# Alfvén waves at low Magnetic Reynolds number

Samy Lalloz1  Laurent Davoust2  François Debray 3  Alban Pothérat 1
<EMAIL_ADDRESS>1 Coventry University, Centre for Fluid and
Complex Systems, Priory Street, Coventry CV1 5FB, UK 2 Grenoble-
INP/CNRS/Université Grenoble-Alpes, SIMaP, Electromagnetic Processing of
Materials (EPM) Laboratory, F-38000 Grenoble, France 3 CNRS LNCMI, 38042
Grenoble, France

###### Abstract

This paper addresses the questions whether Alfvén waves (AW) can be produced
in low-magnetic Reynolds Number ($R\\!m$) liquid metal experiments. AW are
incompressible waves propagating along magnetic fields pervading electrically
conducting fluids typically found in interstellar media, stellar and planetary
interiors. Until now, relatively faint linear waves have been produced in
liquid metals because of the large magnetic dissipation they undergo at
laboratory scales (_i.e._ $R\\!m\ll 1$) and in dense liquid metals. Thus,
these questions first hinge on whether a propagative regime can be identified
and whether the waves can reach sufficient amplitudes to incur nonlinearities.
What is at stake, here, is the possibility of reproducing in relatively simple
laboratory-scale magnetohydrodynamics (MHD) experiments some of the physical
mechanisms of such far remote natural processes as anomalous heating in the
solar corona, oscillations of the Earth inner core or turbulence in the solar
wind.
We study AW forced by an AC electric current in a liquid metal channel
pervaded by a transverse magnetic field. We first derive a wave-bearing
extension of the usual low$-R\\!m$ MHD approximation. With it, we build a
semi-analytical model for electrically-driven oscillations in a channel and
identify two distinct regimes: the first one is a purely diffusive regime
characterised by a sufficiently small oscillation parameter $N_{\omega}$
representing the ratio of the oscillation period to the timescale of diffusive
two-dimensionalisation by the Lorentz force (Sommeria & Moreau, 1982). The
second regime is propagative and governed by a parameter we call the _Jameson
number_ $J\\!a$, in an homage to the pioneering work of Jameson (1964). This
number is built as the ratio of the forcing period to the timescale of AW
propagation across the channel. We show that even in this regime, AW undergo
significant dissipation and become dispersive as their propagation speed
increases in regions of higher transversal velocity gradients. This
configuration is reproduced in the FLOWCUBE experiment (Pothérat & Klein,
2014), with measurements of electric potential on the top and bottom walls.
Both regimes are recovered, with wave attenuation and propagation speeds in
excellent agreement with the model up to $J\\!a\lesssim 0.85$. Near the
$J\\!a=1$ resonance, by contrast, AW are much less attenuated and tend to
propagate slower than the linear model predicts. The patterns of electric
potential and dependence on the forcing amplitude confirm that this
discrepancy is the signature of a nonlinear regime of AW. Hence, we show that
electrically driving AW in a laboratory-scale experiment can produce all three
fundamental building blocks of astro and geophysical AW: propagative,
diffusive and nonlinear processes.

##  1 Introduction

The main purpose of this work is to determine whether Alfvén Waves (AW)
produced in liquid metal experiments can bear relevance to those in stellar or
geophysical systems. Specifically, the question is whether AW can be excited
and can reach sufficiently high intensity to generate complex, possibly
nonlinear, dynamics despite the high dissipation they undergo.

In the absence of dissipation, AW are incompressible, non-dispersive waves
propagating in electrically conducting fluids along a background magnetic
field $\boldsymbol{B}_{0}$ at a phase velocity $V_{A}=B_{0}/\sqrt{\rho\mu}$,
where $\rho$ is the fluid’s density and $\mu$ the magnetic permeability of the
vacuum (Roberts, 1967; Moreau, 1990; Davidson, 2001; Finlay, 2007). In a
medium of finite conductivity $\sigma$, they dissipate over a time scale
$\tau_{d}=L^{2}/\eta$, where $\eta=(\sigma\mu)^{-1}$ is the magnetic
diffusivity. Because of the very large scale $L$ of stellar and geophysical
systems, $\tau_{d}$ is many orders of magnitude (typically 10) greater than
the propagation timescale $\tau_{A}=L/V_{A}$. Hence, AW propagate practically
unimpeded in the very low density stellar and interstellar media, the solar
wind, planetary magnetospheres but also in much denser planetary interiors,
where they play an important role in energy transport and dissipation
(Tsurutani & Ho, 1999; Nakariakov et al., 1999; Jault & Finlay, 2015). The low
dissipation favours large amplitudes and nonlinearities that underpin energy
transfer between them. The solar wind offers a good example of Alfvénic
turbulence, where such transfers operate across a very wide range of
lengthscales (Salem et al., 2012; Howes, 2015). In the Sun, AW are one of the
candidates-reasons for the yet inexplicably high temperatures in the Solar
Corona (Grant et al., 2018; Li et al., 2021). They are also commonly
encountered in magnetised planetary cores, under the form of _torsional AW_
propagating between concentric cylinders aligned with planets rotation
(Bragingsky, 1970; Gillet et al., 2010), or as _magneto-Coriolis_ waves
(Finlay, 2008; Gillet et al., 2022; Varma & Sreenivasan, 2022; Majumder &
Sreenivasan, 2023). Unfortunately, these waves are extremely difficult to
study in their natural environment. Accessibility is an obvious reason, but by
no means the only one: observational data produced by satellites deliver
limited local data. Furthermore, Alfvén waves compete with several other
magneto-mechanical oscillations, for example incompressible oscillations of
solar coronal loops, or magneto-acoustic waves arising out of the medium’s
compressibility (see Nakariakov & Kolotkov (2020) for a review).
Distinguishing AW amongst these, especially with limited observational data,
poses a significant challenge. Numerical simulations are challenging too
because of the extreme Reynolds and magnetic Reynolds numbers at which these
systems operate.

For these reasons, producing carefully controlled Alfvén waves in a laboratory
formed an appealing proposition to understanding their role in natural systems
ever since they were first theorised by Alfvén (1942) in his seminal, yet
remarkably simple paper. The immediate obstacle to such an endeavour arises
from the $10^{8}$ to $10^{10}$ factor between lengthscales of experiments and
natural system, which inflicts just as drastic a reduction in the ratio
$\tau_{d}/\tau_{A}$. While higher magnetic fields linearly reduce $\tau_{A}$,
even the highest magnetic fields available to date (10 T or more) ever regain
3 or 4 orders of magnitude at best. Such is the challenge of keeping this
ratio sufficiently high to observe AW, that it was named after who made the
first attempted (Lundquist, 1949): the Lundquist number $L\\!u=V_{A}L/\eta$.
Lundquist tried to force AW with a conducting disc oscillating across a
background magnetic field in a mercury vessel and measured the intensity of
these oscillations further down the field lines. The amplitude of the
oscillations was weak with a frequency dependence relatively far from the non-
viscous model he tried to match it to. Lehnert (1954)’s subsequent attempt was
based on a similar mechanical principle. Despite improved instrumentation and
control he arrived at a similar result. While Lundquist and Lehnert’s
experiments set milestones as the first laboratory experiments seeking to
produce AW, the most convincing evidence of AW in liquid metals is due to
Jameson (1964). The basis for his success was two new ideas: he showed
theoretically that forcing waves electromagnetically instead of mechanically
led to higher amplitudes and took advantage the spatial inhomogeneity of the
waves too place his probes at the locus where a local resonance maximised
their amplitude. Jameson’s AW precisely matched the linear theory but, their
amplitude was still too low for nonlinear effects to even be noticeable.
Alboussière et al. (2011) tried to venture in this regime by taking advantage
of electromagnets delivering up to $13\,$T in a $160\,$mm diameter bore: these
authors tried to produce a self-interacting wave with an electromagnetic pulse
bouncing against the ends of a vessel filled with Galinstan, and eutectic
alloy liquid at room temperature. AW were produced but their decay was too
fast for the reflected and incident waves to interact. Further recent
experiments focused on different aspect of linear AW: Iwai et al. (2003)
extracted the signature of AW from the pressure fluctuations. Magneto-Coriolis
(Nornberg et al., 2010; Schmitt et al., 2013) and torsional AW were produced
more recently in liquid metal spherical Couette experiments (Tigrine et al.,
2019). The most extreme AW experiment to date is without doubt owed to Stefani
et al. (2021), who produced AW within a small capsule of rubidium subjected to
a $63\,$T pulsed magnetic field of $150\,$ms. In this regime, compressibility
enables these authors to excite a parametric resonance between magneto-
acoustic and Alfvén waves. Although the flow was not purely incompressible,
this mechanism bears relevance to those generating heat in the solar corona
and this experiment is the only liquid metal experiment to have reached a
nonlinear AW regime.

Independently, liquid metal MHD at laboratory scale has developed since the
1960’s with the common assumption that the induced magnetic field is small
enough for the magnetic induction diffusion to be several orders of magnitude
greater than its advection by the flow, _i.e._ that the magnetic Reynolds
number $R\\!m=UL/\eta$ was vanishingly small (except where a dynamo effect was
specifically sought). It became common, but potentially misleading to assume
that the timescale of magnetic field fluctuations is the flow advection
timescale $L/U$, _i.e._
$\partial_{t}\boldsymbol{B}\sim\mathbf{u}\cdot\nabla\boldsymbol{B}$ (Ennayar
et al., 2021). This assumption effectively merges the low$-R\\!m$
approximation with the Quasi-Static MHD (QSMHD) approximation, where magnetic
field fluctuations are smeared out by diffusion. This led to the general
belief that Alfvén waves could not exist in the limit $R\\!m\rightarrow 0$,
even though early works in MHD clearly distinguished the two assumptions
(Jameson, 1964; Moffatt, 1967; Roberts, 1967). This context and the limited
success of liquid metal experiments in producing AW led to the idea that
low-$R\\!m$ liquid metal experiments could not produce AW of relevance to
their natural settings where $L\\!u\gg 1$ and $R\\!m\gg 1$. Plasmas, by
contrast, soon appeared as an alternative to liquid metals due to their
naturally high Lundquist numbers, especially when plasma technology for
nuclear fusion emerged in the 1950’s. The first indisputable evidence of AW
was indeed obtained from a measure of wave velocity in a plasma (Allen et al.,
1959), followed by further experiments (Wilcox et al., 1961; Jephcott &
Stocker, 1962; Woods, 1962), and the generation of linear interferences
between AW (Gekelman et al., 1997). The first nonlinearities in plasmas were
however obtained much more recently: the observation of parametric
instabilities and nonlinear transfers between AW (Carter et al., 2006; Dorfman
& Carter, 2016) achieved an important step towards alfvenic turbulence in the
lab (Howes et al., 2012, 2013). While the compressibility of plasmas makes it
more difficult to isolate AW from other waves, for example magneto-acoustic
waves (Dorfman & Carter, 2013), it also makes them relevant to the solar
corona and the solar wind. Plasmas also pose serious challenges in term of
metrology and require much heavier technological environments than liquid
metals. Liquid metals, by contrast are incompressible, very dense, and bear
close similarities with planetary cores but current experiments with them are
scarce and mostly target the linear regime.

As a result, the current state of understanding of AW is relatively limited,
considering they have been discovered over 80 years ago. To this date there
are no experiment reproducing Alfvenic turbulence or able to reproduce any of
the complex nonlinear effects taking place in solar or geophysical systems.
Even deriving reliable laws for their reflections against walls poses
considerable challenges (Schaeffer et al., 2012). Yet, evidence of their role
in stellar and planetary interiors accumulates in simulations, and
observations (Tomczyk et al., 2007; Gillet et al., 2010; Grant et al., 2018;
Nakariakov & Kolotkov, 2020; Gillet et al., 2022; Li et al., 2021; Schwaiger
et al., 2024). A new form of Helioseismology and planetary seismology using AW
is even emerging as powerful means of probing the interior of the sun and
Jupiter (Hanasoge et al., 2012; Hori et al., 2023). It is also becoming
increasingly clear that wherever AW play a role, they do so through
dissipation and nonlinearity, rather than in a regime of ideal linear AW. In
the solar corona, thin dissipation layers may be the missing heat source that
could explain high temperatures there (Grant et al., 2018), and high
dissipation occurs at small scales are energised by nonlinear transfer from
larger scales (Davila, 1987). In spherical shells representing planetary
interiors, dissipation layers are required to obtain a correct solution
describing the propagation of quasi-2D torsional AW (Luo & Jackson, 2022).
These examples illustrate that the regions where AW play the most crucial role
are much smaller than the planetary or solar scales, so at these scales, the
Local Lundquist number may fall in a range accesible to liquid metal
experiments $1\lesssim L\\!u\lesssim 10^{2}$ (Cattell, 1996; Singh &
Subramanian, 2007). The dissipative behaviour of liquid metal experiments may
therefore not be as irrelevant to astro- and geophysical problems as the
staggering values of $L\\!u$ in these problems suggest. At the same time,
liquid metal technology has made strides since Jameson’s success: experiments
(Klein & Pothérat, 2010; Baker et al., 2018) conducted in very high magnetic
fields (up to 10 T and rising) now offer extensive flow mapping based on
UtraSound Velocimetry, or USV, (Brito et al., 2001; Franke et al., 2010), low-
noise high-precision Electric Potential Velocimetry, or EPV (Kljukin & Thess,
1998; Frank et al., 2001; Baker et al., 2017).
The role of dissipation and nonlinearities, the mounting need to understand
complex AW, and the availability of these technologies prompts us to
reconsider whether AW can be obtained at low$-R\\!m$ and sufficiently far from
the QSMHD regime for AW to incur nonlinear effects. To do this, we take
advantage of electric forcing and transversal homogeneity as Jameson (1964)
did, and implement these ideas in high magnetic fields by adapting the
FLOWCUBE device (Klein & Pothérat, 2010; Pothérat & Klein, 2014; Baker et al.,
2017, 2018; Pothérat & Klein, 2017), which we previously developed to study
MHD turbulence. We seek to answer the following key questions:

  1. 1.

In which conditions do AW propagate at low $R\\!m$, despite diffusion?

  2. 2.

How do dissipative AW differ from the ideal non-dissipative, non-dispersive
AW, especially where they are not homogeneous in planes normal to the field?

  3. 3.

Can a nonlinear regime of AW be reached in liquid metals at low $R\\!m$?

We first revisit the low$-Rm$ approximation, to specifically allow wave
propagation, _i.e._ , outside the QSMHD regime (section 2). Based on this
approximation, a semi-analytical model for the propagation of linear Alfvén
waves in a plane channel normal to a background magnetic field is derived.
Using this model, we analyse the flow in a channel forced by an AC current
injected at a localised electrode imbedded in one of the walls, to identify
the diffusive and the propagative regimes (section 3). Then, a similar flow is
experimentally generated with FLOWCUBE whose principle and electric potential
measurement system are summarised in section 4. By comparing theory and
experiments, we finally identify the diffusive and propagative regimes in the
experiment, and seek nonlinearities where discrepancies between model and
experiment arise (section 5).

##  2 Linear theory of confined Alfvén waves

###  2.1 General configuration and governing equations

Figure 1: Sketch of the 2D axisymmetric geometry considered in the theoretical
model, with a representation of the bottom boundary conditions. The right side
shows the radial distribution of the axial current injected by the electrode
and the left side shows the distribution of the azimuthal magnetic
perturbation induced by the axial current. The black disk represents an
electrode. In an experiment the total current modelled by the theoretical
current distribution would be injected there. The side wall is solid,
impermeable and electrically insulating.

