# Quasi-static magnetoconvection with a tilted magnetic field

Justin A. Nicoski  Department of Physics, University of Colorado, Boulder,
Colorado 80309, USA  Ming Yan  Department of Physics, University of Colorado,
Boulder, Colorado 80309, USA  Michael A. Calkins  Department of Physics,
University of Colorado, Boulder, Colorado 80309, USA

###### Abstract

A numerical study of convection with stress-free boundary conditions in the
presence of an imposed magnetic field that is tilted with respect to the
direction of gravity is carried out in the limit of small magnetic Reynolds
number. The dynamics are investigated over a range of Rayleigh number $Ra$ and
Chandrasekhar numbers up to $Q=2\times 10^{6}$, with the tilt angle between
the gravity vector and imposed magnetic field vector fixed at $45^{\circ}$.
For a fixed value of $Q$ and increasing $Ra$, the convection dynamics can be
broadly characterized by three primary flow regimes: (1) quasi-two-dimensional
convection rolls near the onset of convection; (2) isolated convection columns
aligned with the imposed magnetic field; and (3) unconstrained convection
reminiscent of non-magnetic convection. The influence of varying $Q$ and $Ra$
on the various fields is analyzed. Heat and momentum transport, as
characterized by the Nusselt and Reynolds numbers, are quantified and compared
with the vertical field case. Ohmic dissipation dominates over viscous
dissipation in all cases investigated. Various mean fields are investigated
and their scaling behavior is analyzed. Provided $Ra$ is sufficiently large,
all investigated values of $Q$ exhibit an inverse kinetic energy cascade that
yields strong ‘zonal’ flows. Relaxation oscillations, as characterized by a
quasi-periodic shift in the predominance of either the zonal or non-zonal
component of the mean flow, appear for sufficiently large $Ra$ and $Q$.

##  I Introduction

Convection in the presence of externally imposed magnetic fields, or
magnetoconvection (MC), is important in stars and planetary interiors [1].
Magnetic fields can lead to novel flow regimes relative to non-conducting
fluids. Notably, imposed magnetic fields induce flow anisotropy. The resulting
change in flow structure, however, is dependent on the direction and magnitude
of the imposed magnetic field, implying that a rich variety of dynamics can be
realized in MC. In the context of planets and stars, the magnetic field tends
to be self-generated through dynamo action and is therefore spatially (and
temporally) complex. It is therefore of interest to understand how field
direction, in addition to field magnitude, influences the underlying
convective motions.

The periodic plane layer geometry provides a particularly simple system in
which to study MC. The linear theory of MC for the plane layer is well
established and provides a useful starting point for understanding the
resulting nonlinear dynamics [2]. The constant gravitational field is denoted
by $\boldsymbol{g}=-g{\bf\widehat{z}}$ (where ${\bf\widehat{z}}$ is the unit
vector pointing normal to the planar boundaries) and the uniform imposed
magnetic field is $\mathbf{B}_{0}$. In the present work we focus solely on the
limit in which the induced magnetic field is weak relative to the imposed
field – known as the quasi-static limit [e.g. 3] – we therefore limit our
present discussion to this case. When $\boldsymbol{g}$ and $\mathbf{B}_{0}$
are aligned, which we refer to as vertical MC (VMC), convection is stabilized
and the horizontal scale of the most unstable eigenmodes decreases with
increasing field strength. When $\mathbf{B}_{0}$ is horizontal (HMC), the
preferred mode consists of two-dimensional (2D) convection rolls with their
axes aligned with $\mathbf{B}_{0}$. The general case of a tilted magnetic
field (TMC) is essentially a mixture of these two previous cases in which the
most unstable eigenmodes consist of two-dimensional rolls aligned with the
horizontal component of the imposed magnetic field, but with a horizontal
length scale that decreases with increasing magnetic field strength.

The nonlinear evolution of MC has been investigated both experimentally and
numerically for a variety of field configurations. VMC dynamics has been
studied for both the quasi-static limit, and for the case of arbitrarily large
induced magnetic fields using numerical simulations [e.g. 4]. In the
discussion here we focus on those results pertaining to the quasi-static
limit. The vertical field geometry tends to limit horizontal mixing when the
field strength is sufficiently large given the preference for fluid to move
along the magnetic field direction. In confined geometries, such as cylinders,
distinct flows such as convective wall modes are also present [5, 6, 7]. Heat
and momentum transport, while always weaker than non-magnetic Rayleigh-Bénard
convection (RBC), increases at a rate that is faster than RBC [8, 7]. For
sufficiently strong buoyancy forcing, VMC data appears to approach the
corresponding RBC data [9, 8, 7], which suggests an expected weakening
dynamical role of the imposed magnetic field.

