# Magnetic Induction in Convecting Galilean Oceans

###### Abstract

To date, analyses of magnetic induction in putative oceans in Jupiter’s large
icy moons have assumed uniform conductivity in the modeled oceans. However,
the phase and amplitude response of the induced fields will be influenced by
the increasing electrical conductivity along oceans’ convective adiabatic
temperature profiles. Here, we examine the amplitudes and phase lags for
magnetic diffusion in modeled oceans of Europa, Ganymede, and Callisto. We
restrict our analysis to spherically symmetric configurations, treating
interior structures based on self-consistent thermodynamics, accounting for
variations in electrical conductivity with depth in convective oceans (Vance
., 2018). The numerical approach considers tens of radial layers. The
induction response of the adiabatic conductivity profile differs from that of
an ocean with uniform conductivity set to that at the ice-ocean interface, or
to the mean value of the adiabatic profile, by more than 10% in many cases. We
compare these modeled signals with magnetic fields induced by oceanic fluid
motions that might be used to measure oceanic flows (e.g., Chave, 1983; Tyler,
2011; Minami, 2017). For turbulent convection (Soderlund ., Soderlund .2014),
we find that these signals can dominate induction signal at low latitudes,
underscoring the need for spatial coverage in magnetic investigations. Based
on end-member ocean compositions (Zolotov, 2008; Zolotov Kargel, 2009), we
quantify the residual magnetic induction signals that might be used to infer
the oxidation state of Europa’s ocean and to investigate stable liquids under
high-pressure ices in Ganymede and Callisto. Fully exploring this parameter
space for the sake of planned missions requires electrical conductivity
measurements in fluids at low temperature and to high salinity and pressure.

JGR-Planets

Jet Propulsion Laboratory, California Institute of Technology, Pasadena, USA
Institute for Geophysics, John A. & Katherine G. Jackson School of
Geosciences, The University of Texas at Austin, USA British Geological Survey,
Edinburgh, UK Dept. of Physics, University of Washington, Seattle, USA Dept.
of Earth Sciences, University of Oregon, Eugene, USA

S. D<EMAIL_ADDRESS>
Diffusive induction accounting for adiabatic ocean temperatures is distinct in
phase and amplitude from induction based on electrical conductivity at the
ice-ocean interface

Based on turbulent global convection models, oceanic flows may generate
induced magnetic fields observable by planned spacecraft missions

Determining ocean composition from magnetic induction requires additional
thermodynamic and electrical conductivity data

##  1 Introduction

The jovian system is of particular interest for studying magnetic induction in
icy ocean worlds. Jupiter has a strong magnetic field whose dipole axis is
tilted $9.5^{\circ}$ with respect to its rotation axis (Acuna Ness, 1976),
while the orbits of the Galilean moons lie very nearly in the equatorial plane
of Jupiter. This means that Jupiter?s magnetic field varies in time at the
orbital positions of the satellites. Also, the outer layers of the satellites
themselves are believed to consist mainly of water ice at the surface,
underlain by salty oceans. Brines are good conductors, while ice is a
significant insulator.

Magnetic induction from Jupiter’s diurnal signal sensed by the Galileo mission
provides the most compelling direct observational evidence for the existence
of oceans within Europa and Ganymede (Saur ., 1998; Khurana ., 1998; Kivelson
., 2000; Schilling ., 2007; Hand Chyba, 2007; Khurana ., 2009). The case has
also been made for an induction response from an ocean in Callisto (Zimmer .,
2000), but this interpretation is clouded by possible ionospheric interference
(Liuzzo ., 2015; Hartkorn Saur, 2017).

Longer period signals penetrate more deeply, as penetration of the magnetic
field into the interior is a diffusive process. It is convenient that the skin
depths at the dominant periods of variation experienced by Europa, Ganymede,
and Callisto are comparable to the expected ocean depths, which makes it
possible to probe the properties of their oceans using magnetic induction. The
spectrum of frequencies driving induced magnetic responses includes not just
the orbits of the Galilean satellites and the rotation of Jupiter’s tilted
dipole field, but also their harmonics and natural oscillations (Saur ., 2009;
Seufert ., 2011). Electrical conductivity structure within the subsurface
oceans—for example, from convective adiabatic temperature gradients (Vance .,
2018) and stratification (Vance Goodman, 20091)—will respond at these
frequencies.

Further variations in the magnetic fields arise from the motion of the moons
about Jupiter. Perturbations to the orbits of the moons arise from multiple
sources, including the oblate figure of Jupiter, gravitational interactions
with the other satellites, and even from Saturn and the Sun (Lieske, 1998;
Lainey ., 2006).

Here, we examine the amplitudes and phase lags for magnetic diffusion in
modeled oceans of Europa, Ganymede, and Callisto. We restrict our analysis to
spherically symmetric configurations, treating interior structures based on
self-consistent thermodynamics, which account for variations in electrical
conductivity with depth in convective oceans (Vance ., 2018). In addition, we
consider the generation of induced magnetic fields by oceanic fluid motions
that may bias the interpretation of a satellite’s magnetic behavior if not
accommodated and which, more optimistically, might be used to probe the ocean
flows directly (e.g., Chave, 1983; Tyler, 2011; Minami, 2017). Based on end-
member ocean compositions (Zolotov, 2008; Zolotov Kargel, 2009), we
demonstrate the possibilities for using magnetic induction to infer the
oxidation state of Europa’s ocean and to identify stable liquid layers under
high-pressure ices in Ganymede and Callisto.

In Section 2 we describe a numerical method for computing the induction
response. Section 3 examines the diffusive induction response of Jupiter’s
ocean moons, first describing the frequency content of temporal variations in
Jupiter’s field in the reference frames of the Galilean moons (S 3.1), then
the interior structure models that include layered electrical conductivity
consistent with the modeled compositions (S 3.2). In Section 3.3, we detail
the corresponding amplitude and phase responses of the diffusive magnetic
induction, and finally in Section 3.4, we compare the diffusive fields to the
field imposed by Jupiter. Section 4 describes simulations of oceanic flows (S
4.1) and resulting magnetic induction (S 4.2) that adds to the diffusive
component. Section 5 describes the prospects for detecting these different
signals.

##  2 Induction Response Model

We are interested in the magnetic fields induced within a spherically
symmetric body, in which electrical conductivity is a piece-wise constant
function of distance from the center. We thus assume bounding radii

$\\{r_{1},r_{2},r_{3},\cdots,r_{m}\\}$ (1)

where

$r_{m}=R$ (2)

is the outer radius of the spherical body.

The corresponding conductivity values are

$\\{\sigma_{1},\sigma_{2},\sigma_{3},\cdots,\sigma_{m}\\}$ (3)

We also assume that there is an imposed external magnetic potential,
represented by a sum of terms, each of which has the form

$\Phi[r,\theta,\phi,t]=R\
B_{e}\left(\frac{r}{R}\right)^{n}S_{n,m}[\theta,\phi]\ \exp[-i\ \omega\ t]$
(4)

where $\\{r,\theta,\phi\\}$ are spherical coordinates ($r$ is radius, $\theta$
is colatitude, and $\phi$ is longitude) of the field point, $B_{e}$ is a scale
factor, $S_{n,m}[\theta,\phi]$ is a surface spherical harmonic function of
degree $n$ and order $m$, while $t$ is time and $\omega$ is the frequency of
oscillation of the imposed potential.

Within each layer, the magnetic field vector $B$ must satisfy the differential
equation

$\nabla^{2}B=-k^{2}B$ (5)

where $k$ is a scalar wavenumber given by

$k^{2}=i\ \omega\ \mu_{0}\ \sigma$ (6)

where $\omega$ is frequency, $\sigma$ is electrical conductivity, and the
magnetic constant (permeability of free space) is given by

$\mu_{0}=4\pi\times 10^{-7}N/A^{2}$ (7)

with units $N$ and $A$ being Newton and Ampere.

###  2.1 Radial Basis Functions

The poloidal component of the magnetic field inside the body is given by sums
of terms with the forms

$B_{r}[r,\theta,\phi,t]=\frac{C}{r}\left(F[r]\right)\ n(n+1)\
S_{n,m}[\theta,\phi]\ \exp[-i\ \omega\ t]$ (8)
$B_{\theta}[r,\theta,\phi,t]=\frac{C}{r}\left(\frac{d\ rF[r]}{dr}\right)\ \
\frac{dS_{n,m}[\theta,\phi]\ }{d\theta}\exp[-i\ \omega\ t]$ (9)
$B_{\phi}[r,\theta,\phi,t]=\frac{C}{r\sin[\theta]}\left(\frac{d\
rF[r]}{dr}\right)\ \ \frac{dS_{n,m}[\theta,\phi]\ }{d\phi}\exp[-i\ \omega\ t]$
(10)

where $C$ is a constant, and $F[r]$ is a function of radius, which we need to
determine.

Applying separation of variables to the governing differential equation (5),
one finds that the radial factor $F[r]$ in the solution must satisfy the
ordinary differential equation

$\frac{d^{2}F}{dr^{2}}+\left(\frac{2}{r}\right)\frac{dF}{dr}+(k^{2}-\frac{n(n+1)}{r^{2}})F=0$
(11)

This is a second order equation having two solutions:

$\displaystyle F_{n}^{+}[r]=j_{n}[k\ r]$ (12) $\displaystyle
F_{n}^{-}[r]=y_{n}[k\ r]$ (13)

where $j_{n}[x]$ is a spherical Bessel function of the first kind of order
$n$, and argument $x$, and $y_{n}[x]$ is a spherical Bessel function of the
second kind.

It will also be convenient to define another set of related functions

$\displaystyle G_{n}^{+}[r]$ $\displaystyle=$
$\displaystyle\frac{d}{dr}\left(r\ F_{n}^{+}[r]\right)$ $\displaystyle=$
$\displaystyle(n+1)\ \ j_{n}[k\ r]-\left(k\ r\right)\ j_{n+1}[k\ r]$

and

$\displaystyle G_{n}^{-}[r]$ $\displaystyle=$
$\displaystyle\frac{d}{dr}\left(r\ F_{n}^{-}[r]\right)$ $\displaystyle=$
$\displaystyle(n+1)\ \ y_{n}[k\ r]-\left(k\ r\right)\ y_{n+1}[k\ r]$

In the magnetic induction problem, as applied to the Galilean satellites, the
only case of interest is for an imposed dipole field, where $n=1$. In that
case, the radial basis functions for the radial component of the field, are

$\displaystyle F_{1}^{+}[k\ r]$ $\displaystyle=$ $\displaystyle j_{1}[k\ r]$
$\displaystyle=$ $\displaystyle\frac{\sin[k\ r]-\left(k\ r\right)\cos[k\
r]}{\left(k\ r\right)^{2}}$

and

$\displaystyle F_{1}^{-}[k\ r]$ $\displaystyle=$ $\displaystyle y_{1}[k\ r]$
$\displaystyle=$ $\displaystyle\frac{-\cos[k\ r]-\left(k\ r\right)\sin[k\
r]}{\left(k\ r\right)^{2}}$

In similar fashion, the radial basis functions for the transverse components
are

$\displaystyle G_{1}^{+}[k\ r]$ $\displaystyle=$ $\displaystyle 2\ j_{1}[k\
r]-(k\ r)\ j_{2}[k\ r]$ $\displaystyle=$ $\displaystyle\frac{\left(k\
r\right)\cos[k\ r]-\left(1-k^{2}\ r^{2}\right)\sin[k\ r]}{\left(k\
r\right)^{2}}$

and

$\displaystyle G_{1}^{-}[k\ r]$ $\displaystyle=$ $\displaystyle 2\ y_{1}[k\
r]-(k\ r)\ y_{2}[k\ r]$ $\displaystyle=$ $\displaystyle\frac{\left(k\
r\right)\sin[k\ r]+\left(1-k^{2}\ r^{2}\right)\cos[k\ r]}{\left(k\
r\right)^{2}}$

In both cases, the latter form is singular at the origin ($r=0$), so in the
innermost spherical layer, we only use $F^{+}[k\ r]$ and $G^{+}[k\ r].$ In
other layers, we use linear combinations of $F^{+}$ and $F^{-}$ and linear
combinations of $G^{+}$ and $G^{-}$.

###  2.2 Internal Boundary Conditions

The resulting piece-wise-defined radial functions characterize the radial part
of the magnetic field. The radial component has the form

$F[r]=\left\\{\begin{array}[]{cccc}c_{1}\ F^{+}[k_{1}r]&&\mathrm{if}&0<r\leq
r_{1}\\\ c_{2\ }F^{+}[k_{2}r]&+\ d_{2}\ F^{-}[k_{2}r]&\mathrm{if}&r_{1}<r\leq
r_{2}\\\ c_{3}\ F^{+}[k_{3}r]&+\ d_{3}\ F^{-}[k_{3}r]&\mathrm{if}&r_{2}<r\leq
r_{3}\\\ &&&\\\ c_{m}\ F^{+}[k_{m}r]&+\ d_{m}\
F^{-}[k_{m}r]&\mathrm{if}&r_{m-1}<r\leq r_{m}\end{array}\right.$ (20)

The transverse components yield similar structure, but with $G$ replacing $F$.