We consider a channel of height $h$ (figure 1) filled with an electrically
conducting incompressible fluid of density $\rho$, kinematic viscosity $\nu$
and electric conductivity $\sigma$ and subjected to an axial, homogeneous and
static magnetic field $\boldsymbol{B_{0}}=B_{0}\,\boldsymbol{e_{z}}$. The
domain is bounded by two horizontal solid, impermeable and electrically
insulating walls t $\tilde{z}=0$ and $\tilde{z}=h$, thereafter referred to as
_Hartmann walls_ as they are normal to the magnetic field. The flow is forced
at the bottom wall by injecting an axial AC current density
$\tilde{j}_{z}\left(\tilde{r},\tilde{z}=0,\tilde{t}\right)=\tilde{j}_{w}\left(\tilde{r}\right)\,\cos\left(\omega\,\tilde{t}\right)$
of angular frequency $\omega$ and magnitude $\tilde{j}_{w}$. The amplitude of
the total current injected at the bottom wall is $I_{0}$. Normalising
distances by $h$, time by $2\pi/\omega$, velocity by $u_{0}$, magnetic fields
by $B_{0}$, current density by $j_{0}=I_{0}/h^{2}$, the governing equations
for the velocity $\boldsymbol{u}$, pressure and magnetic disturbance to the
external field $\boldsymbol{b}=\boldsymbol{B}-B_{0}\boldsymbol{e}_{z}$ are the
Navier-Stokes equations, the induction equation, as well as the conservation
of mass and magnetic flux:

$\displaystyle
R\\!_{\nu}\,\partial_{t}\boldsymbol{u}+R\\!e\\{\boldsymbol{u}\cdot\nabla\boldsymbol{u}\\}$
$\displaystyle=\Delta\boldsymbol{u}+H\\!a^{2}\,R\\!m^{-1}\\{\partial_{z}\boldsymbol{b}+\boldsymbol{b}\cdot\nabla\boldsymbol{b}\\},$
(1) $\displaystyle R\\!_{\eta}\partial_{t}\boldsymbol{b}$
$\displaystyle=\Delta\boldsymbol{b}-R\\!m\\{\boldsymbol{u}\cdot\nabla\boldsymbol{b}-\boldsymbol{b}\cdot\nabla\boldsymbol{u}-\partial_{z}\boldsymbol{u}\\},$
(2) $\displaystyle\nabla\cdot\boldsymbol{u}$ $\displaystyle=0,$ (3)
$\displaystyle\nabla\cdot\boldsymbol{b}$ $\displaystyle=0.$ (4)

The five non-dimensional numbers governing this system are respectively the
Reynolds number, the magnetic Reynolds number, the Hartmann number and the
screen parameters for viscous and magnetic diffusion,

$R\\!e=\frac{u_{0}h}{\nu},\ \ R\\!m=\frac{u_{0}h}{\eta},\ \
H\\!a=B_{0}\,h\sqrt{\frac{\sigma}{\rho\nu}},\ \ R\\!_{\nu}=\frac{\omega
h^{2}}{2\pi\nu}\ \ R\\!_{\eta}=\frac{\omega h^{2}}{2\pi\eta},$ (5)

where $\eta=\left(\sigma\mu_{0}\right)^{-1}$ is the magnetic diffusivity.
Here, $R\\!m$ measures the ratio of magnetic field advection by the flow to
magnetic field diffusion. The square of the Hartmann number $H\\!a$ measures
the ratio of Lorentz to viscous forces. Finally, $R\\!_{\nu}$ (resp.
$R\\!_{\eta}$) measures the square ratio af the viscous diffusive (resp.
resistive) penetration depth of boundary oscillations into the domain to the
domain’s size (see for example Batchelor (1967)). The kinematic boundary
conditions at the top and bottom Hartmann walls are no-slip impermeable:

$\boldsymbol{u}(0)=\boldsymbol{u}(1)=0.$ (6)

At the bottom wall ($z=0$) the normal electric current is imposed and the top
wall ($z=1$) is electrically insulating. These conditions are expressed in
terms of the magnetic disturbance $\boldsymbol{b}$ by means of Ampere’s law,

$\displaystyle\boldsymbol{\nabla}\times{\boldsymbol{b}}(z=0)\cdot\boldsymbol{e}_{z}$
$\displaystyle=$ $\displaystyle j_{w},$ (7)
$\displaystyle\boldsymbol{\nabla}\times{\boldsymbol{b}}(z=1)\cdot\boldsymbol{e}_{z}$
$\displaystyle=$ $\displaystyle 0.$ (8)

Integration in the plane of each Hartmann wall respectively leads to
inhomogeneous and homogeneous conditions for $\boldsymbol{b}$ at the bottom
and top walls:

$\boldsymbol{b}_{\perp}(z=0)=\boldsymbol{b}_{w}(\boldsymbol{r}),\qquad\boldsymbol{b}(z=1)=0,$
(9)

where the subscript ⟂ stands for projection in the horizontal plane $(x,y)$
and $\boldsymbol{b}_{\perp}$ is uniquely defined by the choice of
$j_{w}(x,y)$.

###  2.2 A propagative low-$R\\!m$ approximation

We start by simplifying the governing equations in the limit $R\\!m\rightarrow
0$ in way suitable to describe the propagation of waves in liquid metals. In
the limit $R\\!m\rightarrow 0$, the induction equation readily implies that at
the leading order,

$R\\!_{\eta}\partial_{t}\boldsymbol{b}=\Delta\boldsymbol{b}.$ (10)

Indeed, for the fluid motion to actually induce a magnetic field, the
transport term and the magnetic diffusion must balance which implies
$\Delta\boldsymbol{b}=\mathcal{O}(R\\!m)$ and therefore
$\boldsymbol{b}=\mathcal{O}(R\\!m)$ (Roberts, 1967). Whilst only requiring the
assumption $R\\!m\ll 1$, this key result is often associated to the quasi-
static MHD approximation, which carries the additional assumption that
$R\\!_{\eta}\ll 1$ (in our notations). We stress, however, that this result
holds regardless of any assumption made on $R\\!_{\eta}$. In the limit
$R\\!m\rightarrow 0$, the governing equations at $\mathcal{O}(R\\!m)$ and
linearised in the limit of negligible inertia $R\\!e\rightarrow 0$ take the
form:

$\displaystyle\left(R\\!_{\nu}\,\partial_{t}\boldsymbol{-}\Delta\right)\boldsymbol{u}$
$\displaystyle=H\\!a^{2}\,\partial_{z}\boldsymbol{b},$ (11)
$\displaystyle\left(R\\!_{\eta}\,\partial_{t}\boldsymbol{-}\Delta\right)\boldsymbol{b}$
$\displaystyle=\partial_{z}\boldsymbol{u}.$ (12)

Importantly, $R\\!m$ disappears in the linearised low-$R\\!m$ approximation
but three governing non-dimensional numbers are left: the Hartmann number
$H\\!a$ and the two screen parameters $R\\!_{\eta}$ and $R\\!_{\nu}$.
Eliminating either $\boldsymbol{b}$ or $\boldsymbol{u}$ reveals that both
variables obey the same formal equation

$\left(\left(R\\!_{\nu}\,\partial_{t}\boldsymbol{-}\Delta\right)\left(R\\!_{\eta}\,\partial_{t}\boldsymbol{-}\Delta\right)-H\\!a^{2}\partial^{2}_{zz}\right)\\{\boldsymbol{u},\boldsymbol{b}\\}=0.$
(13)

Up to this point, boundary conditions have only been specified at the top and
bottom walls, but the conditions at the lateral boundaries have remained
unspecified. Let us now assume that these allow for a solution of Eq (13) to
be found by separation of variables under the form:

$\\{\boldsymbol{u}_{i},\boldsymbol{b}_{i}\\}\left(r,\theta,z\right)=\\{\boldsymbol{U}^{\perp}_{i}\left(r,\theta\right)\boldsymbol{U}^{z}_{i}\left(r,\theta\right),\boldsymbol{B}^{\perp}_{i}\left(r,\theta\right)\boldsymbol{B}^{z}_{i}\left(r,\theta\right)\\}\,\exp\left({j\epsilon
t}\right),$ (14)

where $\epsilon=\pm 1$. Here, $\boldsymbol{U}^{\perp}$
,$\boldsymbol{B}^{\perp}$, $\boldsymbol{U}^{z}$ ,$\boldsymbol{B}^{z}$ are the
eigenfunctions of the Sturm-Liouville problems respectively associated to the
directions perpendicular and parallel with the magnetic field directions, with
respective eigenvalues $(\lambda_{\perp},\lambda_{z})\in\mathbb{C}^{2}$:

$\displaystyle\left(\Delta_{\perp}+\lambda_{\perp}\right)\\{\boldsymbol{U}^{\lambda_{\perp}},\boldsymbol{B}^{\lambda_{\perp}}\\}$
$\displaystyle=$ $\displaystyle 0,$ (15)
$\displaystyle\left(\partial_{zz}^{2}+\lambda_{z}\right)\\{\boldsymbol{U}^{\lambda_{z}},\boldsymbol{B}^{\lambda_{z}}\\}$
$\displaystyle=$ $\displaystyle 0,$ (16)

with $\Delta_{\perp}=\Delta-\partial^{2}_{zz}$. This decomposition leads to
the dispersion relation:

$\displaystyle\lambda_{z}^{2}+\left(H\\!a^{2}+2\lambda_{\perp}+\epsilon
j\left(R\\!_{\nu}+R\\!_{\eta}\right)\right)\lambda_{z}-R\\!_{\nu}R\\!_{\eta}+\lambda_{\perp}^{2}+\epsilon
j\left(R\\!_{\nu}+R\\!_{\eta}\right)\lambda_{\perp}=0.$ (17)

The boundary conditions at $z=0$ and $z=1$ impose that $\boldsymbol{U}^{z}$
and $\boldsymbol{B}^{z}$ are of the form $C\exp\\{(\kappa+js)z\\}$, where
$\lambda_{z}=(\kappa+js)^{2}$ incorporates real wavenumber $\kappa$ and
spatial attenuation $s$. The eigenvalues $\lambda_{\perp}$ are determined by
the geometry and boundary conditions in the horizontal plane. These are left
unspecified for now, but we shall simply assume that
$\lambda_{\perp}=\kappa_{\perp}^{2}>0$ to cover the most common cases such as
rectangular, periodic or axisymmetric domains. This choice enables us to
introduce a real vertical wavenumber $\kappa\in\mathbb{R}$. Under these
assumptions, the dispersion relation admits four solutions
$\lambda_{z,i}=\pm\left(\kappa_{i}+js_{i}\right)^{2}$, where
$i\in\left\\{1,2\right\\}$, expressed as:

$\displaystyle\kappa_{i}$
$\displaystyle=\pm\frac{1}{2}\bigg{[}-H\\!a^{2}-2\,\kappa_{\perp}^{2}+\epsilon_{i}\,a+\left\\{\left(-H\\!a^{2}-2\,\kappa_{\perp}^{2}+\epsilon_{i}\,a\right)^{2}+\Big{(}-R\\!_{\nu}-R\\!_{\eta}+\epsilon_{i}\,b\Big{)}^{2}\right\\}^{1/2}\bigg{]}^{1/2},$
(18) $\displaystyle s_{i}$
$\displaystyle=\pm\epsilon\frac{1}{2}\bigg{[}H\\!a^{2}+2\,\kappa_{\perp}^{2}-\epsilon_{i}\,a+\left\\{\left(-H\\!a^{2}-2\,\kappa_{\perp}^{2}+\epsilon_{i}\,a\right)^{2}+\Big{(}-R\\!_{\nu}-R\\!_{\eta}+\epsilon_{i}\,b\Big{)}^{2}\right\\}^{1/2}\bigg{]}^{1/2},$
(19)

with

$\displaystyle a=$
$\displaystyle\frac{1}{\sqrt{2}}\bigg{[}H\\!a^{4}+4k_{\perp}^{2}H\\!a^{2}-\left(R\\!_{\nu}-R\\!_{\eta}\right)^{2}+$
$\displaystyle+$
$\displaystyle\left\\{\left[H\\!a^{4}+4k_{\perp}^{2}H\\!a^{2}-\left(R\\!_{\nu}-R\\!_{\eta}\right)^{2}\right]^{2}+4H\\!a^{4}\left(R\\!_{\nu}+R\\!_{\eta}\right)^{2}\right\\}^{1/2}\bigg{]}^{1/2},$
(20) $\displaystyle b=$
$\displaystyle\frac{1}{\sqrt{2}}\bigg{[}-H\\!a^{4}-4k_{\perp}^{2}H\\!a^{2}+\left(R\\!_{\nu}-R\\!_{\eta}\right)^{2}+$
$\displaystyle+$
$\displaystyle\left\\{\left[H\\!a^{4}+4k_{\perp}^{2}H\\!a^{2}-\left(R\\!_{\nu}-R\\!_{\eta}\right)^{2}\right]^{2}+4H\\!a^{4}\left(R\\!_{\nu}+R\\!_{\eta}\right)^{2}\right\\}^{1/2}\bigg{]}^{1/2},$
(21)

and $\epsilon_{i}=\left(-1\right)^{i}$. The two families of solutions $i=1$
and $i=2$ have very different damping and propagation properties. Solutions
from the first family (subscript 1), decays very fast in the $z-$direction, as
the spatial decay rate $s_{1}$ is always greater than the wavenumber
$\kappa_{1}$, and so the corresponding profile mostly follows an exponential
decay away from the top and bottom boundaries, without achieving a complete
oscillation. Physically, this mode represents the Hartmann boundary layers
that develop along the bottom and top walls and we shall refer to it as
_Hartmann mode_ for this reason. As such, $s_{1}\sim H\\!a$ in the limit
$H\\!a\rightarrow\infty$, keeping $\kappa_{\perp}\ll H\\!a$. In contrast,
solutions from the second family (subscript 2) are much less attenuated in the
$z-$direction. Depending on the value of parameters $H\\!a$ and
$\kappa_{\perp}$, a range of values of $R\\!_{\eta}$ may exist such that
$s_{2}<\kappa_{2}$. In other words, an oscillatory solution develops into the
fluid layer, which can describe the propagation of a wave. Hence, we shall
refer to this mode as the _Alfvén mode_. The properties of the Alfvén mode are
best illustrated in the diffusionless limit, where the acceleration term (with
the time derivative) and the Lorentz force (last term of Eq. (13)) balance
each other, as the diffusive terms become negligible compared to them. This
regime is achieved in the limit where $R\\!_{\nu}\rightarrow\infty$ and
$R\\!_{\eta}\rightarrow\infty$, but keeping
$H\\!a^{2}/(R\\!_{\nu}R\\!_{\eta})$ finite for the Lorentz force to remain
finite. This number characterises waves driven a at specific freqency $\omega$
such as in Lunquist, Lehnert and Jameson’s experiments. It can be expressed as
the inverse of a Lundquist number based on the time scale of the oscillation
instead of the magnetic diffusion timescale. However, since Jameson (1964) was
the first to have successfully produced strong Alfvén waves resonances,
precisely by adjusting the forcing frequency, we propose to name this number
after him and define the _Jameson_ number as
$J\\!a=\left(R\\!_{\eta}R\\!_{\nu}\right)^{1/2}/H\\!a=R\\!_{\eta}/(V_{A}h/\eta)=R\\!_{\eta}/L\\!u$,
where $V_{A}=B_{0}/(\rho\mu_{0})^{1/2}$ is the Alfvèn velocity. Hence, in the
diffusionless limit, $L\\!u\rightarrow\infty$, or equivalently, diffusion does
not attenuate the propagation of Alfvén waves (Roberts, 1967). Eq. (13) then
reduces to the purely hyperbolic equation for the propagation of these waves
in an ideal medium:

$\displaystyle\left(\partial^{2}_{tt}-J\\!a^{-2}\partial_{zz}^{2}\right)\\{\boldsymbol{u},\boldsymbol{b}\\}=0.$
(22)

In the absence of viscous diffusion, the governing equations drop to second
order and the no-slip conditions at the top and bottom boundaries need not be
satisfied. For an AC injected current $j_{w}$ with sinusoidal waveform, a
sinusoidal wave propagates through the layer, with wavenumber found as the
asymptotic value of $\kappa_{2}$ in this limit:

$\displaystyle\kappa_{2}\sim\pm
J\\!a=\pm\frac{\left(R\\!_{\eta}R\\!_{\nu}\right)}{H\\!a}^{1/2}.$ (23)

Since the influence of the horizontal geometry (through $\kappa_{\perp}$)
disappears in the diffusionless limit, diffusionless Alfvén waves are non-
dispersive. Hence, for a flow forced by imposing a boundary condition at one
of the Hartmann walls, this boundary condition simply propagates uniformly and
without dispersion along $z$ at speed $V_{A}$. This also illustrates that the
dispersive nature of the waves propagating outside the non-dissipative regime
stems from the magnetic and viscous dissipation.

