# The Transition from Wall Modes to Multimodality in Liquid Gallium
Magnetoconvection

Yufan Xu (徐宇凡<EMAIL_ADDRESS>Department of Earth, Planetary, and Space
Science, University of California, Los Angeles, California 90095, USA  Susanne
Horn (苏珊娜·霍恩)  Centre for Fluid and Complex Systems, Coventry University, CV1
2NL Coventry, UK  Jonathan M. Aurnou（乔纳森·奥诺  Department of Earth, Planetary,
and Space Science, University of California, Los Angeles, California 90095,
USA

###### Abstract

Coupled laboratory-numerical experiments of Rayleigh-Bénard convection (RBC)
in liquid gallium subject to a vertical magnetic field are presented. The
experiments are carried out in two cylindrical containers with diameter-to-
height aspect ratio $\Gamma=1.0$ and $2.0$ at varying thermal forcing
(Rayleigh numbers $10^{5}\lesssim Ra\lesssim 10^{8}$) and magnetic field
strength (Chandrasekhar numbers $0\lesssim Ch\lesssim 3\times 10^{5}$).
Laboratory measurements and numerical simulations confirm that
magnetoconvection in our finite cylindrical tanks onsets via non-drifting
wall-attached modes, in good agreement with asymptotic predictions for a semi-
infinite domain. With increasing supercriticality, the experimental and
numerical thermal measurements and the numerical velocity data reveal
transitions between wall mode states with different azimuthal mode numbers and
between wall-dominated convection to wall and interior multimodality. These
transitions are also reflected in the heat transfer data, which combined with
previous studies, connect onset to supercritical turbulent behaviors in liquid
metal magnetoconvection over a large parameter space. The gross heat transfer
behaviors between magnetoconvection and rotating convection in liquid metals
are compared and discussed.

##  I Introduction

Convection influenced by ambient magnetic fields is called magnetoconvection
(MC), which arises in many areas of fluid dynamics. In geophysics and
planetary physics, motions of turbulent convective flows in planetary liquid
metal outer cores generate planetary-scale magnetic fields via dynamo
processes. Studying the effects of magnetic fields in MC is essential to
understand these processes [e.g., 1, 2, 3, 4]. In astrophysics, MC is
associated with the sunspot umbra on the outer layer of the Sun and other
stars [e.g., 5, 6, 7]. Furthermore, MC has an essential role in numerous
industrial and engineering applications including but not limited to liquid
metal batteries [8, 9], crystal growth [10, 11], nuclear fusion liquid-metal
cooling blanket designs [12, 13, 14], and induction heating and casting [15,
16]. These systems have drastically different ratios between electromagnetic
and inertial forces. Therefore, it is crucial to investigate the effects of a
wide range of magnetic forces in MC systems.

The canonical model of MC is a convection system with an electrically-
conducting fluid layer heated from below, cooled from above, and in the
presence of an external vertical magnetic field [17, 18]. It is most
fundamentally understood in an extended plane layer geometry [e.g., 17, 19,
18]. But there is an increasing interest in MC systems with defined sidewall
boundaries because of their many experimental and industrial applications.
Various numerical and laboratory studies have been carried out in rectangular
[6, 20] and cylindrical geometries [21, 22, 23, 24].

In weakly supercritical, near-onset regimes, MC systems tend to develop steady
wall modes in the sidewall Shercliff boundary layer [25, 26, 20, 24, 22] while
the bulk remains quiescent. As the magnetoconvective supercriticality
increases, convective flows self-organise into multi-cellular bulk flow
structures [18, 24]. Eventually, at very large supercriticalities, the
buoyancy forces dominate and magnetic field effects become subdominant. Large-
scale circulations (LSCs) then form and the heat and momentum transfer
asymptote to that of turbulent RBC [e.g. 27, 24, 28, 29].

The pathway from the onset of convection to fully developed turbulence in
liquid metal MC is not well characterised. To address this deficit, we present
a suite of laboratory-numerical coupled MC experiments in liquid gallium to
investigate how MC transitions from near-onset wall modes to turbulent
multimodality in cylindrical cells. The paper is organised as follows. Section
II introduces control parameters, and reviews established onset predictions
for the magnetoconvection system. Section III presents our experimental setup,
numerical schemes, diagnostics, and the physical properties of liquid gallium.
Section IV compares different theoretical onsets and observations of the
transition to multimodality in a survey with fixed magnetic field strength and
varying convective vigor. Section V shows heat transfer results combining
previous studies of MC and compares the gross heat transfer behaviors between
liquid metal magnetoconvection and liquid metal rotating convection systems.

##  II Control parameters and linear prediction

Laboratory-scale liquid metal magnetoconvection usually has a negligible
induced magnetic field $\bm{b}$ with respect to the external applied magnetic
field $\bm{B}_{0}$, so that $|\bm{b}|\ll|\bm{B}_{0}|$. Moreover, any induced
field is considered temporally invariant, $\partial_{t}\bm{b}\approx 0$. The
magnetic Reynolds number, defined as $Rm=UH/\eta$ [30, 31, 32, 22], is well
below unity, where $U$ and $H$ are the characteristic velocity and length
scales, respectively, and $\eta$ is the magnetic diffusivity. This parameter
represents the ratio between induction and diffusion of the magnetic field.
Thus, the so-called low-$Rm$ quasistatic approximation is valid in most liquid
metal experimental and industrial applications [33, 34, 32, 21]. In the
quasistatic limit, $Rm$ and magnetic Prandtl number $Pm$ formally drop out of
the problem, so it is not necessary to solve the magnetic induction equation
explicitly, and the system is greatly simplified. In addition, the Oberbeck-
Boussinesq approximation is commonly applied in the governing equations for
liquid metal MC systems [e.g. 21, 20, 35, 18, 28].

Four nondimensional control parameters govern quasistatic Oberbeck-Boussinesq
magnetoconvection [20, 18, 28]. The Prandtl number $Pr$ describes the thermo-
mechanical properties of the fluid,

$Pr=\frac{\nu}{\kappa},$ (1)

where $\nu$ is the kinematic viscosity and $\kappa$ is the thermal
diffusivity. In this study, $Pr\approx 0.027$ for liquid gallium. The Rayleigh
number $Ra$ characterises the buoyancy forcing with respect to thermo-viscous
diffusion and is defined as

$Ra=\frac{\alpha g\Delta TH^{3}}{\kappa\nu}.$ (2)

Here, $\alpha$ is the thermal expansion coefficient, $g$ is the magnitude of
the vertically oriented ($\bm{\hat{e}_{z}}$) gravitational acceleration,
$\Delta T$ is the bottom-to-top vertical temperature difference across the
fluid layer, and the characteristic length scale is the layer height $H$. The
Chandrasekhar number $Ch$ denotes the ratio of quasistatic Lorentz forces and
viscous forces,

$Ch=\frac{\sigma B_{0}^{2}H^{2}}{\rho_{0}\nu},$ (3)

where $\sigma$ is the electric conductivity of the fluid, $B_{0}$ is the
magnitude of the applied vertical magnetic field, and $\rho_{0}$ is the mean
density of the fluid. The Chandrasekhar number is the square of the Hartmann
number, $Ch=Ha^{2}$ [e.g. 32, 10, 36]. Additionally, the cylindrical container
has a diameter-to-height aspect ratio

$\Gamma=\frac{D}{H},$ (4)

where $D$ is the diameter of the container. Here, $\Gamma$ is fixed to $1.0$
and $2.0$, respectively.

The onset of convection is controlled by the critical Rayleigh denoted as
$Ra_{crit}$, which characterises the buoyancy forcing needed for a particular
convective mode in the system [37]. Figure 1 shows different $Ra_{crit}$
predictions. Linear analysis has shown that the convection driven by buoyancy
forces must balance the viscous and Joule dissipation [17]. Thus, in general,
the magnetic field inhibits the onset of the convection. Chandrasekhar [17]
derived the onset for the bulk stationary magnetoconvection in an infinite
plane layer ($\infty$). With free-slip (FS) boundaries on both ends, the
dispersion relation expresses the marginal Rayleigh number $Ra_{M}$ as,

$Ra_{M}=\frac{\pi^{2}+a^{2}}{a^{2}}\left[\left(\pi^{2}+a^{2}\right)^{2}+\pi^{2}Ch\right],$
(5)

where $a$ is the characteristic cell aspect ratio [32], defined as $a\equiv\pi
H/L$, where $H$ is the height of the fluid layer, and $2L$ is the horizontal
wavelength of the convection flow, assuming the form of two-dimensional rolls
with each roll having diameter $L$. By minimizing equation (5), setting
$\partial Ra/\partial a=0$, we obtain the critical Rayleigh number for the
bulk stationary magnetoconvection in an infinite layer with free-slip
boundaries on both ends, $Ra^{\infty}_{FS}$, and its critical mode number
$a_{FS}$. In the limit of $Ch\rightarrow\infty$, we have
$Ra^{\infty}_{FS}\rightarrow\pi^{2}Ch$, and
$a_{FS}\rightarrow(\pi^{4}Ch/2)^{1/6}$ [17, 32].

Magnetoconvection with no-slip (NS) rigid horizontal boundaries has a
dispersion relation [38]

$Ra_{M}=\frac{\left(\pi^{2}+a^{2}\right)\left[\left(\pi^{2}+a^{2}\right)^{2}+\pi^{2}Ch\right]}{a^{2}\left[1-4\pi^{2}\delta\left(q_{1}^{2}-q_{2}^{2}\right)/\left(\pi^{2}+q_{1}^{2}\right)\left(\pi^{2}+q_{2}^{2}\right)\right]}$
(6)

where

$q_{1}=\frac{1}{2}\left(\sqrt{Ch+4a^{2}}+\sqrt{Ch}\right),\quad
q_{2}=\frac{1}{2}\left(\sqrt{Ch+4a^{2}}-\sqrt{Ch}\right),$ (7)

and

$\delta=\left(q_{1}\tanh(q_{1}/2)-q_{2}\tanh(q_{2}/2)\right)^{-1}.$ (8)

By assuming a single structure in the vertical direction and minimizing $Ra$
in (6), we obtain the first approximation of critical $Ra$ of
magnetoconvection with two rigid boundaries [38]. Equation (6) also predicts
that $Ra^{\infty}_{NS}\rightarrow\pi^{2}Ch$ and
$a_{NS}\rightarrow(\pi^{4}Ch/2)^{1/6}$ with $Ch\rightarrow\infty$. Thus, both
critical Rayleigh numbers with free-slip boundaries $Ra^{\infty}_{FS}$ and no-
slip boundaries $Ra^{\infty}_{NS}$ asymptote to $\pi^{2}Ch$ above $Ch\gtrsim
10^{4}$, as shown in figure 1a). These asymptotic bulk onset predictions agree
with previous experimental results [19, 21, 18, 24].

Busse [26] theoretically analysed the side wall modes in MC and derived an
asymptotic solution along a straight vertical sidewall in a semi-infinite
domain with free-slip top-bottom boundaries. The critical Rayleigh number
$Ra_{W}$ for these so-called magnetowall modes is,

$Ra_{W}=3\pi^{2}\sqrt{3\pi/2}\left(1+3Ch^{-1/4}\sqrt{3\pi/2}\right)Ch^{3/4}.$
(9)

The asymptotic onset of the wall modes is generally lower than the onset in
the bulk fluid at large $Ch$, since $Ch^{3/4}\ll Ch$ as $Ch\rightarrow\infty$.
These magnetowall modes are non-drifting and extend into the fluid bulk with a
distance that scales as the magnetic boundary layer thickness, which scales
with the Shercliff boundary layer thickness $\delta_{Sh}\sim Ch^{-1/4}$ [39,
20]. The stationary wall modes of MC differ from those found in rotating
convection, where wall modes drift in azimuth [40, 41, 42].

