed to avoid the
discontinuity (Kong et al., 2018), the belt contributes not more than one
percent to $\langle j^{(I)}_{\theta}\rangle$ at any radius, which is less than
the surface fraction it represents. For flow $U_{K}$, the contribution is even
smaller due to the faster decay of the flow amplitude. However, for $U_{G}$
the belt contributes $20$% to the rms current for radii below $0.94\,r_{J}$,
which is a clear sign that the unphysical discontinuity causes problems. In
order to be on the safe side, we will only consider flow model $U_{Z}$ in
connection with estimate $j^{(I)}_{\theta}$ below.

Figure 3: Modified magnetic Reynolds numbers for the three zonal flow and the
two conductivity models explored here. Vertical lines mark where the radii
$r_{1}$ where $\mbox{Rm${}^{(1)}$}=1$. Green lines show profiles for a typical
convective velocity of $10\,$cm/s suggested by scaling laws.

The radius where $\mbox{Rm${}^{(1)}$}=1$, which we will refer to as $r_{1}$ in
the following, roughly marks the point where the approximations discussed
above break down (Wicht et al., 2019). Fig. 3a) illustrates the Rm(1) profiles
that result from combining $\sigma_{F}$ and $\sigma_{Z}$ with rms values for
the three zonal flow models. Tab. 1 compares values at some selected radii.
These modified magnetic Reynolds numbers exceed unity between
$r_{1}=0.957\,r_{J}$ for the combination $\sigma_{Z}$ and $U_{K}$ and
$r_{1}=0.967\,r_{J}$ for $\sigma_{F}$ and $U_{G}$. All $r_{1}$ values are
listed in tab. 2.

Green lines in fig. 3 show Rm(1) profiles for a typical convective velocity of
$10\,$cm/s suggested by scaling laws (e.g. see Duarte et al., 2018). Numerical
simulations show that the velocity increases with radius, an effect not taken
into account here. The comparison of the different Rm(1) profiles suggests
that zonal flow related dynamo action should dominate at least in the outer
$9$% in radius.

For Jupiter’s surface magnetic field we use the JRM09 model by Connerney et
al. (2018), which provides information up to spherical harmonic degree
$\ell=10$. The more recent model by Moore et al. (2018) is only slightly
different. In order to check the impact of smaller scale contributions, we
also tested the numerical model G14 by Gastine et al. (2014), which reproduces
Jupiter’s large scale field and provides harmonics up $\ell=426$. Since it
turned out that the impact of the smaller scales is very marginal, the results
are not shown here.

##  3 Dissipative Heating and Entropy Production in Jupiter

Liu et al. (2008) constrain the depth of the zonal winds by assuming that the
related total Ohmic heating should not exceed the heat flux out of the planet.
Unfortunately, this assumption is not correct, as we will show in the
following. In order to arrive at more meaningful constraints, we start with
reviewing some fundamental considerations.

In a quasi-stationary state, where flow and magnetic field are maintained by
buoyancy and induction against dissipative losses, the conservation of energy
simply states that the heat flux $Q_{o}=Q(r_{o})$ through the outer boundary
is the sum of the flux $Q_{i}=Q(r_{i})$ through the inner boundary and the
total internal heating $H$:

$Q_{o}=Q_{i}+H\;\;.$ (33)

Note that neither viscous nor Ohmic heating contribute to $H$. Since flow and
magnetic field are maintained by the heat flux through the system, they cannot
be counted as net heat sources (Hewitt et al., 1975; Braginsky & Roberts,
1995).

When furthermore also neglecting the effects of helium segregation, core
erosion, or planetary shrinking as potential energy sources, the only
remaining contribution is the slow secular cooling of the planet. The
volumetric heat source is then given by

$h=\tilde{\rho}\,\tilde{T}\,\frac{\partial\tilde{S}}{\partial t}\;\;,$ (34)

where the tilde indicates the hydrostatic, adiabatic, background state
(Braginsky & Roberts, 1995). Assuming that convection maintains an adiabat at
all times, $\partial\tilde{S}/\partial t$ remains homogeneous throughout the
convective region and obeys (Jones, 2014):

$\frac{\partial\tilde{S}}{\partial
t}=\left(Q_{o}-Q_{i}\right)\big{/}\int_{V}\mathrm{d}V\;\tilde{\rho}\,\tilde{T}\;\;.$
(35)

Here, $\int_{V}\mathrm{d}V$ denotes an integration over the whole convective
volume. Note, however, that the thermal evolution could be more complex,
should Jupiter indeed harbor stably stratified regions.