The constants $c_{j}$ and $d_{j}$ are determined by continuity of radial ($r$)
and transverse ($\theta,\phi$) components of the magnetic field across the
boundaries. For each internal boundary, it must hold that

$\displaystyle F[r_{j}]$ $\displaystyle=$ $\displaystyle c_{j}\
F^{+}\left[k_{j}\ r_{j}\right]+d_{j}\ F^{-}\left[k_{j}\ r_{j}\right]$
$\displaystyle=$ $\displaystyle c_{j+1}\ F^{+}\left[k_{j+1\
}r_{j}\right]+d_{j+1}\ F^{-}\left[k_{j+1\ }r_{j}\right]$

to ensure continuity of the radial component of the magnetic field, and
likewise for $G$ to ensure continuity of the transverse components. These
continuity constraints yield two equations at each internal boundary, from
which we can determine the layer coefficients.

The internal boundary conditions are only part of the story. In a model with
$m$ layers, we have $2m-1$ coefficients to determine (recall that $d_{1}=0,$
to avoid singular behavior at the origin), but only $m-1$ internal boundaries,
and thus only $2m-2$ constraints. The external boundary condition provides the
additional information to make the problem evenly determined.

Even without the external boundary condition, a provisional solution is
obtained by setting $c_{1}=1$ and using the internal boundary constraints to
determine the other coefficient values. Using notation similar to that of
Parkinson (1983, page 314), we can write a recursion relation that transforms
the coefficients in the $j$th layer into those for the layer above it

$\left[\begin{array}[]{c}c_{j+1}\\\
d_{j+1}\end{array}\right]=T_{j}[k_{j},k_{j+1},r_{j}]\cdot\left[\begin{array}[]{c}c_{j}\\\
d_{j}\end{array}\right]$ (22)

where the transformation matrix has elements

$T_{j}[k_{j},k_{j+1},r_{j}]=\frac{1}{\alpha_{j}}\left[\begin{array}[]{cc}\beta_{j}&\gamma_{j}\\\
\delta_{j}&\varepsilon_{j}\end{array}\right]$ (23)

with

$\alpha_{j}=F^{+}\left[k_{j+1\ }r_{j}\right]\ast G^{-}\left[k_{j+1\
}r_{j}\right]-F^{-}\left[k_{j+1\ }r_{j}\right]\ast G^{+}\left[k_{j+1\
}r_{j}\right]$ (24)

which is a function of the conductivity in the layer above the boundary only.
The other elements depend on the conductivities on both sides of the boundary

$\beta_{j}=F^{+}\left[k_{j\ \ }r_{j}\right]\ast G^{-}\left[k_{j+1\
}r_{j}\right]-F^{-}\left[k_{j+1\ }r_{j}\right]\ast G^{+}\left[k_{j\ \
}r_{j}\right]$ (25) $\gamma_{j}=F^{-}\left[k_{j\ \ }r_{j}\right]\ast
G^{-}\left[k_{j+1\ }r_{j}\right]-F^{-}\left[k_{j+1\ }r_{j}\right]\ast
G^{-}\left[k_{j\ \ }r_{j}\right]$ (26)

and

$\delta_{j}=F^{+}\left[k_{j+1\ }r_{j}\right]\ast G^{+}\left[k_{j\ \
}r_{j}\right]-F^{+}\left[k_{j\ }\ r_{j}\right]\ast G^{+}\left[k_{j+1\
}r_{j}\right]$ (27) $\varepsilon_{j}=F^{+}\left[k_{j+1\ }r_{j}\right]\ast
G^{-}\left[k_{j\ \ }r_{j}\right]-F^{-}\left[k_{j\ \ }r_{j}\right]\ast
G^{+}\left[k_{j+1\ }r_{j}\right]$ (28)

We thus start in the central spherical layer, with $c_{1}=1$ and $d_{1}=0$,
and then propagate upward through the stack of layers until we have the
coefficients in each of the $m$ layers. This set of layer coefficients, with
the radial basis functions, yields structures as given in equations (22) and
(23).

###  2.3 External Boundary Conditions

The final step is matching the external surface boundary condition. Outside
the sphere, the magnetic field is represented by a scalar potential which is
the sum of an imposed external contribution and an induced internal
contribution. That sum has spatial dependence given by the form

$\Phi[r,\theta,\phi]=R\left(B_{e}\left(\frac{r}{R}\right)^{n}+B_{i}\left(\frac{R}{r}\right)^{n+1}\right)S_{n}[\theta,\phi]$
(29)

We have dropped the subscript $m$ from $S_{n,m}$ because a suitable choice of
axes results in $m=0$ for both external and internal fields for the case of
spherical symmetry we consider here. The vector field is obtained from the
potential via

$B=-\nabla\Phi$ (30)

The radial component of the vector field, evaluated at the surface ($r=R$), is

$B_{r}=-\left(n\ B_{e}-(n+1)B_{i}\right)S_{n}[\theta,\phi]$ (31)

and the tangential components are

$B_{\theta}=-(B_{e}+B_{i})\frac{\partial S_{n}[\theta,\phi]}{\partial\theta}$
(32)

and

$B_{\phi}=-(B_{e}+B_{i})\frac{1}{\sin[\theta]}\frac{\partial
S_{n}[\theta,\phi]}{\partial\phi}$ (33)

Matching these with the corresponding interior components, as given in
equations (8), (9), and (10), but evaluated at the top of the upper-most
layer, we obtain

$-(n\ B_{e}-(n+1)\ B_{i})R=n\ (n+1)\ (c_{m}\ F^{+}[k_{m}\ R]+d_{m}\
F^{-}[k_{m}\ R])$ (34)

and

$-(B_{e}+B_{i})R=\left(c_{m}\ G^{+}[k_{m}\ R]+d_{m}\ G^{-}[k_{m}\ R]\right)$
(35)

From these two equations, we can first solve for $B_{e}$ and $B_{i}$. The
result is

$\widehat{B}_{e}=\frac{-1}{R(2n+1)}\left(c_{m}\ A_{m}+d_{m}\ B_{m}\right)$
(36) $\widehat{B}_{i}=\frac{1}{R(2n+1)}\left(c_{m}\ C_{m}+d_{m}\ D_{m}\right)$
(37)

where we introduce $\widehat{B}_{e}$ and $\widehat{B}_{i}$ to distinguish
solutions in terms of internal properties from the external and induced
magnetic moments. We also define the parameters $A_{m}$, $B_{m}$, $C_{m}$, and
$D_{m}$ by

$\displaystyle A_{m}$ $\displaystyle=$ $\displaystyle\left(n+1\right)\left(n\
F^{+}[k_{m}\ R]+\ G^{+}[k_{m}\ R]\right)$ (38) $\displaystyle B_{m}$
$\displaystyle=$ $\displaystyle\left(n+1\right)\left(n\ F^{-}[k_{m}\ R]+\
G^{-}[k_{m}\ R]\right)$

and

$\displaystyle C_{m}$ $\displaystyle=$ $\displaystyle
n\left(\left(n+1\right)F^{+}[k_{m}\ R]-G^{+}[k_{m}\ R]\right)$ (39)
$\displaystyle D_{m}$ $\displaystyle=$ $\displaystyle
n\left(\left(n+1\right)F^{-}[k_{m}\ R]-G^{-}[k_{m}\ R]\right)$

As previously noted, choice of $c_{1}=1$ permits solution of layer
coefficients $c_{j}$ and $d_{j}$ relative to each other with only knowledge of
the interior properties. We can then solve for $\widehat{B}_{e}$ and
$\widehat{B}_{i}$ in terms of the interior structure quantities $k_{j}$ and
$r_{j}$. We can then conveniently relate this to the magnetic field that will
be induced from the conducting body for a given external field $B_{e}^{\ast}$
by introducing a scale factor:

$S=\frac{B_{e}^{\ast}}{\widehat{B}_{e}}$ (40)

Choosing a normalized value of

$B_{e}^{\ast}=1$ (41)

means that physically correct layer coefficients may be determined by
multiplying the magnitude of the applied external field to the coefficients
$c_{j}^{\ast}$ and $d_{j}^{\ast}$, obtained from

$\left[\begin{array}[]{c}c_{j}^{\ast}\\\ d_{j}^{\ast}\end{array}\right]=S\
\left[\begin{array}[]{c}c_{j}\\\ d_{j}\end{array}\right]$ (42)

For an applied external field $B_{e}^{\ast}$ in real units, the physical
magnetic field within each layer is then given by

$\displaystyle B_{r,j}[r,\theta,\phi,t]$ $\displaystyle=$
$\displaystyle\frac{B_{e}^{\ast}}{r}\left(c_{j}^{\ast}F^{+}[k_{j}r]+d_{j}^{\ast}F^{-}[k_{j}r]\right)n(n+1)S_{n}[\theta,\phi]\
\exp[-i\ \omega\ t]$ $\displaystyle B_{\theta,j}[r,\theta,\phi,t]$
$\displaystyle=$
$\displaystyle\frac{B_{e}^{\ast}}{r}\left(c_{j}^{\ast}G^{+}[k_{j}r]+d_{j}^{\ast}G^{-}[k_{j}r]\right)\frac{dS_{n}[\theta,\phi]\
}{d\theta}\ \exp[-i\ \omega\ t]$ (43) $\displaystyle
B_{\phi,j}[r,\theta,\phi,t]$ $\displaystyle=$
$\displaystyle\frac{B_{e}^{\ast}}{r\sin[\theta]}\left(c_{j}^{\ast}G^{+}[k_{j}r]+d_{j}^{\ast}G^{-}[k_{j}r]\right)\frac{dS_{n}[\theta,\phi]\
}{d\phi}\ \exp[-i\ \omega\ t]$

The ratio of internal and external field strengths at the exterior surface is
given from equations (36) and (37) via

$Q\equiv\frac{\widehat{B}_{i}}{\widehat{B}_{e}}=-\frac{c_{m}^{\ast}\
C_{m}+d_{m}^{\ast}\ D_{m}}{c_{m}^{\ast}\ A_{m}+d_{m}^{\ast}\ B_{m}}$ (44)

In Zimmer . (2000) and Khurana . (2009), this complex ratio is written as the
product of a real magnitude and a phase shift:

$Q=A^{\ast}\exp[i\ \gamma^{\ast}]$ (45)

where $A^{\ast}$ is a positive real number representing amplitude and
$\gamma^{\ast}$ is a real number representing the phase of the induced field
relative to the imposed field.

In the aforementioned previous work, an explicit formula is given for the
result from a 3-layer model, in which the conductivities in the innermost
($j=1$) and outermost ($j=3$) layers are zero, and the middle layer (intended
to represent a salty ocean in Europa) has a finite conductivity. In this
model, there are essentially four free parameters—3 bounding radii
($r_{1},r_{2},r_{3}$) and a middle layer conductivity ($\sigma_{2}$)—that
determine the critical wavenumber ($k_{2}$). We refer to this model as the
ocean-only model.

In our notation, the resulting ratio $Q$ for the ocean-only model is

$\displaystyle Q$ $\displaystyle=$ $\displaystyle\frac{-n}{n+1}\
\frac{j_{n+1}[k_{2}\ r_{1}]\ast y_{n+1}[k_{2}\ r_{2}]-j_{n+1}[k_{2}\
r_{2}]\ast y_{n+1}[k_{2}\ r_{1}]}{j_{n+1}[k_{2}\ r_{1}]\ast y_{n-1}[k_{2}\
r_{2}]-j_{n-1}[k_{2}\ r_{2}]\ast y_{n+1}[k_{2}\ r_{1}]}$ (46)

Because we know the complex phase of the wavenumber $k$, we can use properties
of Bessel functions to solve for the amplitude and phase for the induced
magnetic field. We defined $k^{2}=i\omega\mu\sigma$ (Eq. 6), so
$k=\exp[i\pi/4]\sqrt{\omega\mu\sigma}$. The (real) magnitude of $k$ is
$|k|=\sqrt{\omega\mu\sigma}$, and all layers will have the same complex phase
$\pi/4$. We can therefore express the wavenumber for each layer as

$k_{j}=\kappa_{j}\exp[i\pi/4],\ \ \kappa_{j}=\sqrt{\omega\mu_{j}\sigma_{j}}$
(47)

When $\kappa_{2}r_{2}$ is large,
$j_{n+1}[\kappa_{2}\,r_{2}]=-j_{n-1}[\kappa_{2}\ r_{2}]$ and
$y_{n+1}[\kappa_{2}\,r_{2}]=-y_{n-1}[\kappa_{2}\ r_{2}]$. We can make use of
these relations to note that the amplitude and phase for the induced magnetic
field for a perfectly conducting sphere of radius $r_{2}$ will be $n/(n+1)$
and $0$, respectively. Thus, we can also define an amplitude and phase for the
induction response relative to those for a perfectly conducting sphere of
radius $R$:

$A=A^{\ast}\ \frac{n+1}{n}\,\left(\frac{r_{2}}{R}\right)^{3},\ \ \
\gamma=\gamma^{\ast}$ (48)

A perfectly conducting sphere of radius $R$ therefore has a relative amplitude
of $A=1$ and $\gamma=0$.