Houchens  _et al._ [25] performed a hybrid linear stability analysis combining
the analytical solution for the $\delta_{Ha}\sim Ch^{-1/2}$ Hartmann layers
[43] at top-bottom boundaries and numerical solutions for the rest of the
domain in $\Gamma=1$ and $2$ cylindrical geometries. They also presented a
linear asymptotic analysis for large $Ch$. Their asymptotic solutions for
critical $Ra$ for $\Gamma=1$ and $2$ are, respectively,

$Ra_{cyl,\Gamma=1}=8.302Ch^{3/4};\quad Ra_{cyl,\Gamma=2}=67.748Ch^{3/4}.$ (10)

Figure 1 summarises all the critical Rayleigh predictions mentioned above.
Houchens  _et al._ [25]’s $Ra_{cyl,\Gamma=1}$ values (marked by the purple
dashed line) are approximately an order of magnitude lower than the rest of
the onset predictions that are not aspect-ratio dependent. To test the
validity of these predictions, we combine laboratory experiments and direct
numerical simulations (DNS) to investigate the five different $Ch$ shown by
the vertical dashed lines in figure 1b). The values of these five $Ch$ numbers
and their corresponding critical $Ra$ are summarised in table 1.

$\bm{Ch}$ | $\bm{Ra_{\mathrm{FS}}^{\infty}}$ | $\bm{Ra_{\mathrm{NS}}^{\infty}}$ | $\bm{Ra_{W}}$ | $\bm{Ra_{\mathrm{cyl},\ \Gamma=1}}$ | $\bm{Ra_{\mathrm{cyl},\ \Gamma=2}}$
---|---|---|---|---|---
$1\times 10^{4}$ | $1.20\times 10^{5}$ | $1.25\times 10^{5}$ | $1.06\times 10^{5}$ | $8.30\times 10^{3}$ | $6.77\times 10^{4}$
$4\times 10^{4}$ | $4.46\times 10^{5}$ | $4.54\times 10^{5}$ | $2.66\times 10^{5}$ | $2.35\times 10^{4}$ | $1.92\times 10^{5}$
$1\times 10^{5}$ | $1.08\times 10^{6}$ | $1.09\times 10^{6}$ | $4.94\times 10^{5}$ | $4.67\times 10^{4}$ | $3.81\times 10^{5}$
$3\times 10^{5}$ | $3.15\times 10^{6}$ | $3.17\times 10^{6}$ | $1.05\times 10^{6}$ | $1.06\times 10^{5}$ | $8.68\times 10^{5}$
$1\times 10^{6}$ | $1.03\times 10^{7}$ | $1.03\times 10^{7}$ | $2.45\times 10^{6}$ | $2.63\times 10^{5}$ | $2.14\times 10^{6}$
Table 1: Values of different predicted critical $Ra$ at $Ch=\\{10^{4}$,
$4\times 10^{4}$, $10^{5}$, $3\times 10^{5}$, $10^{6}\\}$, which have been
examined experimentally.

Figure 1: a) Different prediction of critical Rayleigh number ($Ra_{crit}$) as
a function of the Chandrasekhar number. The black dashed curve shows the
$Ra_{cirt}$. For infinite plane MC system with free-slip boundaries at the top
and bottom, $Ra_{crit}=Ra_{FS}^{\infty}$ [17], as shown in the black dashed
curve; for infinite plane MC system with no-slip boundaries,
$Ra_{crit}=Ra_{NS}^{\infty}$ [38], as shown in the blue curve. Note that both
$Ra_{FS}^{\infty}$ and $Ra_{NS}^{\infty}$ asymptote to $\pi^{2}Ch$ as
$Ch\rightarrow\infty$; the grayish-blue curve shows that $Ra_{W}$ is the
asymptotic $Ra_{crit}$ for the wall-mode onset in a half-infinite plane with a
vertical boundary [26]; the purple curve and the purple dotted curve are
Houchens  _et al._ [25]’s predicted $Ra_{crit}$, namely $Ra_{cyl,\ \Gamma=2}$
and $Ra_{cyl,\ \Gamma=1}$, for MC in cylindrical containers with aspect ratio
$1$ and $2$, respectively. b) The zoom-in view of the region circumscribed by
the black rectangular box in panel a). The colored vertical dashed lines
correspond to five $Ch$ numbers employed in our study.

##  III Methods

Our experiments are conducted using UCLA’s RoMag device [44, 45, 35, 29, 28].
Figure 2a) - c) show schematics of the diagnostics and the apparatus. The
container consists of two copper end blocks and a stainless steel sidewall.
Two sets of sidewalls, $\Gamma=1.0$ and $2.0$, have been used in this study to
investigate MC heat transfer from $10^{5}\lesssim Ra\lesssim 10^{8}$. An
external solenoid generates a steady vertical magnetic field,
$0<|\bm{B}_{0}|<800$ Gauss, with a vertical component that varies within $\pm
0.5$% over the field volume [46]. The tank is placed at the center of the
solenoid’s bore. A non-inductively wound electrical resistance pad heats the
bottom of the lower copper end block at a constant rate, $0<P\lesssim
2000\mathrm{W}$, and a thermostated water-cooled heat exchanger maintains a
constant temperature at the top of the upper copper end block. This setup can
reach up to $Ra\approx 10^{9}$ and $Ch=3\times 10^{5}$ in the $\Gamma=1.0$
tank.

Figure 2:  a) Thermometry schematic for the $\Gamma=2R/H=2.0$ tank
($R=98.6\,\mathrm{mm}$): there are six thermistors in the top lid at $0.71R$,
six thermistors at the midplane sidewall, and six thermistors in the bottom
lid at $0.71R$. These thermistors at different heights align with each other
azimuthally. The thermometry is in a similar layout on the aspect ratio one
tank ($\Gamma=1.0$). b) Photo of the convection cell of the RoMag device. c)
Schematics of the convection cell of the RoMag device. For further device
details, see Xu  _et al._ [28].

Twelve thermistors in total are placed inside the top and bottom boundaries
about $28.9\ \mathrm{mm}$ radially inwards from the sidewall, as shown in
figure 2a). These end-block thermistors are used to measure the heat transfer
efficiency of the system, characterised by the Nusselt number,

$Nu=\frac{qH}{\lambda\Delta T},$ (11)

where $q=4P/(\pi D^{2})$ is the heat flux, $P$ is the heating power, and
$\lambda=31.4\ \mathrm{W/(m\cdot K)}$ is the thermal conductivity of gallium
[47]. The Nusselt number describes the total to conductive heat transfer ratio
across the fluid layer, and $Nu=1$ corresponds to the conductive state. The
vertical temperature difference across the fluid layer, $\Delta T$, is
indirectly controlled by the constant basal heat flux. Six thermistors are
attached to the sidewall midplane to detect wall modes and any thermal
imprints of the bulk fluid structures at the sidewall.

We have also conducted direct numerical simulations (DNS) using the finite
volume code goldfish [48, 49, 50, 42, 51, 52, 53]. The nondimensional
equations governing quasi-static, Oberbeck-Boussinesq [54, 55]
magnetoconvection are: ∇ ⋅~u &= 0,
D_~t ~u = - ∇ ~p + PrRa ∇^2 ~u \+ PrRa Ek2 ~u ×^e_z \+ Ch2Pr Ra ~j×^e_z \+ ~T
^e_z,
D_~t ~T = 1Ra Pr ∇^2 ~T,
∇ ⋅~j= 0~j= - ∇ ~Φ+ (~u×^e_z) } ∇^2 ~Φ = ∇ ⋅(~u ×^e_z), where
$\widetilde{\bm{u}}$ denotes nondimensional velocity, $\widetilde{T}$ the
temperature, $\widetilde{p}$ the pressure, $\widetilde{\bm{j}}$ the current
density, and $\widetilde{\Phi}$ the electrostatic potential. The scales used
for the nondimensionalisation are the free-fall speed $U_{f\\!f}=\sqrt{\alpha
g\Delta TH}$ [56], the temperature difference between top and bottom $\Delta
T$, the reference pressure $\rho_{0}U_{f\\!f}^{2}$, the reference current
density $\sigma B_{0}U_{f\\!f}$ and the reference potential $B_{0}HU_{f\\!f}$.

Our non-linear DNS solves these equations in a cylindrical domain $(r,\phi,z)$
with $\Gamma=2.0$. The sidewall is assumed to be perfectly thermally
insulating, $\partial_{r}T\mid_{\text{r=R}}=0$, and the top and bottom plates
are isothermal with $T_{t}=-0.5$ and $T_{b}=0.5$, respectively. All boundaries
are assumed to be impermeable and no-slip, $\bm{u}\mid_{\text{wall}}=0$, and
electrically insulating, $\bm{j}\mid_{\text{wall}}=0$, i.e. the current forms
closed loops inside the domain. The DNS control parameters are set to
$Pr=0.027$, $Ch=4.0\times 10^{4}$, and $Ra=\\{1.5\times 10^{5},\ 2.0\times
10^{5},\ 3.0\times 10^{5},\ 4.0\times 10^{5},\ 7.0\times 10^{5},\ 1.0\times
10^{6},\ 1.5\times 10^{6},\ 4.0\times 10^{6}\\}$. The numerical mesh
resolution is $N_{r}\times N_{\phi}\times N_{z}=240\times 256\times 240$. This
choice of mesh was verified by running simulations at twice the resolution for
the highest $Ra$ for a shorter time, indicating the grid independence of the
solution.

##  IV Results

###  IV.1 Comparing onset predictions

Figure 3: a) $Nu-Ra$ survey with various $Ch$ from current study and one set
of $Ch=10^{6}$ data by King and Aurnou [46]. Different colors correspond to
different $Ch$ values. Different shapes correspond to different aspect ratios
and sets of experiments, marked in the legend of panel b). Error bars are
shown on the experimental data from the current study. The dashed line is the
heat transfer scaling acquired from non-magnetic Rayleigh-Bénard convection
data in the $\Gamma=1.0$ tank, marked by dark blue circles. See equation
(V.2a). Panel b)-d) show ratios of convective heat transfer to conduction
($Nu-1$) versus the reduced bifurcation parameter [57, 42] using three
different predicted critical Rayleigh numbers: b) infinite-plane stress-free
critical $Ra$ defined in Chandrasekhar [17]; c) magnetowall mode critical $Ra$
by Busse [26]. d) Houchens  _et al._ [25]’s critical $Ra$ for magnetowall
modes in two different aspect ratios, $\Gamma=1$ and $2$. The linear fit in
panel c) uses data close to onset at $\varepsilon_{W}\leq 5$. The $\Gamma=1.0$
data in panel d) are shown in the smaller subplot, which only appears at
$\varepsilon\gtrsim 25$.

The validity and accuracy of the onset predictions discussed in Section II are
tested here using both laboratory and DNS data at $10^{5}\lesssim Ra\lesssim
10^{8}$, $0\lesssim Ch\lesssim 3\times 10^{5}$, and in $\Gamma=1.0$ and $2.0$
cylindrical cells.

Figure 3a) shows measurements of heat transfer efficiency, $Nu$, as a function
of the buoyancy forcing $Ra$. (All the detailed measurement data are provided
in the tables in the Appendix). Vertical error bars based on heat loss and
accuracy of the thermometry, are shown in the lab data from this study. We
also include King and Aurnou [46]’s $Ch=10^{6}$ data (red stars) made in the
same $\Gamma=1.0$ experimental setup used in this study. Figure 3b) - d) show
convective heat transfer data ($Nu-1$) as a function of supercriticality of
the convection, as described by the reduced bifurcation parameter
$\varepsilon=(Ra-Ra_{crit})/Ra_{crit}$, following the convention of [40, 57,
42]. Three different $Ra_{crit}$ are examined in panel b) to d): for
convection in an infinite plane layer with two rigid boundaries,
$Ra_{NS}^{\infty}$ (6), wall-attached convection, $Ra_{W}^{\infty}$ (9), and
convection in a cylinder with aspect ratio 1 and 2, $Ra_{cyl}$ (10). If the
$Ra_{crit}$ prediction is accurate, the onset of convection occurs at
$\varepsilon=0$, and $Nu$ and follows an approximate linear scaling for
sufficiently small $\varepsilon$.

Figure 3b) presents the convective heat transfer data, $Nu-1$, as a function
of the reduced bifurcation parameter $\varepsilon_{NS}=(Ra-
Ra_{NS}^{\infty})/Ra_{NS}^{\infty}$ calcuated using eq. eq:ns. The laboratory-
numerical $Nu-1$ data exceeds 0 at $\varepsilon_{NS}<0$. This implies that the
$Ra_{NS}^{\infty}$ predictions do not capture the onset of MC in our system.
Moreover, the increased scatter and variation in $Nu$ for different $Ch$ as
$\varepsilon_{NS}$ increases suggest a low correlation between the data and
the expected linear $\varepsilon_{NS}$ scaling. As the infinite-plane
$Ra_{crit}$ is associated with the bulk onset of convection, our heat transfer
data implies that the MC flow does not initiate in the fluid bulk.