In order to get a handle on dissipative heating, one has to consider the local
heat equation

$\tilde{\rho}\tilde{T}\left(\frac{\partial s}{\partial
t}+\mbox{$\boldsymbol{U}$}\cdot\nabla s\right)=\nabla\cdot\left(k\nabla
T\right)+h+\varphi\;\;,$ (36)

where $\varphi$ denotes the volumetric dissipative heat source, and $k$ is the
thermal conductivity. When assuming a steady state and adopting the anelastic
approximation $\nabla\cdot(\tilde{\rho}\mbox{$\boldsymbol{U}$})=0$, the
integration over the shell between the inner boundary $r_{i}$ and radius $r$
yields

$Q_{D}(r)+Q_{A}(r)=Q_{i}+\int_{r_{i}}^{r}\mathrm{d}r^{\prime}\int_{F}\mathrm{d}F\;\left(h+\varphi+\tilde{\rho}\;s\;U_{r}\frac{\partial\tilde{T}}{\partial
r}\right)\;\;.$ (37)

The left hand side is the total flux through the boundary at $r$, i.e. the sum
of the diffusive contribution

$Q_{D}(r)=-\int_{F}\mathrm{d}F\;k\frac{\partial T}{\partial r}$ (38)

and the advective contribution

$Q_{A}(r)=\int_{F(r)}\mathrm{d}F\;\tilde{\rho}\,\tilde{T}\,s\,U_{r}\;\;.$ (39)

The right hand side of eqn. (37) reflects the influx through the lower
boundary $Q_{i}$ plus three volumetric contributions: the slow secular
cooling, the dissipative heating, and the adiabatic cooling. Writing the
adiabatic cooling in terms of $Q_{A}$ yields the relation

$\displaystyle Q_{D}(r)+Q_{A}(r)=Q_{i}$ $\displaystyle+$
$\displaystyle\int_{r_{i}}^{r}\mathrm{d}r^{\prime}\int_{F}\mathrm{d}F\;\left(h+\varphi\right)$
$\displaystyle-$
$\displaystyle\int_{r_{i}}^{r}\mathrm{d}r^{\prime}Q_{A}(r^{\prime})\,\big{/}\,D_{T}(r^{\prime})\;\;,$

where $D_{T}=-\tilde{T}/(\partial\tilde{T}/\partial r)$ is the thermal scale
height.

Integrating eqn. (3) over the whole convective volume and using eqn. (33)
reveals that the total dissipative heating $\varPhi_{T}$ is balanced by the
total adiabatic cooling:

$\varPhi_{T}=\int_{V}\mathrm{d}V\;\varphi=\int_{r_{i}}^{r_{o}}\mathrm{d}r^{\prime}\;Q_{A}(r^{\prime})\,\big{/}\,D_{T}(r^{\prime})\;\;.$
(41)

The total adiabatic cooling is actually identical to the buoyancy power $P$
that drives convection and thus the dynamo mechanism. Multiplying the buoyancy
term in the Navier-Stokes equation with velocity and integrating over the
convective volume to yield the total convective power input indeed gives the
same expression (Braginsky & Roberts, 1995). Eqn. (41) thus simply states that
dissipation is balanced by the power input $P$ to the system, a fact used in
many scaling laws to establish how the rms magnetic field strength or rms
velocity scale with $P$ (Christensen & Aubert, 2006; Christensen et al., 2009;
Davidson, 2013; Yadav et al., 2013).

Eqn. (41) requires to know $Q_{A}$ at each radius. Since $Q_{A}$ itself
depends on the distribution of dissipative heat sources, however, an
additional condition is required. Assuming that Ohmic heating and adiabatic
cooling not only cancel globally but, at least roughly, also at each radius
offers a simple solution used in most scaling laws (though never stated
explicitely). With the exception of thin thermal boundary layers, the heat
flux is then dominated by the advective contribution, so that

$Q_{A}(r)\approx Q_{i}+\int_{r_{i}}^{r}\mathrm{d}r\int_{F}\mathrm{d}F\;h\;\;.$
(42)

Adopting the interior model by Nettelmann et al. (2012) and French et al.
(2012) and the observed flux $Q_{o}=3.35\mbox{$\times 10^{17}$}\,$W from the
planet’s interior (Guillot & Gautier, 2015) allows calculating $h$ via eqn.
(34). Because the inner core occupies only $10$% in radius, $Q_{i}$ can be
neglected. When, for example, assuming that $h$ also describes the cooling of
the rocky core, $Q_{i}$ is two orders of magnitude smaller than $Q_{o}$.