##  3 Diffusive Induction in Jupiter’s Ocean Moons

###  3.1 Spectral Content of the Imposed Magnetic Field Variations

Temporal variations in the magnetic field occur in the reference frames of
Jupiter’s satellites. Figure 1 shows the strongest components, arising from
the orbital and synodic periods and their harmonics. Seufert . (2011)
determined the frequency spectra for the time-varying magnetic perturbations
applied to each of the four Galilean moons based on the VIP4 model of J.
Connerney . (1998) and the Jovian current sheet model of Khurana (1997).
Seufert . (2011) also examined the frequency spectra of magnetic perturbations
from dynamic migration of the Jovian magnetopause based on solar wind data
from the Ulysses spacecraft, which we do not consider here.

To calculate the frequencies, we first compute the magnetic field using the
JRM09 Jupiter field model accounting for Juno measurements (JEP. Connerney .,
2018) and using the plasma sheet model from Khurana (1997). We then compute
the field at the orbital positions of the moons using the most recent and up-
to-date NAIF-produced spice kernels and three years of data covering the
duration of the Europa Clipper mission (tour 17F12v2). Finally, we compute the
Fourier transform of the entire data sets to determine the induction
frequencies.

The temporal variations in imposed magnetic field at each satellite depend on
the orbits of the satellites and the magnetic field of Jupiter. To find them,
we compute Jupiter’s magnetic field in a Jupiter-centered coordinate system
from a spherical harmonic series representation of the magnetic potential,
which is a variant of Eq. 4:

$\Phi[r,\theta,\phi]=R\sum_{n=1}\left(\frac{R}{r}\right)^{n+1}\sum_{m=0}^{n}P_{n,m}[\sin[\theta]]\
\left(g_{n,m}\cos[m\phi]+h_{n,m}\sin[m\phi]\right).$ (49)

The magnetic field vector is the negative gradient of the scalar potential

$\displaystyle B$ $\displaystyle=$ $\displaystyle-\nabla\Phi$ (50)
$\displaystyle=$ $\displaystyle-\left\\{\frac{\partial\Phi}{\partial
x},\frac{\partial\Phi}{\partial y},\frac{\partial\Phi}{\partial z}\right\\}$
(51)

The mean radius is $R=71,492$ km. The rotation rate of Jupiter, as defined in
the System III longitude (Seidelmann Divine, 1977), is
$\omega=870.536^{\circ}$/day.

Figure 1: Europa: Variations in orbital parameters over time introduce
magnetic fluctuations at multiple frequencies beyond the Jupiter rotation and
satellite orbital frequencies. The different vector components contain unique
information at multiple frequencies resulting from the harmonics and beats of
the orbital and rotational oscillations. Figure 2: Ganymede: Variations in
orbital parameters over time introduce magnetic fluctuations at multiple
frequencies beyond the Jupiter rotation and satellite orbital frequencies.
Figure 3: Callisto: Variations in orbital parameters over time introduce
magnetic fluctuations at multiple frequencies beyond the Jupiter rotation and
satellite orbital frequencies.

###  3.2 Electrical Conductivity in Adiabatic Galilean Oceans

Fluid temperature, pressure, and salt content determine the electrical
conductivity of an aqueous solution, and thus dictate the magnetic induction
responses of the Galilean oceans. The amplitude and phase of the magnetic
fields induced by the oceans depend on the conductive properties of the
oceans, which are influenced by the composition of the dissolved salts. With
sufficient prior knowledge of the ice thickness and hints to the ocean’s
composition—for example, from geological and compositional measurements by the
Europa Clipper (Buffington ., 2017)—magnetic induction studies can provide
information on the amounts and compositions of the salts that link to global
thermal and geochemical processes. On Europa, the flux of surface-generated
oxygen to the ocean may have created oxidizing (acidic) conditions (Hand
Chyba, 2007; Pasek Greenberg, 2012; Vance ., 2016) permitting the presence of
dissolved MgSO4 in addition to NaCl (Zolotov, 2008; Zolotov Kargel, 2009).

Depth-dependent electrical conductivity can arise from melting or freezing at
the ice–ocean interface, and from dissolution and precipitation within the
ocean or at the water–rock interface. Even for oceans with uniform salinity,
as is typically assumed, conductivity will increase with depth along the
ocean’s convective adiabatic profile because the greater temperature and
pressure increase the electrical conductivity. Figure 4 depicts this variation
for Europa and Ganymede, based on forward models of Vance . (2018) that use
available thermodynamic and geophysical data to explore the influences of the
ocean, rock layer, and any metallic core on the radial structures of known icy
ocean worlds. For each ocean, we consider a nominal 10 wt% MgSO4 salinity, as
investigated in previous work. The published equation of state and electrical
conductivity data are adequate for the pressures in the largest moon,
Ganymede, up to 1.6 GPa (Vance ., 2018). The pressure conditions in Europa’s
ocean are low enough ($<200$ MPa) that the equation of state for seawater
(McDougall Barker, 2011) provides plausible values of conductivity for
salinity of 35 ppt less. For Europa, the respective radial models of
electrical conductivity for oceans containing seawater and MgSO4 are
consistent with compositions linked to chemically reducing and oxidizing model
oceans cited above.

Figure 4: Adiabatic ocean temperature (left) and electrical conductivity
(right). Convecting oceans with MgSO4 (dashed lines) are warmer. Standard
seawater (mostly NaCl; dot–dashed lines) creates colder oceans and lower
electrical conductivities. Thicker ice (blue), corresponds to colder adiabatic
profiles in the underlying oceans, which also lowers electrical conductivity.
Open and closed circles correspond to the inferred depth to the upper boundary
of the silicate layer for the saline and pure water oceans, respectively.
Conductivities in the liquid regions are several orders of magnitude larger
than in the ice and rock. Adapted from Vance . (2018).

Radial conductivity profiles for Europa (Fig. 4; top) illustrate the coupling
to temperature and composition. We consider ice thicknesses of 5 and 30 km
(magenta and blue curves, respectively) as representative extremes. Seawater
(dot–dashed lines), though less concentrated than the modeled composition of
MgSO4 (dashed lines), has a stronger melting point suppression, leading to an
overall colder ocean for the same thickness of ice. Adiabats for pure water
(solid lines) are shown for comparison. The lower temperature for seawater
combines with the different electrical conductivity for the different
dissolved ions to create distinct profiles unique to ocean composition and ice
thickness (upper right).

Larger Ganymede (Fig. 4; bottom) also has distinct conductivity profiles for
both ice thickness and ocean composition. They reveal an additional nuance to
deep planetary oceans that can influence the induction response. Although
electrical conductivity generally increases with depth, it begins to decrease
at the greatest depths for the warm Ganymede ocean (right-most curve). This
inflection occurs because the ocean achieves GPa+ pressures, at which the
packing of water molecules begins to inhibit the charge exchange of the
dissolved ions (Schmidt Manning, 2017).

Dense brines may also reside at the base of the high-pressure ices on
Ganymede, and even between them (Journaux ., 2013, 2017; S. Vance ., 2014;
Vance ., 2018). Although more detailed modeling of the coupled geochemical and
geodynamic regimes is needed, this scenario seems consistent with recent
simulations of two-phase convection in high-pressure ices (Choblet ., 2017).
These simulations imply that fluids should occur at the water-rock interface
through long periods of the evolution of even of large icy world containing
high-pressure ices. If such a fluid layer exists under the high-pressure ice,
it will create an induction response at low frequencies, as discussed below.

###  3.3 Amplitude and Phase Lag of the Diffusive Response

The normalized surface induction response for Europa, Ganymede, and Callisto,
shown in Fig. 5, are based on the adiabatic ocean electrical conductivity
profiles shown in Fig. 4, assuming spherical symmetry (Section 2). Warmer and
thus thicker oceans (magenta curves) have larger amplitude responses,
corresponding to overall higher values of the conductance. The induction
signatures for the adiabatic ocean profile are nearly equal to those of oceans
with uniform conductivity equal to the mean of the adiabatic model (Section
2). These signatures differ, however, from those of an ocean with uniform
conductivity based on the temperature and electrical conductivity at the
ice–ocean interface.

For Europa, the induction signatures for modeled oxidized (10 wt% MgSO4) and
reduced (seawater) oceans are nearly identical in their amplitude responses.
However, the two ocean models show phase separation of a few degrees at the
orbital frequency of 3.6$\times 10^{-6}$ Hz (85.23 hr period).

Local enhancements in the ocean conductivity can have a discernible induction
response. For Ganymede, we simulated a second ocean layer at the water–rock
interface at a depth of 900 km, under 530 km of ice VI (Vance ., 2018),
modeled as a 10-km-thick high-conductivity region (20 S/m) corresponding to a
nearly saturated MgSO4 solution, consistent with (Hogenboom ., 1995) and
(Calvert ., 1958). The influence of such a layer (dotted lines in Fig. 5) is a
$\sim$4% increase in the amplitude response and a corresponding $\sim$7%
decrease in the phase response around 2.3$\times 10^{-7}$ Hz. A $\sim$1%
decrease in amplitude is also seen at frequencies of 0.93$\times 10^{-6}$ Hz
and 1.6$\times 10^{-6}$ Hz.

For Callisto, there is a small range of conditions under which oceans may be
present. Salty oceans considered by Vance . (2018) have thicknesses of 20 and
132 km. For the thinner ocean, a 96 km layer of high-pressure ice underlies
the ocean. The depicted state is likely transient, as ice III is buoyant in
the modeled 10wt% MgSO4 composition, and an upward snow effect should hasten
the transfer of heat from the interior. Simulating a subsequent stage with ice
III above the ocean awaits improved thermodynamic data, and will be discussed
in future work. The present simulations illustrate the effect of the greater
skin depth for the thicker and deeper ocean in terms of a higher amplitude
response at lower frequencies and phase curve also shifted in the direction of
lower frequencies.

Figure 5: Normalized magnetic induction amplitudes (left) and phases (right)
for the conductivity profiles in Fig. 4, at frequencies including the
induction peaks noted in Fig. 1 (vertical red lines).

###  3.4 Mean Diffusive Response Relative to the Imposed Field

For the sake of comparing the passive induction responses of Europa, Ganymede
and Callisto with fields induced by oceanic flows, we introduce the residual
field, $B_{R}$. This quantity allows us to quickly examine the frequency
dependent induction response for a given interior model, accounting for both
the amplitude ($A$) and phase shift ($\phi$). For the geometric mean frequency
components of Jupiter’s field ($|B|=\sqrt{B^{2}_{x}+B^{2}_{y}+B^{2}_{z}}$), we
define $B_{R}$ as

$B_{R}=|B|(\cos{\phi}-A)$ (52)

More information can be gained by examining the directional components of
Jupiter’s field (Figure 1).

Figures 6, 7, and 8 show the spectra of residual fields for Europa, Ganymede,
and Callisto, respectively. Subpanels in each figure isolate the peak
responses at the main driving frequencies shown in Figure 1. Tables 1, 2, and
3 include the corresponding data. Figures S1-S3 illustrate possible errors
arising from analyses assuming a uniform conductivity of the ocean. They plot
the deviations (in percent) between the residual fields ($B_{R}$) of the
adiabatic oceans (Figure 4) and the equivalent responses obtained by giving
the oceans uniform conductivity, either as the equivalent mean value or the
value at the top of the ocean (i.e. at the ice–ocean interface).