Figure 3c) tests Busse [26]’s asymptotic onset predictions for magnetowall
modes as a function of $\varepsilon_{W}=(Ra-Ra_{W})/Ra_{W}$ calculated using
eq:Busse. The nonzero $Nu-1$ data start approximately at the origin of the
graph, being only slightly below $\varepsilon_{W}=0$ and show a good data
collapse up to moderately high supercriticalities of $\varepsilon\lesssim 10$.
Thus, our data provide evidence that in our system the onset of convection
occurs in the form of wall-attached modes. This is further quantified by two
different linear least-square fits for the nonzero $Nu-1$ data for
$\varepsilon_{W}\leq 5$. The first fit ‘fit1’ (blue line) makes no assumptions
on the onset and yields $Nu-1=0.419\ \varepsilon_{W}+0.490$. The second ‘fit2’
(red line) is forced to pass through the origin, i.e. it assumes the onset
prediction $\varepsilon_{W}=(Ra-Ra_{W})/Ra_{W}$ is exact and yields
$Nu-1=0.582\ \varepsilon_{W}$. Thus, in agreement with the theoretical
predictions and as also found by Zürner  _et al._ [24], our MC onset data is
consistent with the wall mode predictions. Furthermore, both linear fits hold
well up to $\varepsilon_{W}\lesssim 10$, suggesting that the dynamics and the
heat transfer in our system are largely controlled by linear magnetowall modes
within this supercriticality range [cf. 40, 57, 42, 58].

Figure 3d) tests the hybrid theoretical-numerical predictions of Houchens  _et
al._ [25] for magnetowall modes in cylindrical geometries by plotting $Nu-1$
versus $\varepsilon_{cyl}=(Ra-Ra_{cyl})/Ra_{cyl}$ calculated using eq:cyl. The
underlying assumptions for these predictions best match the experimental and
numerical setup. Therefore, they should best capture the measured onset of
convection. The $\Gamma=2.0$ data (green and yellow hues) show an excellent
agreement with theory. The $\Gamma=1.0$ data (inset, red and orange hues) does
not have a low enough supercriticality ($\varepsilon_{cyl}>28$) to reliably
test the exact $Ra_{cyl}$ value. Thus, we are currently unable to disambiguate
which magnetowall mode onset predictions are more accurate, even though
Houchens  _et al._ [25]’s $\Gamma=1.0$ predictions differ from Busse [26]’s by
nearly an order of magnitude.

Figure 4: Temperature and velocity fields from $\Gamma=2.0$ laboratory
experiments at $Ch=4.0\times 10^{4}$. The vertical columns show cases at a)
$Ra=2.0\times 10^{5}$, b) $Ra=4.0\times 10^{5}$, c) $Ra=7.0\times 10^{5}$, d)
$Ra=1.5\times 10^{6}$, and e) $Ra=4.0\times 10^{6}$, respectively ($Ra$ is
only approximate for the laboratory cases; their exact values are given in
tables 2 and 3). The first row shows the azimuthal-temporal temperature
contours at the midplane interpolated by lab data over $5$ thermal diffusion
times, $\tau_{\kappa}=H^{2}/\kappa$. The color represents the dimensionless
temperature, $(T-\overline{T})/\Delta T$, where $\overline{T}$ is the mean
temperature obtained by averaging the top and bottom temperatures. The second
row consists of snapshots of the normalised DNS temperature field
$\widetilde{T}$ and the third row presents snapshots of normalised DNS
vertical velocity fields $\widetilde{u}_{z}$ at the same moment in time as the
temperature field. The vertical black dashed line between b) and c) separates
between cases below bulk onset (based on $Ra_{NS}^{\infty}$) to the left and
above bulk onset to the right. The vertical green dash-dotted line between c)
and d) indicates the transition from an azimuthal mode number of $m=3$ to
$m\leq 2$ seen in the laboratory cases.

###  IV.2 Transition to multimodality

Figure 5: a) Azimuthal mode number $m$ as a function of $Ra$ at $Ch=4\times
10^{4}$, $\Gamma=2.0$ case for both lab (green triangles) and numerical data
(red triangles). b) Nusselt number $Nu$ as a function of $Ra$ for both
laboratory experimental and numerical data. c) $Nu$-$Ra$ Data plotted on
detrended curves. Here, $\widetilde{Nu}$ is $Nu$ normalised by the linear fits
(12, 13) of each respective data set in panel b). In parallel to figure 4, the
black dashed line between b) and c) indicates the predicted
$Ra_{NS}^{\infty}$, whereas the green dash-dotted line between c) and d) marks
the average $Ra$ between two adjacent laboratory data with $m=3$ and $m=2$.
The kinks in $Nu$-$Ra$ data in b) are shown as the fluctuations around
$\widetilde{Nu}=1$ in the $\widetilde{Nu}-Ra$ trend, likely associated with
mode switching.

We analysed temperature and velocity field data to elucidate both the wall
modes and the transition to multi-modal flow at higher supercriticality.
Figure 4 shows temperature and velocity fields of five laboratory numerical
cases with $Ch\simeq 4\times 10^{4}$ in the $\Gamma=2.0$ tank. Each column
represents the laboratory case (top row) and its corresponding numerical case
(middle and bottom rows) at a similar $Ra$. The detailed parameters are given
in table 2 and 3 in the appendix. The top rows show nondimensional temperature
$(T-\overline{T})/\Delta T$ Hovmöller diagrams, a time evolution of the
sidewall midplane temperature field. The temperature fields $T$ are
interpolated by the mid-plane thermistor data taken $60^{\circ}$ apart in
azimuth. The mean temperature $\overline{T}$ is measured by averaging the top
and bottom boundaries’ temperature; the vertical temperature difference across
the fluid layer is denoted as $\Delta T$. The second and third rows of figure
4 show snapshots of numerical 3D isosurfaces of the dimensionless temperature
fields $\widetilde{T}=(T-\left<T\right>)/\Delta T$ and corresponding vertical
velocity fields $\widetilde{\bm{u}}_{z}$ at the same instant in the time. The
mean temperature $\left<T\right>$ is calculated by averaging the temperature
fields over the entire domain.

The velocity and temperature fields in figure 4 all show magnetowall modes,
manifesting as azimuthally alternating upwelling warm and downwelling cold
patches located close to the sidewall. The snapshots of the DNS velocity
fields in figures 4a) and b) further reveal that the magnetowall modes have a
two-layer, ‘nose-like’ flow pattern attached to the sidewall with alternating
$\pm\widetilde{\bm{u}}_{z}$. Liu  _et al._ [20] observed similar structures in
their simulations in a rectangular box at $Ra=10^{7},\ Ch=4\times 10^{6}$.
They found that these noses scale approximately with a Shercliff boundary
layer thickness $\delta_{Sh}\propto Ch^{-1/4}$ [20].

Figure 4a) to b) show that these noses also grow gradually towards the
interior as the supercriticality increases, while the interior remains
otherwise quiescent. This “Pinocchio effect” persist until $Ra\gtrsim
Ra_{NS}^{\infty}$, when the bulk fluid starts convecting from the top and
bottom boundaries and then interacts with the inward-extended wall modes.

Figure 4c) shows this extending nose behavior for $Ra=7\times 10^{5}$, which
is just above the bulk onset $Ra>Ra_{NS}^{\infty}$. The DNS velocity field
visualises how two noses with positive/negative $u_{z}$ (pink/blue) connect
across the entire diameter of the tank via the convecting
upwelling/downwelling fluid in the interior. The laboratory and numerical
temperature field on the sidewall agree perfectly and show that close to the
sidewall the wall modes are virtually unaffected by this interior dynamics. In
total there are six alternating cold and hot patches along the sidewall
azimuth, i.e., the azimuthal mode number is $m=3$. The magnetowall mode number
$m$ is defined as the number of repeating azimuthal structures along the
lateral surface.

Figure 4d) and e) show that these nonlinear interactions become more
complicated and chaotic as $Ra$ increases further. The nose-like structures
interact and impinge on each other. The nonlinear behaviour also affects the
flow close to the sidewall as visible in the temperature Hovmöller diagram
from sidewall thermometry in d) for $Ra=1.5\times 10^{6}$ and even more so in
e) for $Ra=4.0\times 10^{6}$.

For $Ra=4.0\times 10^{6}$ (Figure 4e), the experimental temperature Hovmöller
diagram shows that the magnetowall modes are transient between $m=1$ and $m=2$
in a chaotic sequence. The velocity field of the DNS further demonstrates that
the bulk flow dominates the dynamics, and, hence the flow for $Ra=4.0\times
10^{6}$ significantly differs from the ones at lower $Ra$ shown in figure 4a)
to d).

There is, however, a small discrepancy between the number of azimuthal
structures between the lab and DNS data, being $m=2$ and $m=3$, respectively
for $Ra=1.5\times 10^{6}$ (Figure 4d). This may be because $m$ is sensitive to
small changes in $Ra$ and $\Gamma$, and there are slight differences in
parameters between the lab and the DNS, or because of the sidewall boundary
conditions which are not perfectly adiabiatic in the experiment. It is also
possible that the DNS snapshots do not capture fully equilibrated flow
patterns whilst the lab experiments revealed more averaged dynamics of MC, as,
unlike the DNS, they can be run for many thermal diffusion times.

Figure 5a) shows how the time-averaged azimuthal mode numbers $m$ observed in
the laboratory experiment and the DNS velocity fields depend on the Rayleigh
number $Ra$. The $m$ values generally decrease with increasing $Ra$, which
qualitatively agrees with previous studies [20, 22, 24]. For $Ra<10^{4}$, we
only present DNS data in this plot and no lab data due to a combination of
both precision and spatial aliasing issues of the sidewall thermistor array.
The temperature variation between each wall mode structure near the midplane
is $\lesssim 0.2\ \mathrm{K}$, which is too small to be resolved by our
thermometers. Additionally, with only six thermistors evenly spaced at the
azimuth, we can only resolve up to $m=3$ according to the Nyquist-Shannon
sampling theorem. Thus, even though the first-row temperature contour in
figure 5b) shows an $m=3$ structure, it was omitted in figure 5a) and only the
$m=4$ from the velocity field from the DNS is shown.

The changes in mode number also affect the global heat transport. Figure 5b)
shows that $Nu$ increases monotonically with $Ra$, but not at a constant rate.
Instead, kinks exist in the $Nu$-$Ra$ trends in both lab experiments and DNS
for $Ch=4\times 10^{4}$ (and $Ch=10^{5}$, see figure 3a), a phenomenon which
has not been reported in previous MC experiments [21, 24]. To further
investigate this behaviour, we normalised $Nu$ by power laws obtained by
separate fits to the $Ch=4\times 10^{4}$ laboratory and DNS $Nu$-$Ra$ data
sets. For the laboratory data, the best fit is

$\widetilde{Nu}=Nu/(0.0029Ra^{0.493}),$ (12)

whereas for the DNS, it is found that

$\widetilde{Nu}=Nu/(0.0014Ra^{0.541}).$ (13)

Figure 5c) shows the normalised $\widetilde{Nu}$. The non-monotonicity of the
trend manifests as fluctuations around $\widetilde{Nu}\approx 1$ with an
amplitude of approximately $0.05$. The increase after the first local minimum
in the experimental $\widetilde{Nu}$ data curve coincides with the bulk onset,
$Ra=Ra_{NS}^{\infty}$, and is marked by the vertical black dashed line. This
suggests that bulk convection enhances heat transfer efficiency. The decrease
after the first local maximum in the experimental $\widetilde{Nu}$ data curve
coincides with the change of mode numbers from $m=3$ to $m=2$ observed in the
laboratory cases (cf. figure 4) and is marked by the green dash-dotted line.
The transition to a smaller mode number appears to suppress the heat transfer
efficiency temporarily. A similar behaviour was observed in Horanyi  _et al._
[59]’s liquid metal Rayleigh-Bénard convection experiments. The second
enhancement in $\widetilde{Nu}$ after the second local minimum happens when
the highly-nonlinear flow structures in the bulk fluid start to dominate the
convective dynamics. This corresponds to flow behaviors somewhere between
$Ra=1.5\times 10^{6}$ (figure 4d) and $Ra=4\times 10^{6}$ (figure 4e). The DNS
data in figure 5c) match the first enhancement near $Ra=Ra_{NS}^{\infty}$.
Because no mode switch from $m=3$ to $m=2$ was found in the DNS, no kink shows
up in the $Nu$-$Ra$ trend in the DNS data at this point.