Plugging eqn. (42) into eqn. (41) finally allows calculating the total
dissipative heating:

$\varPhi_{T}=1.20\mbox{$\times 10^{18}$}\,\mbox{W}\;\;.$ (43)

The result reveals that dissipative heating can in fact exceed the heat flux
out of Jupiter’s interior by a factor of $3.6$. Gastine et al. (2014) came up
with a power estimate that is about $50$% smaller because they used a
simplified formula provided by Christensen et al. (2009).

Considering the entropy rather than the heat balance avoids the need to come
up with an additional condition (Hewitt et al., 1975; Gubbins et al., 1979;
Braginsky & Roberts, 1995; Gubbins et al., 2003). Dividing the heat equation
eqn. (36) by temperature and integrating over the convective volume yields the
entropy budget

$\frac{Q_{o}}{\tilde{T}_{o}}=\frac{Q_{i}}{\tilde{T}_{i}}\;+\;\int_{V}\mathrm{d}V\left(\frac{h}{\tilde{T}}\;+\;k\;\left|\frac{\partial
T/\partial r}{\tilde{T}}\right|^{2}\;+\;\frac{\varphi}{\tilde{T}}\right)\;\;,$
(44)

where we have once more used the anelastic approximation
$\nabla\cdot(\tilde{\rho}\mbox{$\boldsymbol{U}$})=0$. When assuming that the
temperature profile stays close to the adiabat, the total dissipative entropy
production $\varTheta$ can thus be approximated by:

$\varTheta_{T}=\int_{V}\mathrm{d}V\frac{\varphi}{\tilde{T}}\approx\frac{Q_{o}}{\tilde{T}_{o}}-\frac{Q_{i}}{\tilde{T}_{i}}-\int_{V}\mathrm{d}V\frac{h}{\tilde{T}}-\int_{V}\mathrm{d}Vk\;\left|\frac{\partial\tilde{T}/\partial
r}{\tilde{T}}\right|^{2}\;\;.$ (45)

An upper bound for the total dissipative heating can be derived when assuming
that $\tilde{T}_{i}$ is the highest temperature in the system (Hewitt et al.,
1975; Currie & Browning, 2017):

$\varPhi\;<\;\tilde{T}_{i}\int_{V}\mathrm{d}V\frac{\varphi}{\tilde{T}}\;<\;\frac{\tilde{T}_{i}}{\tilde{T}_{o}}\;Q_{o}\;\;.$
(46)

Using once more the internal model by Nettelmann et al. (2012) puts the upper
bound at $10^{2}\,Q_{o}$ for Jupiter, which is at least consistent with
estimate (43).

When complementing the internal model with the thermal conductivity profile by
French et al. (2012), we can quantify the different terms in Jupiter’s entropy
budget (45). Because of the strong temperature contrast between the outer
boundary and the deeper convective region, the entropy flux through the outer
boundary clearly dominates. The total dissipative entropy production is thus
given by:

$\varTheta_{T}\approx Q_{o}/\tilde{T}_{o}=2.0\mbox{$\times
10^{15}$}\,\mbox{W/K}\;\;.$ (47)

The second largest term in eqn. (45), the entropy due to the secular cooling,
is already two orders of magnitude smaller at $3.0\mbox{$\times
10^{13}$}\,$W/K. The two remaining terms, entropy flux through the inner
boundary and the diffusive entropy flux down the adiabat, are only of order
$10^{11}\,$W/K.

Since the magnetic diffusivity is about $10^{6}$ times larger than its viscous
counterpart in planetary dynamo regions, Ohmic heating by far dominates. We
can use the current density estimates to predict the Ohmic heating due to the
zonal flows above radius $r$:

$\varPhi_{O}(r)=\int_{r}^{r_{J}}\;\mathrm{d}r^{\prime}\;\int_{F}\mathrm{d}F\;\frac{\mbox{$\boldsymbol{j}$}^{2}}{\sigma}\;\;.$
(48)

The conditions

$\varPhi_{O}(r)\leq\varPhi_{T}=1.20\mbox{$\times 10^{18}$}\,\mbox{W}$ (49)

provides a possible constraint for the depth of the zonal winds in Jupiter.