Figure 6: Europa:Residual field ($B_{R}$) of the diffusive induction response. Thick lines are higher salinities (10wt% and 3.5wt%, respectively) for oceans with aqueous MgSO4 (magenta and blue $--$) and seawater (cyan dash-dot). Thinner lines are for oceans with 10% of those concentrations. The lower pane shows responses at the strongest inducing frequencies in Figure 1. Filled symbols are for the higher concentrations. Upward triangles are for thicker ice (30 km) and downward triangles are for thinner ice (5 km). Figure 7: Ganymede: Residual field ($B_{R}$) of the diffusive induction response. Thick lines are higher salinities (10wt%) for oceans with aqueous MgSO4 (magenta and blue $--$). Thinner lines are for oceans with 1wt% MgSO4. The dotted line is for the case with a 30-km-thick oceanic layer underneath the high-pressure ice. The lower pane shows responses at the strongest inducing frequencies in Figure 1. Filled symbols are for the higher concentrations. Upward triangles are for thicker ice ($\sim 100$ km) and downward triangles are for thinner ice ($\sim 30$ km) Figure 8: Callisto: Residual field ($B_{R}$) of the diffusive induction response. Thick lines are higher salinities (10wt%) for oceans with aqueous MgSO4 (magenta and blue $--$). Thinner lines are for oceans with 1wt% MgSO4. The lower pane shows responses at the strongest inducing frequencies in Figure 1. Filled symbols are for the higher concentrations. Upward triangles are for thicker ice ($\sim 130$ km) and downward triangles are for thinner ice ($\sim 100$ km). | $T_{b}$ | $T_{mean}$ | $D_{I}$ | $D_{ocean}$ | $B_{R}$
---|---|---|---|---|---
| (K) | (K) | (km) | (km) | (nT)
Europa |  |  |  $f$ ($\times 10^{-6}$Hz) | 3.25 | 24.73 | 49.46
MgSO4 1Wt% | 270.4 | 271.5 | 31 | 120 | 0.841 | 21.862 | 2.715
$<\sigma>=$ 0.4227 S m-1 | 270.4 | 271.5 | 31 | 120 | 0.823 | 21.417 | 2.654
$\sigma_{top}=$ 0.3847 S m-1 | 270.4 | 271.5 | 31 | 120 | 0.769 | 21.304 | 2.650
| 273.1 | 274.3 | 6 | 147 | 0.791 | 16.892 | 1.980
$<\sigma>=$ 0.4640 S m-1 | 273.1 | 274.3 | 6 | 147 | 0.755 | 15.964 | 1.900
$\sigma_{top}=$ 0.4107 S m-1 | 273.1 | 274.3 | 6 | 147 | 0.702 | 16.122 | 1.928
MgSO4 10Wt% | 269.8 | 271.3 | 30 | 127 | 1.591 | 18.741 | 1.983
$<\sigma>=$ 3.4478 S m-1 | 269.8 | 271.3 | 30 | 127 | 1.539 | 18.234 | 1.961
$\sigma_{top}=$ 3.0763 S m-1 | 269.8 | 271.3 | 30 | 127 | 1.536 | 18.686 | 2.008
| 272.7 | 274.5 | 5 | 154 | 1.233 | 10.477 | 0.982
$<\sigma>=$ 3.8547 S m-1 | 272.7 | 274.5 | 5 | 154 | 1.167 | 9.800 | 0.935
$\sigma_{top}=$ 3.3197 S m-1 | 272.7 | 274.5 | 5 | 154 | 1.173 | 10.634 | 1.000
Seawater 0.35165 Wt% | 270.0 | 271.1 | 31 | 120 | 0.763 | 21.749 | 2.719
$<\sigma>=$ 0.3734 S m-1 | 270.0 | 271.1 | 31 | 120 | 0.746 | 21.112 | 2.645
$\sigma_{top}=$ 0.3339 S m-1 | 270.0 | 271.1 | 31 | 120 | 0.684 | 21.026 | 2.636
| 272.5 | 273.6 | 6 | 146 | 0.712 | 16.850 | 2.029
$<\sigma>=$ 0.3945 S m-1 | 272.5 | 273.6 | 6 | 146 | 0.678 | 16.046 | 1.926
$\sigma_{top}=$ 0.3415 S m-1 | 272.5 | 273.6 | 6 | 146 | 0.614 | 15.921 | 1.947
Seawater 3.5165 Wt% | 268.2 | 269.7 | 31 | 122 | 1.559 | 19.524 | 2.091
$<\sigma>=$ 2.9548 S m-1 | 268.2 | 269.7 | 31 | 122 | 1.523 | 18.989 | 2.052
$\sigma_{top}=$ 2.6476 S m-1 | 268.2 | 269.7 | 31 | 122 | 1.510 | 19.349 | 2.098
| 270.8 | 272.3 | 5 | 148 | 1.205 | 11.538 | 1.079
$<\sigma>=$ 3.1457 S m-1 | 270.8 | 272.3 | 5 | 148 | 1.138 | 10.805 | 1.024
$\sigma_{top}=$ 2.7346 S m-1 | 270.8 | 272.3 | 5 | 148 | 1.140 | 11.350 | 1.068
Table 1: Europa: Residual fields ($B_{R}$) at the main inducing frequencies in Fig 1. For the different ocean compositions and thicknesses of the upper ice I lithosphere ($D_{I}$; Figure 4, the adiabatic response is given first, followed by the response for the ocean with uniform conductivity set to the mean of the adiabatic ocean ($\langle\sigma\rangle$), and then for the case with uniform conductivity set to the value at the ice-ocean interface ($\sigma_{top}$). | $T_{b}$ | $T_{mean}$ | $D_{I}$ | $D_{ocean}$ | $B_{R}$
---|---|---|---|---|---
| (K) | (K) | (km) | (km) | (nT)
Ganymede |  |  |  $f$ ($\times 10^{-6}$Hz) | 1.62 | 26.37 | 52.74
MgSO4 1Wt% | 270.7 | 279.0 | 25 | 442 | 0.265 | 9.580 | 0.517
$<\sigma>=$ 0.5166 S m-1 | 270.7 | 279.0 | 25 | 442 | 0.243 | 8.753 | 0.477
$\sigma_{top}=$ 0.3890 S m-1 | 270.7 | 279.0 | 25 | 442 | 0.229 | 9.601 | 0.516
| 261.5 | 266.1 | 93 | 272 | 0.212 | 16.389 | 1.007
$<\sigma>=$ 0.3295 S m-1 | 261.5 | 266.1 | 93 | 272 | 0.203 | 15.626 | 0.967
$\sigma_{top}=$ 0.2608 S m-1 | 261.5 | 266.1 | 93 | 272 | 0.175 | 15.906 | 0.999
MgSO4 10Wt% | 270.1 | 278.2 | 28 | 455 | 0.226 | 5.286 | 0.309
$<\sigma>=$ 4.0541 S m-1 | 270.1 | 278.2 | 28 | 455 | 0.209 | 4.991 | 0.290
$\sigma_{top}=$ 3.1056 S m-1 | 270.1 | 278.2 | 28 | 455 | 0.226 | 5.325 | 0.306
| 260.0 | 263.5 | 96 | 282 | 0.316 | 12.202 | 0.762
$<\sigma>=$ 2.3476 S m-1 | 260.0 | 263.5 | 96 | 282 | 0.304 | 11.919 | 0.750
$\sigma_{top}=$ 1.9483 S m-1 | 260.0 | 263.5 | 96 | 282 | 0.304 | 12.174 | 0.761
30 km 20 S m-1 layer | 260.0 | 263.5 | 96 | 282 | 0.332 | 12.156 | 0.765
Table 2: Ganymede: Residual fields ($B_{R}$) at the main inducing frequencies in Fig 1. For the different ocean compositions and thicknesses of the upper ice I lithosphere ($D_{I}$; Figure 4, the adiabatic response is given first, followed by the response for the ocean with uniform conductivity set to the mean of the adiabatic ocean ($\langle\sigma\rangle$), and then for the case with uniform conductivity set to the value at the ice-ocean interface ($\sigma_{top}$). | $T_{b}$ | $T_{mean}$ | $D_{I}$ | $D_{ocean}$ | $B_{R}$
---|---|---|---|---|---
| (K) | (K) | (km) | (km) | (nT)
Callisto |  |  |  $f$ ($\times 10^{-6}$Hz) | 0.69 | 26.60 | 27.29 | 27.99
MgSO4 1Wt% | 257.4 | 259.6 | 99 | 132 | 0.012 | 0.085 | 9.201 | 0.052
$<\sigma>=$ 0.2307 S m-1 | 257.4 | 259.6 | 99 | 132 | 0.012 | 0.084 | 8.990 | 0.050
$\sigma_{top}=$ 0.1965 S m-1 | 257.4 | 259.6 | 99 | 132 | 0.010 | 0.083 | 8.926 | 0.050
| 250.8 | 250.9 | 128 | 21 | 0.001 | 0.024 | 2.688 | 0.015
$<\sigma>=$ 0.0895 S m-1 | 250.8 | 250.9 | 128 | 21 | 0.001 | 0.025 | 2.740 | 0.016
$\sigma_{top}=$ 0.0874 S m-1 | 250.8 | 250.9 | 128 | 21 | 0.001 | 0.024 | 2.689 | 0.015
MgSO4 10Wt% | 255.7 | 256.9 | 99 | 130 | 0.063 | 0.083 | 8.875 | 0.050
$<\sigma>=$ 1.5256 S m-1 | 255.7 | 256.9 | 99 | 130 | 0.062 | 0.082 | 8.763 | 0.049
$\sigma_{top}=$ 1.3789 S m-1 | 255.7 | 256.9 | 99 | 130 | 0.058 | 0.082 | 8.822 | 0.049
| 250.0 | 251.5 | 129 | 18 | 0.004 | 0.072 | 7.778 | 0.044
$<\sigma>=$ 0.6025 S m-1 | 250.0 | 251.5 | 129 | 18 | 0.005 | 0.072 | 7.781 | 0.044
$\sigma_{top}=$ 0.6062 S m-1 | 250.0 | 251.5 | 129 | 18 | 0.005 | 0.072 | 7.790 | 0.044
Table 3: Callisto: Residual fields ($B_{R}$) at the main inducing frequencies
in Fig 1. For the different ocean compositions and thicknesses of the upper
ice I lithosphere ($D_{I}$; Figure 4, the adiabatic response is given first,
followed by the response for the ocean with uniform conductivity set to the
mean of the adiabatic ocean ($\langle\sigma\rangle$), and then for the case
with uniform conductivity set to the value at the ice-ocean interface
($\sigma_{top}$).

##  4 Magnetic Induction from Oceanic Fluid Flows

Another component of the induced magnetic response might occur in the icy
Galilean satellites, arising not from Jupiter’s changing magnetic field, but
from charges moving with oceanic fluid flows. Such induced magnetic fields are
typically neglected because they are expected to be relatively weak. On Earth,
ocean currents induce fields on the order of 100 nT in a background field of
about 40,000 nT; these fields are observable by space-based magnetometers and
have been used to monitor ocean currents (Constable Constable, 2004; Tyler .,
2003). If there are oceanic flow-driven induction signals present in the icy
Galilean satellites, and if the spatial or temporal structures of these
induction signals allow them to be separated from the contributions driven by
variations in Jupiter’s magnetic field, it would permit characterization of
the ocean flows themselves as has been done for the Earth’s ocean (e.g.,
Chave, 1983; Tyler ., 2003; Grayver ., 2016; Minami, 2017). Conversely, if
such induced signals are present but the analysis does not accommodate that
fact, then the recovered electrical conductivity estimates will be biased and
inaccurate.

While Tyler (2011) discusses the possibility of magnetic remote sensing to
detect resonant ocean tides on Europa in the limits of shallow water equations
and thin-shell electrodynamics, we are not aware of any studies that have
examined magnetic induction signatures due to other flows or for other
satellites (e.g., Lemasquerier ., 2017; Gissinger Petitdemange, 2019; Rovira-
Navarro ., 2019; Soderlund, 2019). Here, we focus on global fluid motions that
may be driven by convection within the oceans of Europa, Ganymede, and
Callisto, followed by estimates of the induction response that may be expected
from these flows.

###  4.1 Oceanic Fluid Motions

The majority of ocean circulation studies have focused on hydrothermal plumes
at Europa, with global models being developed relatively recently (Vance
Goodman, 20092; Soderlund ., Soderlund .2014; Soderlund, 2019). Thermal
convection in Europa’s ocean is expected in order to efficiently transport
heat from the deeper interior that arises primarily from radiogenic and tidal
heating in the mantle. Moreover, by estimating the extent to which rotation
will organize the convective flows, Europa’s ocean was predicted to have
quasi-three-dimensional turbulence (Soderlund ., Soderlund .2014; Soderlund,
2019). As shown in Figure 9, this turbulence generates three-jet zonal flows
with retrograde (westward) flow at low latitudes, prograde (eastward) flow at
high latitudes, and meridional overturning circulation. Upwelling at the
equator and downwelling at middle to high latitudes from this circulation
effectively forms a Hadley-like cell in each hemisphere.

Application of these calculations to Ganymede suggests convection is expected
within its ocean as well and may have similar convective flows, although there
is significantly more uncertainty in the predicted convective regime
(Soderlund, 2019). Convection in Callisto’s potential ocean may be in the
double-diffusive regime if the ocean’s composition is nearly saturated (Vance
., 2018). However, considering thermal convection as an upper bound,
application of the scaling arguments in Soderlund (2019) to Callisto suggest
similar ocean flows may be expected here as well.

The nominal ocean model shown in Figure 9 is, therefore, applicable to all
three ocean worlds considered here. As described in Soderlund (2019), the
model was carried out using the MagIC code (Wicht, 2002) with the SHTns
library for the spherical harmonics transforms (Schaeffer, 2013) and is
characterized by the following dimensionless input parameters: shell geometry
$\chi=r_{i}/r_{o}=0.9$, Prandtl number $Pr=\nu/\kappa=1$, Ekman number
$E=\nu/\Omega D^{2}=3.0\times 10^{-4}$, and Rayleigh number $Ra=\alpha g\Delta
TD^{3}/\nu\kappa$, where $r_{i}$ and $r_{o}$ are the inner and outer radii of
the ocean, $D=r_{o}-r_{i}$ is ocean thickness, $\Omega$ is rotation rate,
$\nu$ is kinematic viscosity, $\kappa$ is thermal diffusivity, $\alpha$ is
thermal expansivity, $g$ is gravitational acceleration, and $\Delta T$ is
superadiabatic temperature contrast. The boundaries are impenetrable, stress-
free, and isothermal.