##  V Discussion

###  V.1 Wall modes stability and the cellular flow regime

In our laboratory-numerical experiments, the magnetowall modes are stationary
and do not drift over dynamically long time scales ($\gg 5\tau_{\kappa}$), in
contrast to the drifting wall modes in rotating convection systems [e.g., 40].
This is because the quasi-static Lorentz force ($f_{L}\propto B_{0}^{2}$) does
not break the system’s azimuthal symmetry, unlike the Coriolis force [40]).
The stationarity of magnetowall modes has been confirmed in both numerical
simulations [20] and laboratory experiments [24]. Furthermore, Liu  _et al._
[20] showed that the magnetowall modes can inject jets into the bulk. This
phenomenon was also found in the numerical MC simulations of Akhmedagaev  _et
al._ [22], where strong, axially invariant wall mode injections were
accompanied by a net azimuthal drift of the flow field with random
orientations. We believe that the collisional interaction of the jets in a
small aspect ratio cylinder ($\Gamma=1.0$), rather than any innate azimuthal
motion of magnetowall modes, is responsible for the drifting motions observed
by Akhmedagaev  _et al._ [22].

The fully three-dimensional flow fields from our DNS facilitated the
investigation of the bulk flow patterns in this study. Thus, we are also able
to compare multiple cases at similar parameters with Zürner  _et al._ [24] who
inferred the interior structure solely from linewise Ultrasonic Doppler
Velocimetry (UDV) and pointwise temperature measurements along the sidewalls
and within the top and bottom plate. Our identified flow structures and
corresponding flow changes match well with their observations. Specifically,
what they denoted as the ’cellular regime’ corresponds to our case with
extended wall mode noses with interior bulk modes. Our $Ra=7\times 10^{5}$ and
$Ra=1.5\times 10^{6}$ cases (figure 4c, d) resemble the inferred ’3-cell’ and
’4-cell’ patterns of figure 3 in Zürner  _et al._ [24]. Our $Ra=1.5\times
10^{6},\ m=2$ thermal data in figure 4d) also agrees with their ’2 cell’
pattern on the sidewall. Moreover, the transition range from the ’cellular
regime’ to the non-rotating LSC regime in their experiment occurred
approximately at $Ra\gtrsim 4\times 10^{6}$ for $Ch=4\approx 10^{4}$, which is
consistent with our observation of a more chaotic interior and unsteady and
irregular wall mode behaviour, as shown in figure 4e).

###  V.2 The $Nu$ vs. $Ra$ MC party

Figure 6 presents a broad compilation of laboratory MC heat transfer
measurements in different aspect ratios and geometries, all in the presence of
an external vertical magnetic field. Cioni  _et al._ [21] (open circles)
studied liquid mercury in a $\Gamma=1.0$ cylindrical cell up to $Ch\approx
4\times 10^{6}$ and $Ra\approx 3\times 10^{9}$. Aurnou and Olson [60] (open
triangles pointing right) carried out near onset liquid gallium experiments in
a $\Gamma=8$ rectangular cell. Burr and Müller [61] (open triangles pointing
left) investigated sodium-potassium alloy in a $20:10:1$ rectangular cell.
King and Aurnou [46] studied liquid gallium MC in a $\Gamma=1.0$ cylinder on
the same device (RoMag) as this study. Zürner  _et al._ [24] studied both heat
and momentum transfer behaviors of liquid GaInSn in a $\Gamma=1$ cylinder. The
results from our current $\Gamma=\\{1.0,\ 2.0\\}$ cylindrical liquid gallium
experiments and simulations are demarcated by the filled symbols.

Figure 6: Collection of $Nu$-$Ra$ data from this study and previous MC
laboratory experiments in liquid metal [17, 21, 60, 61, 46, 24]. Color
represents $Ch$. The filled symbols mark the data in this study. The five-
pointed stars at $Nu=1$ mark the $Ra_{W}$ (9) for all different $Ch>0$. The
two asterisk symbols from left to right mark $Ra^{\infty}_{NS}$ for
$Ch=2\times 10^{6}$ and $4\times 10^{6}$, respectively, and corresponding to
the Cioni  _et al._ [21]’s two $Ch$ data set. The non-filled color symbols are
selected heat transfer data from prior liquid metal laboratory experiments.
All data displayed here are included in appendix tables 2 \- 5.

In addition, we have included the Nusselt number data for $Ch=0$, i.e. pure
Rayleigh-Bénard convection (RBC). Best fits to the RBC cases yield

$\displaystyle Nu_{0}$ $\displaystyle\approx(0.191\pm 0.088)Ra^{0.248\pm
0.025}\quad\text{for}\ \Gamma=1.0,$ (14) $\displaystyle Nu_{0}$
$\displaystyle\approx(0.176\pm 0.081)Ra^{0.246\pm 0.028}\quad\text{for}\
\Gamma=2.0,$ (15)

which are in good agreement with previous studies on the same device (RoMag)
[62, 46, 35, 47]. The differences between these two scaling laws lie within
their error bars but may be due to the different tank aspect ratios [63, 35,
47].

As discussed in section IV, our data show that the onset of MC in a cylinder
occurs via wall modes. The five-point stars at $Nu=1$ mark the magnetowall
mode onset predictions $Ra_{W}$ eq:Busse for the different $Ch$. Our near-
onset data at $Ch=10^{5}$ (yellow triangles) is in good agreement with the
onset prediction by Busse [26] (yellow star). However, Cioni  _et al._ [21]’s
heat transfer data at $Ch=2\times 10^{6}$ and $Ch=4\times 10^{6}$ have
$Ra_{crit}\approx 3Ra_{W}$. The lowest $Nu$ data from Cioni  _et al._ [21]’s
$Ch=2\times 10^{6}$ and $Ch=4\times 10^{6}$ align well with the bulk onset
prediction, $Ra^{\infty}_{NS}$. Zürner  _et al._ [24] have analysed the
$Nu$-$Ra$ trends and also found a large deviation between the experimental
results of Cioni  _et al._ [21] and King and Aurnou [46]. This discrepancy is
likely due to Cioni  _et al._ [21]’s thermometry setup, which used a single
thermistor at the center of each top and bottom boundary to measure $\Delta
T$. This setup was not designed to characterise wall modes and could only
detect the convective heat transfer occurring near the center of the tank.
Thus, top and bottom end wall temperature measurements nearer to the sidewall
are required in order to detect the onset of wall modes and to measure their
contributions to the total heat transfer [cf. 22, 24, 29].

###  V.3 Comparison between magnetoconvection and rotating convection in
liquid metal

The goal of this work is to provide a better understanding of the pathway from
convective onset to multimodal turbulence in liquid metal magnetoconvection.
Thus far, we have compared our laboratory-numerical data with the results of
other MC studies. Here we expand on this by comparing our MC data against
rotating convection data. Although the Lorentz and Coriolis forces both act to
constrain the convection in these systems [64], their data are rarely closely
compared since the vast majority of rotating convection (RC) studies are
carried out in moderate to high Prandtl fluids (non-metals), whereas MC
studies are nearly always made using low $Pr$ liquid metals [cf. 65].

Figure 7:  a) Normalised Nusselt number ($Nu/Nu_{0}$) versus magnetic Rossby
number ($Ro_{m}$) for MC lab data in $\Gamma=1.0$ and $2.0$ tanks. Here,
$Nu_{0}$ are the best-fit power laws in (V.2) and $Ro_{m}$ is the inverse of
interaction parameter $N$, as defined in (16). Symbols in thick black outlines
represent $\Gamma=2.0$ data, and those in thin grey outlines are $\Gamma=1.0$
data. The color of the symbols indicates $\mathrm{log_{10}}(Ch)$. The white
symbols are subcritical cases according to wall mode onset ($Ra<Ra_{W}$). The
symbol shapes do not contain information but help differentiate different
$Ch$. b) Rotating convective heat transfer data adapted from the $\Gamma=1.0$
liquid gallium experiments of King and Aurnou [63]. The color indicates
$\mathrm{log_{10}}(Ek^{-1})$. The vertical axis shows a reduced Nusselt number
$Nu/Nu_{0}=Nu/(0.185Ra^{0.25})$, following [63]. The horizontal axis is
convective Rossby number $Ro_{c}$, as defined in (16).

Figure 7 shows a side-by-side comparison of the convective heat transfer
efficiency $Nu/Nu_{0}$ as a function of the normalised buoyancy forcing in a)
our present liquid gallium $\Gamma=1.0$ and $2.0$ MC experiments and b) the
liquid gallium $\Gamma=1.0$ rotating convection data from King & Aurnou (2013)
[63]. The thicker outlined symbols in panel a) demarcate the $\Gamma=2.0$ MC
cases. The liquid gallium convection data in Figure 7 was all obtained using
the same experimental apparatus and setup. The fill color in a) denotes
$\log_{10}(Ch)$, whereas it denotes $\log_{10}(Ek^{-1})$ in panel b). The
Ekman number, $Ek=\nu/2\Omega H^{2}$, is the ratio of viscous and Coriolis
forces in rotating systems and $\Omega$ is the RC system’s angular rotation
rate.

The best co-collapse of the $Nu/Nu_{0}$ data sets was found when the buoyancy
force was normalised by the appropriate constraining force, that being Lorentz
in MC and Coriolis forces in RC. In MC, this non-dimensional ratio is called
the magnetic Rossby number, $Ro_{m}$, which formally describes the ratio of
convective inertia and the Lorentz force:

$Ro_{m}=\frac{\text{Inertia}}{\text{Lorentz}}=Re_{f\\!f}Ch^{-1}=\sqrt{\frac{RaCh^{-2}}{Pr}},$
(16)

where $Re_{f\\!f}=U_{f\\!f}H/\nu$. In the MHD literature, the reciprocal of
this ratio, which is called the interaction parameter $N=Ro_{m}^{-1}$ is often
employed [e.g., 28]. In RC, this non-dimensional ratio is called the
convective Rossby number, $Ro_{c}$,

$Ro_{c}=\frac{\text{Inertia}}{\text{Coriolis}}=Re_{f\\!f}Ek=\sqrt{\frac{RaEk^{2}}{Pr}},$
(17)

which shows up as a collapse parameter in a broad array of rotating convection
problems [e.g. 66, 56, 67]. Lorentz forces dominate in MC when $Ro_{m}=1/N$ is
small; Coriolis forces dominate in RC when $Ro_{c}$ is small. When these
Rossby numbers exceed unity, buoyancy-driven inertial forces should be
dominant and the convection is expected to be effectively unconstrained on all
available length scales in the system.

Comparing Figures 7a) and 7b), it is clear that the liquid metal MC and RC
data have similar gross morphologies. The $Nu/Nu_{0}$ is near unity and
effectively flat for both $Ro_{m}\gtrsim 1$ and $Ro_{c}\gtrsim 1$. Thus, when
the constraining Lorentz or Coriolis forces become subdominant to inertia in
either system, the heat transfer is similar to that found in unconstrained RBC
experiments.

The basic structures of MC and RC data are also similar at $Ro_{m}\lesssim 1$
and $Ro_{c}\lesssim 1$: the normalised heat transfer trends relatively sharply
downwards with decreasing Rossby number. However, the detailed structures of
the low $Ro_{m}$ and low $Ro_{c}$ data differ substantively. The data fall off
more steeply with Rossby in the rotating case, then it greatly flattens out in
the lowest $Nu/Nu_{0}$ RC cases. The differences in slope may be due to the
difference in Ekman pumping (EP) effects in both systems [68], although heat
transfer enhancement by EP is typically weak in metals since it is hard to
modify the thermal boundary layers in low $Pr$ flows.

Alternatively, these differences may be caused by the differences in critical
$Ra$ values and their scalings. For instance, in the parameter ranges explored
in Figure 7, oscillatory bulk convection first onsets in RC [47], whereas it
is the wall modes that develop first in MC. Further, the bulk
magnetoconvective onset scales asymptotically as $Ra_{crit}\sim Ch^{1}$
whereas bulk oscillatory convective onset asymptotically scales as
$Ra_{crit}\sim Ek^{-4/3}$. This $1/3$ difference in the scaling exponents may
imply that the available range of $Nu/Nu_{0}$ will be larger in the RC cases.
Further, the flat tail in the lowest $Nu/Nu_{0}$ RC data is likely due to the
low convective heat transfer efficiency of oscillatory rotating convection.