The dissipative entropy production related to the Ohmic heating is given by

$\varTheta_{O}(r)\approx\int_{r}^{r_{J}}\;\mathrm{d}r^{\prime}\;\int_{F}\mathrm{d}F\;\frac{\mbox{$\boldsymbol{j}$}^{2}}{\sigma\,\tilde{T}}\;\;.$
(50)

This can be used for the alternative depth constraint

$\varTheta_{O}(r)\leq\varTheta_{T}=2.0\mbox{$\times
10^{15}$}\,\mbox{W/K}\;\;.$ (51)

##  4 Dynamo Action in Jupiter’s SDCR

###  4.1 Electric Currents and Locally Induced Field

Figure 4: (a) Rms current density estimates for flow model $U_{Z}$ and
conductivity model $\sigma_{F}$. (b) Estimates of rms radial and horizontal
LIF. The profile of $\langle\tilde{B}_{r}\rangle$ has been included for
comparison. Vertical lines mark the radii where $\mbox{Rm${}^{(1)}$}=1$ for
the different flow models. Current density estimates
$\mbox{$\boldsymbol{j}$}^{(O)}$ and conductivity model $\sigma_{F}$ have been
used.

We start with discussing the current estimates for the different zonal flow
and conductivity model combinations. Fig. 4a) compares rms values of integral
estimates $\mbox{$\boldsymbol{j}$}^{(I)}$ and Ohm’s law estimates
$\mbox{$\boldsymbol{j}$}^{(O)}$ for conductivity model $\sigma_{F}$ and flow
$U_{Z}$. The integral estimates of the latitudinal currents are at least 40
times larger than their azimuthal counterparts for all conductivity and flow
model combinations. When using $\mbox{$\boldsymbol{j}$}^{(O)}$ as outer
boundary condition for $\mbox{$\boldsymbol{j}$}^{(I)}$ (blue line in fig. 4),
both estimates remain very similar down to about $0.98\,r_{J}$. At
$0.97\,r_{J}$, however, $\mbox{$\boldsymbol{j}$}^{(I)}$ is already about $50$%
larger than $\mbox{$\boldsymbol{j}$}^{(O)}$, and at $0.96\,r_{J}$ the
difference has increased to about $250$%. When assuming a vanishing outer
boundary current for $\mbox{$\boldsymbol{j}$}^{(I)}$, the differences are even
larger: $\mbox{$\boldsymbol{j}$}^{(I)}$ (green line) is $3.5$ times larger
than $\mbox{$\boldsymbol{j}$}^{(O)}$ at $0.97\,r_{J}$ and about $6$ times
larger at $0.96\,r_{J}$.

Estimates of the rms horizontal and radial LIF are shown in fig. 4b), based on
$\mbox{$\boldsymbol{j}$}^{(O)}$ and eqn. (20) for the horizontal and on eqn.
(26) for the radial components. The radial LIF is between two and three orders
of magnitude smaller than its horizontal counterpart. The rougher estimates
(23) and (28), based on Rm(1) and Rm(2) respectively, provide values that are
less than a factor two smaller and can thus safely be used for order of
magnitude assessments. They correctly predict that the rms azimuthal LIF
reaches the level of the background field at $r_{1}$ and also that the ratio
of radial to azimuthal LIF is about
$\mbox{Rm${}^{(2)}$}/\mbox{Rm${}^{(1)}$}=D_{\lambda}/r_{J}$. At $r_{1}$, the
rms radial LIF is thus roughly three orders of magnitude smaller than the
background field or the horizontal LIF. Tab. 2 lists the relative rms radial
LIF (column 7) at $r_{1}$ (column 3) for all $\sigma$ and flow combinations
when using $\mbox{$\boldsymbol{j}$}^{(O)}$.