The model outputs, such as the velocity field, are also non-dimensional. For
example, the Rossby number $Ro=U/\Omega D$ is the ratio of rotational
$\Omega^{-1}$ to inertial $D/U$ timescales that allows the dimensional flow
speeds to be determined: $U=\Omega DRo$ using ocean thickness $D$ as the
length scale and rotation rates $\Omega=[2.1\times 10^{-5},1.0\times
10^{-5},4.4\times 10^{-6}]$ s-1 for Europa, Ganymede, and Callisto,
respectively. Following Table 1, Europan ocean thicknesses of $120-154$ km are
considered. This range of liquid ocean thicknesses extends to $272-455$ km for
Ganymede (Table 2) and $18-132$ km for Callisto (Table 3), given the larger
uncertainties on their internal structures. We therefore assume the following
mean parameter values in Figure 9: $D_{Europa}=135$ km, $D_{Ganymede}=360$ km,
and $D_{Callisto}=75$ km, with the ranges considered in Table 4. Flows are
fastest for Ganymede and Europa, where the zonal jets can reach m/s speeds,
the mean latitudinal flows have peak speeds of tens of cm/s, and the mean
radial flows are $\sim 10$ cm/s.

Figure 9: Mean flow fields in our nominal global ocean model from Soderlund
(2019), averaged over 18 planetary rotations and all longitudes. a) Zonal
velocity field where red denotes prograde flows and blue denotes retrograde
flows. b) Theta velocity field where red denotes away from the north pole and
blue denotes toward the north pole. c) Radial velocity field where red denotes
upwelling flows and blue denotes downwelling flows.

###  4.2 Generation of Induced Magnetic Fields

The magnetic induction equation can be used to estimate the components of the
magnetic field ${\bf B}$ induced by ocean currents with velocity ${\bf u}$ and
those arising from changes in the externally imposed field:

$\frac{\partial{\bf B}}{\partial t}=\nabla\times({\bf u}\times{\bf
B})-\nabla\times(\eta\nabla\times{\bf B})$ (53)

where $\eta=(\mu_{0}\sigma)^{-1}$ is the magnetic diffusivity. Here, the first
term represents the evolution of the magnetic field, the second term
represents magnetic induction, and the third term represents magnetic
diffusion.

Neglecting variations in oceanic electrical conductivity with depth and
assuming an incompressible fluid, equation 53 simplifies to

$\frac{\partial{\bf B}}{\partial t}=({\bf B}\cdot\nabla){\bf u}-({\bf
u}\cdot\nabla){\bf B}+\eta\nabla^{2}{\bf B},$ (54)

after also expanding the induction term and utilizing $\nabla\cdot{\bf B}=0$
and $\nabla\cdot{\bf u}=0$. Let us decompose the total magnetic field into
jovian imposed ${\bf F}$ and the satellite’s induced ${\bf b}$ field
components:

${\bf B}={\bf F}+{\bf b}$ (55)

with $|{\bf F}|\gg|{\bf b}|$. The induction equation then becomes

$\frac{\partial{\bf b}}{\partial t}=-\frac{\partial{\bf F}}{\partial t}+({\bf
F}\cdot\nabla){\bf u}-({\bf u}\cdot\nabla)({\bf F}+{\bf
b})+\eta\nabla^{2}({\bf F}+{\bf b})$ (56)

Here, the first term is the evolution of the induced magnetic field, the
second term is induction due to variations in Jupiter’s magnetic field, the
third term is induction due to oceanic fluid motions, the fourth term is
advection of the field by ocean flows, and the fifth and sixth terms are
diffusion of the Jovian and induced fields.

Let us next assume that the Jovian field can be approximated by ${\bf
F}=F_{o}{\bf\hat{z}}$, where $F_{o}$ is constant and homogeneous and
${\bf\hat{z}}$ is aligned with the rotation axis, in which case equation 56
further simplifies to:

$\frac{\partial{\bf b}}{\partial t}=F_{o}\frac{\partial{\bf u}}{\partial
z}-({\bf u}\cdot\nabla){\bf b}+\eta\nabla^{2}{\bf b}.$ (57)

We will also focus on the quasi-steady induction signal generated by ocean
flows rather than the rapidly varying contribution that could be difficult to
distinguish from other magnetic field perturbations. Towards this end, the
induced magnetic field and velocity fields are decomposed into mean and
fluctuating components: ${\bf b}=\overline{\bf b}+{\bf b^{\prime}}$ and ${\bf
u}=\overline{\bf u}+{\bf u^{\prime}}$. Inserting this into equation 57 and
using Reynolds averaging yields

$\frac{\partial\overline{\bf b}}{\partial t}=F_{o}\frac{\partial\overline{\bf
u}}{\partial z}-(\overline{\bf u}\cdot\nabla)\overline{\bf b}-\overline{({\bf
u^{\prime}}\cdot\nabla){\bf b^{\prime}}}+\eta\nabla^{2}\overline{\bf b}.$ (58)

Next, we focus on the radial and latitudinal components because the zonal flow
($\overline{u_{\phi}}$) is nearly invariant in the z-direction (Figure 9a),
noting also that azimuthally oriented (toroidal) magnetic fields would not be
detectable by spacecraft:

$\frac{\partial\overline{b_{r}}}{\partial
t}=F_{o}\frac{\partial\overline{u_{r}}}{\partial z}-(\overline{{\bf
u}}\cdot\nabla)\overline{b_{r}}-\overline{({\bf
u^{\prime}}\cdot\nabla)b_{r}^{\prime}}+\eta\nabla^{2}\overline{b_{r}}$ (59)
$\frac{\partial\overline{b_{\theta}}}{\partial
t}=F_{o}\frac{\partial\overline{u_{\theta}}}{\partial z}-(\overline{{\bf
u}}\cdot\nabla)\overline{b_{\theta}}-\overline{({\bf
u^{\prime}}\cdot\nabla)b_{\theta}^{\prime}}+\eta\nabla^{2}\overline{b_{\theta}}$
(60)

Using simple scaling arguments, the second and third terms on the right sides
are likely small compared to the first term since $|F|\gg|b|$ (assuming
similar characteristic flow speeds and length scales) such that

$\frac{\partial\overline{b_{r}}}{\partial t}\approx
F_{o}\frac{\partial\overline{u_{r}}}{\partial
z}+\eta\nabla^{2}\overline{b_{r}}$ (61)
$\frac{\partial\overline{b_{\theta}}}{\partial t}\approx
F_{o}\frac{\partial\overline{u_{\theta}}}{\partial
z}+\eta\nabla^{2}\overline{b_{\theta}}.$ (62)

In the steady state limit and approximating the gradient length scales as $D$
and flow speeds as $U_{r}$ and $U_{\theta}$, the magnetic fields induced by
ocean currents can be estimated as:

$\frac{F_{o}U_{r}}{D}\sim\frac{\eta b_{r}}{D^{2}}\mbox{ such that
}b_{r}\sim\frac{F_{o}U_{r}D}{\eta}=\mu_{o}\sigma DU_{r}F_{o}$ (63)
$\frac{F_{o}U_{\theta}}{D}\sim\frac{\eta b_{\theta}}{D^{2}}\mbox{ such that
}b_{\theta}\sim\frac{F_{o}U_{\theta}D}{\eta}=\mu_{o}\sigma DU_{\theta}F_{o}.$
(64)

The resulting induced magnetic fields are then stronger for larger electrical
conductivities, ocean thicknesses, flow velocities, and satellites closer to
the host planet, since $F_{o}$ decreases with distance.

Table 4 summarizes the ambient Jovian conditions at Europa, Ganymede, and
Callisto as well as the relevant characteristics of their oceans, and the
computed upper bounds on the induced magnetic field strengths. Here, we assume
flow speeds typical of the global, steady overturning cells due to their
temporal persistence and large spatial scale, which we hypothesize will
produce the strongest induced magnetic signature that would be detectable at
spacecraft altitudes. We find that the theta magnetic field components are
larger than the radial components by roughly a factor of five, reaching $\sim
200$ nT for both Europa and Ganymede (higher salt content, thinner ice shell
models); estimates can be an order of magnitude weaker in the lower salt
content, thicker ice shell models). The radial variations correspond to
signals up to 33% (Ganymede) and 8% (Europa) of the ambient Jovian field,
which could be detectable with future missions. The signature at Callisto is
small ($\lesssim 1$ nT). In addition, we predict the fields to be strongest
near the equator where large vertical gradients in the convective flows exist
(Figure 9b-c).

| $\sigma$ | $D$ | $U_{r}$ | $U_{\theta}$ | $F_{o}$ | $b_{r}$ | $b_{\theta}$
---|---|---|---|---|---|---|---
| [S/m] | [km] | [m/s] | [m/s] | [nT] | [nT] | [nT]
Europa |  |  |  |  |  |  |
MgSO4 1 Wt%, Thicker ice shell | 0.4 | 120 | 0.08 | 0.38 | 420 | 2 | 10
MgSO4 1 Wt%, Thinner ice shell | 0.5 | 147 | 0.09 | 0.46 | 420 | 3 | 18
MgSO4 10 Wt%, Thicker ice shell | 3.4 | 127 | 0.08 | 0.40 | 420 | 18 | 91
MgSO4 10 Wt%, Thinner ice shell | 3.9 | 154 | 0.10 | 0.49 | 420 | 32 | 155
Seawater 0.35 Wt%, Thicker ice shell | 0.4 | 120 | 0.08 | 0.38 | 420 | 2 | 10
Seawater 0.35 Wt%, Thinner ice shell | 0.4 | 146 | 0.09 | 0.46 | 420 | 3 | 14
Seawater 3.5 Wt%, Thicker ice shell | 3.0 | 122 | 0.08 | 0.38 | 420 | 15 | 73
Seawater 3.5 Wt%, Thinner ice shell | 3.1 | 148 | 0.09 | 0.47 | 420 | 22 | 114
Ganymede |  |  |  |  |  |  |
MgSO4 1 Wt%, Thicker ice shell | 0.3 | 272 | 0.08 | 0.41 | 120 | 1 | 5
MgSO4 1 Wt%, Thinner ice shell | 0.5 | 442 | 0.13 | 0.66 | 120 | 4 | 22
MgSO4 10 Wt%, Thicker ice shell | 2.3 | 282 | 0.08 | 0.42 | 120 | 8 | 41
MgSO4 10 Wt%, Thinner ice shell | 4.1 | 455 | 0.14 | 0.68 | 120 | 39 | 191
Callisto |  |  |  |  |  |  |
MgSO4 1 Wt%, Thicker ice shell | 0.09 | 21 | 0.003 | 0.01 | 35 | $\ll 1$ | $\ll 1$
MgSO4 1 Wt%, Thinner ice shell | 0.2 | 132 | 0.02 | 0.09 | 35 | 0.02 | 0.1
MgSO4 10 Wt%, Thicker ice shell | 0.6 | 18 | 0.002 | 0.01 | 35 | $\ll 1$ | $\ll 1$
MgSO4 10 Wt%, Thinner ice shell | 1.5 | 130 | 0.02 | 0.09 | 35 | 0.2 | 0.8
Table 4: Assumed properties and resulting calculated upper bounds on the
strengths of the magnetic fields induced by oceanic fluid flows. Ambient
magnetic field strengths, $F_{o}$, from Showman Malhotra (1999); radial and
theta flow speeds, $U_{r}$ and $U_{\theta}$ with $U=\Omega DRo$, from Figure
9; ocean thicknesses, $D$, from Vance . (2018); and electrical conductivity,
$\sigma$, from Figure 4. These signals are anticipated to be largest near the
equator where $U_{\theta}$ and $U_{r}$ are strongest, as indicated in Figure
9b-c.

The simplified approach shown above gives an order of magnitude estimate of
the maximum induced field. Future work will assess the implications of these
assumptions through more detailed calculations. For example, we have assumed a
homogeneous and constant Jovian field; however, the magnetic environment
throughout the orbit close in to Jupiter may be highly variable and the
external field is affected by the presence of heavy ions and a variable
magnetosphere dynamics throughout a single orbit (e.g., Schilling ., 2008).
The temporal and spatial variation of the ambient field is expected to be
significant and the influence of these variations on ocean flow-driven
magnetic field signatures remains to be explored. Kinematic models that
directly solve the coupled momentum and induction equations are also an
exciting avenue to refine these estimates.

##  5 Discussion and Conclusions

The inverse problem of reconstructing the full induction response is complex
and is discussed in detail elsewhere (e.g., Khurana ., 2009, and Cochrane et
al. in progress). Here, we focus instead on how the adiababic conductivity
profile of the ocean affects the induction response relative to the mean case
that is usually considered in space physics analyses (e.g., Kivelson ., 2000),
and relative to the isothermal case often considered in analyses of interior
structure (e.g., Schubert ., 2004).