Thus, the gross structures of the two data compilations are similar in Figure
7. We hypothesise that their differences in our current data are likely due to
the various modal onset phenomena, as are clearly present in Figure 5, that
alter the low Rossby branches of each figure panel. However, it may be that
differences in MC and RC supercritical dynamics better explain these data [cf.
18, 69]. Regardless of the root cause, these low Rossby differences have thus
far thwarted our attempts to create a unified plot in which all the
$Nu/Nu_{0}$ data are simultaneously collapsed [cf. 70].

###  V.4 Summary

We have conducted a suite of laboratory thermal measurements of liquid gallium
magnetoconvection in cylindrical containers of aspect ratios $\Gamma=1.0$ and
$2.0$. Our data allow us to characterise liquid metal MC from wall mode onset
to multimodality. We performed a fixed $Ch=4\times 10^{4}$ survey of direct
numerical simulation for the same system in a $\Gamma=2.0$ cylindrical
geometry. Both laboratory and numerical methods obtained similar heat transfer
behaviors, with possible subtle differences in flow morphology. Together with
previous studies, our liquid metal heat transfer data comprise a convective
heat transfer survey over six orders of magnitude in both $Ra$ and in $Ch$
(Figure 6).

Busse [26]’s asymptotic solutions for magnetowall modes best collapse all our
MC heat transfer data, whereas the hybrid theoretical-numerical solutions by
Houchens  _et al._ [25] captures the exact onset for $\Gamma=1.0$, but the
onset for $\Gamma=2.0$ remains unverified. Better theoretical onset
predictions are needed for liquid metal MC in a cylindrical cell as a function
of $\Gamma$. This differs from liquid metal rotating convection where accurate
theoretical predictions currently exist for low-$Pr$ fluids in cylindrical
geometries [71, 72].

The MC flow morphology was characterised experimentally using a sidewall
thermistor array as well as the DNS temperature and velocity fields. The onset
of convection was verified to occur in the form of stationary (non-drifting)
magnetowall modes. These magnetowall modes develop nose-like protuberances
that extend into the fluid bulk with increasing supercriticality. At Rayleigh
numbers beyond the critical value for steady bulk convection, the noses
interact with the interior bulk modes, likely resulting in the apparent cell-
like flow patterns observed by Zürner  _et al._ [24]. Our data show that MC
convective heat transport is sensitive to the flow morphology, with the
Nusselt number $Nu$-$Ra$ data containing distinct kinks at these points where
the dominant convection mode appears to change.

Lastly, liquid metal heat transfer trends in magnetoconvection were compared
with rotating convection. The gross behavior of the heat transfer is
controlled by the magnetic and convective Rossby numbers, $Ro_{m}$ and
$Ro_{c}$, in the respective systems, with the normalised heat transport
$Nu/Nu_{0}$ approaching the RBC scaling as $Ro_{m}$ and $Ro_{c}$ approach
unity from below. The detailed trends at Rossby values less than unity show
clear differences between MC and RC. We have not yet deduced a scheme by which
it is possible to collapse all the liquid metal MC and RC data in a unified
way.

## Acknowledgments

We gratefully acknowledge the support of the NSF Geophysics Program (EAR
awards #1853196 and #2143939) and S.H. also thanks the EPSRC (grant
#EP/V047388/1). Thanks also go to Jewel Abbate and Taylor Lonner for providing
assistance with the experimental setup of RoMag, and to Meredith Plumley for
generously sharing her extracted $Nu$-$Ra$ data from King & Aurnou (2015) and
Burr & Muller (2011).