conduct. | flow | $r_{1}\big{/}r_{J}$ | $r_{10}\big{/}r_{J}$ | $r_{\varPhi}\big{/}r_{J}$ | $r_{\varTheta}\big{/}r_{J}$ | $\langle\hat{B}_{r}\rangle\big{/}\langle\tilde{B}_{r}\rangle$ | $\varPhi_{O}\big{/}\varPhi_{T}$ | $\varTheta_{O}\big{/}\varTheta_{T}$
---|---|---|---|---|---|---|---|---
$\sigma_{F}$ | $U_{G}$ | $0.967$ | $0.960$ | $0.961$ | $0.955$ | $2.8\mbox{$\times 10^{-3}$}$ | $1.2\mbox{$\times 10^{-1}$}$ | $2.3\mbox{$\times 10^{-2}$}$
| $U_{Z}$ | $0.965$ | $0.957$ | $0.956$ | $0.948$ | $2.2\mbox{$\times 10^{-3}$}$ | $8.0\mbox{$\times 10^{-2}$}$ | $1.5\mbox{$\times 10^{-2}$}$
| $U_{K}$ | $0.963$ | $0.954$ | $0.946$ | $0.929$ | $3.0\mbox{$\times 10^{-3}$}$ | $4.8\mbox{$\times 10^{-2}$}$ | $9.0\mbox{$\times 10^{-3}$}$
$\sigma_{Z}$ | $U_{G}$ | $0.960$ | $0.956$ | $0.956$ | $0.953$ | $1.8\mbox{$\times 10^{-3}$}$ | $1.5\mbox{$\times 10^{-1}$}$ | $2.5\mbox{$\times 10^{-2}$}$
| $U_{Z}$ | $0.958$ | $0.954$ | $0.953$ | $0.949$ | $1.6\mbox{$\times 10^{-3}$}$ | $8.9\mbox{$\times 10^{-2}$}$ | $1.5\mbox{$\times 10^{-2}$}$
| $U_{K}$ | $0.957$ | $0.952$ | $0.948$ | $0.942$ | $1.7\mbox{$\times 10^{-3}$}$ | $2.9\mbox{$\times 10^{-2}$}$ | $4.9\mbox{$\times 10^{-3}$}$
Table 2: While the first two columns define the model, columns 3 to 6 list
different radii: radius $r_{1}$ where $\mbox{Rm${}^{(1)}$}=1$, $r_{10}$ where
$\mbox{Rm${}^{(1)}$}=10$, $r_{\varPhi}$ where $\varPhi_{O}=\varPhi_{T}$, and
$r_{\varTheta}$ where $\varTheta_{O}=\varTheta_{T}$. Column 7 provides the
ratio of rms radial LIF to rms radial background field at $r_{1}$. Columns 8
and 9 give ratios $\varPhi_{O}/\varPhi_{T}$ and $\varTheta_{O}/\varTheta_{T}$
at $r_{1}$, respectively. Current density estimates
$\mbox{$\boldsymbol{j}$}^{(O)}$ have been used.  Figure 5: Maps of (a) the
radial surface field in the Jupiter field model JRM09, (b) the radial LIF at
$r_{1}=0.965\,r_{J}$, and (c) the local Ohmic heating
$q=\int_{r}^{r_{J}}dr\,\mbox{$\boldsymbol{j}$}^{2}/\sigma$ at
$r=r_{1}=0.965\,r_{J}$. Flow model $U_{Z}$, conductivity $\sigma_{F}$, and
current density estimates $\mbox{$\boldsymbol{j}$}^{(O)}$ (for panel c) have
been used. Outward (inward) directed field is shown in red (blue).

Wicht et al. (2019) demonstrate that the Ohm’s-law based estimates not only
provide good rms but also decent local values for their Jupiter-like dynamo
simulations. Fig. 5 shows the radial surface field of JRM09 in panel a) and
the radial LIF for $\sigma_{F}$ and $U_{Z}$ at $r_{1}$ in panel b). A very
distinct pattern of localized field patches can be found where the fast zonal
jets around the equator interact with the strong blue patch in the JRM09
model.

Figure 6: (a) $U_{Z}$ profiles at different depths; the grey background color
highlights where surface jets are prograde. (b) azimuthal rms radial LIF and
(c) heat flux $q=\int_{r}^{r_{J}}dr\,\mbox{$\boldsymbol{j}$}^{2}/\sigma$ at
three different depths. Conductivity model $\sigma_{F}$, flow $U_{Z}$, and
estimate $\mbox{$\boldsymbol{j}$}^{(O)}$ have been used.

The zonal flow pattern remains recognizable in the LIF, as is clearly
demonstrated in fig. 6, which compares zonal flow profiles in panel a) with
the azimuthal rms of the radial LIF in panel b). Due to the flow geometry, the
currents and LIF show a depth-dependent phase shift relative to the surface
jets. The equatorial jet, which is so prominent at the surface, contributes
very little to dynamo action, since it does not reach down to depths where the
electrical conductivity is more significant.