Differences between the adiabatic and mean conductivity cases have less
dependence on frequency (Tables 1-3 and Figures S1, S3, and S5). For Europa,
the nominal oceans with ice shells 5- and 30-km thick have errors of about 6%
and 3%, respectively, and amount to nearly a 1 nT difference for the largest
signals that exceed 20 nT. For Ganymede, the nominal oceans with ice shells
$\sim$25- and $\sim$100-km thick have errors of about 7% and 3%, and are also
nearly 1 nT for the largest signals that exceed 10 nT. For Callisto, the
induction response of the mean conductivity ocean for ice shells of $\sim$100-
and $\sim$130-km thickness is within about 2% of the response for the
adiabatic ocean, less than 0.3 nT for the largest signals that approach 10 nT.

The induction response of the adiabatic ocean differs from that of the
equivalent ocean with the conductivity of fluid at the ice-ocean interface.
The greater mismatch of conductivities of the lower part of the ocean causes
large differences in amplitude and phase at lower frequencies (i.e. for larger
skin depths). For Europa, this means that the lower-frequency mean-motion
signal (3.2$\times 10^{-6}$ Hz; Table 1) differs by more than 15% for the
warmer lower-salinity oceans, or about 0.1 nT. For Ganymede, the differences
at the mean-motion frequency (1.62$\times 10^{-6}$ Hz; Table 2) can approach
25%, which amounts to 0.04 nT. For Callisto, the differences at the mean-
motion frequency (6.9$\times 10^{-7}$ Hz; Table 3) approach 20%, which amounts
to only 2 pT for the small predicted residual field based on the mean field.
By contrast, the higher-frequency diurnal signals differ by less than 5%.

Based on the circulation models and upper bound induced magnetic field
estimates described in Section 4, flow-induced fields may be a prominent
component of the magnetic fields measured in the low latitudes for Europa and
Ganymede. The peak flow-induced magnitude is 30-40 nT (Table 4) compared with
Jovian-induced residual fields of less than 20 nT for both Europa (Table 1)
and Ganymede (Table 2).

###  5.1 Implications for future missions

The Europa Clipper mission will conduct multiple ($>$40) flybys of Europa, and
will investigate its induction response with the goal of constraining the
ocean conductivity to within $\pm 0.5$ S m-1 and ice thickness to within
$\pm$2 km (Buffington ., 2017). The flybys at high latitudes will allow the
Europa Clipper investigation to isolate flow-induced fields from the diffusive
response, and possibly to derive constraints on currents in the ocean. With
independent constraints on ice thickness obtained from the Radar for Europa
Assessment and Sounding: Ocean to Near-surface (REASON) and Europa Imaging
System (EIS) investigations (Steinbrügge ., 2018), it may be possible to
constrain the ocean’s temperature and thus the adiabatic structure for the
best-fit ocean composition inferred from compositional investigations. The
analyses provided here (Figure 6 and Table 1) indicate that a sensitivity of
1.5 nT is probably insufficient to distinguish between end-member MgSO4 and
NaCl oceans, but might be sufficient to distinguish between order-of-magnitude
differences in salinity.

The JUpiter ICy moons Explorer (JUICE) will execute two Europa flybys and nine
Callisto flybys, and will orbit Ganymede (Grasset ., 2013). The magnetic field
investigation seeks to determine the induction response to better than 0.1 nT.
The Europa flybys might aid the Europa Clipper investigation in constraining
the composition of the ocean. At Ganymede, the magnetic field investigation
will not be sufficient to discern the presence of a basal liquid layer at the
ice VI-rock interface. Although the ability to discern between ocean
compositions could not be assessed owing to insufficient electrical
conductivity data at high pressures, it seems likely that useful constraints
could be derived based on the signal strengths at Ganymede, if laboratory-
derived electrical conductivity data for relevant solutions under pressure
became available. At Callisto, 0.1 nT accuracy may only allow sensing of the
induction response to Jupiter’s synodic field, which might be sufficient to
infer the thickness and salinity of an ocean if adequate temporal coverage is
obtained to confirm the phase of the response.

###### Acknowledgements.

Work by JPL co-authors was partially supported by the Jet Propulsion
Laboratory, Caltech, and by the Icy Worlds node of NASA’s Astrobiology
Institute (13-13NAI7_2-0024). This work was partially supported by NASA’s
Europa Clipper project. Work by MJS was supported by the NASA Earth and Space
Science Fellowship Program - Grant 80NSSC18K1236. Work by KMS was also
supported by NASA Grant NNX14AR28G. The Matlab scripts and associated data
needed to compute the results shown here are currently being archived. The
data and scripts for radial structure models and diffusive induction will be
placed on github (https://github.com/vancesteven/PlanetProfile) and Zenodo.
The MagIC code is publicly available at the https://magic-
sph.github.io/contents.html website. All global convection model data were
first published in Soderlund et al. (2019) and are available therein. ©2020.
All rights reserved.