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##  VI Appendix

$\bm{Ch}$ | $\bm{Ra}$ | $\bm{Nu}$ | $\bm{Ch}$ | $\bm{Ra}$ | $\bm{Nu}$ | $\bm{Ch}$ | $\bm{Ra}$ | $\bm{Nu}$
---|---|---|---|---|---|---|---|---
0 | 1.93E+06 | 6.13 | 1.00E+04 | 5.64E+05 | 3.41 | 3.98E+04 | 1.93E+06 | 3.52
0 | 1.02E+06 | 5.16 | 1.01E+04 | 6.87E+05 | 3.74 | 4.00E+04 | 2.79E+06 | 4.48
0 | 4.86E+05 | 4.27 | 1.00E+04 | 9.30E+05 | 4.16 | 4.02E+04 | 4.03E+06 | 5.33
0 | 2.70E+05 | 3.75 | 1.02E+04 | 1.39E+06 | 4.89 | 4.15E+04 | 5.40E+06 | 6.06
0 | 1.52E+05 | 3.37 | 1.03E+04 | 2.15E+06 | 5.79 | 9.40E+04 | 1.65E+05 | 1.20
0 | 1.06E+05 | 3.16 | 1.04E+04 | 2.58E+06 | 6.23 | 9.40E+04 | 2.10E+05 | 1.15
0 | 3.81E+04 | 3.24 | 1.04E+04 | 3.23E+06 | 6.62 | 9.42E+04 | 3.39E+05 | 1.07
0 | 5.69E+04 | 3.99 | 1.06E+04 | 4.37E+06 | 7.42 | 9.44E+04 | 5.81E+05 | 1.28
0 | 5.26E+06 | 8.31 | 1.05E+04 | 5.40E+06 | 8.11 | 9.49E+04 | 8.26E+05 | 1.52
1.06E+04 | 9.98E+04 | 1.70 | 1.09E+04 | 6.01E+06 | 8.28 | 9.47E+04 | 1.09E+06 | 1.75
1.06E+04 | 1.29E+05 | 1.89 | 3.87E+04 | 9.24E+04 | 1.47 | 9.48E+04 | 1.28E+06 | 2.00
1.01E+04 | 1.51E+05 | 2.01 | 3.72E+04 | 1.56E+05 | 1.27 | 9.70E+04 | 1.63E+06 | 2.36
1.00E+04 | 1.64E+05 | 2.25 | 3.88E+04 | 1.96E+05 | 1.24 | 9.73E+04 | 1.79E+06 | 2.47
1.01E+04 | 1.70E+05 | 2.17 | 3.89E+04 | 2.87E+05 | 1.38 | 9.76E+04 | 1.92E+06 | 2.59
1.01E+04 | 2.17E+05 | 2.10 | 3.89E+04 | 4.48E+05 | 1.67 | 9.76E+04 | 1.99E+06 | 2.63
1.01E+04 | 2.13E+05 | 2.14 | 3.91E+04 | 6.07E+05 | 2.09 | 9.81E+04 | 2.35E+06 | 2.89
1.01E+04 | 2.30E+05 | 2.19 | 3.92E+04 | 7.86E+05 | 2.45 | 9.91E+04 | 3.51E+06 | 3.57
1.01E+04 | 2.50E+05 | 2.33 | 3.91E+04 | 9.43E+05 | 2.72 | 1.01E+05 | 4.96E+06 | 4.37
1.01E+04 | 2.52E+05 | 2.32 | 3.92E+04 | 1.33E+06 | 2.90 | 1.04E+05 | 6.61E+06 | 5.03
1.01E+04 | 2.96E+05 | 2.53 | 3.93E+04 | 1.33E+06 | 2.90 | 1.05E+05 | 8.08E+06 | 5.52
1.01E+04 | 4.23E+05 | 3.00 | 3.97E+04 | 1.95E+06 | 3.49 |  |  |
Table 2: Current study. Liquid gallium. $Pr=0.027$, $\Gamma=2$. $\bm{Ch}$ | $\bm{Ra}$ | $\bm{Nu}$ | $\bm{Ch}$ | $\bm{Ra}$ | $\bm{Nu}$ | $\bm{Ch}$ | $\bm{Ra}$ | $\bm{Nu}$
---|---|---|---|---|---|---|---|---
4.00E+04 | 1.5E+05 | 1.000 | 4.00E+04 | 4.0E+05 | 1.477 | 4.00E+04 | 1.50E+06 | 3.302
4.00E+04 | 2.0E+05 | 1.004 | 4.01E+04 | 7.0E+05 | 2.165 | 4.00E+04 | 4.0E+06 | 5.202
4.00E+04 | 3.0E+05 | 1.241 | 4.00E+04 | 1.0E+06 | 2.635 |  |  |
Table 3: Current study. DNS. $Pr=0.025$, $\Gamma=2$. $\bm{Ch}$ | $\bm{Ra}$ | $\bm{Nu}$ | $\bm{Ch}$ | $\bm{Ra}$ | $\bm{Nu}$ | $\bm{Ch}$ | $\bm{Ra}$ | $\bm{Nu}$
---|---|---|---|---|---|---|---|---
0 | 1.76E+06 | 6.95 | 1.11E+04 | 5.45E+07 | 16.40 | 4.43E+04 | 5.45E+07 | 16.40
0 | 1.78E+06 | 6.83 | 3.83E+04 | 1.88E+06 | 4.53 | 9.13E+04 | 4.63E+06 | 4.61
0 | 2.28E+06 | 7.17 | 3.84E+04 | 2.92E+06 | 5.28 | 9.13E+04 | 4.60E+06 | 4.61
0 | 4.86E+06 | 8.61 | 3.86E+04 | 3.83E+06 | 6.03 | 9.17E+04 | 5.98E+06 | 5.16
0 | 8.61E+06 | 9.98 | 3.86E+04 | 4.69E+06 | 6.57 | 9.21E+04 | 7.07E+06 | 5.54
0 | 1.33E+07 | 11.37 | 3.89E+04 | 5.53E+06 | 7.07 | 9.15E+04 | 8.40E+06 | 5.95
0 | 2.06E+07 | 13.10 | 3.85E+04 | 6.55E+06 | 7.61 | 9.10E+04 | 9.55E+06 | 6.29
9.55E+03 | 1.11E+06 | 5.66 | 3.86E+04 | 7.46E+06 | 8.04 | 9.12E+04 | 1.10E+07 | 6.69
9.35E+03 | 1.40E+06 | 6.10 | 3.86E+04 | 8.43E+06 | 8.48 | 9.19E+04 | 1.24E+07 | 7.06
9.60E+03 | 2.84E+06 | 7.53 | 3.91E+04 | 9.74E+06 | 8.99 | 9.53E+04 | 1.86E+07 | 8.53
9.65E+03 | 3.76E+06 | 8.21 | 3.86E+04 | 9.79E+06 | 8.92 | 9.44E+04 | 2.03E+07 | 8.83
9.61E+03 | 4.53E+06 | 8.66 | 3.87E+04 | 1.05E+07 | 9.19 | 9.55E+04 | 2.40E+07 | 9.46
9.60E+03 | 5.52E+06 | 9.04 | 3.88E+04 | 1.19E+07 | 9.59 | 9.62E+04 | 2.74E+07 | 9.99
9.55E+03 | 6.39E+06 | 9.37 | 3.89E+04 | 1.26E+07 | 9.80 | 9.65E+04 | 3.05E+07 | 10.45
9.58E+03 | 7.57E+06 | 9.70 | 3.88E+04 | 1.36E+07 | 10.09 | 9.82E+04 | 3.38E+07 | 10.99
9.62E+03 | 8.67E+06 | 10.04 | 3.92E+04 | 1.41E+07 | 10.31 | 9.89E+04 | 3.67E+07 | 11.41
9.75E+03 | 1.09E+07 | 10.52 | 3.90E+04 | 1.49E+07 | 10.40 | 1.02E+05 | 4.57E+07 | 12.60
9.77E+03 | 1.15E+07 | 10.68 | 3.93E+04 | 1.64E+07 | 10.83 | 2.88E+05 | 5.98E+06 | 3.43
9.94E+03 | 1.16E+07 | 10.88 | 3.95E+04 | 1.66E+07 | 10.75 | 2.75E+05 | 9.07E+06 | 4.13
9.78E+03 | 1.32E+07 | 11.07 | 3.97E+04 | 1.82E+07 | 11.10 | 2.88E+05 | 1.24E+07 | 4.67
9.92E+03 | 1.55E+07 | 11.49 | 3.95E+04 | 1.80E+07 | 11.22 | 2.90E+05 | 1.56E+07 | 5.38
1.00E+04 | 1.88E+07 | 12.04 | 3.96E+04 | 1.94E+07 | 11.58 | 2.90E+05 | 2.07E+07 | 6.24
1.01E+04 | 2.18E+07 | 12.53 | 3.98E+04 | 2.24E+07 | 12.19 | 2.90E+05 | 2.31E+07 | 6.61
1.03E+04 | 2.78E+07 | 13.31 | 4.09E+04 | 2.81E+07 | 13.24 | 2.89E+05 | 2.52E+07 | 6.84
1.03E+04 | 3.05E+07 | 13.68 | 4.13E+04 | 3.07E+07 | 13.69 | 2.96E+05 | 2.96E+07 | 7.34
1.07E+04 | 3.91E+07 | 14.67 | 4.26E+04 | 3.90E+07 | 14.82 | 2.97E+05 | 3.34E+07 | 7.83
1.09E+04 | 4.68E+07 | 15.53 | 4.31E+04 | 4.67E+07 | 15.55 | 2.88E+05 | 2.51E+07 | 6.86
Table 4: Current study. Liquid gallium. $Pr=0.027$, $\Gamma=1$.  $\bm{Ch}$ | $\bm{Ra}$ | $\bm{Nu}$ | $\bm{Ch}$ | $\bm{Ra}$ | $\bm{Nu}$ | $\bm{Ch}$ | $\bm{Ra}$ | $\bm{Nu}$
---|---|---|---|---|---|---|---|---
0 | 8.06E+06 | 8.676 | 0 | 2.53E+09 | 48.527 | 2.00E+06 | 4.99E+08 | 15.051
0 | 1.27E+07 | 9.620 | 0 | 2.67E+09 | 51.986 | 2.00E+06 | 6.77E+08 | 17.400
0 | 1.82E+07 | 10.667 | 7.22E+05 | 2.97E+07 | 4.103 | 2.00E+06 | 1.10E+09 | 21.055
0 | 2.35E+07 | 11.930 | 7.22E+05 | 5.06E+07 | 5.545 | 2.00E+06 | 1.57E+09 | 24.800
0 | 3.71E+07 | 13.228 | 7.22E+05 | 7.22E+07 | 6.936 | 2.00E+06 | 1.83E+09 | 26.111
0 | 3.81E+07 | 12.562 | 7.22E+05 | 9.08E+07 | 8.382 | 2.00E+06 | 2.07E+09 | 27.000
0 | 5.30E+07 | 14.542 | 7.22E+05 | 1.14E+08 | 9.703 | 3.93E+06 | 3.76E+07 | 1.000
0 | 6.83E+07 | 15.986 | 7.22E+05 | 1.67E+08 | 12.033 | 3.93E+06 | 4.97E+07 | 1.578
0 | 1.08E+08 | 17.879 | 7.22E+05 | 2.21E+08 | 14.050 | 3.93E+06 | 6.41E+07 | 1.940
0 | 1.54E+08 | 18.989 | 7.22E+05 | 2.85E+08 | 15.578 | 3.93E+06 | 6.74E+07 | 1.875
0 | 2.09E+08 | 21.055 | 7.22E+05 | 4.17E+08 | 18.989 | 3.93E+06 | 9.37E+07 | 2.930
0 | 2.44E+08 | 21.055 | 7.22E+05 | 5.81E+08 | 20.696 | 3.93E+06 | 1.24E+08 | 3.998
0 | 2.77E+08 | 22.556 | 7.22E+05 | 9.66E+08 | 25.445 | 3.93E+06 | 1.24E+08 | 4.000
0 | 3.40E+08 | 23.347 | 7.22E+05 | 1.34E+09 | 29.455 | 3.93E+06 | 1.55E+08 | 5.001
0 | 4.98E+08 | 25.011 | 2.00E+06 | 1.80E+07 | 1.120 | 3.93E+06 | 1.86E+08 | 5.940
0 | 5.11E+08 | 25.665 | 2.00E+06 | 2.14E+07 | 1.448 | 3.93E+06 | 2.58E+08 | 7.691
0 | 5.80E+08 | 26.564 | 2.00E+06 | 2.62E+07 | 1.970 | 3.93E+06 | 3.33E+08 | 9.290
0 | 6.59E+08 | 27.259 | 2.00E+06 | 4.14E+07 | 2.958 | 3.93E+06 | 4.18E+08 | 10.667
0 | 7.67E+08 | 27.972 | 2.00E+06 | 6.06E+07 | 3.960 | 3.93E+06 | 5.53E+08 | 13.800
0 | 8.72E+08 | 28.458 | 2.00E+06 | 9.34E+07 | 5.358 | 3.93E+06 | 5.97E+08 | 13.228
0 | 9.90E+08 | 28.952 | 2.00E+06 | 9.58E+07 | 5.180 | 3.93E+06 | 7.90E+08 | 15.400
0 | 1.28E+09 | 31.284 | 2.00E+06 | 1.17E+08 | 6.419 | 3.93E+06 | 1.22E+09 | 19.319
0 | 1.57E+09 | 32.103 | 2.00E+06 | 1.47E+08 | 7.490 | 3.93E+06 | 1.69E+09 | 22.600
0 | 1.87E+09 | 35.904 | 2.00E+06 | 2.10E+08 | 9.375 | 3.93E+06 | 1.97E+09 | 23.147
0 | 2.29E+09 | 36.843 | 2.00E+06 | 2.78E+08 | 11.000 | 3.93E+06 | 2.24E+09 | 24.600
0 | 2.35E+09 | 40.155 | 2.00E+06 | 3.50E+08 | 12.455 | 3.93E+06 | 2.54E+09 | 25.887
0 | 2.41E+09 | 43.764 | 2.00E+06 | 4.99E+08 | 15.600 |  |  |
Table 5: Cioni  _et al._ [21]. Liquid mercury. $Pr=0.025$, $\Gamma=1$.  $\bm{Ch}$ | $\bm{Ra}$ | $\bm{Nu}$ | $\bm{Ch}$ | $\bm{Ra}$ | $\bm{Nu}$ | $\bm{Ch}$ | $\bm{Ra}$ | $\bm{Nu}$
---|---|---|---|---|---|---|---|---
0 | 7.21E+02 | 1.00 | 0 | 5.76E+03 | 1.27 | 670 | 1.52E+03 | 1.00
0 | 8.18E+02 | 1.01 | 0 | 6.13E+03 | 1.34 | 670 | 2.74E+03 | 1.01
0 | 1.12E+03 | 1.01 | 0 | 7.09E+03 | 1.38 | 670 | 4.24E+03 | 1.00
0 | 1.30E+03 | 1.01 | 0 | 6.57E+03 | 1.40 | 670 | 6.00E+03 | 1.01
0 | 1.46E+03 | 1.02 | 0 | 8.49E+03 | 1.44 | 670 | 7.04E+03 | 1.01
0 | 1.86E+03 | 1.03 | 0 | 9.10E+03 | 1.48 | 670 | 9.17E+03 | 1.01
0 | 2.12E+03 | 1.03 | 0 | 1.05E+04 | 1.49 | 670 | 8.26E+03 | 1.01
0 | 2.27E+03 | 1.04 | 0 | 1.07E+04 | 1.54 | 670 | 1.16E+04 | 1.03
0 | 2.50E+03 | 1.05 | 0 | 1.11E+04 | 1.55 | 670 | 1.03E+04 | 1.02
0 | 3.35E+03 | 1.11 | 0 | 1.31E+04 | 1.61 | 670 | 1.25E+04 | 1.03
0 | 3.52E+03 | 1.14 | 0 | 1.12E+04 | 1.63 | 670 | 1.34E+04 | 1.04
0 | 3.77E+03 | 1.16 | 0 | 1.20E+04 | 1.64 | 670 | 1.37E+04 | 1.05
0 | 4.15E+03 | 1.20 | 0 | 1.30E+04 | 1.68 | 670 | 1.44E+04 | 1.05
0 | 4.71E+03 | 1.23 | 0 | 1.32E+04 | 1.71 | 670 | 1.48E+04 | 1.06
0 | 4.55E+03 | 1.25 | 0 | 1.48E+04 | 1.71 | 670 | 1.53E+04 | 1.08
0 | 5.22E+03 | 1.25 |  |  |  |  |  |
Table 6: Aurnou and Olson [60]. Liquid gallium. $Pr=0.023$, $\Gamma=6$.  $\bm{Ch}$ | $\bm{Ra}$ | $\bm{Nu}$ | $\bm{Ch}$ | $\bm{Ra}$ | $\bm{Nu}$ | $\bm{Ch}$ | $\bm{Ra}$ | $\bm{Nu}$
---|---|---|---|---|---|---|---|---
0 | 1.74E+03 | 1.05 | 0 | 2.65E+04 | 1.80 | 400 | 4.92E+04 | 1.48
0 | 2.29E+03 | 1.06 | 0 | 3.64E+04 | 1.95 | 400 | 6.13E+04 | 1.60
0 | 3.08E+03 | 1.10 | 0 | 4.59E+04 | 2.08 | 400 | 7.39E+04 | 1.69
0 | 4.72E+03 | 1.23 | 0 | 5.58E+04 | 2.19 | 400 | 8.46E+04 | 1.78
0 | 6.92E+03 | 1.35 | 400 | 9.27E+03 | 1.00 | 1600 | 3.22E+04 | 1.01
0 | 9.19E+03 | 1.39 | 400 | 1.49E+04 | 1.05 | 1600 | 4.56E+04 | 1.07
0 | 1.07E+04 | 1.45 | 400 | 2.11E+04 | 1.12 | 1600 | 6.22E+04 | 1.18
0 | 1.49E+04 | 1.59 | 400 | 2.68E+04 | 1.20 | 1600 | 7.60E+04 | 1.29
0 | 1.92E+04 | 1.67 | 400 | 3.65E+04 | 1.30 | 1600 | 9.10E+04 | 1.40
Table 7: Burr and Müller [61]. Liquid Na-K alloy. $0.017<Pr<0.021$, rectangular box $20:10:1$.  $\bm{Ch}$ | $\bm{Ra}$ | $\bm{Nu}$ | $\bm{Ch}$ | $\bm{Ra}$ | $\bm{Nu}$ | $\bm{Ch}$ | $\bm{Ra}$ | $\bm{Nu}$
---|---|---|---|---|---|---|---|---
9.46E+03 | 2.37E+06 | 7.23 | 4.74E+04 | 8.44E+06 | 8.40 | 2.85E+05 | 1.09E+07 | 4.89
9.50E+03 | 3.30E+06 | 7.78 | 9.35E+04 | 3.81E+06 | 4.50 | 2.85E+05 | 1.34E+07 | 5.29
9.54E+03 | 4.32E+06 | 8.31 | 9.39E+04 | 5.02E+06 | 5.11 | 9.35E+05 | 7.39E+06 | 2.31
9.60E+03 | 5.95E+06 | 8.94 | 9.43E+04 | 6.33E+06 | 5.67 | 9.40E+05 | 9.80E+06 | 2.61
9.58E+03 | 7.53E+06 | 9.42 | 9.49E+04 | 8.25E+06 | 6.46 | 9.46E+05 | 1.23E+07 | 2.91
9.58E+03 | 1.06E+07 | 10.20 | 9.48E+04 | 1.01E+07 | 7.05 | 9.54E+05 | 1.59E+07 | 3.37
9.74E+03 | 2.80E+07 | 13.00 | 9.47E+04 | 1.33E+07 | 8.11 | 9.54E+05 | 1.90E+07 | 3.76
1.01E+04 | 5.08E+07 | 15.00 | 9.53E+04 | 1.89E+07 | 9.56 | 9.57E+05 | 2.53E+07 | 4.29
4.68E+04 | 3.10E+06 | 5.52 | 9.96E+04 | 5.40E+07 | 14.10 | 9.67E+05 | 3.48E+07 | 5.25
4.70E+04 | 4.11E+06 | 6.24 | 2.80E+05 | 5.23E+06 | 3.27 | 1.02E+06 | 8.85E+07 | 8.84
4.72E+04 | 5.13E+06 | 6.99 | 2.81E+05 | 6.70E+06 | 3.83 | 1.24E+06 | 1.70E+08 | 11.90
4.75E+04 | 6.81E+06 | 7.81 | 2.83E+05 | 8.37E+06 | 4.28 |  |  |
Table 8: King and Aurnou [46]. Liquid gallium. $Pr=0.025$, $\Gamma=1$.  $\bm{Ch}$ | $\bm{Ra}$ | $\bm{Nu}$ | $\bm{Ch}$ | $\bm{Ra}$ | $\bm{Nu}$ | $\bm{Ch}$ | $\bm{Ra}$ | $\bm{Nu}$
---|---|---|---|---|---|---|---|---
1.73E+02 | 4.18E+06 | 7.59 | 1.73E+04 | 2.51E+07 | 11.80 | 2.12E+05 | 3.34E+07 | 8.84
1.73E+02 | 6.27E+06 | 8.10 | 1.73E+04 | 3.34E+07 | 12.30 | 2.12E+05 | 5.85E+07 | 12.00
1.73E+02 | 1.05E+07 | 9.19 | 1.74E+04 | 5.78E+07 | 15.50 | 4.32E+05 | 1.05E+07 | 3.18
1.73E+02 | 1.67E+07 | 11.00 | 1.74E+04 | 5.82E+07 | 15.50 | 4.32E+05 | 1.05E+07 | 3.67
1.72E+02 | 3.34E+07 | 12.50 | 1.73E+04 | 5.84E+07 | 15.00 | 4.32E+05 | 1.05E+07 | 3.18
1.74E+02 | 5.82E+07 | 15.70 | 3.89E+04 | 1.05E+07 | 7.67 | 4.32E+05 | 1.05E+07 | 3.41
1.78E+02 | 5.91E+07 | 15.70 | 6.92E+04 | 6.26E+06 | 5.48 | 4.32E+05 | 1.67E+07 | 4.98
4.32E+03 | 4.18E+06 | 7.49 | 6.92E+04 | 1.05E+07 | 6.62 | 4.33E+05 | 3.34E+07 | 7.20
4.32E+03 | 6.27E+06 | 7.71 | 6.92E+04 | 1.67E+07 | 8.55 | 4.33E+05 | 5.86E+07 | 10.10
4.32E+03 | 1.05E+07 | 9.23 | 6.92E+04 | 2.51E+07 | 9.77 | 7.31E+05 | 1.67E+07 | 3.63
4.33E+03 | 1.67E+07 | 10.90 | 6.93E+04 | 3.34E+07 | 11.00 | 7.31E+05 | 3.34E+07 | 5.90
4.33E+03 | 2.51E+07 | 11.60 | 6.93E+04 | 3.34E+07 | 10.80 | 7.32E+05 | 5.86E+07 | 8.89
4.33E+03 | 3.34E+07 | 12.30 | 6.94E+04 | 5.84E+07 | 14.40 | 1.11E+06 | 1.67E+07 | 2.97
4.34E+03 | 5.81E+07 | 15.20 | 6.94E+04 | 5.85E+07 | 14.00 | 1.11E+06 | 3.34E+07 | 4.75
1.73E+04 | 4.18E+06 | 5.99 | 6.94E+04 | 5.85E+07 | 14.40 | 1.11E+06 | 3.34E+07 | 5.12
1.73E+04 | 6.27E+06 | 7.28 | 6.94E+04 | 5.86E+07 | 14.30 | 1.11E+06 | 3.34E+07 | 4.88
1.73E+04 | 1.05E+07 | 9.07 | 2.12E+05 | 1.05E+07 | 5.05 | 1.11E+06 | 5.85E+07 | 7.63
1.73E+04 | 1.67E+07 | 10.90 | 2.12E+05 | 1.67E+07 | 6.75 |  |  |
Table 9: Zürner  _et al._ [24]. Liquid GaInSn. $Pr=0.029$, $\Gamma=1$.