Figure 7: Power spectra of rms radial field contributions (Mauersberger-Lowes)
for JRM09, the downward continued $\tilde{B}_{r}$ and the radial LIF
$\hat{B}_{r}$ at $r_{1}=0.965\,r_{J}$. (a) shows the spherical harmonic degree
spectrum, while (b) shows the harmonic order spectrum. The LIF has been
amplified with $10^{3}$. Flow $U_{Z}$ and conductivity $\sigma_{F}$ have been
used.

Fig. 7 compares spherical harmonic power spectra of the background radial
field and the radial LIF. As already apparent from the map shown in fig. 5,
the LIF is dominated by smaller scale contributions. The spherical harmonic
degree spectrum results from the convolution of the complex latitudinal zonal
flow structure with the background field. At $r_{1}$, the dipole contribution
in the LIF is about $10^{-4}$ times smaller than the respective background
field contribution. For degree $\ell=10$, the ratio has increased to
$10^{-2}$. The spectrum peaks at $\ell=12$ but has also significant
contributions from even higher degrees.

The spherical harmonic order spectrum, shown in panel b) of fig. 7, is very
different. The action of the axisymmetric zonal flow on $\tilde{B}_{r}$
excites no additional harmonic orders so that the spectrum remains confined to
$m\leq 10$. The LIF spectrum is rather flat but has no axisymmetric
contribution. At $m=10$, the rms LIF amplitude reaches roughly $25\,$% of the
background field.

The results for the conductivity model $\sigma_{F}$ presented so far can
roughly be scaled to model $\sigma_{Z}$ by multiplying with the conductivity
ratio $\sigma_{Z}/\sigma_{F}$. Around $0.97\,r_{J}$, the LIF is two orders of
magnitude weaker, and the difference decreases with depth, reaching about one
order of magnitude around $0.96\,r_{J}$. Where $\mbox{Rm${}^{(1)}$}=1$, on the
other hand, the LIF reaches comparable values for both conductivity models.
The different flow models yield very similar LIF pattern, albeit with the
different amplitudes indicated in fig. 4.

###  4.2 Ohmic Heating and Entropy Constraint

We now use the electric current estimates to calculate Ohmic heating and
entropy production. Panel c) of fig. 5 shows the map of Ohmic heat flux
density $q=\int_{r}^{r_{J}}dr\,\mbox{$\boldsymbol{j}$}^{2}/\sigma$ at radius
$r_{1}$ when using $\mbox{$\boldsymbol{j}$}^{(O)}$, $\sigma_{F}$, and $U_{Z}$.
The currents induced by interaction between the fierce zonal jets close to the
equator and the strong blue patch in JRM09 not only yield a highly localized
LIF but also intense local heating. While the action of various other zonal
jets reaches a lower level, the related pattern remains roughly recognizable
in the form of thin heating bands. The azimuthal mean of $q$, shown in Panel
c) of fig. 6, clearly illustrates the correlation between heating and the
zonal jets. Like for the LIF, there is a depth-dependent phase shift between
the observed surface zonal wind profile and the Ohmic heating pattern.

Figure 8: Profiles of (a) Ohmic heating and (b) entropy production in the
layer above radius $r$ for current estimate $\mbox{$\boldsymbol{j}$}^{(O)}$.
In (a) the solid horizontal line shows the total convective power of
$1.2\mbox{$\times 10^{18}$}\,$W, while the dotted horizontal line shows the
heat flow of $Q_{o}=3.35\mbox{$\times 10^{17}$}\,$W out of Jupiter’s interior.
In (b) the horizontal line indicates the total dissipative entropy production
predicted by the entropy flux $\varTheta_{T}=Q_{o}/T_{o}=2.0\mbox{$\times
10^{15}$}\,\mbox{W/K}$ through the outer boundary. Vertical lines mark the
radii $r_{1}$ where $\mbox{Rm${}^{(1)}$}=1$ (see fig. 3).