## References

  * Acuna Ness (1976) Acuna_1976Acuna, MH. Ness, NF.  1976Jun.  The main magnetic field of Jupiter The main magnetic field of jupiter.  Journal of Geophysical Research81162917–2922.  http://dx.doi.org/10.1029/JA081i016p02917 10.1029/ja081i016p02917
  * Buffington . (2017) buffington2017evolutionBuffington, B., Lam, T., Campagnola, S., Ludwinski, J., Ferguson, E., Bradley, B.Siddique, F.  2017\.  EVOLUTION OF TRAJECTORY DESIGN REQUIREMENTS ON NASA’S PLANNED EUROPA CLIPPER MISSION Evolution of trajectory design requirements on NASA’s planned Europa Clipper mission.  Proceedings of the 68th International Astronautical Congress (IAC)IAC-17-C1.7.8.
  * Calvert . (1958) calvert1958determinationCalvert, R., Cornelius, JA., Griffiths, VS. Stock, DI.  1958\.  The determination of the electrical conductivities of some concentrated electrolyte solutions using a transformer bridge The determination of the electrical conductivities of some concentrated electrolyte solutions using a transformer bridge.  The Journal of Physical Chemistry62147–53.
  * Chave (1983) Chave83Chave, AD.  1983\.  On the theory of electromagnetic induction in the Earth by ocean currents On the theory of electromagnetic induction in the Earth by ocean currents.  J. Geophys. Res.883531–3542.
  * Choblet . (2017) Choblet_2017Choblet, G., Tobie, G., Sotin, C., Kalousová, K. Grasset, O.  2017Mar.  Heat transport in the high-pressure ice mantle of large icy moons Heat transport in the high-pressure ice mantle of large icy moons.  Icarus285252–262.  http://dx.doi.org/10.1016/j.icarus.2016.12.002 10.1016/j.icarus.2016.12.002
  * J. Connerney . (1998) connerney1998newConnerney, J., Acuna, M., Ness, N. Satoh, T.  1998\.  New models of Jupiter’s magnetic field constrained by the Io flux tube footprint New models of Jupiter’s magnetic field constrained by the Io flux tube footprint.  Journal of Geophysical Research: Space Physics103A611929–11939.
  * JEP. Connerney . (2018) Connerney_2018Connerney, JEP., Kotsiaros, S., Oliversen, RJ., Espley, JR., Joergensen, JL., Joergensen, PS.et al.  2018Mar.  A New Model of Jupiter’s Magnetic Field From Juno’s First Nine Orbits A new model of Jupiter’s magnetic field from Juno’s first nine orbits.  Geophysical Research Letters4562590–2596.  http://dx.doi.org/10.1002/2018GL077312 10.1002/2018gl077312
  * Constable Constable (2004) ConstableConstable04Constable, S. Constable, CG.  2004\.  Observing geomagnetic induction in magnetic satellite measurements and associated implications for mantle conductivity Observing geomagnetic induction in magnetic satellite measurements and associated implications for mantle conductivity.  Geochem. Geophys. Geosys.5Q01006.
  * Gissinger Petitdemange (2019) GissingerPetitdemange19Gissinger, C. Petitdemange, L.  2019\.  A magnetically driven equatorial jet in Europa’s ocean A magnetically driven equatorial jet in Europa’s ocean.  Nature Astron.1.
  * Grasset . (2013) grasset2013jupiterGrasset, O., Dougherty, M., Coustenis, A., Bunce, E., Erd, C., Titov, D.others  2013\.  JUpiter ICy moons Explorer (JUICE): An ESA mission to orbit Ganymede and to characterise the Jupiter system Jupiter icy moons explorer (juice): An esa mission to orbit ganymede and to characterise the jupiter system.  Planetary and Space Science781–21.
  * Grayver . (2016) GrayverEA16Grayver, AV., Schnepf, NR., Kuvshinov, AV., Sabaka, TJ., Manoj, C. Olsen, N.  2016\.  Satellite tidal magnetic signals constrain oceanic lithosphere-asthenosphere boundary Satellite tidal magnetic signals constrain oceanic lithosphere-asthenosphere boundary.  Science Adv.29e1600798.
  * Hand Chyba (2007) hand2007empiricalHand, K. Chyba, C.  2007\.  Empirical constraints on the salinity of the europan ocean and implications for a thin ice shell Empirical constraints on the salinity of the europan ocean and implications for a thin ice shell.  Icarus1892424–438.
  * Hartkorn Saur (2017) Hartkorn_2017Hartkorn, O. Saur, J.  2017Nov.  Induction signals from Callisto’s ionosphere and their implications on a possible subsurface ocean Induction signals from Callisto’s ionosphere and their implications on a possible subsurface ocean.  Journal of Geophysical Research: Space Physics1221111,677–11,697.  http://dx.doi.org/10.1002/2017JA024269 10.1002/2017ja024269
  * Hogenboom . (1995) hogenboom1995magnesiumHogenboom, DL., Kargel, JS., Ganasan, JP. Lee, L.  199506\.  Magnesium Sulfate-Water to 400 MPa Using a Novel Piezometer: Densities, Phase Equilibria, and Planetological Implications Magnesium sulfate-water to 400 MPa using a novel piezometer: Densities, phase equilibria, and planetological implications.  Icarus1152258–277.  http://www.Sciencedirect.com/Science/article/B6WGF-45NJJ40-14/2/5f176b700065df4efb2e1eba3abd3224
  * Journaux . (2013) Journaux_2013Journaux, B., Daniel, I., Caracas, R., Montagnac, G. Cardon, H.  2013Sep.  Influence of NaCl on ice VI and ice VII melting curves up to 6GPa, implications for large icy moons Influence of NaCl on ice VI and ice VII melting curves up to 6GPa, implications for large icy moons.  Icarus2261355–363.  http://dx.doi.org/10.1016/j.icarus.2013.05.039 10.1016/j.icarus.2013.05.039
  * Journaux . (2017) Journaux_2017Journaux, B., Daniel, I., Petitgirard, S., Cardon, H., Perrillat, JP., Caracas, R. Mezouar, M.  2017Apr.  Salt partitioning between water and high-pressure ices. Implication for the dynamics and habitability of icy moons and water-rich planetary bodies Salt partitioning between water and high-pressure ices. implication for the dynamics and habitability of icy moons and water-rich planetary bodies.  Earth and Planetary Science Letters46336–47.  http://dx.doi.org/10.1016/j.epsl.2017.01.017 10.1016/j.epsl.2017.01.017
  * Khurana . (2009) khurana2009electromagneticKhurana, K., Kivelson, M., Hand, K. Russell, C.  2009\.  Electromagnetic induction from Europa’s ocean and the deep interior Electromagnetic induction from Europa’s ocean and the deep interior.  Europa, Edited by Robert T. Pappalardo, William B. McKinnon, Krishan K. Khurana; with the assistance of René Dotson with 85 collaborating authors. University of Arizona Press, Tucson, 2009. The University of Arizona space science series ISBN: 9780816528448, p. 5711571\.
  * Khurana . (1998) khurana1998inducedKhurana, K., Kivelson, M., Stevenson, D., Schubert, G., Russell, C., Walker, R. Polanskey, C.  1998\.  Induced magnetic fields as evidence for subsurface oceans in Europa and Callisto Induced magnetic fields as evidence for subsurface oceans in Europa and Callisto.  Nature3956704777–780.
  * Khurana (1997) Khurana_1997Khurana, KK.  1997Jun.  Euler potential models of Jupiter’s magnetospheric field Euler potential models of Jupiter’s magnetospheric field.  Journal of Geophysical Research: Space Physics102A611295–11306.  http://dx.doi.org/10.1029/97JA00563 10.1029/97ja00563
  * Kivelson . (2000) kivelson2000galileoKivelson, M., Khurana, K., Russell, C., Volwerk, M., Walker, R. Zimmer, C.  2000\.  Galileo Magnetometer Measurements: A Stronger Case for a Subsurface Ocean at Europa Galileo magnetometer measurements: A stronger case for a subsurface ocean at Europa.  Science2891340–1343.
  * Lainey . (2006) lainey2006srgLainey, V., Duriez, L. Vienne, A.  2006\.  Synthetic representation of the Galilean satellites’ orbital motions from L1 ephemerides Synthetic representation of the Galilean satellites’ orbital motions from L1 ephemerides.  A&A456783–788.
  * Lemasquerier . (2017) LemasquerierEA17Lemasquerier, D., Grannan, AM., Vidal, J., Cébron, D., Favier, B., Le Bars, M. Aurnou, JM.  2017\.  Libration-driven flows in ellipsoidal shells Libration-driven flows in ellipsoidal shells.  J. Geophys. Res. Planets12291926–1950.
  * Lieske (1998) lieske1998galileanLieske, J.  1998\.  Galilean satellite ephemerides E5 Galilean satellite ephemerides E5.  Astronomy and Astrophysics Supplement Series1292205–217.
  * Liuzzo . (2015) Liuzzo_2015Liuzzo, L., Feyerabend, M., Simon, S. Motschmann, U.  2015Nov.  The impact of Callisto’s atmosphere on its plasma interaction with the Jovian magnetosphere The impact of Callisto’s atmosphere on its plasma interaction with the jovian magnetosphere.  Journal of Geophysical Research: Space Physics120119401–9427.  http://dx.doi.org/10.1002/2015JA021792 10.1002/2015ja021792
  * McDougall Barker (2011) mcdougall2011gettingMcDougall, TJ. Barker, PM.  2011\.  Getting started with TEOS-10 and the Gibbs Seawater (GSW) oceanographic toolbox Getting started with TEOS-10 and the Gibbs Seawater (GSW) oceanographic toolbox.  SCOR/IAPSO WG1271–28.
  * Minami (2017) Minami17Minami, T.  2017\.  Motional induction by tsunamis and ocean tides: 10 years of progress Motional induction by tsunamis and ocean tides: 10 years of progress.  Surv. Geophys.381097–1132.
  * Pasek Greenberg (2012) pasek2012acidificationPasek, MA. Greenberg, R.  2012\.  Acidification of Europa’s Subsurface Ocean as a Consequence of Oxidant Delivery Acidification of Europa’s subsurface ocean as a consequence of oxidant delivery.  Astrobiology122151–159.
  * Rovira-Navarro . (2019) RoviraNavarroEA19Rovira-Navarro, M., Rieutord, M., Gerkema, T., Maas, LR., van der Wal, W. Vermeersen, B.  2019\.  Do tidally-generated inertial waves heat the subsurface oceans of Europa and Enceladus? Do tidally-generated inertial waves heat the subsurface oceans of Europa and Enceladus?  Icarus321126–140.
  * Saur . (2009) Saur_2009Saur, J., Neubauer, FM. Glassmeier, KH.  2009Dec.  Induced Magnetic Fields in Solar System Bodies Induced magnetic fields in solar system bodies.  Space Science Reviews1521-4391–421.  http://dx.doi.org/10.1007/s11214-009-9581-y 10.1007/s11214-009-9581-y
  * Saur . (1998) Saur_1998Saur, J., Strobel, DF. Neubauer, FM.  1998Aug.  Interaction of the Jovian magnetosphere with Europa: Constraints on the neutral atmosphere Interaction of the jovian magnetosphere with europa: Constraints on the neutral atmosphere.  Journal of Geophysical Research: Planets103E919947–19962.  http://dx.doi.org/10.1029/97JE03556 10.1029/97je03556
  * Schaeffer (2013) Schaeffer13Schaeffer, N.  2013\.  Efficient spherical harmonic transforms aimed at pseudospectral numerical simulations Efficient spherical harmonic transforms aimed at pseudospectral numerical simulations.  Geochem. Geophys. Geosy.143751–758.
  * Schilling . (2008) schilling2008influenceSchilling, N., Neubauer, F. Saur, J.  2008\.  Influence of the internally induced magnetic field on the plasma interaction of Europa Influence of the internally induced magnetic field on the plasma interaction of europa.  Journal of Geophysical Research: Space Physics113A3.
  * Schilling . (2007) schilling2007timeSchilling, N., Neubauer, FM. Saur, J.  2007\.  Time-varying interaction of Europa with the jovian magnetosphere: Constraints on the conductivity of Europa’s subsurface ocean Time-varying interaction of Europa with the jovian magnetosphere: Constraints on the conductivity of Europa’s subsurface ocean.  Icarus192141–55.
  * Schmidt Manning (2017) schmidt2017pressureSchmidt, C. Manning, C.  2017\.  Pressure-induced ion pairing in MgSO4 solutions: Implications for the oceans of icy worlds Pressure-induced ion pairing in MgSO4 solutions: Implications for the oceans of icy worlds.  Geochemical Perspectives Letters366-74.
  * Schubert . (2004) schubert2004interiorSchubert, G., Anderson, J., Spohn, T. McKinnon, W.  2004\.  Interior composition, structure and dynamics of the Galilean satellites Interior composition, structure and dynamics of the Galilean satellites.  Jupiter: The Planet, Satellites and Magnetosphere281–306.
  * Seidelmann Divine (1977) seidelmann1977evaluationSeidelmann, P. Divine, N.  1977\.  Evaluation of Jupiter longitudes in System III (1965) Evaluation of jupiter longitudes in system iii (1965).  Geophysical Research Letters4265–68.
  * Seufert . (2011) seufert2011multiSeufert, M., Saur, J. Neubauer, FM.  2011\.  Multi-frequency electromagnetic sounding of the Galilean moons Multi-frequency electromagnetic sounding of the Galilean moons.  Icarus2142477–494.
  * Showman Malhotra (1999) ShowmanMalhotra99Showman, AP. Malhotra, R.  1999\.  The galilean satellites The galilean satellites.  Science286543777–84.
  * (39) soderlund2014oceanSoderlund, K., Schmidt, B., Wicht, J. Blankenship, D.  2014\.  Ocean-driven heating of Europa’s icy shell at low latitudes Ocean-driven heating of Europa’s icy shell at low latitudes.  Nature Geoscience7116–19.
  * Soderlund (2019) Soderlund19Soderlund, KM.  2019\.  Ocean dynamics of outer solar system satellites Ocean dynamics of outer solar system satellites.  Geophys. Res. Lett.https://doi.org/10.1029/2018GL081880.
  * Steinbrügge . (2018) Steinbrugge_2018Steinbrügge, G., Schroeder, D., Haynes, M., Hussmann, H., Grima, C. Blankenship, D.  2018Jan.  Assessing the potential for measuring Europa’s tidal Love number h2 using radar sounder and topographic imager data Assessing the potential for measuring Europa’s tidal Love number h2 using radar sounder and topographic imager data.  Earth and Planetary Science Letters482334–341.  http://dx.doi.org/10.1016/j.epsl.2017.11.028 10.1016/j.epsl.2017.11.028
  * Tyler (2011) Tyler11Tyler, RH.  2011\.  Magnetic remote sensing of Europa’s ocean tides Magnetic remote sensing of Europa’s ocean tides.  Icarus211906–908.
  * Tyler . (2017) Tyler_2017Tyler, RH., Boyer, TP., Minami, T., Zweng, MM. Reagan, JR.  2017Nov.  Electrical conductivity of the global ocean Electrical conductivity of the global ocean.  Earth, Planets and Space691.  http://dx.doi.org/10.1186/s40623-017-0739-7 10.1186/s40623-017-0739-7
  * Tyler . (2003) TylerEA03Tyler, RH., Maus, S. Luhr, H.  2003\.  Satellite observations of magnetic fields due to ocean tidal flow Satellite observations of magnetic fields due to ocean tidal flow.  Science299239–241.
  * S. Vance . (2014) Vance_2014Vance, S., Bouffard, M., Choukroun, M. Sotin, C.  2014Jun.  Ganymede׳s internal structure including thermodynamics of magnesium sulfate oceans in contact with ice Ganymede׳s internal structure including thermodynamics of magnesium sulfate oceans in contact with ice.  Planetary and Space Science9662–70.  http://dx.doi.org/10.1016/j.pss.2014.03.011 10.1016/j.pss.2014.03.011
  * Vance Goodman (20091) vance2009oceanographyVance, S. Goodman, J.  20091\.  Europa Europa.  RTPWBMK. Khurana (), ( 459-482).  Arizona University Press.
  * Vance Goodman (20092) VanceGoodman09Vance, S. Goodman, JC.  20092\.  Oceanography of an ice-covered moon Oceanography of an ice-covered moon.  RT. Pappalardo, WB. McKinnon KK. Khurana (), Europa Europa ( 459-482).  Tucson: University of Arizona Press.
  * Vance . (2016) Vance_2016Vance, SD., Hand, KP. Pappalardo, RT.  2016May.  Geophysical controls of chemical disequilibria in Europa Geophysical controls of chemical disequilibria in Europa.  Geophysical Research Letters43104871–4879.  http://dx.doi.org/10.1002/2016GL068547 10.1002/2016gl068547
  * Vance . (2018) Vance_2018Vance, SD., Panning, MP., Stähler, S., Cammarano, F., Bills, BG., Tobie, G.et al.  2018Nov.  Geophysical investigations of habitability in ice-covered ocean worlds Geophysical investigations of habitability in ice-covered ocean worlds.  Journal of Geophysical Research: Planets.  http://dx.doi.org/10.1002/2017JE005341 10.1002/2017je005341
  * Wicht (2002) Wicht02Wicht, J.  2002\.  Inner-core conductivity in numerical dynamo simulations Inner-core conductivity in numerical dynamo simulations.  Phys. Earth Planet. Int.132281–302.
  * Zimmer . (2000) zimmer2000subsurfaceZimmer, C., Khurana, KK. Kivelson, MG.  2000\.  Subsurface oceans on Europa and Callisto: Constraints from Galileo magnetometer observations Subsurface oceans on Europa and Callisto: Constraints from Galileo magnetometer observations.  Icarus1472329–347.
  * Zolotov (2008) zolotov2008oceanicZolotov, M.  2008\.  Oceanic composition on Europa: Constraints from mineral solubilities Oceanic composition on europa: Constraints from mineral solubilities.  Lunar and Planetary Institute Science Conference Abstracts Lunar and planetary institute science conference abstracts ( 39, 2349).
  * Zolotov Kargel (2009) zolotov2009chemicalZolotov, MY. Kargel, J.  2009\.  On the chemical composition of Europa’s icy shell, ocean, and underlying rocks On the chemical composition of Europa’s icy shell, ocean, and underlying rocks.  Europa, edited by RT Pappalardo, WB McKinnon, and K. Khurana, University of Arizona Press, Tucson, AZ431–458.


# Passive Haptic Rehearsal for Accelerated Piano Skill Acquisition

Tan Gemicioglu, Noah Teuscher, Brahmi Dwivedi, Soobin Park, Emerson Miller
<EMAIL_ADDRESS>Georgia Institute
of TechnologyAtlanta, GAUSA ,  Celeste Mason<EMAIL_ADDRESS>University of HamburgHamburgGermany ,  Caitlyn Seim<EMAIL_ADDRESS>Stanford UniversityStanford, CaliforniaUSA and  Thad Starner<EMAIL_ADDRESS>Georgia Institute of TechnologyAtlanta, GAUSA

(2022)

###### Abstract.

Passive haptic learning (PHL) uses vibrotactile stimulation to train piano
songs using repetition, even when the recipient of stimulation is focused on
other tasks. However, many of the benefits of playing piano cannot be acquired
without actively playing the instrument. In this position paper, we posit that
passive haptic rehearsal, where active piano practice is assisted by separate
sessions of passive stimulation, is of greater everyday use than solely PHL.
We propose a study to examine the effects of passive haptic rehearsal for
self-paced piano learners and consider how to incorporate passive rehearsal
into everyday practice.

Haptic, Tactile, Wearable, Passive Training, PHL, Piano

††copyright: ccby††journalyear: 2022††conference: CHI Conference on Human
Factors in Computing Systems; April 29-May 5, 2022; New Orleans, LA,
USA††booktitle: CHI Conference on Human Factors in Computing Systems (CHI
’22), April 29-May 5, 2022, New Orleans, LA, USA††ccs: Human-centered
computing Human computer interaction (HCI)††ccs: Human-centered computing
Haptic devices

Figure 1. (a) Passive haptic rehearsal gloves. (b) Web app for active and
passive practice. (c) Active practice on light-up keyboard.

Teaser image featuring passive haptic rehearsal gloves, web app with sheet
notes displayed, and person playing light-up keyboard for active session.

##  1\. Introduction

Playing piano improves mental health and has a positive impact on welfare
(McCaffrey and Edwards, 2016). However, the learning process can take
significant time between learning theory and technique, practicing new piano
pieces and rehearsing previously learned ones. Passive haptic learning (PHL)
offers an alternative method of rehearsal, speeding the learning process and
increasing retention (Donchev et al., 2021).

Previous PHL studies have involved almost entirely passive training, with
users only actively playing on a keyboard for evaluation (Seim et al., 2015).
This research prioritized internal validity, focusing on narrow conditions:
simple, short songs, rigid pedagogical methods, and inexperienced students. If
PHL is to be a successful piano education tool, it must be usable by real
piano students learning complicated songs and engaging in multiple forms of
practice through various mediums. We expand upon existing work by integrating
passive training into a piano learning program we hope to be suitable for
songs and students of all levels. We propose passive haptic rehearsal, in
which students learn songs through a combination of active practice and
passive training sessions, to reinforce developing skills.