###  2.3 The Quasi-Static MHD limit

While the set of governing equations (11-12) potentially supports waves, it
only does so when the magnetic field oscillations are not fully damped by
magnetic diffusion. By constrast, the waveless regime where wavers are
overdamped takes place in the quasi-static (QS) limit,where magnetic diffusion
acts much faster than the timescale of the variations of the induced magnetic
field, _i.e._ $\mu_{0}\sigma h^{2}\ll\omega$, or $R\\!_{\eta}\ll 1$. As such
$|R\\!_{\eta}\partial_{t}\boldsymbol{b}|\ll|\Delta\boldsymbol{b}|$, so the
induction equation (12) expresses that the magnetic field instantly adapts to
the flow field. In the Navier-Stokes equation, on the other hand, since no
assumption is made on $H\\!a$ or $R\\!_{\nu}$,
$R\\!_{\nu}\,R\\!m=\mathcal{O}(R\\!m)$, $H\\!a^{2}\partial_{z}b\sim
H\\!a^{2}\,R\\!m=\mathcal{O}(R\\!m)$ so all terms in Eq. (24) are
$\mathcal{O}(R\\!m)$ and must be retained. In other words, while the resistive
screen parameter $R\\!_{\eta}$ must be $0$ in the QS limit, the viscous screen
parameter $R\\!_{\nu}$ and the Hartmann number $H\\!a$ may retain finite
values. Since, $R\\!_{\nu}$ remains finite whilst $R\\!_{\eta}$ vanishes, this
approximation requires that $P\\!m=R\\!_{\eta}/R\\!_{\nu}\rightarrow 0$, and
so applies to MHD flows of liquid metals or conducting electrolytes (Andreev
et al., 2013; Aujogue et al., 2016; Moudjed et al., 2020) but not necessarily
to plasmas. In the end, the linearised low-$R\\!m$ QSMHD equations take the
form

$\displaystyle\left(R\\!_{\nu}\,\partial_{t}\boldsymbol{-}\Delta\right)\boldsymbol{u}$
$\displaystyle=$ $\displaystyle H\\!a^{2}\,\partial_{z}\boldsymbol{b},$ (24)
$\displaystyle\Delta\boldsymbol{b}$ $\displaystyle=$
$\displaystyle-\partial_{z}\boldsymbol{u},$ (25)

and the governing equation for $\boldsymbol{u}$ and $\boldsymbol{b}$
simplifies to:

$\left(\left(R\\!_{\nu}\,\partial_{t}\boldsymbol{-}\Delta\right)\Delta+H\\!a^{2}\partial_{zz}^{2}\right)\\{\boldsymbol{u},\,\boldsymbol{b}\\}=0,$
(26)

with the following dispersion relation

$\displaystyle\lambda_{z}^{2}+\left(H\\!a^{2}+2\,\lambda_{\perp}+\epsilon
jR\\!_{\nu}\right)\lambda_{z}+\epsilon
jR\\!_{\nu}\lambda_{\perp}+\lambda_{\perp}^{2}=0.$ (27)

Formally, Eq. (26) expresses the eigenvalue problem for the dissipation
operator $\Delta-H\\!a^{2}\Delta^{-1}\partial_{zz}^{2}$, with $\epsilon
jR\\!_{\nu}$ as the eigenvalue (Pothérat & Alboussière, 2003, 2006). The
corresponding eigenfunctions offer a minimal basis for the representation of
MHD flows in the QSMHD approximation (Dymkou & Pothérat, 2009; Pothérat &
Dymkou, 2010; Kornet & Pothérat, 2015; Pothérat & Kornet, 2015). In the QSMHD
approximation, the Lorentz force acts to diffuse momentum of transversal
lengthscale $l_{\perp}$ along magnetic field over distance $l_{z}$ in
timescale $\tau_{2D}=(\rho/(\sigma B^{2}))(l_{z}/l_{\perp})^{2}$. Structures
of sufficiently large scale for this process to overcome viscous dissipation
and inertia become quasi-two dimensional (Sommeria & Moreau, 1982; Klein &
Pothérat, 2010; Pothérat & Klein, 2014; Baker et al., 2018). Sice waves do not
propagate in the quasi-static limit comparing solutions from the QSMHD
equations to those the propagative low$-R\\!m$ equations derived in section
2.2 provides us with an effective way to detect wave propagation.

##  3 Electrically-driven waves

###  3.1 Wave driven by injecting current with a single electrode

We now turn to the more specific case where the flow is forced by injecting
electric current at one or several localised electrodes embedded in the bottom
wall, which is directly relevant to the experiments. We first consider a
single electrode, located at the centre of a cylindrical container of non-
dimensional radius $r_{e}$. In the actual experiment, the current is fully
localised within the radius of the electrode (typically 0.5 mm) and drops
abruptly outside it. Mathematically, this would impose a discontinuity in
$j_{w}$. To circumvent the numerical issues that would ensue, we therefore
model the current injected at a single electrode located at $r=0$ by a sharp
enough Gaussian distribution, on the basis that the impact of this change on
the flow is limited, at least in the QS limit (Baker et al., 2015):

$\boldsymbol{j}_{w}(r)=\frac{1}{\pi\,r_{0}^{2}}\,\mathrm{exp}(-(r/r_{0})^{2})\,\boldsymbol{e}_{z},$
(28)

where $r_{0}=\tilde{r}_{0}/h$ is the dimensionless radius of the electrode.
The polar frame of reference ($\boldsymbol{e}_{r}$,$\boldsymbol{e}_{\theta}$)
is centred on the electrode. Assuming that the magnetic perturbation is
axisymmetric ($\partial_{\theta}=0$) and the azimuthal magnetic perturbation
vanishes at $r=0$, the Biot-Savart law yields the inhomogeneous Dirichlet
boundary condition for the magnetic perturbation (figure 1):

$\boldsymbol{b}_{\perp}\left(r,z=0,t\right)=b_{\theta}^{w}\left(r\right)\cos(t)\,\boldsymbol{e}_{\theta}=\left(2\pi\right)^{-1}\,\frac{1-\exp\left(-r^{2}/{r_{0}}^{2}\right)}{r}\cos\left(t\right)\boldsymbol{e}_{\theta}.$
(29)

In the general case, a similar condition
$\boldsymbol{u}_{\perp}(r,z=0,t)=u^{w}_{\theta}(r,z=0,t)\boldsymbol{e}_{\theta}$
can be imposed on the velocity at the bottom wall to force the flow with a
moving wall, as Lundquist (1949) did experimentally. Here, for the purpose of
modelling the experimental setup (see section 4), both Hartmann walls are
considered as fixed, impermeable, with no slip solid walls and the top wall is
assumed electrically insulated:

$\displaystyle\boldsymbol{u}_{\perp}(r,z=1,t)=\boldsymbol{u}_{\perp}(r,z=1,t)=0,\qquad\boldsymbol{b}_{\perp}(r,z=1,t)=0.$
(30)

Finally, the problem is closed with lateral boundary conditions that define
the Sturm Liouville problems (14). Since the upper wall is electrically
insulated, we assume that the current injected at the electrode escapes
radially at infinity (Baker et al., 2015). Axisymmetry imposes that for
$r_{e}/r_{0}\gg 1$, $j_{r}\sim r^{-1}$ so the corresponding boundary condition
for the Sturm-Liouville problem (14) is $\lim_{r\rightarrow\infty}j_{r}=0$.
Since, however we will solve the problem numerically, the boundary condition
at infinity is approximated by one at finite radius $r_{e}\gg r_{0}$.
Axisymmetric solutions of the Sturm-Liouville problems (14) are Bessel
functions of the first kind $J_{1}(\kappa_{\perp}^{i}r)$, where
$\kappa_{\perp}^{i}$ is $J_{1}$’s $i^{\rm th}$ root scaled by the
dimensionless radius of the vessel $r_{e}$. With these conditions in the
horizontal plane, the general form for $b_{\theta}$ and $u_{\theta}$ takes the
form

$\displaystyle\\{u_{\theta},b_{\theta}\\}=\sum_{i=1}^{N_{\perp}}\\{u_{\theta}^{i},b_{\theta}^{i}\\}J_{1}\left(\kappa_{\perp}^{i}r\right),$
(31)

with

$\displaystyle\\{u^{i}_{\theta},b^{i}_{\theta}\\}\left(z,t\right)$
$\displaystyle=\left\\{\mathrm{e}^{s_{1}^{i}\,z}\left(\left\\{U_{1}^{i},B_{1}^{i}\right\\}\cos\left(t+\kappa_{1}^{i}\,z\right)-\left\\{U_{2}^{i},B_{2}^{i}\right\\}\sin\left(t+\kappa_{1}^{i}\,z\right)\right)\right.$
$\displaystyle+\mathrm{e}^{-s_{1}^{i}\,z}\left(\left\\{U_{3}^{i},B_{3}^{i}\right\\}\cos\left(t-\kappa_{1}^{i}\,z\right)-\left\\{U_{4}^{i},B_{4}^{i}\right\\}\sin\left(t-\kappa_{1}^{i}\,z\right)\right)$
$\displaystyle+\mathrm{e}^{s_{2}^{i}\,z}\left(\left\\{U_{5}^{i},B_{5}^{i}\right\\}\cos\left(t+\kappa_{2}^{i}\,z\right)-\left\\{U_{6}^{i},B_{6}^{i}\right\\}\sin\left(t+\kappa_{2}^{i}\,z\right)\right)$
$\displaystyle+\left.\mathrm{e}^{-s_{2}^{i}\,z}\left(\left\\{U_{7}^{i},B_{7}^{i}\right\\}\cos\left(t-\kappa_{2}^{i}\,z\right)-\left\\{U_{8}^{i},B_{8}^{i}\right\\}\sin\left(t-\kappa_{2}^{i}\,z\right)\right)\right\\}.$
(32)

The real coefficients $\left\\{U_{l}^{i},B_{l}^{i}\right\\}_{l=1..8}$ are
found by expanding the boundary condition (29) as

$\displaystyle
b_{w}\left(r,t\right)=\sum_{i=1}^{N_{\perp}}\left[\dfrac{2}{{\left(r_{e}\,\mathrm{J_{2}}\left(\kappa_{\perp}^{i}\,r_{e}\right)\right)}^{2}}\,\displaystyle{\int_{0}^{r_{e}}}\,\zeta\,b_{\theta}^{w}\left(\zeta\right)\,\mathrm{J_{1}}\left(\kappa_{\perp}^{i}\dfrac{\zeta}{r_{e}}\right)\mathrm{d\zeta}\right]\,J_{1}(\kappa_{\perp}^{i}r)\cos\left({t}\right).$
(33)

From the above equation, vectors $B_{w}^{i}$ and $U_{w}^{i}$ are defined for
each mode $\kappa_{\perp}^{i}$ so as to express the boundary conditions on the
magnetic and velocity perturbations for each term of the Bessel-Fourier
series:

$\displaystyle B_{w}^{i}$
$\displaystyle=\dfrac{2}{{\left(r_{e}\,\mathrm{J_{2}}\left(\kappa_{\perp}^{i}\,r_{e}\right)\right)}^{2}}\,\displaystyle{\int_{0}^{r_{e}}}\,\zeta\,b_{\theta}^{w}\left(\zeta\right)\,\mathrm{J_{1}}\left(\kappa_{\perp}^{i}\dfrac{\zeta}{r_{e}}\right)\mathrm{d\zeta}\,\begin{bmatrix}1&0&0&0\end{bmatrix}^{\mathrm{T}},$
(34)

with

$\displaystyle U_{w}^{i}$
$\displaystyle=\begin{bmatrix}0&0&0&0\end{bmatrix}^{\mathrm{T}}.$ (35)

Then, using the boundary conditions (34, 35), the linearised Navier-Stokes
equation (11) and Eq.(15, 16) leads to the linear system that determine
$B_{l}^{i}$ and $U_{l}^{i}$:

$\displaystyle U_{l}^{i}$
$\displaystyle={\begin{bmatrix}N_{1}^{i}&N_{2}^{i}\\\
H\\!a^{-2}{D\\!z^{i}_{1}}^{-1}\left[R\\!_{\nu}D\\!t-M_{1}^{i}\right]N_{1}^{i}&H\\!a^{-2}{D\\!z^{i}_{2}}^{-1}\left[R\\!_{\nu}D\\!t-M_{2}^{i}\right]N_{2}^{i}\\\
\end{bmatrix}}^{-1}\begin{bmatrix}U_{w}^{i}\\\ B_{w}^{i}\\\ \end{bmatrix},$
(36) $\displaystyle B_{l}^{i}$
$\displaystyle=H\\!a^{-2}{\begin{bmatrix}{D\\!z^{i}_{1}}^{-1}\left[R\\!_{\nu}D\\!t-M_{1}^{i}\right]&0\\\
0&{D\\!z^{i}_{2}}^{-1}\left[R\\!_{\nu}D\\!t-M_{2}^{i}\right]\\\
\end{bmatrix}}U_{l}^{i}.$ (37)

The block-arrays represent the magnetic boundary conditions, the kinematic
boundary conditions and the space and time derivatives, expressed as

$\displaystyle N_{1}^{i}$ $\displaystyle=\begin{bmatrix}1&0&1&0\\\ 0&1&0&1\\\
\mathrm{e}^{s_{1}^{i}}\cos\kappa_{1}^{i}&-\mathrm{e}^{s_{1}^{i}}\sin\kappa_{1}^{i}&\mathrm{e}^{-s_{1}^{i}}\cos\kappa_{1}^{i}&\mathrm{e}^{-s_{1}^{i}}\sin\kappa_{1}^{i}\\\
-\mathrm{e}^{s_{1}^{i}}\sin\kappa_{1}^{i}&-\mathrm{e}^{s_{1}^{i}}\cos\kappa_{1}^{i}&\mathrm{e}^{-s_{1}^{i}}\sin\kappa_{1}^{i}&-\mathrm{e}^{-s_{1}^{i}}\cos\kappa_{1}^{i}\\\
\end{bmatrix},$ $\displaystyle N_{2}^{i}$
$\displaystyle=\begin{bmatrix}1&0&1&0\\\ 0&1&0&1\\\
\mathrm{e}^{s_{2}^{i}}\cos\kappa_{2}^{i}&-\mathrm{e}^{s_{2}^{i}}\sin\kappa_{2}^{i}&\mathrm{e}^{-s_{2}^{i}}\cos\kappa_{2}^{i}&\mathrm{e}^{-s_{2}^{i}}\sin\kappa_{2}^{i}\\\
-\mathrm{e}^{s_{2}^{i}}\sin\kappa_{2}^{i}&-\mathrm{e}^{s_{2}^{i}}\cos\kappa_{2}^{i}&\mathrm{e}^{-s_{2}^{i}}\sin\kappa_{2}^{i}&-\mathrm{e}^{-s_{2}^{i}}\cos\kappa_{2}^{i}\\\
\end{bmatrix},$ (38)

$\displaystyle D\\!z^{i}_{1}$
$\displaystyle=\begin{bmatrix}s_{1}^{i}&-\kappa_{1}^{i}&0&0\\\
\kappa_{1}^{i}&s_{1}^{i}&0&0\\\ 0&0&-s_{1}^{i}&\kappa_{1}^{i}\\\
0&0&-\kappa_{1}^{i}&-s_{1}^{i}\\\
\end{bmatrix},\;D\\!z^{i}_{2}=\begin{bmatrix}s_{2}^{i}&-\kappa_{2}^{i}&0&0\\\
\kappa_{2}^{i}&s_{1}^{i}&0&0\\\ 0&0&-s_{2}^{i}&\kappa_{2}^{i}\\\
0&0&-\kappa_{2}^{i}&-s_{2}^{i}\\\ \end{bmatrix},$ $\displaystyle D\\!t$
$\displaystyle=\begin{bmatrix}0&-1&0&0\\\ 1&0&0&0\\\ 0&0&0&-1\\\ 0&0&1&0\\\
\end{bmatrix},$ (39)

$\displaystyle M_{1}^{i}$
$\displaystyle=\begin{bmatrix}{s_{1}^{i}}^{2}-{\kappa_{1}^{i}}^{2}-{\kappa_{\perp}^{i}}^{2}&-2\kappa_{1}^{i}s_{1}^{i}&0&0\\\
2\kappa_{1}^{i}s_{1}^{i}&{s_{1}^{i}}^{2}-{\kappa_{1}^{i}}^{2}-{\kappa_{\perp}^{i}}^{2}&0&0\\\
0&0&{s_{1}^{i}}^{2}-{\kappa_{1}^{i}}^{2}-{\kappa_{\perp}^{i}}^{2}&-2\kappa_{1}^{i}s_{1}^{i}\\\
0&0&2\kappa_{1}^{i}s_{1}^{i}&{s_{1}^{i}}^{2}-{\kappa_{1}^{i}}^{2}-{\kappa_{\perp}^{i}}^{2}\\\
\end{bmatrix},$ $\displaystyle M_{2}^{i}$
$\displaystyle=\begin{bmatrix}{s_{2}^{i}}^{2}-{\kappa_{2}^{i}}^{2}-{\kappa_{\perp}^{i}}^{2}&-2\kappa_{2}^{i}s_{2}^{i}&0&0\\\
2\kappa_{2}^{i}s_{2}^{i}&{s_{2}^{i}}^{2}-{\kappa_{2}^{i}}^{2}-{\kappa_{\perp}^{i}}^{2}&0&0\\\
0&0&{s_{2}^{i}}^{2}-{\kappa_{2}^{i}}^{2}-{\kappa_{\perp}^{i}}^{2}&-2\kappa_{2}^{i}s_{2}^{i}\\\
0&0&2\kappa_{2}^{i}s_{2}^{i}&{s_{2}^{i}}^{2}-{\kappa_{2}^{i}}^{2}-{\kappa_{\perp}^{i}}^{2}\\\
\end{bmatrix}.$

The expressions of $U_{w}^{i}$ and $B_{w}^{i}$ can be easily adapted to
reflect different types of electrical or mechanical forcing. For example,
waves can be forced purely mechanically with an oscillatory rotation of
electrically insulating top or bottom walls (or both), or electromagnetically
as in the present study. Former experiments based on the generic geometry of
an axisymmetric vessel subject to an axial magnetic field involve more complex
types of forcing at the bottom wall: Lundquist (1949) mechanically forced a
wave, using a disk with radial strips at the bottom of the vessel. While the
actuation was mechanical and modelled as such by Lundquist, it is possible
that the fluid between the strips acted as a solid electrical conductor, whose
motion induces electromagnetic forcing. Lehnert (1954) indeed used such
indirect magnetic forcing, by driving a copper disk in oscillatory rotation,
thereby inducing an axial current. Whether electromagnetic or mechanical both
experiments would be modelled by a different radial distribution of velocity
or magnetic field at the boundary than the one in the present study. In any
case, the question of which boundary conditions best represent these two
experiments remains one to clarify, as significant discrepancies exist between
models and experiments.