Panel a) of fig. 8 compares the Ohmic heating profiles $\varPhi_{O}(r)$ for
the different zonal flow and electrical conductivity models. Because of the
extremely low conductivity, heating remains negligible in outer two percent in
radius. When using $\mbox{$\boldsymbol{j}$}^{(O)}$, the outermost radius where
$\varPhi_{O}$ reaches the level of $\varPhi_{T}$ is $r_{\varPhi}=0.950\,r_{J}$
for flow $U_{G}$ and both conductivity models. When using $U_{K}$ and
$\sigma_{F}$, the Ohmic heating always remains below $\varPhi_{T}$. Results
based on $\mbox{$\boldsymbol{j}$}^{(I)}$ (not shown) are less sensitive to the
differences between the three flow models at depth and are generally similar
to the results for $U_{G}$ and $\mbox{$\boldsymbol{j}$}^{(O)}$.

The different $r_{\varPhi}$ values where $\varPhi_{O}=\varPhi_{T}$ have been
marked by vertical lines in fig. 8 and are listed in column $5$ of tab. 2. All
are located below the radii $r_{1}$ where $\mbox{Rm${}^{(1)}$}=1$ for the
respective model combinations (column 3) and thus in a region where the
approximations employed here break down. The maximum Ohmic heating reached at
$r_{1}$ remains nearly one order of magnitude below $\varPhi_{T}$ (see column
8 of tab. 2).

Similar inferences hold for the entropy production shown in panel b) of fig.
8. The entropy condition is less strict than the power-based heat condition,
and the radii $r_{\varTheta}$ where the different models exceed the threshold
$\varTheta_{T}$ (column 6 of tab. 2) are somewhat deeper than respective
$r_{\varPhi}$ values. The largest value of $r_{\varTheta}=0.955$ is found for
the combination $\mbox{U}_{G}$ and $\sigma_{Z}$. The combination of $U_{K}$
and $\sigma_{F}$, on the other hand, yields the deepest value of
$r_{\varTheta}=0.929$.

The exploration of numerical dynamo simulations by Wicht et al. (2019) suggest
that $\mbox{$\boldsymbol{j}$}^{(O)}$ may provide an acceptable estimate for a
limited region below $r_{1}$, at least down to where $\mbox{Rm${}^{(1)}$}=5$.
Column 4 of tab. 2 demonstrates that even the radius $r_{10}$ where
$\mbox{Rm${}^{(1)}$}=10$ lies deeper than $r_{\varPhi}$ for most flow and
conductivity combinations. The only exceptions are the results for the
geostrophic flow. This could indicated that strictly geostrophic flows would
indeed violate the heating contraint.

##  5 Discussion and Conclusion

The dominance of Ohmic dissipation in the outer few percent of Jupiter’s
radius leads to simple quasi-stationary dynamo action. This can be exploited
for estimating the electric currents and the Locally Induced Fields with
surprisingly high quality (Wicht et al., 2019), once a conductivity profile, a
surface magnetic field model, and flow model are given. Here we explored two
conductivity profiles, used the new Juno-based JRM09 field model, and tested
two zonal flow models suggested from inversions of Juno gravity measurements.
A geostrophic zonal flow model was also considered as a third option.

The estimates roughly apply to the upper four percent in radius, or roughly
$3000\,$km, where the modified magnetic Reynolds number Rm(1) is smaller than
one. The radial LIF in this quasi-strationary dynamo region typically reaches
rms values in the order of $\mu$T with peak values up to $15\,\mu$T. Could
such a small contribution be measured by the Juno magnetometer? The instrument
has been designed to provide a nominal vector accuracy of $1$ in $10^{4}$.
Since the surface field reaches peak values of about $2\,m$T, the LIF could
indeed be detectable.

One would still have to separate the LIF from contributions produced deeper in
the planet. What should help with this task, is the distinct pattern imprinted
by the zonal flows, which also leads to a distinct magnetic spectrum. The LIF
spectrum peaks at degree $\ell=12$ and has significant contributions at even
higher degrees. At $\ell=10$, the largest degree provided by JRM09, the LIF
amounts to about $1$% of the background field, which seems smaller than the
estimated JRM09 precision (Connerney et al., 2018). Updated future models,
based on a larger number of Juno orbits, will provide smaller scale details
and increase the chances of identifying the LIF. Another possibility is a
dedicated analysis of measurements around the ’big blue spot’ in JRM09, where
inductions effects are particularly strong.