##  2\. Related Work

Existing work in piano skill acquisition primarily focuses on learning
assisted by a piano instructor. We focus instead on self-paced piano learning
where the user learns without any direct human assistance. Consistent feedback
can encourage learners’ progress (Costa-Giomi et al., 2005), and the process
can be gamified by providing more incentives as the learner spends time
practicing piano, ensuring continuity and regular practice (Raymaekers et al.,
2014). New piano interfaces seek to speed learning and reduce the cognitive
load demanded by practice. Visual cues, such as light-up keys, new piano
notations (Rogers et al., 2014) and finger position tracking (Marky et al.,
2021) have enabled such advances in active learning.

Passive learning, in contrast, is ”caught, rather than taught” (Krugman and
Hartley, 1970) and can enable learning even in the absence of motivation and
effort (Zukin and Snyder, 1984). PHL uses vibrotactile stimulation to teach
piano passively using intensive repetition of instructional cues applied
directly to the fingers (Huang et al., 2008). Surprisingly, learning occurs
whether or not the tactile stimulation is accompanied by audio of the notes
playing (Markow, 2012). Early studies extensively compared passive and active
learning for piano and confirmed that PHL did not occupy the user’s attention
during learning (Kohlsdorf and Starner, 2010; Huang et al., 2010). Newer
research has expanded from simple melodies to complete, two-handed chorded
piano pieces (Seim et al., 2015). Furthermore, recent work shows that
passively learned note sequences can be recalled with greater accuracy than
actively learned sequences when recall is supported by audiovisual cues
(Donchev et al., 2021).

##  3\. Passive Piano Learning Framework

Our proposed framework includes a pair of vibrotactile haptic feedback gloves,
a Bluetooth-capable tablet such as a Samsung Galaxy Tab A8 and a Casio LK-S250
keyboard. The Casio keyboard can illustrate how to play a song using light-up
keys and can capture the user’s practice sessions using MIDI. We use soft,
faux leather driving gloves with sizes varying for each user. The PHL gloves
contain five vibration motors (Precision Microdrives 310-103) at the base of
the fingers and a 3D-printed case at the back of the palm to protect the
electronics from sweat, dust and other damage.

###  3.1. Integrating PHL with Active Practice

In addition to the 30 minute active practice session each day, learners wear
the PHL gloves for 2.5 hours. Using vibration motors at the base of each
finger, the gloves repeatedly stimulate the user’s fingers in the order and
rhythm in which the piano keys are played. The gloves are fingerless so as not
to interfere with the user’s everyday tasks. Since no audio is played and the
stimulation is relatively subtle, the user can ignore the gloves. The gloves
are self-contained in that they do not need a continuous connection to an
external device to operate. The right glove controls the left through
Bluetooth LE. Battery life is around three hours.

Testing the learner before and after active sessions also provides data on
passive rehearsal. The test at the end of the previous active session provides
a baseline to compare to the test at the beginning of the current active
session to determine how much “forgetting” has happened between sessions.
Without passive stimulation, we expect significant degradation of performance,
but in pilot testing with passive rehearsal, learners seem to retain or even
improve upon the benefits of their previous active learning session (Huang et
al., 2008).

###  3.2. Performance Evaluation

To evaluate the performance of a user, we treat the notes played as a temporal
sequence $P$ for calculating $Eval(P)$. We use the Needleman–Wunsch algorithm
(Needleman and Wunsch, 1970), a dynamic programming algorithm for optimal
global sequence alignment, for matching the note sequence played to the note
sequence in the original piece. Previous work utilized Dynamic Time Warping
for sequence alignment which ignores the error of multiple presses of the same
key (the error between the key-press sequences “C A A B” and “C A B” is zero)
(Huang et al., 2010; Seim et al., 2015). Needleman-Wunsch is “stricter” when
assigning error costs. In addition to alignment errors, we have included a
timing error in our new algorithm to measure rhythmic inaccuracies. Once the
two sequences are matched, a timing error is applied to matched key presses in
which the time difference between user played and original key presses crosses
a pre-determined threshold.

Original Times: | $t_{1}$ | $t_{2}$ | - | $t_{3}$ | $t_{4}$ | - | $t_{5}$ | $t_{6}$ | -
---|---|---|---|---|---|---|---|---|---
Original Notes: | C | B | B | A | E | E | F | C | -
Performed Notes: | C | B | - | A | E | C | F | C | C
Performed Times: | $t_{1}^{\prime}$ | $t_{2}^{\prime}$ |  -↑  Deletion | $t_{3}^{\prime}$ | $t_{4}^{\prime}$ |  -↑  Substitution | $t_{5}^{\prime}$ | $t_{6}^{\prime}$ |  -↑  Insertion

For the above example, assuming $m$ total matched notes and the timing error
threshold to be $T$:

$\text{Timing Cost = Timing cost of matched notes =
}\sum_{i=0}^{m}1(|t_{i}-t_{i}^{\prime}|\geq T)$ Alignment Cost = Deletion cost
+ Insertion cost + Substitution cost $=1+1+1=3$

Alignment and timing can be weighted to give an overall evaluation of the
performance; with weights $W_{a}$ and $W_{t}$ as the weights of alignment cost
and timing cost respectively:

$\text{Total Cost }=W_{a}*\text{Alignment Cost}+W_{t}*\text{Timing Cost}$

Key presses that occur in rapid succession are extracted as chords and treated
as unordered sequences.

##  4\. Proposed Study

Since we focus on self-paced piano learning, we wish to recruit hobbyist piano
learners who are practicing in their own time without formal piano lessons. We
plan to reach these participants by advertising to people who watch content
from online piano instructors on video streaming platforms. 16 participants
will be recruited and paired by skill level based on the improvements they
achieve while practicing a previously unseen piece actively for half an hour.

For the study, we chose a pair of songs from the composer Bartok: Children at
Play and Young Men’s Dance. Our participants likely have not heard or learned
these songs before, and a disqualification criteria during recruitment will be
if they have learned these songs previously. The songs are of a difficulty
level and length such that they are not trivial to learn but our target
subjects should be able to adequately learn them in a two week period.

The study will follow a counter-balanced Latin Square design for four
conditions: two each for starting one of two piano pieces and starting with
the sham or functional gloves. Over the course of 2 weeks, participants will
be required to log a total of 3 hours of practice per day: 30 minutes of
active practice and 2.5 hours of passive practice.

We will bring subjects into the lab three times throughout the study for more
in-depth evaluations: once at the beginning of the study, once at the midway
point (two weeks), and once at the conclusion. These evaluations will consist
of recording their unaided performance of each of our curriculum songs. Likert
scale and preference surveys and free responses about their progress will also
be compared between the conditions.

The experiment is designed to be both within-subjects and between subjects.
Each pair will be assigned the same song. One learner will be using a fully
operational glove while the other will be given a similar glove and be told
that the vibrations are set at a level below perception (when in fact they
will not be vibrating at all). Using the daily pre and post active session
tests, we will compare learning and “forgetting” rates between subjects. After
two weeks, the pair will change to the second song, and we will switch the
sham and active glove condition. In this manner we can compare each
participant’s performance to themselves between the sham and active condition.
Note that, because of the pairing, participants must be recruited and
randomized in teams of eight to maintain counterbalancing. ANOVAs and
permutation tests will be used to determine if the learning rates differ
statistically.

Figure 2. Sessions scheduled on day $d$, where we measure
progress from active practice as $Eval(R_{d}^{2})-Eval(R_{d}^{1})$ and
progress from passive rehearsal as $Eval(R_{d+1}^{1})-Eval(R_{d}^{2})$

##  5\. Conclusion

This work presents a new framework for PHL by designing a curriculum that
combines active learning with passive haptic rehearsal. We focus our efforts
on hobbyist piano learners as they may most appreciate the benefits of the PHL
gloves. In order to examine the effects of passive haptic rehearsal, we
propose a controlled, randomized study to test the augmented piano learning
process in the users’ own environments over the course of four weeks.

##  6\. Acknowledgments

This research was supported by the National Science Foundation, Partnerships
for Innovation grant #2122797 and the Georgia Research Alliance, Inc. based in
Atlanta, Georgia.

## References

  * (1)
  * Costa-Giomi et al. (2005) Eugenia Costa-Giomi, Patricia J. Flowers, and Wakaha Sasaki. 2005.  Piano Lessons of Beginning Students Who Persist or Drop Out: Teacher Behavior, Student Behavior, and Lesson Progress.  _Journal of Research in Music Education_ 53, 3 (Oct. 2005), 234–247.  https://doi.org/10.1177/002242940505300305 Publisher: SAGE Publications Inc.
  * Donchev et al. (2021) Rumen Donchev, Erik Pescara, and Michael Beigl. 2021\.  Investigating Retention in Passive Haptic Learning of Piano Songs.  _Proceedings of the ACM on Interactive, Mobile, Wearable and Ubiquitous Technologies_ 5, 2 (June 2021), 60:1–60:14.  https://doi.org/10.1145/3463513
  * Huang et al. (2008) Kevin Huang, Ellen Yi-Luen Do, and Thad Starner. 2008.  PianoTouch: A wearable haptic piano instruction system for passive learning of piano skills. In _2008 12th IEEE International Symposium on Wearable Computers_. 41–44.  https://doi.org/10.1109/ISWC.2008.4911582 ISSN: 2376-8541.
  * Huang et al. (2010) Kevin Huang, Thad Starner, Ellen Do, Gil Weiberg, Daniel Kohlsdorf, Claas Ahlrichs, and Ruediger Leibrandt. 2010.  Mobile music touch: mobile tactile stimulation for passive learning. In _Proceedings of the 28th international conference on Human factors in computing systems - CHI ’10_. ACM Press, Atlanta, Georgia, USA, 791.  https://doi.org/10.1145/1753326.1753443
  * Kohlsdorf and Starner (2010) Daniel Kohlsdorf and Thad Starner. 2010.  Mobile Music Touch: The effect of primary tasks on passively learning piano sequences. In _International Symposium on Wearable Computers (ISWC) 2010_. 1–8.  https://doi.org/10.1109/ISWC.2010.5665877 ISSN: 2376-8541.
  * Krugman and Hartley (1970) Herbert E. Krugman and Eugene L. Hartley. 1970.  Passive Learning from Television.  _Public Opinion Quarterly_ 34, 2 (Jan. 1970), 184–190.  https://doi.org/10.1086/267788
  * Markow (2012) Tanya Thais Markow. 2012\.  _Mobile music touch: using haptic stimulation for passive rehabilitation and learning_.  Ph. D. Dissertation. Georgia Institute of Technology.
  * Marky et al. (2021) Karola Marky, Andreas Weiß, Andrii Matviienko, Florian Brandherm, Sebastian Wolf, Martin Schmitz, Florian Krell, Florian Müller, Max Mühlhäuser, and Thomas Kosch. 2021\.  Let’s Frets! Assisting Guitar Students During Practice via Capacitive Sensing. In _Proceedings of the 2021 CHI Conference on Human Factors in Computing Systems_ _(CHI ’21)_. Association for Computing Machinery, New York, NY, USA, 1–12.  https://doi.org/10.1145/3411764.3445595
  * McCaffrey and Edwards (2016) Tríona McCaffrey and Jane Edwards. 2016.  ”Music Therapy Helped Me Get Back Doing”: Perspectives of Music Therapy Participants in Mental Health Services.  _Journal of Music Therapy_ 53, 2 (2016), 121–148.  https://doi.org/10.1093/jmt/thw002
  * Needleman and Wunsch (1970) Saul B. Needleman and Christian D. Wunsch. 1970.  A general method applicable to the search for similarities in the amino acid sequence of two proteins.  _Journal of molecular biology_ 48 3 (1970), 443–53.
  * Raymaekers et al. (2014) Linsey Raymaekers, Jo Vermeulen, Kris Luyten, and Karin Coninx. 2014.  Game of tones: learning to play songs on a piano using projected instructions and games. In _CHI ’14 Extended Abstracts on Human Factors in Computing Systems_ _(CHI EA ’14)_. Association for Computing Machinery, New York, NY, USA, 411–414.  https://doi.org/10.1145/2559206.2574799
  * Rogers et al. (2014) Katja Rogers, Amrei Röhlig, Matthias Weing, Jan Gugenheimer, Bastian Könings, Melina Klepsch, Florian Schaub, Enrico Rukzio, Tina Seufert, and Michael Weber. 2014\.  P.I.A.N.O.: Faster Piano Learning with Interactive Projection. In _Proceedings of the Ninth ACM International Conference on Interactive Tabletops and Surfaces_ _(ITS ’14)_. Association for Computing Machinery, New York, NY, USA, 149–158.  https://doi.org/10.1145/2669485.2669514
  * Seim et al. (2015) Caitlyn Seim, Tanya Estes, and Thad Starner. 2015\.  Towards Passive Haptic Learning of piano songs. In _2015 IEEE World Haptics Conference (WHC)_. 445–450.  https://doi.org/10.1109/WHC.2015.7177752
  * Zukin and Snyder (1984) Cliff Zukin and Robin Snyder. 1984.  Passive Learning: When the Media Environment Is the Message.  _Public Opinion Quarterly_ 48, 3 (Jan. 1984), 629–638.  https://doi.org/10.1086/268864