For the purpose of modelling the present experiment, where the flow is mapped
through electric potential measurements at the Hartmann walls, we need
theoretical expressions for these quantities. The electric field
$\boldsymbol{E}$ and the electric potential gradient $\boldsymbol{\nabla}\phi$
along the transversal directions can be deduced from the magnetic and velocity
fields solutions. The radial electric field $E_{r}$ is directly obtained from
$u_{\theta}$ and $b_{\theta}$ using the Ohm’s law
$\boldsymbol{j}=\boldsymbol{E}+\boldsymbol{u}\times{\boldsymbol{e}}_{z}$ and
Ampère’s law:

$E_{r}=-\partial_{z}b_{\theta}-u_{\theta},\\\ $ (41)

where the electric field is normalized by $E_{0}=\sigma u_{0}B_{0}$. Defining
coefficients $E^{i}_{l}$ for the electric field in the same way as $U^{i}_{l}$
and $B^{i}_{l}$, these satisfy:

$E_{l}^{i}=-U_{l}^{i}-\begin{bmatrix}D\\!z^{i}_{1}&0\\\ 0&D\\!z^{i}_{2}\\\
\end{bmatrix}B_{l}^{i}.$ (42)

The electric potential gradient is directly measured experimentally, and
related to the magnetic vector potential $\boldsymbol{A}$, where
$\boldsymbol{b}=\nabla\times\boldsymbol{A}$ through Faraday’s law, and
Coulomb’s gauge $\nabla\cdot\boldsymbol{A}=0$:

$\displaystyle-\partial_{r}\phi$ $\displaystyle=$ $\displaystyle
E_{r}+\partial_{t}A_{r}$ (43) $\displaystyle\boldsymbol{\Delta}A_{r}$
$\displaystyle=$ $\displaystyle-\partial_{z}b_{\theta}.$ (44)

Defining coefficients $\Phi^{i}_{l}$ for $\partial_{r}\phi$ as before, we
finally obtain:

$\Phi^{i}_{l}=-\begin{bmatrix}D\\!z^{i}_{1}-D\\!t{M_{1}^{i}}^{-1}D\\!z^{i}_{1}&0\\\
0&D\\!z^{i}_{2}-D\\!t{M_{2}^{i}}^{-1}D\\!z^{i}_{2}\\\
\end{bmatrix}B^{i}_{l}-U^{i}_{l}.$ (45)

From these coefficients, the radial potential gradient is expressed as

$\displaystyle\partial_{r}\phi\left(r,z,t\right)=\sum_{i=1}^{N_{\perp}}\partial_{r}\phi^{i}\left(z,t\right)J_{1}\left(\kappa_{\perp}^{i}r\right),$
(46)

with

$\displaystyle\partial_{r}\phi^{i}\left(z,t\right)=$
$\displaystyle\left\\{\mathrm{e}^{s_{1}^{i}\,z}\left(\Phi_{1}^{i}\cos\left(t+\kappa_{1}^{i}\,z\right)-\Phi_{2}^{i}\sin\left(t+\kappa_{1}^{i}\,z\right)\right)\right.$
$\displaystyle+\mathrm{e}^{-s_{1}^{i}\,z}\left(\Phi_{3}^{i}\cos\left(t-\kappa_{1}^{i}\,z\right)-\Phi_{4}^{i}\sin\left(t-\kappa_{1}^{i}\,z\right)\right)$
$\displaystyle+\mathrm{e}^{s_{2}^{i}\,z}\left(\Phi_{5}^{i}\cos\left(t+\kappa_{2}^{i}\,z\right)-\Phi_{6}^{i}\sin\left(t+\kappa_{2}^{i}\,z\right)\right)$
$\displaystyle+\left.\mathrm{e}^{-s_{2}^{i}\,z}\left(\Phi_{7}^{i}\cos\left(t-\kappa_{2}^{i}\,z\right)-\Phi_{8}^{i}\sin\left(t-\kappa_{2}^{i}\,z\right)\right)\right\\}.$
(47)

Using the same procedure, the QSMHD solution is obtained for each set of
parameters $H\\!a$, $R\\!_{\nu}$, $R\\!_{\eta}$ using Eq. (2.3) instead of Eq.
(2.2).

###  3.2 Quantities for flow diagnosis

Being now in a position to express solutions semi-analytically, we define
three quantities -represented schematically in figure 2\- to best characterise
how oscillations imposed at the bottom wall propagate, damp and rotate across
the layer, in both theory and experiments:

Figure 2: Illustration of the different diagnosis quantities for the
oscillating flow: the local phase shift
$\varphi\left(r,z\right)=\varphi_{0}\left(r,z\right)-\varphi_{0}\left(r,z=0\right)$,
the local attenuation coefficient
$\alpha\left(r,z\right)=\ln\left(|\widehat{\nabla_{u}\phi}\left(r,z\right)|/|\widehat{\nabla_{u}\phi}\left(r,z=0\right)\right)|$
and the horizontal polar angle $\Theta\left(r,\theta,z,t\right)$ of the
electric potential gradient $\boldsymbol{\nabla}\phi$.

  * •

The local phase shift with respect to the bottom wall, at
$(\boldsymbol{r},z)=(x,y,z)$, based on electric potential gradients along a
horizontal unit vector $\hat{\boldsymbol{e}}_{u}$ (thereafter
${\boldsymbol{e}}_{r}$ or ${\boldsymbol{e}}_{x}$):

$\varphi\left(\boldsymbol{r},z\right)=\varphi_{0}\left(\boldsymbol{r},z\right)-\varphi_{0}\left(\boldsymbol{r},z=0\right),$
(48)

where
$\varphi_{0}\left(\boldsymbol{r},z\right)=\arg\left\\{\int_{-\infty}^{\infty}{\boldsymbol{e}}_{r}\cdot\hat{\boldsymbol{e}}_{u}\cdot\nabla\phi\left(\boldsymbol{r},z,t\right)e^{-jt}\mathrm{d}t\right\\}$
is the phase of $\partial_{u}\phi$, obtained analytically from Eq. (46).

  * •

The local attenuation coefficient for $\partial_{u}\phi$:

$\alpha\left(\boldsymbol{r},z\right)=\ln\left(\frac{|\widehat{\nabla_{u}\phi}|\left(\boldsymbol{r},z\right)}{|\widehat{\nabla_{u}\phi}|\left(\boldsymbol{r},z=0\right)}\right),$
(49)

where
$|\widehat{\nabla_{u}\phi}|\left(r,z\right)=2\left|\int_{-\infty}^{\infty}\hat{\boldsymbol{e}}_{u}\cdot\boldsymbol{\nabla}\phi\left(r,z,t\right)e^{-jt}\mathrm{d}t\right|$
is the amplitude of the oscillation of $\partial_{r}\phi$ at unit angular
frequency (dimensionally $\omega$), also calculated analytically from Eq.
(46).

  * •

The horizontal polar angle $\Theta$ of $\boldsymbol{\nabla}\phi$ with respect
to the origin

$\Theta\left(\boldsymbol{r},\theta,z,t\right)=-\arctan\left(\frac{\sin\theta\,{\boldsymbol{e}}_{r}\cdot\boldsymbol{\nabla}\phi\left(\boldsymbol{r},z,t\right)+\cos\theta\,{\boldsymbol{e}}_{\theta}\cdot\boldsymbol{\nabla}\phi\left(\boldsymbol{r},z,t\right)}{\cos\theta\,{\boldsymbol{e}}_{r}\cdot\boldsymbol{\nabla}\phi\left(\boldsymbol{r},z,t\right)-\sin\theta\,{\boldsymbol{e}}_{\theta}\cdot\boldsymbol{\nabla}\phi\left(\boldsymbol{r},z,t\right)}\right).$
(50)

The first two quantities are defined so as to return the spatial attenuation
and phase shift of a pure sinusoidal planar wave of the form $\exp\left(\alpha
z\right)\sin\left(\kappa z-\omega t\right)$. For axisymmetric flows,$\Theta$
reduces to $\theta$ and thus is not of interest, but will come in hand in
section 4, where the flow is forced with four electrodes and thus no longer
axisymmetric.

###  3.3 Numerical solver

An in-house Matlab code was developed to numerically solve the axisymmetric
one-electrode problem, _i.e._ to evaluate time-dependent fields $u_{\theta}$,
$b_{\theta}$, $E_{r}$, $\partial_{r}\phi$ and $E_{r}$ on a discrete grid. The
code first solves for $s_{i},\kappa_{zi}$ in (18) and (19) and then solves the
$N_{\perp}$ systems of 16 equations (36-37) to determine coefficients
$U_{l}^{i}$ and $B_{l}^{i}$. Because $|s_{i}|$ and $\kappa_{1}$ scale with
$H\\!a$ and appear as exponential arguments, the variable-precision arithmetic
package in Matlab was used to ensure a sufficient accuracy in computing the
solutions, with the number of digits set to 320. The code takes as input
$R\\!_{\eta}$, $R\\!_{\nu}$ and $H\\!a$, the radial location $r$, the number
of transverse modes $N_{\kappa_{\perp}}$, the outer radius $r_{e}$ and the
requested time-interval for evolution.

The discretisation in time and in the $z$ direction are for display only and
do not affect the precision of the result. Similarly the precision of the
solution for each mode $\kappa_{\perp}^{i}$ depends only on the numerical
precision set. On the other hand, the approximation of the boundary condition
from a Fourier-Bessel expansion leads to two errors: First, the finite number
of modes in the series $N_{\perp}$ incurs a discretisation error. Second, an
error on current conservation appears by setting boundary condition $J_{r}=0$
at a finite radius $r_{e}$ instead of infinity. We assess both type errors
relative to a high resolution run at $N_{\perp}^{\rm max}=2500$ for
$r_{e}=\left\\{0.66,1.32,2.65\right\\}$ respectively as:

$\varepsilon\left(r_{e},N_{\kappa_{\perp}}\right)=\left.\frac{\parallel\partial_{r}\phi\left(N_{\kappa_{\perp}}\right)-\partial_{r}\phi\left(N_{\kappa_{\perp}}^{\rm
max}\right)\parallel}{\parallel\partial_{r}\phi\left(N_{\kappa_{\perp}}^{\rm
max}\right)\parallel}\right.,$ (51)

and

$\varepsilon_{b}\left(r_{e}\right)=\frac{\parallel
b_{w}\left(r,t=0,r_{e}\right)-b^{w}_{\theta}\parallel}{\parallel
b^{w}_{\theta}\parallel},$ (52)

where $\parallel\cdot\parallel$ represents the $\mathcal{L}^{2}$ norm.
Convergence tests performed with the propagative low-$R\\!m$ model show that
$\varepsilon_{b}\sim r_{e}^{-2}$ for $r_{e}\gtrsim 0.6$ and $\varepsilon\sim
N_{\perp}^{-54}$ for the sets $\left\\{N_{\perp}\gtrsim
10^{3},r_{e}=2.65\right\\}$ and $\left\\{N_{\perp}\gtrsim
600,r_{e}=1.34\right\\}$. Based on this result, a good compromise between
numerical cost and precision is reached by setting $r_{e}$ and $N_{\perp}$ so
as to ensure $\varepsilon_{b}\leq 1\%$ on $b_{w}$ for any $r$, and
$\varepsilon\leq 10^{-10}$ for all forthcoming calculations. The same test is
conducted on the QSMHD model, with similar results, so that forthcoming
comparison between the two models are conducted with the same number of modes
$N_{\perp}$.

###  3.4 Propagative resonance versus oscillating diffusive maxima

We first seek to identify the parameter regimes where propagation occurs and
the purely diffusive ones. To this end, we seek solutions of the one-electrode
problem with both the propagative low-$R\\!m$ model and the QSMHD model which
cannot produce waves for $H\\!a\in\left\\{2.23,2.66,3.18,3.80\right\\}\times
10^{4}$ and $R\\!_{\eta}\in[4.3\times 10{-5},5.1\times 10^{1}]$. These values
are chosen to match the experimentally accessible range. Waves are expected
where discrepancies exist between the models. Figure 3 represents the
variations of the attenuation coefficient $\alpha\left(r=0.1,z=1\right)$ with
$R\\!_{\eta}$, renormalised with two different parameters designed to
highlight the purely diffusive regime and the propagative regime respectively.

(a)

(b)

Figure 3:  $\alpha$ against $N\\!_{\omega}^{-1}$ (a) and $J\\!a$ (b) for
$H\\!a=\left\\{1.9\times 10^{4},2.66\times 10^{4},3.18\times 10^{4},3.8\times
10^{4}\right\\}$. Solid lines ( ): propagative low-$R\\!m$ model, dashed lines
( ): QSMHD model. Calculations are performed at $r=0.1$. The attenuation
coefficient $\alpha$ is calculated at $z=1$. Figure 4: Contours of
$|\widehat{\nabla_{r}\phi}|$ at $H\\!a=3.8\times 10^{4}$ and for
$J\\!a=\big{\\{}0.25$ (a,e); $0.5$ (b,f); $0.75$ (c,g); $1$ (d,h)$\big{\\}}$,
(a-d) propagative low$-R\\!m$ model, (e-h) QSMHD model. The vertical dash-
dotted line ( ) is located at $r=0.1$ and the vertical dashed line ( ) at
$r=0.4$.

The first parameter $N\\!_{\omega}=H\\!a^{2}/R\\!_{\nu}$ represents the ratio
between the Lorentz force and the acceleration term, present in both models.
As such, it does not account for propagative phenomena. We call it
_oscillation parameter_ by analogy with the interaction parameter that
measures the ratio of Lorentz force to the inertial terms in QSMHD (Moreau,
1990). The solutions of the QSMHD and the propagative Low-$R\\!m$ models
collapse for all $H\\!a$ numbers in the limit $N\\!_{\omega}\rightarrow 0$ and
exhibit a maximum value at $N\\!_{\omega}^{-1}=1\times 10^{-3}$. As such this
maximum does not stem from propagative phenomena. The solution of both models
starts to diverge around this maximum (marked (1a-1d) on figure 3), with a
relative error from $3\%$ to $10\%$ for $2.23\times 10^{4}\leq H\\!a\leq
3.18\times 10^{4}$. The two models diverge much more significantly around a
second set of maxima (marked (2a-2d) on figure 3), that only exists in the
propagative solution. As such, this second set of maxima necessarily results
from a propagative process.

Figure 5:  Contours of $\alpha(r,z)$ at $H\\!a=3.8\times 10^{4}$ and for
$J\\!a=\big{\\{}0.25$ (a,e); $0.5$ (b,f); $0.75$ (c,g); $1$ (d,h)$\big{\\}}$,
(a-d) propagative low-$R\\!m$ model, (e-f) QSMHD model (e-f). The vertical
dash-dotted line ( ), indicates the radial position $r=0.1$ where values of
$\alpha$ are taken for figure 3. The solid line ( ) is the locus of
$\alpha(r,z)=0$.