# Multi-wavelength Study of Magnetic Field and Turbulence in the Monogem
Pulsar TeV halo

Sunil Malik Institute fur Physik und Astronomie Universitat Potsdam, Golm Haus
28, D-14476 Potsdam, Germany  Deutsches Elektronen-Synchrotron DESY,
Platanenallee 6, 15738 Zeuthen, Germany<EMAIL_ADDRESS>Ka Ho Yuen
Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545,
USA<EMAIL_ADDRESS>Huirong Yan Institute fur Physik und Astronomie
Universitat Potsdam, Golm Haus 28, D-14476 Potsdam, Germany  Deutsches
Elektronen-Synchrotron DESY, Platanenallee 6, 15738 Zeuthen, Germany
<EMAIL_ADDRESS>(Corresponding author)

###### Abstract

Magnetic fields are ubiquitous in the interstellar medium, including extended
objects such as supernova remnants (SNRs) and Pulsar Wind Nebulae (PWNe). Its
turbulent characteristics govern the diffusion of cosmic rays and the multi-
wavelength emission from PWNe. However, the geometry and turbulence nature of
the magnetic fields in the ambient region of PWN is still unknown. Recent
gamma-ray observations from HAWC and synchrotron observations suggest a highly
suppressed diffusion coefficient compared to the mean interstellar value. In
this letter, we present the first direct observational evidence that the local
mean magnetic field is nearly aligned toward the line of sight (LoS) with an
inclination angle $\theta_{\lambda}<10^{\circ}$ employing a recently developed
statistical recipe known as ‘Y-parameter’. Furthermore, we report that the
magnetic field fluctuations are mostly dominated by compressible modes, with a
2D correlation length of approximately $3\ {\rm pc}$ in the vicinity of
Monogem PWN region. Our study highlights the pivotal role of magnetic field
and turbulence in unraveling the physical processes in TeV halos and cosmic
ray transport.

Magnetic field – polarization, Stokes parameters, objects: Pulsar wind
nebulae, general – interstellar medium, Turbulence – techniques: synchrotron
radiation, Cosmic Ray emission, Star formation, Particle Transport

##  1 Introduction

The interstellar medium (ISM) and extended objects such as pulsar wind nebulae
(PWNe) and supernova remnants (SNRs) are permeated by the magnetic fields.
This magnetic field exhibits a complex structure, spanning a wide range of
strengths from $\mu{\rm G}$ to mG (Reynolds et al., 2012). The turbulent
characteristics of this magnetic field play a crucial role in facilitating
various physical processes, including particle transport and acceleration
(Schlickeiser, 2002; Yan & Lazarian, 2002; Yan et al., 2012; Lemoine, 2022;
Yan, 2022). Magnetohydrodynamic (MHD) turbulence is classified into the
incompressible Alfvénic mode and the compressible modes (Cho & Lazarian, 2003;
Lazarian & Pogosyan, 2012; Makwana & Yan, 2020), and it interacts
significantly with cosmic rays, governing their scattering and diffusion
across different phases of the ISM as well as in the ambient regions of
extended objects (Yan & Lazarian, 2004, 2008). Therefore, to comprehensively
understand cosmic ray diffusion and multi-wavelength emissions in the vicinity
of PWNe, it is essential to investigate the underlying magnetic field
configuration and the nature of the associated turbulence.

Recent studies (Liu et al., 2019b; Breuhaus et al., 2022; Mukhopadhyay &
Linden, 2022) have investigated the TeV’s halo observations from PWNe using a
theoretical spectrum generated through a combination of synchrotron radiation
and inverse-Compton (IC) scattering from the same population of energetic
electrons. The authors suggest that the suppressed diffusion in the ambient
region of PWNe can be attributed to the topology of the local magnetic fields.

However, the direct observation of the orientation and strength of the
magnetic field in the ambient halo region of pulsars has proven to be
challenging. To address this issue, our recent study has employed turbulence
mapping theory based on synchrotron emission statistics, presenting a novel
statistical approach using the Y-parameter (Malik et al., 2023). This
technique has the potential to unveil the 3D geometry of magnetic fields and
their dominant MHD turbulence characteristics. In this paper, we utilize the
Y-parameter technique to provide the first observational investigation to
constrain the magnetic field direction within these halos.

This paper is structured as follows: In Section 2, we examine the
characteristics of the PWN TeV halo, with particular emphasis on the Monogem
halo. Our primary analysis of diagnosing the local magnetic field inclination
angle, utilizing the recently developed turbulence anisotropy analysis and the
multi-points structure function, is described in Section 3. The discussions
and summary of our findings are presented in Section 4 and 5, respectively.

##  2 Characteristic of PWN TeV halo

The HAWC collaboration recently made notable discoveries of potential TeV
sources, namely 2HWC J0635+180 and 2HWC J0700+143 (Abeysekara et al., 2017a),
in the nearby PWN Geminga and PSR B0656+14 (Monogem) at significance levels of
13.1$\sigma$ and 8.1$\sigma$ respectively (Abeysekara et al., 2017a, b). These
sources are located at a relatively short distance from Earth, with Geminga at
$250\ {\rm pc}$ and Monogem at $288\ {\rm pc}$. The diffusive propagation of
particles within these halos extends to approximately $10-50\ {\rm pc}$, which
points to a much slower diffusion coefficient compared to the Galactic mean,
presenting a challenge in comprehending the underlying mechanisms responsible
for the observed high-energy emissions (Abeysekara et al., 2017b).

In Liu et al. (2019b), the authors proposed a promising model for explaining
the TeV observations of the Geminga PWN using anisotropic diffusion. In the
case of sub-Alfvénic turbulence, the diffusion coefficients perpendicular and
parallel to the mean magnetic field show significant differences (i.e.,
$D_{\perp}\sim D_{||}M_{A}^{4}$) (where $D_{\perp}$, $D_{\parallel}$ is
diffusion coefficient in perpendicular and parallel direction of the magnetic
field, respectively, $M_{A}$ represents Alfvénic Mach Number) (Yan & Lazarian,
2008; Giacinti & Sigl, 2012; López-Coto & Giacinti, 2018; Maiti et al., 2022).
The authors of this theoretical model inferred that local magnetic fields with
small inclination angles relative to the line of sight could account for the
observed TeV emissions and the phenomenon of suppressed diffusion. Currently,
direct measurements of the magnetic field strength and orientation are not
feasible due to the absence of synchrotron polarized emissions in the vicinity
of the Geminga PWN.

To get the first direct observational evidence, in this study, we focus on
Monogem PWN (PSR B0656+14) which is observed in multi-wavelength from radio to
gamma rays (Uyaniker et al., 1999; Johnston et al., 2006, 2007; Abeysekara et
al., 2017b). It has a period (P) of approximately $385\ {\rm ms}$, period
derivative ($\dot{\rm P}\sim 5.4\times 10^{-14}\ {\rm s}\ {\rm s}^{-1}$)
corresponding111See e.g. Lorimer & Kramer (2004) for the formulae used to
derive $\rm{\tau}$, $\dot{{\rm E}}_{rot}$ and $\rm{{\rm B}_{surf}}$ using
${\rm P}$ and ${\rm\dot{\rm P}}$ to a spin-down age ($\tau$) of approximately
$111\ {\rm kyr}$, a spin-down power (${\rm E}_{rot}$) of $3.8\times 10^{34}$
erg ${\rm s}^{-1}$, and a surface magnetic field (${\rm B}_{surf}$) of
approximately $4.65\times 10^{12}$ G.

##  3 3D magnetic field from turbulence analysis

###  3.1 Synchrotron Polarisation Observations

In order to investigate the strength and morphology of magnetic fields in the
Monogem PWN TeV halo region, we utilized the Effelsberg 100-m 21 cm
POL222https://www3.mpifr-bonn.mpg.de/survey.html survey. This survey,
conducted at medium Galactic latitudes, involves radio continuum and
polarization observations. The data were obtained using the Effelsberg 100-m
telescope at a frequency of $1.4\ {\rm GHz}$, with an angular resolution of
$9.35^{{}^{\prime}}$ (Uyaniker et al., 1999).

We specifically employed $5\degree\times 5\degree$ synchrotron polarization
observations of the Monogem (B0656+14) pulsar region, as depicted in Fig. 1.
The observation map was overlaid with streamlines computed using the Line
Integral Convolution (LIC) algorithm, which indicates the inferred direction
of the magnetic field based on the polarization direction (Cabral & Leedom,
1993; Zöckler et al., 1997). The figure exhibits a significant increase in
linearly polarized flux in and around the vicinity of the Monogem Pulsar,
which is indicated by the black star symbol. The observed variations in
streamlining suggest a 2D Alfvén Mach number in the pictorial plane
approximately $M_{A,2D}\approx 1.0$. Utilizing this map, we examined the
magnetic field topology in the surrounding region of the Monogem PWN.