Our analysis of Jupiter’s heat balance shows that Ohmic heating can
significantly exceed the heat flux $Q_{o}$ out of the planet’s interior. Using
the interior model by Nettelmann et al. (2012) and French et al. (2012)
suggests a total dissipative heating of
$\varPhi_{T}=3.58\,Q_{o}=1.20\mbox{$\times 10^{18}$}$W.

It would be interesting to repeat this assessment for the newer Jupiter models
that include stably stratified regions (Debras & Chabrier, 2019). However, the
most important constraint is the knowledge of $Q_{o}$, and the somewhat
different distribution of internal heat sources implied by the newer models
can only have a limited effect.

While the total Ohmic heating remains typically one order of magnitude below
$\varPhi_{T}$, we find extreme lateral variations. Peak values in the Ohmic
heating density reach $25\,$W/m2 around the ’blue spot’ in the JRM09, which is
nearly five times larger than the mean heat flux density from Jupiter’s
interior. These peak values are reached at the bottom of the quasi-stationary
region, i.e. at a depth of $3000\,km$. This is much deeper than any (current)
remote instrument could reach for. For example MWR, the micro-wave instrument
on Juno, hopes to detect temperature radiation from up to $1\,$kbar, which
corresponds to a depth of about $600\,$km. However, the local heating may
trigger convective plumes that rise to shallower depths and thus become
detectable.

We also estimated the entropy flux out of Jupiter’s interior to
$2.0\mbox{$\times 10^{15}$}\,$W/K. The entropy produced by zonal-wind related
Ohmic heating in the quasi-stationary region does not exceed this value for
any model combination. This means that neither Ohmic heating nor the entropy
production offer any reliable constraint on the depth of the zonal winds.

Below the quasi-stationary region, electric fields become a significant
contribution to Ohm’s law, tend to oppose induction effects, and lead to
weaker electric currents than predicted by our approximations. Wicht et al.
(2019) demonstrate that the currents remain roughly constant below the depth
where $\mbox{Rm${}^{(1)}$}\approx 5$ in their numerical simulations. However,
this may be different in Jupiter where the magnetic Reynolds numbers reach
values orders of magnitude higher than in their computer models.

Fig. 3 demonstrates that Rm(1) increases to a value of at least $10^{3}$ at
$0.90\,r_{J}$. This is a consequence of the electrical conductivity profiles
that easily overcompensate the depth-decrease in zonal flow velocities
indicated by Juno gravity measurements. The zonal flows may thus actually play
a larger role for dynamo action below than in the quasi-stationary region.
While the gravity data convincingly show that the zonal winds must be
significantly weaker below about $0.96\,r_{J}$, they cannot uniquely constrain
their structure or amplitude at this depth.

It has been speculated that the fast observed zonal winds may remain confined
to a thin weather layer, where differential solar heating and also moist
convection could significantly contribute to the dynamics (see for example
Showman (2007) or Thomson & McIntyre (2016)). Kong et al. (2018) show that the
gravity signal can then be explained by an independent zonal flow system that
reaches down to about $0.7\,r_{J}$ with typical amplitudes of about $1\,$m/s
and has larger latitudinal scales than the surface winds. The strongest local
dynamo action happens towards the bottom of the quasi-stationary region where
models $U_{K}$ and $U_{Z}$ reach velocities of about $10\,m/s$. The currents
and magnetic fields induced by this alternative flow model should thus be
roughly an order of magnitude weaker than for $U_{K}$ or $U_{Z}$.
Consequently, Ohmic heating and entropy production would be two orders of
magnitude lower and play practically no role for the global power or entropy
budgets.

Below $0.96\,r_{J}$, full 3d numerical simulations would be required to model
the zonal-wind related dynamo action. However, since they cannot be run at
altogether realistic parameters and generally yield a much simpler zonal wind
pattern, the results must be interpreted with care (Gastine et al., 2014;
Jones, 2014; Duarte et al., 2018; Dietrich & Jones, 2018). These simulation
suggest that even the weaker zonal winds at depth would significantly shear
the large scale field produced by the deeper primary dynamo action. The
resulting strong longitudinal (toroidal) flux bundles are converted into
observable radial field by the small scale convective flows present in this
region. The combined action of primary and secondary dynamo typically yields a
radial surface field that is characterized by longitudinal banded structures
and large scale patches with wavenumber one or two, resulting in a morphology
is often reminiscent of the recent Juno-based field model JRM09 (Gastine et
al., 2014; Duarte et al., 2018; Dietrich & Jones, 2018).

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