On figure 3(b), the attenuation coefficients are plotted against the Jameson
number $J\\!a=\left(R\\!_{\eta}R\\!_{\nu}\right)^{1/2}/H\\!a$. All maxima from
the second set(2a-2d) collapse at the same value of $J\\!a=0.5$, for all
Hartmann numbers. Dimensionally, the period associated with these maxima
therefore scales with the propagation time of Alfvén waves across the channel,
and so suggests a form of resonance due to Alfvén waves.

Hence two maxima corresponds to reduced attenuation of the oscillations across
the channel: one is purely diffusive (henceforth referred to as _oscillating-
diffusive_), and the other one is due to Alfvén wave resonances (henceforth
_propagative_). To understand the precise mechanism behind each of them, we
now inspect the spatial distribution of the local amplitude
$|\widehat{\nabla_{r}\phi}|$, the attenuation $\alpha(r,z)$ and the phase
$\varphi(r,z)$ of the electric potential gradients. Solutions from the QSMHD
and propagative low-$R\\!m$ models are represented on figures 4, 5 and 6 for
different values of the Jameson numbers $J\\!a=$ 0.25 (a,e), 0.5 (b,f), 0.75
(c,g) and 1 (d,h).

###  3.5 Phenomenology of the oscillating diffusive maxima

Figure 6:  Contours of $\varphi$ at $H\\!a=3.8\times 10^{4}$ and for
$J\\!a=\big{\\{}0.25$ (a,e); $0.5$ (b,f); $0.75$ (c,g); $1$ (d,h)$\big{\\}}$:
(a-d) propagative low$-R\\!m$ model, (e-h) QSMHD model. The solid line ( )
marks the $\varphi=0$ iso-value and the dotted line ( ) the $\varphi=-\pi$
iso-value. Figure 7: Contours of $|\widehat{\nabla_{r}\phi}|$ from the QSMHD
model at $H\\!a=3.8\times 10^{4}$ and for $N\\!_{\omega}^{-1}=\big{\\{}0.75$
(a), 1 (b), 1.25 (c) $\big{\\}}\times 10^{-3}$.

Understanding the mechanism underpinning the oscillating diffusive maxima
requires a closer look at the region near $r=0$, where velocities are small.
Since in the limit $r\rightarrow\infty$, $\boldsymbol{u}\rightarrow 0$, the
flow reaches a maximum velocity at a radius $r_{\rm m}$ between the close and
far regions, also visible on the profiles of electric potential gradients from
figure 4. The fluid is accelerated in the region $r\leq r_{\rm m}(z)$. In the
limit $R\\!_{\nu}\rightarrow 0$, this region is governed by the balance
between viscous friction in horizontal planes and the Lorentz forces:

$\nu\Delta_{\perp}\sim{\nu}{H\\!a^{2}}\Delta_{\perp}^{-1}\partial_{zz}^{2},$
(53)

which, for a lengthscale along $z$ scaling with z, leads to the classical
scaling for the thickness of vortex cores in high magnetic fields:
$\tilde{r}_{\rm m}^{(\nu)}\sim hH\\!a^{-1/2}(\tilde{z}/h)^{1/2}$. Crucially,
the vortex core becomes thicker as the axial distance to the injection point
increases (Sommeria, 1988).

For $R\\!_{\nu}$ sufficiently large, the acceleration term $\partial_{t}$
overcomes viscous friction to balance the Lorentz force and so $\tilde{r}_{\rm
m}^{(\omega)}(\tilde{z})\sim
h(H\\!a^{2}/R\\!_{\nu})^{-1/2}\tilde{z}/h=N\\!_{\omega}^{-1/2}\tilde{z}/h$.
The transition between these two regimes takes place when $\tilde{r}_{\rm
m}^{(\omega)}\sim\tilde{r}_{\rm m}^{(\nu)}$, _i.e._ when $R\\!_{\nu}\sim
H\\!a$ at $\tilde{z}=h$. Since the oscillating diffusive maximum is located at
$R\\!_{\nu}/H\\!a=H\\!aN\\!_{\omega}^{-1}\gtrsim 10\gg 1$ for all values of
$H\\!a$ considered, it is well within the regime dominated by the oscillating
term. In this regime too, the vortex core becomes wider at larger distance
from the electrode. However, when $z$ reaches the axial wavelength of the
oscillations $2\pi/\kappa$, the second derivative $\partial_{zz}$ saturates
(at $(\tilde{\kappa})^{2}/(4\pi^{2})$) and so does the radius of the vortex
core. This $z-$variation implies that the point $r=\tilde{r}/h=0.1$ on the
bottom wall is always outside of the region where oscillations are strong,
whereas for $\tilde{r}>\tilde{r}_{\rm m}$, the point $r=0.1$ on the top wall
is inside it. Hence, when $R\\!_{\nu}$ varies from 0, the ratio of amplitudes
of the oscillations at the top and bottom is the highest when $\tilde{r}_{\rm
m}^{(\omega)}$ coincides with the location of the probe $r=0.1$, and so
explains the existence of the oscillating diffusive maxima. The contours plots
of $|\widehat{\nabla_{r}\phi}|$ in figure 7 obtained with the QSMHD model for
three different $N\\!_{\omega}$ values, _i.e._
$N\\!_{\omega}=\left\\{0.75,\,1,\,1.25\right\\}\times 10^{-3}$, illustrate
these effect and that the vortex radius at the top plate increases with
$N\\!_{\omega}^{-1}$.

###  3.6 Phenomenology of the propagative maxima

Figure 8:  $\alpha$ from the propagative low-$R\\!m$ model against $J\\!a$ for
$r=\left\\{0.1,0.4\right\\}$ at $H\\!a=3.8\times 10^{4}$.

The propagative resonance is governed by the reflection of Alfvén waves on the
Hartmann walls, as each propagative resonance corresponds to values of
$R\\!_{\eta}$ (or of the frequency) for which the wavelength
$l_{A}=V_{A}/2\pi\omega$ is an integer fraction of the channel height $h$ or,
equivalently, $J\\!a=\\{1/4,1/2,3/4,1...\\}$.
Far from the electrode, the radial velocity and potential gradients are small
compared to the axial ones. This is made visible through the horizontal
orientation of $\varphi$ isolines as $r$ increases (figure 6). In areas of low
radial gradient, such as near $r=0.4$, propagative resonances appear at
$J\\!a=\left\\{1/4,3/4\right\\}$ _i.e._ respectively
$l_{A}=\left\\{1h/4,3h/4\right\\}$. This is illustrated on figure 8, which
shows the variations of $\alpha$ with $J\\!a$ for two radial locations
$r=\left\\{0.1,0.4\right\\}$ at $H\\!a=3.8\times 10^{4}$. The maximum in
$\alpha$ results from the co-existence of a minimum of
$|\widehat{\nabla_{r}\phi}|$ at the bottom wall and a maximum at the top wall.
This is also visible from the location of the node and anti-node of
$|\widehat{\nabla_{r}\phi}|$ along the dashed line ( ) $r=0.4$ in figures 4b
and 4c. This case corresponds to theory and observations of Jameson (1964),
whose resonances also appears at $J\\!a=1/4$ and $3/4$, in our notations.

Figure 9: $\varphi$ from the propagative low-$R\\!m$ model against $J\\!a$ at
$r=0.1$ for $H\\!a=\left\\{1.9\times 10^{4},2.66\times 10^{4},3.18\times
10^{4},3.8\times 10^{4}\right\\}$. The dash-dotted line ( ) $\varphi$ is for
an ideal unbounded wave.

Close to the electrode, by contrast, radial gradients increase significantly.
Comparing the values of $\alpha$ at $r=0.1$ with those at $r=0.4$ in figure 5,
shows that the attenuation coefficient decreases in absolute value in this
region. Hence waves are more strongly damped in areas of higher transversal
gradients, because of the extra Joule dissipation that these incur. In this
region, _e.g._ at $r=0.1$, the propagative resonance appears at $J\\!a=0.5$
(marked (2a-2d) on figure 3(b)). Plots of $\varphi$ against $J\\!a$ at $r=0.1$
for different values of $H\\!a$ on figure 9 reveal that the phase shift
between the two Hartmann walls is $-\pi$ at this resonance. Thus, propagative
resonances in strong radial gradients take place in phase opposition, which is
confirmed by the two anti nodes of $|\widehat{\nabla_{r}\phi}|$ located at the
Hartmann walls for this $J\\!a$ value (see figure 4b).

Lastly, an ideal AW propagating between the Hartmann walls would incur a phase
shift with a linear dependence on $J\\!a$. From figure 9, by contrast,
$\varphi$ exhibits a nonlinear dependence on $J\\!a$. The discrepancy to
linearity is particularly pronounced when $J\\!a\lesssim 0.8$. This regime
overlaps the regime of wave propagation for $J\\!a\gtrsim 0.1$, so in this
regime, dissipation causes AW to become dispersive. This effect however
progressively vanishes at higher values of $J\\!a$. Hence, in this limit, a
nearly dispertionless propagative regime is recovered, despite the
dissipation.

##  4 Experimental methods

###  4.1 The FLOWCUBE experimental device

The experimental device is an upgraded version of the FLOWCUBE facility (Klein
& Pothérat, 2010; Pothérat & Klein, 2014; Baker et al., 2017). It consists of
a modular cuboid vessel of inner height $h=10$ cm and width $L=15$ cm.The side
walls are electrically insulated and made of polycarbonate. The top and bottom
Hartmann walls in contact with the fluid are printed circuit boards (PCBs)
made of an FR4 epoxy layer and a ROGER 4003 ceramic layer acting as an
insulator, all mounted on polyamide-coated brass frame. Both are fitted with
injection electrodes to drive the flow, and probes to measure electric
potentials. The working liquid metal hermetically enclosed inside the vessel
is and eutectic alloy of gallium, indium and tin that is liquid at room
temperatuer, of electric conductivity $\sigma=3.4\times 10^{6}\,\mathrm{S/m}$,
density $\rho=6400\,\mathrm{kg\,m^{-3}}$, kinematic viscosity $\nu=3.7\times
10^{-7}\mathrm{m^{2}\,s^{-1}}$ (_i.e._ that the magnetic Prandtl number
$P\\!m=R\\!_{\eta}/R\\!_{\nu}=\nu/\eta=1.6\times 10^{-6}$). The procedures
described in Baker et al. (2017) ensure hermetic filling of and good
electrical contact between the metal and both set of probes and electrodes.
FLOWCUBE is subjected to a static and uniform axial magnetic field
$B_{0}\,\boldsymbol{e_{z}}$ provided by the 12 MW M10 resistive electromagnet
of $376$ mm diameter bore at the LNCMI (CNRS) in Grenoble. The field
homogeneity over the vessel is about $5\%$. The range of magnetic fields used
$B_{0}=\left\\{5.87,7,8.37,10\right\\}\,$T corresponds to Hartmann numbers
$H\\!a=\left\\{1.9\times 10^{4},2.66\times 10^{4},3.18\times 10^{4},3.8\times
10^{4}\right\\}$.

###  4.2 Flow forcing mechanism

Figure 10: Sketch of the face of the bottom Hartmann wall in contact with the
GaInTn alloy The injection pattern used during the experiments is represented
coloured circles. The red circles are connected to the phase of the AC power
supply and the blue ones to the neutral. The blue and red electrodes are in
phase opposition. Tail circles represent active potential probes.

As in Sommeria (1986), Klein et al. (2009) and Baker et al. (2017), the flow
is driven by injecting a prescribed current through an array of electrodes
fitted flush at the bottom Hartmann wall located at $z=0$. These are arranged
on a 16x16 square lattice spaced $5$ mm apart and a ring lattice of 8
electrodes per side spaced $15$ mm apart (figure 10). Each of the electrodes
is made of 1 mm diameter copper wire and surface-coated with a thin layer of
gold to warrant the electrical contact with the liquid metal. An AC electric
current is injected through a subset of these electrodes. In this paper, the
corresponding forcing pattern is a near-central square of $2\times 2$
electrodes spaced by $\tilde{d}_{0}=0.02\,$m (non-dimensionally,
${d_{0}}=\tilde{d}_{0}/{h}=0.2$), with adjacent electrodes of opposite
polarities (figure 10). The injected current is generated by a HP 33120A
function generator feeding into two superimposed KEPCO 400 W BOP 20-20M
amplifiers. It is sinusoidal, with a frequency range
$\omega/\left(2\pi\right)\in\left[1,1330\right]\,$Hz corresponding to
$R\\!_{\eta}\in\left[4.4\times 10^{-2},5.8\times 10^{1}\right]$. Different RMS
current intensities $|I_{0}|$ per electrode were investigated, from $0.1\,$A
to $6.5\,$A per electrode. To overcome current variations incurred by
fluctuations in the contact resistance between metal and electrodes (of the
order of $1\times 10^{-2}\Omega$), a constant ohmic resistance of $2\Omega\pm
0.25\%$ is added in series to each electrode (Pothérat & Klein, 2014).
Injecting AC current in this way drives an oscillatory flow of controlled
frequency, amplitude, and transversal lengthscale given by the pitch
$\tilde{d}_{0}$ between electrodes.

###  4.3 Measurement of electrical potentials

The flow is diagnosed using two arrays of electric potential probes fitted
flush at the surface of the each Hartmann wall. They are arranged on a $2.5$
mm lattice made of a central square of 14$\times$14 probes, and two
perpendicular arrays of 4$\times$48 probes spanning the $x$ and $y$ directions
of the Hartmann walls (figure 10). The arrays of both walls are aligned
opposite each other along $\boldsymbol{e}_{z}$. Additionally one of four
potential probes located at the outer ring of the wall serves as reference for
potential measurements. Each potential probe is made of $0.2\,$mm diameter
copper wire touching the metal at the wall surface. From these potential
probes, the electric signals are gathered through PCBs embedded into the
Hartmann walls and routed externally to two National Instrument PXIe-4303 data
acquisition modules housed in a PXI express chassis clocked at $10\,$MHz. Each
module synchronously records the signals of 32 probes to 24-bit precision at a
sampling frequency of up to 52.2 kHz per channel, with a low RMS noise
($3.2\,\mathrm{\mu\\!V}$). Such high acquisition frequencies are required to
accurately capture phase shift between the Hartmann walls of oscillations
expected to fall into either the oscillating diffusive regime whose timescale
is $\tau_{2D}=(\rho/(\sigma B_{0}^{2}))(h/d_{0})^{2}$ or the propagative
regime whose timescale is $\tau_{A}=h/V_{A}$. As an example, at the highest
magnetic field of $B_{0}=10$T, $\tau_{2D}\approx 5\times 10^{-4}$s and
$\tau_{A}\approx 10^{-3}$s.

$B_{0}$ [T] | 5.87 | 7 | 8.37 | 10
---|---|---|---|---
$\omega_{min}/2\pi$ [Hz] | 1 | 1 | 1 | 1
$\omega_{max}/2\pi$ [Hz] | 781 | 933 | 1115 | 1330
$I_{0}$ [Arms] | 1.40 | 1.40 | 1.40 | 1.40
$H\\!a$ | $2.23\times 10^{4}$ | $2.66\times 10^{4}$ | $3.18\times 10^{4}$ | $3.80\times 10^{4}$
${R\\!_{\eta}}_{min}$ | $4.36\times 10^{-2}$ | $4.36\times 10^{e-2}$ | $4.36\times 10^{-2}$ | $4.36\times 10^{-2}$
${R\\!_{\eta}}_{max}$ | $3.41\times 10^{1}$ | $4.07\times 10^{1}$ | $4.85\times 10^{1}$ | $5.79\times 10^{1}$
${R\\!_{\nu}}_{min}$ | $2.69\times 10^{5}$ | $2.69\times 10^{5}$ | $2.69\times 10^{5}$ | $2.69\times 10^{5}$
${R\\!_{\nu}}_{max}$ | $2.10\times 10^{8}$ | $2.50\times 10^{8}$ | $2.99\times 10^{8}$ | $3.57\times 10^{8}$
${R\\!e}_{0}$ | $1.33\times 10^{4}$ | $1.33\times 10^{4}$ | $1.33\times 10^{4}$ | $1.33\times 10^{4}$
Table 1: Range of control parameters and dimensionless numbers investigated to
test the theory developed in section 3.1.