Figure 1: The synchrotron polarisation emission map of Monogem (B0656+14) PWN
($5\degree\times 5\degree$) with the center indicated by star symbol at ${\rm
R.A.(J2000)}={\rm 06^{h}59^{m}48^{s}.02}\ (104.94^{\circ}),{\rm
Dec.(J2000)}=14\degree 14^{{}^{\prime}}19^{{}^{\prime\prime}}.4\
(14.23^{\circ})$ using Effelsberg 100-m telescope observation at $1.4\ {\rm
GHz}$ from Uyaniker et al. (1999). Here the color bar represents the flux in
mJy/beam. The streamline, computed via the Line Integral Convolution (LIC)
algorithm (Cabral & Leedom 1993, Zöckler et al. 1997), represents the
direction of the magnetic field inferred from the synchrotron polarisation
observations.

###  3.2 Correlation length from Structure-Function Analysis

Figure 2: The multi-point structure function for the total polarised
intensity. Different colors correspond to different points’ structure
functions. The vertical and horizontal lines indicate the plateau stage. These
different structure functions are estimated by using Eq. 1 & 2.

In polarisation observations, the large-scale magnetic fields can coexist with
the small-scale turbulent components. In recent studies (Hildebrand et al.,
2009; Pattle et al., 2017; Cho, 2019) several approaches have been discussed
to address this issue, including the fitting method and wavelet transformation
(Grebenev et al., 1995). These techniques heavily rely on data coarsening. One
widely accepted and commonly used method for assessing fluctuations is through
two-point structure function (SF), defined as follows:

${\rm SF^{2pt}_{2}}({\bf R})=\langle|X({\bf R}^{\prime}+{\bf R})-X({\bf
R})|^{2}\rangle_{{\bf R}^{\prime}}$ (1)

where $X$ represents an observable in the plane of the sky. ${\rm
SF^{2pt}_{2}}$ will be an increasing function of ${\bf R}$ when we are in the
regime of ${\bf R}<{\rm L_{s}}$, here ${\rm L_{s}}$ represents the correlation
length of small-scale fluctuations. It will attain a plateau for ${\bf
R}\gtrsim{\rm L_{s}}$. However, the shape of ${\rm SF^{2pt}_{2}}$ gets
distorted in the presence of large-scale gradients in the observables. In this
case, three and more point structure functions can be utilized to remove these
large-scale variations (Cho, 2019). The three points SF is expressed as;

${\rm SF^{3pt}_{2}}({\bf R})=\frac{1}{3}\langle|X({\bf R}^{\prime}-{\bf
R})-2X({\bf R}^{\prime})+X({\bf R}^{\prime}+{\bf R})|^{2}\rangle_{{\bf
R}^{\prime}}.$ (2)

The two-point structure function is represented by the red line in Fig 2. From
the plot, it is evident that our region near Monogem PWN exhibits both large-
scale gradients and small-scale fluctuations in the magnetic field. The three
points ($3\ {\rm pts}$) SF is depicted by the blue line in Fig.2. We want to
emphasize that evaluating structure functions with more than two points limits
us to smaller length scales, as evident in our SF plot in Fig.2.

Consequently, we determined that the region possesses a characteristic
correlation length of approximately $3\ {\rm pc}$ for the local MHD
turbulence, based on the synchrotron map. Furthermore, we can employ this 2D
correlation length, $\tt R\sim 3\ {\rm pc}$, to conduct a turbulence
anisotropy analysis known as the ‘Y-parameter’ (Malik et al., 2023).

Figure 3:  Left panel: This figure shows the correlation function distribution
for the $I+Q\propto B_{x}^{2}$. Right panel: It is same as the left but for
$I-Q\propto B_{y}^{2}$. Here we have drawn a vertical arrow to indicate the
direction of the mean B-field in the $5\degree\times 5\degree$ region of
interest around the Monogem pulsar.

###  3.3 Line of sight inclination angle of B-field from Y-parameter Analysis

To measure the relative angle between the mean magnetic field and the line of
sight, we employ a recently developed technique called the ”y-parameter
analysis” (Malik et al., 2023). It was suggested in earlier literature that
the anisotropy of the Stokes parameters can be used to infer the geometric
properties of the magnetic field. The global correlation function, represented
as $D_{X}$, can effectively capture the anisotropy of a given observable,
denoted as $X$, which can be expressed as follows:

$D_{X}({\bf R})=\langle(X({\bf R}^{\prime})X({\bf R}^{\prime}+{\bf
R}))^{2}\rangle_{{\bf R}^{\prime}}.$ (3)

Malik et al. (2023) suggest that the ‘Y-parameter”, which is expressed as:

$Y=\frac{\text{Anisotropy}(D_{I+Q})}{\text{Anisotropy}(D_{I-Q})}=\frac{v/h(D_{I+Q})}{v/h(D_{I-Q})}.$
(4)

can be used to trace the inclination angle of the magnetic field. Here, $v$
and $h$ represent the vertical and horizontal dimensions relative to B⟂. This
statistical parameter, derived from polarized synchrotron radiation, captures
the variations in the embedded magnetic fields resulting from turbulence.
Consequently, it enables us to investigate the geometry of the 3D magnetic
field and the characteristics of the underlying MHD turbulence. We established
a statistical criterion of ${\rm Y}\sim 1.5\pm 0.5$ (with ${\rm Y}>1.5$
indicating the A mode and ${\rm Y}<1.5$ indicating the C mode) to identify the
dominant fraction of MHD turbulence modes. Remarkably, the Y-parameter for
different modes exhibited contrasting trends (either decreasing or increasing)
with respect to the local mean magnetic field inclination angle, denoted as
$\theta_{\lambda}$.

To estimate the mean-field inclination angle for the vicinity of the Monogem
TeV halo, we applied the technique described earlier to synchrotron
polarization observations (Fig.1) within a ($5^{\circ}\times 5^{\circ}$)
region around the Monogem pulsar. We initially analyzed the correlation
function of $I+Q\propto B_{x}^{2}$ and $I-Q\propto B_{y}^{2}$ and plotted the
results in Fig. 3. These correlation function plots clearly depict the
difference in horizontal and vertical lengths relative to the mean magnetic
field direction. To quantify the anisotropies, we performed 2D Gaussian
fittings on these distributions by keeping the fitting radius within the 2D
correlation length of 3.0 pc and obtained ${\rm Y}\approx 0.83$. According to
our previous investigation in Malik et al. (2023), a ${\rm Y}<1.0$ implies a
mean magnetic field inclination angle of $\theta_{\lambda}<10^{\circ}$ and
indicates that the local turbulence is predominantly of compressible nature.

These results clearly indicate that the vicinity of the PWNe exhibits a mean
magnetic field that is potentially aligned with the line of sight, along with
the presence of compressible turbulence. This alignment is crucial for the
propagation of cosmic rays (Yan & Lazarian, 2008). Importantly, our analysis
provides direct observational confirmation for the speculation that the
magnetic fields in such TeV halos are not in the extreme limit, as suggested
in the existing literature. This finding offers reassurance regarding the
effectiveness of the Y-parameter technique in retrieving 3D magnetic fields in
various environments, including the ISM and extended objects such as PWNe and
SNRs.

##  4 Discussion

As discussed in Section 1, MHD turbulence plays a crucial role in the
acceleration and scattering of cosmic rays in the ISM and other extended
objects. The recent discovery of TeV halos around middle-aged pulsars, such as
Geminga and Monogem (PSR B0656+14), has raised intriguing questions,
particularly the origin of the slower diffusion (Abeysekara et al., 2017b). In
a recent study by Liu et al. (2019b), it was proposed that sub-Alfvenic
turbulence combined with a local mean magnetic field aligned with the line of
sight can account for the missing X-ray flux as well as the slow diffusion
since $D_{\perp}=D_{\parallel}M_{A}^{4}$. Therefore, to obtain a comprehensive
understanding of these observations, it is crucial to directly investigate the
3D magnetic field strength, orientation, and the underlying MHD turbulence.

To investigate the magnetic field topology in extended objects such as TeV
halos associated with PWNe and supernova remnants, direct observational
evidence is currently lacking. A recent study by Liu et al. (2019a) has
suggested an upper limit of $0.8\ \mu{\rm G}$ for the magnetic field in the
Geminga PWN, which is lower than the average galactic magnetic field strength
($\sim 5.5\ \mu G$, Sofue et al. 2019). The absence of radio-polarized
emission in the vicinity of Geminga has hindered direct observations of the
magnetic field strength and associated turbulence in that region.

In this study, we employ radio synchrotron polarization statistics from a
($5\degree\times 5\degree$) region around the Monogem PWN to infer the nature
of the 3D magnetic field and its turbulence using our Y-parameter technique,
as described in Malik et al. (2023). Our analysis yielded a value of ${\rm
Y\sim 0.83}$ based on the 2D coherence length scale estimated using the
structure-function in Fig.2. Following the criteria established in Malik et
al. (2023), we have identified two potential scenarios: one with
$\theta_{\lambda}\lesssim 10^{\circ}$ dominated by compressible modes, and the
other with $\theta_{\lambda}>60^{\circ}$ characterized by Alfv’enic
turbulence. Analysis of simulated MHD datacubes favours the first scenario
over the others.

Furthermore, in the Monogem TeV halo, the required diffusion coefficient is
also significantly suppressed compared to the ISM value, similar to the case
of the Geminga TeV halo. This suppressed diffusion, combined with the small
$\theta_{\lambda}$ angle, suggests that cross-field transport with
$D_{\perp}\propto D_{\|}M_{A}^{4}$ serves as the underlying mechanism for the
reduced diffusion (Yan & Lazarian, 2008). Assuming $D_{\|}\simeq D_{ISM}\sim
3.8\times 10^{28}$, the average ISM diffusion coefficient as adopted in Liu et
al. (2019b), we estimate a local Alfvénic Mach number $M_{A}\sim 0.2$. The
results obtained using various diagnostics are summarized in the table below.

Table 1: We have listed all the estimated quantities in this study. Diagnostics | Measure | Values
---|---|---
Y-parameter | $\theta_{\lambda}$ | $<10^{\circ}$
Structure Function | $L_{coh,2D}$ |  $\sim 3.0$ pc
Polarisation Angle | $M_{A,2D}$ | 1.0
${D_{\perp}/D_{\|}}$ | $M_{A,3D}$ | $\gtrsim 0.2$

Additionally, in the halo-like environment, there are limited methods to gain
insight into the strength of the parallel magnetic field component ($B_{\|}$).
The rotation and dispersion measures of the pulsar might offer some clues
about the $B_{\|}$333Using RM of $22.73\pm 8.0\ {\rm rad}\ {\rm m}^{-2}$ and
DM of $13.7\pm 0.2$ pc cm-3, we can estimate the $B_{\|}\sim 2.04\pm
0.72\mu$G. component of the magnetic field, but are also affected by
foreground contamination. In any case, the magnetic field orientation can not
be inferred from the observations of pulsars themselves. Therefore, our study
facilitates a pivotal advancement in the understanding of magnetism and the
underlying physical process governing the TeV-PeV halos associated with PWN
and SNRs.

##  5 Conclusions

In this letter, we have presented the first observation analysis of magnetic
field 3D geometry and underlying MHD turbulence nature in the TeV halo
associated with the Monogem PWN using various probes from multi-wavelength
observations. Our structure-function analysis shows that the region has a
pictorial plane correlation length of $\sim 3\ {\rm pc}$ for the local MHD
turbulence. Further, the Y-parameter analysis using synchrotron polarisation
observations with ${\rm Y}\sim 0.83$ suggests that the local mean magnetic
field has an inclination angle $\theta_{\lambda}<10^{\circ}$ and is dominated
by compressible turbulence fluctuations, both of which contribute to the
suppressed diffusion of cosmic rays as observed by HAWC (Abeysekara et al.,
2017b).

Thus our study underpins the crucial role of the magnetic field in
understanding the physical processes involved in recent and future detections
of such extraordinary high-energy emissions in the Milky Way.

## Acknowledgments

SM would like to thank Parth Pavaskar, Siqi Zhao, and Bingqiang Qiao for the
helpful discussions. The research presented in this article was partially
supported by the Laboratory Directed Research and Development program of Los
Alamos National Laboratory under project number(s) 20220700PRD1.

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