###  4.4 Experimental procedure

FLOWCUBE offers four dimensional control parameters: the injection scale
$\tilde{d_{0}}$, the RMS value of the electric current injected per electrode
$I_{0}$, the magnitude of the magnetic field $B_{0}$ and the angular frequency
of the injected current $\omega$. In this study, only the last three
parameters were varied so as to independently control non-dimensional
parameters $R\\!e_{0}$, $H\\!a$ and $R\\!_{\eta}$
($P\\!m=R\\!_{\eta}/R\\!_{\nu}=1.6\times 10^{-6}$ being set by fluid
properties).
$R\\!e_{0}=I_{0}/\big{[}2\pi\nu\left(\sigma\rho\nu\right)^{1/2}\big{]}$ is the
Reynolds number based on the forcing current intensity, as defined in Klein &
Pothérat (2010). While variations of $H\\!a$ and $R\\!_{\eta}$, all summarised
in table 1, enable us to test the linear theory in section 3.1, $I_{0}$ and so
$R\\!e_{0}$ were varied for the specific purpose of assessing the linearity of
the wave forcing, and thereby the limits of the linear theory.
Measurements are performed at a constant magnetic field $B_{0}$, _i.e._ at
constant $H\\!a$. Starting from a flow at rest, the electric potentials are
first recorded during $30$s. Then the AC power supply is switched on at a set
current angular frequency $\omega$ and at a RMS value per electrode of the
current $I_{0}$. Then, electric potentials are recorded for $30$s, to ensure a
relative convergence error on $\phi$ lower than $10^{-2}$ for all cases. The
current supply is then switched off. The last three steps are repeated for
different forcing frequencies to span values of $R\\!_{\eta}$ and the whole
procedure is repeated for different values of the magnetic field in order to
span different values of $H\\!a$.

##  5 Experimental results and 4-electrode theoretical model

###  5.1 Methodology for the comparison between theory and experiment

To enable direct comparison between experimental data and the prediction of
the low-$R\\!m$ model, the latter is extended to the case of a 4-electrode
configuration representing the experimental current injection pattern. Since
the model is linear, this is done by translating the solution for one forcing
electrode to the location of each of the four forcing electrodes and
superimposing the four solutions so obtained, weighing them by the sign of the
curren they carry. The solution is then expressed in a common Cartesian frame
whose origin is at the centre of the top left electrode of the injection
pattern on figure 10. In this frame, we use two attenuation coefficients and
phase shifts defined in a similar way as in $\S$3.2: $\alpha_{x}$ and
$\varphi_{x}$ are based on $\boldsymbol{\nabla}\phi\cdot{\boldsymbol{e}}_{x}$
while $\alpha$ and $\varphi$ are based on $\|\boldsymbol{\nabla}\phi\|$.
Gradients are evaluated by second order central finite differences using
signals from adjacent probes, so $\alpha_{x}$ and $\varphi_{x}$ require the
signals of two probes while $\alpha$ and $\varphi$ require four.

###  5.2 Oscillating diffusive regime versus propagative regime

(a)

(b)

Figure 11:  $\alpha_{x}$ _vs._ $N\\!_{\omega}^{-1}$ (a) and $J\\!a$ (b) for
$H\\!a=\left\\{1.9\times 10^{4},2.66\times 10^{4},3.18\times 10^{4},3.8\times
10^{4}\right\\}$ and $\left(x,y,z\right)=\left(0.1,0.012,1\right)$. Solid
lines and markers represent the propagative low-$R\\!m$ model and measurements
respectively. White markers ($\vartriangleright,\square,\vartriangleleft$)
highlight the cases at $J\\!a=0.1,0.5$ and 1 respectively, all at
$H\\!a=3.8\times 10^{4}$, studied in more detail on figures 16, 17 and 18.

(a)

(b)

Figure 12:  $\varphi_{x}$ against $N\\!_{\omega}^{-1}$ (a) and $J\\!a$ (b) for
$H\\!a=\left\\{1.9\times 10^{4},2.66\times 10^{4},3.18\times 10^{4},3.8\times
10^{4}\right\\}$ and at $\left(x,y,z\right)=\left(0.1,0.012,1\right)$. Solid
lines and markers represent the propagative low-$R\\!m$ model and measurements
respectively. The dash-dotted line ( ) $\varphi_{x}$ is for an ideal unbounded
wave. White markers ($\vartriangleright,\square,\vartriangleleft$) highlight
the cases at $J\\!a=0.1,0.5$ and 1 respectively, all at $H\\!a=3.8\times
10^{4}$, studied in more detail on figures 16, 17 and 18.

To track the existence of the diffusive and propagative flow regimes
identified in $\S$ 3.4, we first analyse how these quantities vary with the
screen parameter $R\\!_{\nu}$: figures 11 and 12 show
$\alpha_{x}(x=0.1,y=0.012,z=1)$ and $\varphi_{x}(x=0.1,y=0.012,z=1)$ from both
theory and experiments, plotted against $J\\!a$ and $N\\!_{\omega}^{-1}$. The
variations of both $\alpha_{x}$ and $\varphi_{x}$ are very similar to those
observed for the single-electrode model in $\S$ 3. The attenuation coefficient
exhibits a plateau in the limit $N_{\omega}\rightarrow 0$, a maximum around
$N\\!_{\omega}^{-1}=1\times 10^{-3}$, followed by a sharp decay at higher
values, with a second peak in the higher frequencies. Similarly, the phase
shift $\varphi_{x}$ displays two distinct sequences of monotonic decay at low
and high screen parameter $R\\!_{\nu}$, that are qualitatively very similar to
those observed for a single electrode. The collapse of both experimental and
theoretical data into a single curve for $(N\\!_{\omega})^{-1}\lesssim 5\times
10^{-3}$ on figures 11a and 12a confirms that the variations of $\alpha_{x}$
within this range reflect the spatial attenuation of diffusive oscillations.
Similarly, the alignment of resonant peaks at $J\\!a=0.5$ and 1 on figure 11b
is a clear signature of resonant waves characteristic of the propagative
regime. This point is one of the main results of this work as it shows that
intense MHD waves akin to Alfvén waves can be electrically driven at low-
Magnetic Reynolds number.
The agreement between theory and experiment is generally excellent, but two
distinct types of discrepancies deserve further comments: First, an offset
separates the collapsed theoretical from the collapsed experimental values of
$\alpha_{x}$ (figure 11) resulting in a $\simeq 15\%$ difference in the value
of the peak in the diffusive regime and $\simeq 15\%$ in its position. For the
propagative peaks, these discrepancies are respectively of $\simeq 10\%$ and
$\simeq 3\%$. However, the experimental data still shows excellent collapse in
the diffusive regime and the propagative peak are all well aligned to
$J\\!a=0.5$ and 1. Further calculations not reported here show that the actual
value of $\alpha_{x}$ is quite sensitive to the choice of distribution of
injected current across the electrode $j_{w}(\boldsymbol{r})$. While our
simplified model of a Gaussian distribution (28) is numerically convenient and
roughly reflects the localisation of the injected current at an electrode, it
is very different from the complex distribution of current that occurs in the
experiment as a result of the interaction between the liquid phase and the
solid conductor that forms the electrode (Herreman et al., 2019). Hence, this
offset can be attributed to our simplified model of the current distribution
and we argue that this minor difference does not detract from the excellent
agreement between the physical mechanisms taking place in FLOWCUBE and those
captured by the model.
The second discrepancy takes place in the high $R\\!_{\nu}$ limit, within the
propagative regime: for $J\\!a>0.85$, both the attenuation and phase shift
depart significantly in value and in behaviour from the model. The excellent
agreement between theory and model elsewhere suggests that the model misses a
physical mechanism in this regime. Since waves propagate more freely in this
regime, they reach higher amplitudes and may thus interact nonlinearly, a
process that the linear model cannot capture and that requires more specific
experimental scrutiny.

---
Figure 16:  Snapshot contours of $\Theta$ for $J\\!a=0.1$ (a), $J\\!a=0.5$ (b)
and $J\\!a=1$ (c). $H\\!a=3.8\times 10^{4}$ for all cases. For each value of
$J\\!a$, instantaneous contours of $\Theta$ are plotted at the top wall (i,ii)
and at the bottom wall (iii,iv), from experimental data (i,iii) and from the
model (ii,iv). The full coloured circles show the location of two of the four
injection electrodes -in phase opposition- while the empty circles show their
virtual projection on the top wall. The yellow stars highlights focus points.

###  5.3 Linear and nonlinear flow patterns

Having now identified the diffusive and propagative regimes in the
experimental data, we seek to understand the discrepancy between theory and
experimental at high values of the Jameson number. To this end, the topology
of the flow is investigated using the potential gradients’ angle with
${\boldsymbol{e}}_{x}$ at the Hartmann walls, $\Theta(z=0)$ and $\Theta(z=1)$.
This quantity has the advantage of ignoring the amplitude of the potential
gradient and so reflects solely the topology of the flow. We focus on three
different cases: one in the oscillating diffusive regime, one well captured by
the model within the propagative regime, and one within the propagative regime
but with significant discrepancy between model and experiment. We set
$H\\!a=3.8\times 10^{4}$ for all three cases corresponding to
$J\\!a=\left\\{0.1,0.5,1\right\\}$ respectively. These cases are highlighted
by three different white-filled symbols in figures 11 and 12. Instantaneous
experimental and theoretical contours of $\Theta$ at the top and bottom walls
($z=0$ and $z=1$) are plotted on figure 16 for all three cases.

For $J\\!a=0.1$ and 0.5 (figures 16a and 16b), isolines of $\Theta$ from
experiments are topologically consistent with those from the model. The main
difference is that positive iso-values of $\Theta$ swell slightly toward the
positive $x-$direction in the experimental data at $J\\!a=0.5$. While this
could be due to nonlinearities, the limited spatial resolution in the
experiments and the added uncertainty of comparing instantaneous snapshots
make it difficult to reach a definite conclusion regarding the origin of this
effect based on these snapshots alone.

For $J\\!a=1$, the experimental contours of $\Theta$ are consistent with
theory at the bottom wall $z=0$. At the top wall $z=1$, by contrast, two focus
points and at least one saddle appear respectively at
$\left(x,y\right)=\left(0.38,-0.14\right)$ and
$\left(x,y\right)=\left(0.48,-0.11\right)$ (figure 16c). Further critical
points must exist on the top wall to satisfy the topological constraints on
their numbers (Hunt et al., 1978) but these fall outside of the spatial
visualisation window available to us. Because of these, the topology found in
the experimental data differs fundamentally from that predicted by the model.
This new topological structure is not compatible with the linear prediction,
and implies an underpinning nonlinear mechanism. Furthermore, the two focus
points are located in the region of the isoline swelling observed for the case
$J\\!a=0.5$, suggesting that the topological change may ensue from small
nonlinear deviations to the linear prediction at lower values of $J\\!a$. This
also suggests that nonlinearities are favoured at higher values of $J\\!a$ and
especially near the resonance $J\\!a=1$, _i.e._ further into the propagative
regime, and where the amplitude of the waves is greatest: these conditions are
most favourable to the occurrence of nonlinearities and the resonance points
to a $J\\!a$ nonlinear self-interaction of the wave. Whether an instability
mechanism is involved is an open question: instabilities of forced electro-
mechanical oscillations were indeed observed, but at $H\\!a<10$, _i.e._
outside of the propagative regime of relevance here (Thomas et al., 2010). In
any case, this change in topology corresponds to the discrepancy observed on
§5.2 for $J\\!a>0.85$ and so confirms that these discrepancies reflect a
nonlinear regime of Alfvén waves, not accounted for in the model.

###  5.4 Influence of horizontal gradients on the diffusion of oscillations
and wave propagation

Figure 17:  Snapshot contours of $\alpha$ for $J\\!a=0.1$ (a,b), $J\\!a=0.5$
(c,d) and $J\\!a=1$ (e,f). $H\\!a=3.8\times 10^{4}$ for all cases. For each
value of $J\\!a$, contours of $\Theta$ are plotted from experimental data
(a,c,e) and from the model (b,d,f). The coloured circles show the location of
two electrodes in phase opposition out of the four injection electrodes.
Figure 18:  Snapshot contours of $\varphi_{x}$ for $J\\!a=0.1$ (a,b),
$J\\!a=0.5$ (c,d) and $J\\!a=1$ (e,f). $H\\!a=3.8\times 10^{4}$ for all cases.
For each value of $J\\!a$, the contours of $\Theta$ are plotted from
experimental data (a,c,e) and from the model (b,d,f). The coloured circles
show the location of two electrodes in phase opposition out of the four
injection electrodes.

A key result of the theoretical study in §3.4 was that both the diffusion of
oscillations and wave propagation are influenced by the transversal gradients
of electric potential. We shall now track and quantify this effect
experimentally.The measurements in FLOWCUBE deliver attenuation and phase
shifts between the Hartmann walls ($z=0$ and $z=1$) only. Hence, we plot the
experimental and theoretical contours of these quantities, $\alpha(x,y,z=1)$
and $\varphi_{x}(x,y,z=1)$, on figures 17 and 18 respectively, again focusing
on the case displaying the most prominent wave propagation at high $J\\!a$,
_i.e._ $H\\!a=3.8\times 10^{4}$ and for $J\\!a\in\\{0.1,0.5,1\\}$.

Experimental contours of $\alpha$ and $\varphi_{x}$ exhibit topologically
equivalent patterns to their theoretical counterparts except for $J\\!a=1$
(within the same limitations on the spatial resolution in the experiment and
the error incurred by comparing snapshot as in the previous section). The
gradients of these quantities visibly decrease with the distance to the
neighbouring electrode in all cases but the experimental one at $J\\!a=1$ too.
The value of $\alpha$ itself (figure 17) increases with the transversal
distance form each electrode, both in the oscillating diffusive regime (figure
17a-b) and in the linear propagative regime (figure 17c-f) in both theory and
experiment. Experimental contours of $\varphi_{x}$ are consistent with the
model for $J\\!a=0.1$ and 0.5 (figure 18). In both cases, the magnitude of
$\varphi_{x}$ increases with the distance to the neighbouring electrode. These
observations concur to show that the high transversal gradients in the
vicinity of the electrodes increase the attenuation of the oscillations but
accelerate their diffusion and their propagation along ${\boldsymbol{e}}_{z}$.

For $J\\!a=1$, the experimental contours of both $\alpha$ and $\varphi_{x}$
deviate from the model. This case, highlighted by marker ($\vartriangleleft$)
on figures 11 and 12, corresponds to the regime where nonlinear wave
interactions incur a change in the flow topology at the top wall (figure 16c),
a phenomenon that the linear theory cannot capture. In the experimental data,
the locus of the emerging nonlinear pattern coincides with a locally enhanced
phase shift compared to the model, indicating that the nonlinear pattern
travels along ${\boldsymbol{e}}_{z}$ at a different velocity from the
fundamental mode driven by the forcing.

To further quantify the dependence of the attenuation and phase shift on the
horizontal gradients of electric potential, we take advantage of the near-
monotonic dependence of these gradients on the transversal distance to the
nearest electrode in the $\left(x,y\right)$ plane, which we denote $r^{*}$.
Based on this, the lengthscale of transversal gradients can be approximated by
$r^{*}$. We focus on the linear propagative regime where experiments and
theory agree well _e.g._ $J\\!a=0.5$. Figures 19 and 20 respectively show the
variations of $\alpha$ and $|\varphi_{x}|$ with $r^{*}$ from the experiment
and the model. Since these fields are not axisymmetric about either of the
electrodes, these variations also depend on the azimuthal angle with respect
to the closest electrode, especially for points that are nearly equidistant
from two electrodes. The theoretical range of variation of $\alpha$ and
$\varphi_{x}$ spanned in this way is represented on the figures by a red zone.
The experimental data shows a greater spread attributed to the local
measurement error, of around $4\%$, and to the offset in damping rate incurred
by our simplified model of the current distribution at the electrode (see $\S$
5.2). Nevertheless, the experimental values of $\alpha$ are consistent with
the model. In both cases, $\alpha$ increases with $r^{*}$. These variations do
not appear to follow a simple scaling law. Indeed, the values of $r^{*}$ fall
in an intermediate range between the regime of low transversal gradient
investigated by Jameson (1964) and a regime of much higher transversal
gradient where waves are highly damped. In the high damping regime, horizontal
gradients are sufficiently strong to incur magnetic dissipation comparable to
the axial ones, and so considerably increase the damping of oscillations along
${\boldsymbol{e}}_{z}$. This phenomenon takes place in the deep blue area very
close to the electrode in figure 5(c). The vertical contours of $\alpha$
obtained from the single electrode theory show that this region extends over
the entire height of the vessel. In the QSMHD limit, this region becomes the
classical viscous vortex core of inertialess thickness $\sim H\\!a^{1/2}$
studied in detail by Sommeria (1988).

Figure 19:  $\alpha$ _vs._ $r^{*}$ for $J\\!a=0.5$ and $H\\!a=3.8\times
10^{4}$. $r^{*}$ is the distance in the $(x,y)$ plane between a given
measurement location and the nearest injection electrode. The solid line and
markers correspond to the propagative low$-R\\!m$ model and measurements
respectively. The error bars show the local measurements errors for $\alpha$,
of $4\%$.

Similarly, the magnitude of the phase shift $\varphi_{x}$ increases with
$r^{*}$ in both experiment and model, without following any obvious scaling
law (figure 20). It should also noticed that the scattering from experiments
is more important for $|\varphi_{x}|$ than for $\alpha$. The scattering can be
attributed to several factors: first, using a single component of the
potential gradient $\varphi_{x}$ incurs errors when the gradient is close to
normal to that direction. Second, the most scattered points fall mostly in the
region where discrepancies in flow topologies were observed between theory and
experiment. This suggest that the scattering may arise out of nonlinear
effects.

The variations of $\alpha$ and $\phi$ quantify the previous observation that
stronger transversal gradients damp the waves whilst accelerating their
propagation. Additionally, they show that the dissipative waves observed in
FLOWCUBE sit at an intermediate regime of dissipation. As such, less
dissipative waves may be obtained in a different geometry minimising
transversal gradient. Jameson (1964) had indeed identified this effect and
used it to maximise the intensity of the waves he could observe within the
physical limitations of his experiment and despite the relatively low magnetic
field available to him. The quality of the results he obtained in this manner,
compared to other experiments available in these times (Lundquist, 1949;
Bostick & Levine, 1952; Lehnert, 1954; Allen et al., 1959) is indeed a tribute
to his deep understanding of the subtleties of these waves.

Figure 20:  $\varphi_{x}$ against $r^{*}$ for $J\\!a=0.5$ and $H\\!a=3.8\times
10^{4}$. $r^{*}$ is the distance in the (x,y) plane between a given
measurement location and the nearest injection electrode. The solid line and
the light red coloured area exhibit model’s prediction while markers show
measurements.

###  5.5 Linearity and nonlinearity of the experimental waves

The discrepancies between model and experiment at high Jameson numbers
($J\\!a\gtrsim 0.85$) in §5.2 revealed significant nonlinearities in this
regime. Smaller discrepancies between theoretical and experimental contours of
the potential gradient angle may further suggest that these nonlinearities
develop at lower Jameson numbers ($J\\!a\simeq 0.5$). Hence, notwithstanding
the otherwise excellent agreement between theory and experiment, the question
arises as to which regime of controlled parameters sees nonlinearities
affecting either the oscillating diffusive regime or the propagative one. In
particular, how much of the experimental data may fall within this regime? To
answer these questions, we assess the linearity of electric potential
gradients with respect to the forcing intensity. Dimensionally, the forcing
intensity is controlled by the total current injected in the flow, and
nondimensionally, it is measured by $R\\!e_{0}$. Nonlinearities are assessed
using the average intensity of the electric potential gradients, and their
relative nonlinearity is defined as

$\epsilon_{\rm
NL}=\left|1-\frac{\left\langle\left|\widehat{\nabla_{x}\phi}(R\\!e_{0})\right|\right\rangle_{x,y}}{R\\!e_{0}}\lim_{R\\!e_{0}\rightarrow
0}{\frac{R\\!e_{0}}{\left\langle\left|\widehat{\nabla_{x}\phi}(R\\!e_{0})\right|\right\rangle_{x,y}}}\right|,$
(54)

where $|\widehat{\nabla_{x}\phi}|$ refers to the amplitude of the oscillations
of $\partial_{x}\phi$ while $\left\langle\cdot\right\rangle_{x,y}$ denotes the
operator for spatial averaging in the $\left(x,y\right)$ plane. For flows
purely within the linear regime, $\epsilon_{\rm NL}=0$. To assess this
quantity, the purely linear solution in the limit $R\\!e_{0}\rightarrow 0$ is
approximated by the experimental data obtained at the lowest value of
$R\\!e_{0}$, _i.e._ $R\\!e_{0}=3.75\times 10^{3}$. The relative nonlinearity
of the potential gradient $\epsilon_{\rm NL}$ is plotted at the bottom and top
walls for $H\\!a=\left\\{3.18\times 10^{4},3.8\times 10^{4}\right\\}$ and
$J\\!a=\left\\{0.1,0.5,1\right\\}$ (figure 21).

(a)

(b)

Figure 21:  Normalised $\left\langle|\widehat{\nabla_{x}\phi}|\right\rangle$
against $R\\!e_{0}$ at the top wall (a) and the bottom wall (b) for
$H\\!a=\left\\{3.18\times 10^{4},3.8\times 10^{4}\right\\}$ and
$J\\!a=\left\\{0.1,0.5,1\right\\}$. For each case of the set
$\left\\{H\\!a,J\\!a\right\\}$,
$\left\langle|\widehat{\nabla_{x}\phi}|\right\rangle$ is normalised by its
value at the lowest experimental $R\\!e_{0}$.

In all cases, $\epsilon_{\rm NL}$ smoothly increases with $R\\!e_{0}$. Thus,
nonlinearities are always present, albeit in vanishingly small amount in the
limit $R\\!e_{0}\rightarrow 0$. This concurs to the observation of small
discrepancies in the contours of the angle of electric potential gradients
$\Theta$ (figure 16b-c), even in regimes where theoretical values of both the
attenuation coefficient and phase shift agree well with the experimental ones.
At the bottom wall, $\epsilon_{\rm NL}$ remains below $6\%$ for all cases
(figure 21(b)). A saturation in the variations of $\epsilon_{\rm
NL}(R\\!e_{0})$ however takes places around $R\\!e_{0}\simeq 2\times 10^{4}$,
and only in the propagative regime $J\\!a\geq 0.5$ (for
$\left\\{H\\!a=3.8\times 10^{4},J\\!a=0.5\right\\}$ and
$\left\\{H\\!a=3.8\times 10^{4},J\\!a=1\right\\}$ (figure 21(b)). This is
indicative of a change in nonlinear dynamics with a possible dominance of
nonlinear wave interaction. By contrast, flows within the diffusive regime
remain essentially linear for the entire range of forcing spanned within the
experiment. Data at the top wall (figure 21(a)) reveal the same phenomenology
but with significantly greater nonlinearity, up to $\simeq 40\%$. This also
concurs with the observations of the contours of $\Theta$, where the
discrepancy between theory and experiment was found greater near the top wall.
This further supports the idea that nonlinearities arise from nonlinear wave
interaction. Indeed, these interactions are impeded near the bottom wall where
patterns of injected current are imposed. However, they grow as they propagate
away from the forcing region, and so are more likely observed near the top
wall.

Except for the present section, all experimental data shown in this work were
obtained for $R\\!e_{0}=1.33\times 10^{4}$ (highlighted with dashed lines on
figure 21(a) and 21(b)). For this forcing intensity, $\epsilon_{\rm NL}$
remains lower than $\simeq 12\%$ for all cases but $H\\!a=3.8\times
10^{4},J\\!a=1$. Hence, it can be concluded that the phenomenology discussed
throughout this work is linear in all regimes where experiment and theory
agree. Nevertheless, the data at higher $R\\!e_{0}$ discussed in this section
clearly shows that nonlinear regimes are easily reached within FLOWCUBE’s
operational range of parameters.

##  6 Conclusion

We studied an electrically driven oscillating flow confined between two walls
perpendicular to an externally imposed magnetic field, in view of identifying
the conditions in which diffusive, propagative and nonlinear processes occur
in low-$R\\!m$ magnetohydrodynamics. The underlying question at stake is
whether Alfvén waves observed in astro- and geophysical systems can be studied
in detail in laboratory-scale experiments using liquid metals. To address this
problem, we developed two linear models, one based on a propagative extension
of the usual low-$R\\!m$ MHD approximation, and one based on the Quasi-Static
MHD (QSMHD) approximation. We also adapted the FLOWCUBE experimental device
(Klein & Pothérat, 2010) to directly track these regimes in a rectangular
vessel filled with liquid metal. The propagative low-$R\\!m$ approximation
differs from the usual QSMHD approximation in that the time scale of local
oscillations is different from the flow turnover time. In the limit of
vanishing Reynolds number. It is therefore governed by three non-dimensional
parameters: the usual Hartmann number $H\\!a=Bh(\sigma/(\rho\nu)^{1/2})$ and
two screen parameters, a viscous one, $R\\!_{\nu}=\omega h^{2}/(2\pi\nu)$ and
a resistive one $R\\!_{\eta}=\omega h^{2}/(2\pi\eta)$. The linearised QSMHD
approximation is recovered in the limit $R\\!_{\eta}\rightarrow 0$, keeping
$R\\!_{\nu}$ finite.

Combining theory and experiments enabled us to identify and characterise three
different regimes of forced oscillations. The first regime falls within the
QSMHD approximation. It is captured by both linear models and experiments and
occurs at low screen parameters $R\\!_{\nu}$ or $R\\!_{\eta}$. In this regime,
the Lorentz force acts exclusively so as to diffuse momentum along magnetic
field lines. In the limit $R\\!_{\eta}\rightarrow 0$, this diffusion process
drives flows towards a quasi-two dimensional state (Sommeria & Moreau, 1982).
Hence, we called this regime _oscillating diffusive_ , and found that it is
characterised by an oscillating parameter $N\\!_{\omega}=2\pi\sigma
B_{0}^{2}/{\rho\omega}$ built out of the ratio of the oscillation timescale
$2\pi/\omega$ to the two-dimensionalisation timescale at the scale of the box,
associated to the Lorentz force, $\tau_{2D}=\rho/(\sigma B_{0}^{2})$, first
introduced by Sommeria & Moreau (1982). The second regime occurs at higher
values of $R\\!_{\eta}$. It is captured by experiments in excellent agreement
with the linear propagative low-$R\\!m$ model, but not by the QSMHD model.
This linear _propagative_ regime is dominated by the propagation of Alfvén
waves. Their resonance across the channel occurs at values of $R\\!_{\eta}$
such that the ratio between the propagation time of Alfvén waves across the
channel and the oscillation period is either 1/4, 1/2, 3/4 or 1. The second
regime is characterised by values of this ratio, which we named the _Jameson
number_ $J\\!a=\omega h/\left(2\pi V_{A}\right)$ below $\simeq 0.85$. The
existence of this regime is one of the chief results of this study as it
establishes that MHD waves akin the Alfvén waves but different from ideal AW
can propagate at low $R\\!m$. The third regime occurs at the highest values of
$R\\!_{\nu}$, corresponding to $J\\!a\gtrsim 0.85$ and therefore remains
propagative in nature. It emerges where wave amplitudes are too high for the
linear model to remain valid (most noticeably near the resonance $J\\!a=1$).
Indeed, experimental data departs from the linear model, and displays a clear
nonlinear behaviour. Hence we name this regime the _nonlinear propagative
regime_.

The particular type of AW we found in the propagative regime, further exhibit
unique propagation properties so far unobserved at low-$R\\!m$. In the linear
propagative regime, both the model and the experimental data display a
dependence of the phase velocity on $R\\!_{\eta}$, so unlike ideal AW, these
waves are dispersive. Furthermore, the configuration involves a spatially
inhomogeneous forcing in the planes perpendicular to the magnetic field with
electric potential gradients decaying away from the points of current
injection. This provided us an opportunity to analyse how such inhomogeneity
affects wave propagation. Indeed, both model and experiment clearly show that
the propagation of these dispersive Alfvén waves depends on the transversal
gradients: they are both locally accelerated and more strongly damped in
stronger gradients. Such variations of propagation velocity, never observed
before at low-$R\\!m$, is sometimes referred to as _phase mixing_. It locally
increases the viscous and resistive wave dissipation responsible for a heating
process that may play a role in the Sun’s corona heating (Heyvaerts & Priest,
1983; McLaughlin et al., 2011; Prokopyszyn et al., 2019).

Outside the linear propagative regime, nonlinearities manifest themselves in
several ways: First, contour maps of electric potential reveal a change in the
flow topology, noticeable near the top Hartmann wall, _i.e._ further away from
the forcing point. This, together with the occurrence near a resonance of
these nonlinearities, supports the hypothesis of nonlinear self interaction of
Alfvén waves. Additionally, varying the forcing intensity expressed by
Reynolds number $R\\!e_{0}$ for different values of $J\\!a$ values shows
stronger departure from the linear model at higher amplitudes, another
signature of nonlinearity. The discrepancy from linearity was further found to
increase continuously from very small $R\\!e_{0}$ and so suggests that
nonlinearities are always present, rather than ignited near a bifurcation as
the forcing amplitude increases. The presence of nonlinearities driven by wave
self-interaction is another major result and so answers one of the key
questions that motivated this work: nonlinear energy transfers can be
reproduced in liquid metal experiments. This paves the way for the study of a
second mechanism directly relevant to processes expected to take place in
astrophysical systems, especially the solar corona (Davila, 1987; Holst et
al., 2014).

The three regimes found in this work show that all three key ingredients of
astro- and geophysical AW can be reproduced in a small-scale experiment:
diffusion, propagation and nonlinearity. Additionally, we have been able to
characterise the inhomogeneous properties of dispersive AW, another key
feature of the complex dynamics of AW in these systems. These results open a
new spectrum of opportunities to model such practically inaccessible objects
as planetary interiors or the solar corona in the relative comfort of a small-
scale laboratory, using liquid metals. We are certainly a long way from
bringing the physics of the sun into a box filled with liquid metal. But we
are almost an equally long way from having exhausted the possibilities offered
by the increasingly high intensity of magnetic fields available to liquid
metal experiments, and the range of possibilities offered by electrically
driven AW: varying the forcing intensity, the forcing geometry, the shape size
of the box offer as many opportunities to seek more intense nonlinearities,
different types of inhomogeneities and of nonlinear effects. Each of these may
hold the key to some aspects of how AW behave in the Sun or in planetary
interiors.

## References

  * Alboussière et al. (2011) Alboussière, Thierry, Cardin, Philippe, Debray, François, La Rizza, Patrick, Masson, Jean-Paul, Plunian, Franck, Ribeiro, Adolfo & Schmitt, Denys 2011 Experimental evidence of Alfvén wave propagation in a Gallium alloy. Physics of Fluids 23 (9), 096601, number: 9 Publisher: American Institute of Physics.
  * Alfvén (1942) Alfvén, H. 1942 Existence of Electromagnetic-Hydrodynamic Waves. Nature 150 (3805), 405–406, number: 3805.
  * Allen et al. (1959) Allen, T. K., Baker, William R., Pyle, Robert V. & Wilcox, John M. 1959 Experimental generation of plasma Alfvèn waves. Phys. Rev. Lett. 2, 383–384.
  * Andreev et al. (2013) Andreev, Oleg, Kolesnikov, Yurii & Thess, André 2013 Visualization of the Ludford column. Journal of Fluid Mechanics 721, 438–453, publisher: Cambridge University Press.
  * Aujogue et al. (2016) Aujogue, Kélig, Pothérat, Alban, Bates, Ian, Debray, François & Sreenivasan, Binod 2016 Little Earth Experiment: An instrument to model planetary cores. Review of Scientific Instruments 87 (8), 084502.
  * Baker et al. (2015) Baker, Nathaniel T., Pothérat, Alban & Davoust, Laurent 2015 Dimensionality, secondary flows and helicity in low-rm MHD vortices. J. Fluid Mech. 779, 325–350.
  * Baker et al. (2018) Baker, Nathaniel T., Pothérat, Alban, Davoust, Laurent & Debray, François 2018 Inverse and direct energy cascades in three-dimensional magnetohydrodynamic turbulence at low magnetic Reynolds number. Phys. Rev. Lett. 120 (22), 224502, number: 22 Publisher: American Physical Society.
  * Baker et al. (2017) Baker, Nathaniel T., Pothérat, Alban, Davoust, Laurent, Debray, François & Klein, Rico 2017 Controlling the dimensionality of low-Rm MHD turbulence experimentally. Exp. Fluids 58 (7), 79.
  * Batchelor (1967) Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge Mathematical Library 683\. Cambridge: Cambridge University Press.
  * Bostick & Levine (1952) Bostick, Winston H. & Levine, Morton A. 1952 Experimental demonstration in the laboratory of the existence of magneto-hydrodynamic waves in ionized helium. Phys. Rev. 87, 671–671.
  * Bragingsky (1970) Bragingsky, S.I. 1970 Torsional magnetohydrodynamic vibrations in the Earth’s core and variations in day length. Geomagn. Aeron. 10 (1), 3–12.
  * Brito et al. (2001) Brito, D., Nataf, H. C., Cardin, P., Aubert, J. & P., Masson J. 2001 Ultrasonic doppler velocimetry in liquid gallium. Exp. Fluids 31 (6), 653–663.
  * Carter et al. (2006) Carter, T. A., Brugman, B., Pribyl, P. & Lybarger, W. 2006 Laboratory observation of a nonlinear interaction between shear alfvén waves. Phys. Rev. Lett. 96, 155001.
  * Cattell (1996) Cattell, CA 1996 Experimental evaluation of the Lundquist number for the Earth’s magnetopause and magnetotail. Journal of Geophysical Research: Space Physics 101 (A12), 27309–27316.
  * Davidson (2001) Davidson, P. A. 2001 An Introduction to Magnetohydrodynamics. Cambridge University Press.
  * Davila (1987) Davila, Joseph M. 1987 Heating of the Solar Corona by the Resonant Absorption of Alfven Waves. ApJ 317, 514.
  * Dorfman & Carter (2013) Dorfman, S. & Carter, T. A. 2013 Nonlinear excitation of acoustic modes by large-amplitude Alfvén waves in a laboratory plasma. Phys. Rev. Lett. 110, 195001.
  * Dorfman & Carter (2016) Dorfman, S. & Carter, T. A. 2016 Observation of an Alfvén wave parametric instability in a laboratory plasma. Phys. Rev. Lett. 116, 195002.
  * Dymkou & Pothérat (2009) Dymkou, V. & Pothérat, A. 2009 Spectral methods based on the least dissipative modes for wall bounded MHD flows. Theoretical and Computational Fluid Dynamics 23 (6), 535–555.
  * Ennayar et al. (2021) Ennayar, H, Karcher, C & Boeck, T. 2021 Numerical simulation of Stokes’ second problem affected by magnetic field. In Proceedings of the 10th international conference Electromagnetic Processing of Materials Riga, Latvia, June 14-16.
  * Finlay (2007) Finlay, Christopher 2007 Alfvén Waves, pp. 3–6. Dordrecht: Springer Netherlands.
  * Finlay (2008) Finlay, Christopher C. 2008 Course 8 waves in the presence of magnetic fields, rotation and convection. Les Houches Summer School Proceedings 88, 403 – 450, cited by: 27.
  * Frank et al. (2001) Frank, M., Barleon, L. & Müller, U. 2001 Visual analysis of two-dimensional magnetohydrodynamics. Phys. Fluids 13 (8), 2287–2295.
  * Franke et al. (2010) Franke, Sven, Büttner, Lars, Czarske, Jürgen, Räbiger, Dirk & Eckert, Sven 2010 Ultrasound doppler system for two-dimensional flow mapping in liquid metals. Flow Measurement and Instrumentation 21 (3), 402–409.
  * Gekelman et al. (1997) Gekelman, Walter, Vincena, Stephen, Leneman, David & Maggs, James 1997 Laboratory experiments on shear Alfvén waves and their relationship to space plasmas. Journal of Geophysical Research: Space Physics 102 (A4), 7225–7236.
  * Gillet et al. (2022) Gillet, Nicolas, Gerick, Felix, Jault, Dominique, Schwaiger, Tobias, Aubert, Julien & Istas, Mathieu 2022 Satellite magnetic data reveal interannual waves in Earth’s core. Proceedings of the National Academy of Sciences 119 (13), e2115258119.
  * Gillet et al. (2010) Gillet, N., Jault, D. & Canet, E. et al. 2010 Fast torsional waves and strong magnetic field within the Earth’s core. Nature 465, 74–77.
  * Grant et al. (2018) Grant, S.D.T., Jess, D.B., Zaqarashvili, T.V., Beck, C., Soccas-Navarro, H., Achawanden, M. J., Keys, P. H., Christian, D. J., Houston, S. J. & Hewitt, R. L. 2018 Alfvén wave dissipation in the solar chromosphere. Nature Phys 14, 480–483.
  * Hanasoge et al. (2012) Hanasoge, Shravan, Birch, Aaron, Gizon, Laurent & Tromp, Jeroen 2012 Seismic probes of solar interior magnetic structure. Physical review letters 109 (10), 101101.
  * Herreman et al. (2019) Herreman, W., Nore, C., Ziebell Ramos, P., Cappanera, L., Guermond, J.-L. & Weber, N. 2019 Numerical simulation of electrovortex flows in cylindrical fluid layers and liquid metal batteries. Physical Review Fluids 4 (11), 113702.
  * Heyvaerts & Priest (1983) Heyvaerts, J & Priest, Eric R 1983 Coronal heating by phase-mixed shear Alfvèn waves. Astronomy and Astrophysics 117 (2), 220–234.
  * Holst et al. (2014) Holst, B. van der, Sokolov, I. V., Meng, X., Jin, M., W. B. Manchester, IV, Tóth, G. & Gombosi, T. I. 2014 Alfvèn wave solar model (AWSoM): Coronal heating. The Astrophysical Journal 782 (2), 81.
  * Hori et al. (2023) Hori, Kumiko, Jones, Chris A, Antuñano, Arrate, Fletcher, Leigh N & Tobias, Steven M 2023 Jupiter’s cloud-level variability triggered by torsional oscillations in the interior. Nature Astronomy 7 (7), 825–835.
  * Howes (2015) Howes, G. G. 2015 A dynamical model of plasma turbulence in the solar wind. Phil. Trans. R. Soc. A. p. 3732014014520140145\.
  * Howes et al. (2012) Howes, G. G., Drake, D. J., Nielson, K. D., Carter, T. A., Kletzing, C. A. & Skiff, F. 2012 Toward astrophysical turbulence in the laboratory. Phys. Rev. Lett. 109, 255001.
  * Howes et al. (2013) Howes, G. G., Nielson, K. D., Drake, D. J., Schroeder, J. W. R., Skiff, F., Kletzing, C. A. & Carter, T. A. 2013 Alfvèn wave collisions, the fundamental building block of plasma turbulence. III. Theory for experimental design. Physics of Plasmas 20 (7), 072304.
  * Hunt et al. (1978) Hunt, J. C. R, Abell, C. J., Peterka, J. A. & Woo, H. 1978 Kinematical studies of the flows around free or surface-mounted obstacles; applying topology to flow visualization. J. Fluid Mech. 86 (1), 179–200.
  * Iwai et al. (2003) Iwai, K., Shinya, K. & Takashi, K. Moreau, R. 2003 Pressure change accompanying Alfvèn waves in a liquid metal. Magnetodrodynamics 39 (3), 245–250.
  * Jameson (1964) Jameson, Antony 1964 A demonstration of Alfvén waves Part 1. Generation of standing waves. Journal of Fluid Mechanics 19 (4), 513–527, number: 4.
  * Jault & Finlay (2015) Jault, D. & Finlay, C.C. 2015 8.09 - waves in the core and mechanical core–mantle interactions. In Treatise on Geophysics (Second Edition), Second edition edn. (ed. Gerald Schubert), pp. 225–244. Oxford: Elsevier.
  * Jephcott & Stocker (1962) Jephcott, DF & Stocker, PM 1962 Hydromagnetic waves in a cylindrical plasma: an experiment. Journal of Fluid Mechanics 13 (4), 587–596.
  * Klein & Pothérat (2010) Klein, R. & Pothérat, A. 2010 Appearance of three dimensionality in wall-bounded MHD flows. Phys. Rev. Lett. 104 (3), 034502, number: 3 Publisher: American Physical Society.
  * Klein et al. (2009) Klein, R, Pothérat, A & Alferenok, Artem 2009 Experiment on a confined electrically driven vortex pair. Physical review. E 79, 016304.
  * Kljukin & Thess (1998) Kljukin, A. & Thess, A. 1998 Direct measurement of the stream-function in a quasi-two-dimensional liquid metal flow. Experiments in Fluids 25 (4), 298–304.
  * Kornet & Pothérat (2015) Kornet, Kacper & Pothérat, Alban 2015 A method for spectral DNS of low Rm channel flows based on the least dissipative modes. Journal of Computational Physics 298, 266–279.
  * Lehnert (1954) Lehnert, Bo 1954 Magneto-Hydrodynamic Waves in Liquid Sodium. Physical Review 94 (4), 815–824, number: 4.
  * Li et al. (2021) Li, Xinyu, Beloborodov, Andrei M. & Sironi, Lorenzo 2021 Fast dissipation of colliding Alfvèn waves in a magnetically dominated plasma. The Astrophysical Journal 915 (2), 101.
  * Lundquist (1949) Lundquist, S. 1949 Experimental Investigations of Magneto-Hydrodynamic Waves. Physical Review 76 (12), 1805–1809, number: 12.
  * Luo & Jackson (2022) Luo, Jiawen & Jackson, Andrew 2022 Waves in the Earth’s core. i. mildly diffusive torsional oscillations. Proc. R. Soc. A. p. 4782021098220210982.
  * Majumder & Sreenivasan (2023) Majumder, Debarshi & Sreenivasan, Binod 2023 The role of magnetic waves in tangent cylinder convection. Physics of the Earth and Planetary Interiors 344, 107105.
  * McLaughlin et al. (2011) McLaughlin, J. A., De Moortel, I. & Hood, A. W. 2011 Phase mixing of nonlinear visco-resistive Alfvén waves. Astronomy & Astrophysics 527, A149.
  * Moffatt (1967) Moffatt, H. K. 1967 On the suppression of turbulence by a uniform magnetic field. J. Fluid Mech. 28, 571–592.
  * Moreau (1990) Moreau, R. 1990 Magnetohydrodynamics. Kluwer Academic Publisher.
  * Moudjed et al. (2020) Moudjed, B., Pothérat, A. & Holdsworth, M. 2020 PIV mapping of pressure and velocity fields in the plane magnetohydrodynamic Couette flow. Experiments in Fluids 61 (12), 255.
  * Nakariakov & Kolotkov (2020) Nakariakov, Valery M. & Kolotkov, Dmitrii Y. 2020 Magnetohydrodynamic waves in the solar corona. Annual Review of Astronomy and Astrophysics 58, 441–481.
  * Nakariakov et al. (1999) Nakariakov, V. M., L., Ofman., DeLuca, E. E., B., Roberts & Davila, J. M. 1999 Trace observation of damped coronal loop oscillations: Implications for coronal heating. Science 285 (5429), 862–864.
  * Nornberg et al. (2010) Nornberg, MD, Ji, H, Schartman, E, Roach, A & Goodman, J 2010 Observation of magnetocoriolis waves in a liquid metal Taylor-Couette experiment. Physical review letters 104 (7), 074501.
  * Pothérat & Klein (2017) Pothérat, A. & Klein, R. 2017 Do magnetic fields enhance turbulence at low magnetic Reynolds number ? Phys. Rev. Fluids 2 (6), 063702.
  * Pothérat & Alboussière (2003) Pothérat, A. & Alboussière, T. 2003 Small scales and anisotropy in low$-{R}m$ magnetohydrodynamic turbulence. Physics of Fluids 15 (10), 3170–3180.
  * Pothérat & Alboussière (2006) Pothérat, Alban & Alboussière, Thierry 2006 Bounds on the attractor dimension for low-$Rm$ wall-bound magnetohydrodynamic turbulence. Physics of Fluids 18 (12), 125102.
  * Pothérat & Dymkou (2010) Pothérat, Alban & Dymkou, Vitali 2010 Direct numerical simulations of low-Rm MHD turbulence based on the least dissipative modes. Journal of Fluid Mechanics 655, 174–197, publisher: Cambridge University Press.
  * Pothérat & Klein (2014) Pothérat, Alban & Klein, Rico 2014 Why, how and when MHD turbulence at low becomes three-dimensional. Journal of Fluid Mechanics 761, 168–205, publisher: Cambridge University Press.
  * Pothérat & Kornet (2015) Pothérat, Alban & Kornet, Kacper 2015 The decay of wall-bounded MHD turbulence at low. Journal of Fluid Mechanics 783, 605–636, publisher: Cambridge University Press.
  * Prokopyszyn et al. (2019) Prokopyszyn, A. P. K., Hood, A. W. & De Moortel, I. 2019 Phase mixing of nonlinear Alfvén waves. Astronomy & Astrophysics 624, A90.
  * Roberts (1967) Roberts, P. H. 1967 An introduction to magnetohydrodynamics. New York: American Elsevier Pub. Co., open Library ID: OL14117453M.
  * Salem et al. (2012) Salem, Chadi S, Howes, GG, Sundkvist, D, Bale, SD, Chaston, CC, Chen, CHK & Mozer, FS 2012 Identification of kinetic alfvén wave turbulence in the solar wind. The Astrophysical Journal Letters 745 (1), L9.
  * Schaeffer et al. (2012) Schaeffer, Nathanaël, Jault, Dominique, Cardin, Philippe & Drouard, Marie 2012 On the reflection of Alfvén waves and its implication for Earth’s core modelling. Geophysical Journal International 191 (2), 508–516.
  * Schmitt et al. (2013) Schmitt, Denys, Cardin, Philippe, La Rizza, Patrick & Nataf, H-C 2013 Magneto–coriolis waves in a spherical Couette flow experiment. European Journal of Mechanics-B/Fluids 37, 10–22.
  * Schwaiger et al. (2024) Schwaiger, T., Gillet, N., Jault, D., Istas, M. & Mandea, M. 2024 Wave-like motions and torques in Earth’s core as inferred from geomagnetic data: A synthetic study. Physics of the Earth and Planetary Interiors 346, 107104.
  * Singh & Subramanian (2007) Singh, KAP & Subramanian, Prasad 2007 An evaluation of possible mechanisms for anomalous resistivity in the solar corona. Solar Physics 243, 163–169.
  * Sommeria (1986) Sommeria, Joël 1986 Experimental study of the two-dimensional inverse energy cascade in a square box. J. Fluid Mech. 170, 139–168.
  * Sommeria (1988) Sommeria, J. 1988 Electrically driven vortices in a strong magnetic field. J. Fluid Mech. 189, 553–569.
  * Sommeria & Moreau (1982) Sommeria, Joël & Moreau, René 1982 Why, how, and when, MHD turbulence becomes two-dimensional. J. Fluid Mech. 118, 507–518, publisher: Cambridge University Press.
  * Stefani et al. (2021) Stefani, F., Forbriger, J., Gundrum, Th., Herrmannsdörfer, T. & Wosnitza, J. 2021 Mode conversion and period doubling in a liquid rubidium Alfvén-wave experiment with coinciding sound and Alfvén speeds. Phys. Rev. Lett. 127, 275001.
  * Thomas et al. (2010) Thomas, A., Bassom, A. & Davies, C. 2010 Linear stability of a Stokes layer with an imposed axial magnetic field, the. J. Fluid Mech. 662, 320–328.
  * Tigrine et al. (2019) Tigrine, Zahia, Nataf, Henri-Claude, Schaeffer, Nathanaël, Cardin, Philippe & Plunian, Franck 2019 Torsional Alfvén waves in a dipolar magnetic field: experiments and simulations. Geophysical Journal International 219 (Supplement_1), S83–S100.
  * Tomczyk et al. (2007) Tomczyk, S., McIntosh, S. W., Keil, S. L., Judge, P. G., Schad, T., Seeley, D. H. & Edmondson, J. 2007 Alfvén waves in the solar corona. Science 317 (5842), 1192–1196.
  * Tsurutani & Ho (1999) Tsurutani, Bruce T & Ho, Christian M 1999 A review of discontinuities and Alfvén waves in interplanetary space: Ulysses results. Reviews of Geophysics 37 (4), 517–541.
  * Varma & Sreenivasan (2022) Varma, Aditya & Sreenivasan, Binod 2022 The role of slow magnetostrophic waves in the formation of the axial dipole in planetary dynamos. Physics of the Earth and Planetary Interiors 333, 106944\.
  * Wilcox et al. (1961) Wilcox, John M, DeSilva, Alan W & Cooper, William S 1961 Experiments on Alfvén-wave propagation. Phys. Fluids 4 (12), 1506–1513.
  * Woods (1962) Woods, LC 1962 Hydromagnetic waves in a cylindrical plasma. Journal of Fluid Mechanics 13 (4), 570–586.