Studies of HMC show that 2D rolls persist over a significant range of
parameter space and can yield heat transport that is more efficient than RBC
[10, 11, 12, 13]. The flow eventually transitions to anisotropic 3D convection
that can exhibit rich time dependent motions [14, 15, 16]. In confined
geometries the lateral walls can have a significant influence on the dynamics
due to the formation of Hartmann boundary layers [11]. These boundary layers
stabilize the convection and lead to an increase in the temperature gradient
required to initiate convection.

In comparison to VMC and HMC, less is presently known about the nonlinear
behavior of TMC. Previous 2D studies of TMC have found mean flows and
traveling wave solutions that are generated by the broken symmetry associated
with the tilted magnetic field [17]. This previous work suggests that the mean
flows tend to travel in the direction of the tilt, though the dependence of
the amplitude of this mean flow on the input parameters, namely the imposed
field strength, has not been investigated in detail. In addition, to our
knowledge the efficiency of heat and momentum transport has been unexplored
for TMC.

Rotating convection (RC) has been studied in great detail due to its relevance
for planetary and stellar applications [18]. Like MC, RC with a vertical
rotation vector stabilizes the convection and yields anisotropic structures
[2]. The inverse kinetic energy cascade is prevalent in RC provided that the
influence of rotation is strong; the resulting flows are characterized by
vortices that span the horizontal length of the system and are approximately
invariant in the direction of the rotation axis [19, 20, 21, 22]. When the two
horizontal dimensions are unequal, the inverse cascade is instead manifested
by the presence of large scale horizontally-directed jets [23]. Recent studies
of RC with a tilted rotation axis find that both jets and vortices are
present, depending on the tilt of the rotation axis (and likely also the
relative importance of rotation and inertia) [24, 25].

In the present work we carry out a systematic investigation of three-
dimensional TMC using direct numerical simulations. Flow regimes are
delineated, and heat and momentum transport are quantified over a range of
imposed field strengths and buoyancy forcing. When possible, a comparison is
made with recent VMC simulations; we find that the primary difference between
TMC and VMC is the presence of magnetically constrained turbulent states in
the former. We find that TMC, like rotating convection, yields an inverse
cascade of kinetic energy, characterized by energetic (relative to the
convection) mean flows that tend to be dominated by a meandering, alternating
jet structure. In some cases we also find that these jets can become unstable
and give rise to relaxation oscillations.

##  II Methods

Figure 1: Geometry used in the present study. The imposed magnetic field
points in the direction
$\widehat{{\boldsymbol{\eta}}}=\eta_{1}{\bf\widehat{x}}+\eta_{3}{\bf\widehat{z}}$,
where $\eta_{1}=\sin\theta$ and $\eta_{3}=\cos\theta$. For the present study
the angle is fixed to $\theta=45^{\circ}$.

We employ a periodic plane layer geometry, as shown in Fig. 1. The fluid layer
has depth $H$ and the spatially uniform imposed magnetic field vector is given
by

$\mathbf{B}_{0}=B_{0}\widehat{{\boldsymbol{\eta}}},$ (1)

where we define the vector

$\widehat{{\boldsymbol{\eta}}}=\eta_{1}\,{\bf\widehat{x}}+\eta_{3}\,{\bf\widehat{z}},$
(2)

with $\eta_{1}=\sin\theta$ and $\eta_{3}=\cos\theta$. The angle $\theta$ is
measured relative to the vertical and fixed in the present study to
$\theta=45^{\circ}$.

The fluid is Oberbeck-Boussinesq with density $\rho$, kinematic viscosity
$\nu$, thermal diffusivity $\kappa$, thermal expansion coefficient $\alpha$,
magnetic diffusivity $\lambda$, and vacuum permeability $\mu$. The dimensional
temperature difference between the bottom and top boundaries is denoted by
$\Delta\mathcal{T}>0$. The governing equations are non-dimensionalized with
the viscous diffusion timescale $H^{2}/\nu$ and are given by

$D_{t}\mathbf{u}=-\nabla
P+\frac{Ra}{Pr}T\,{\bf\widehat{z}}+Q\,\widehat{{\boldsymbol{\eta}}}\cdot\nabla\mathbf{b}+\nabla^{2}\mathbf{u},$
(3)
$0=\widehat{{\boldsymbol{\eta}}}\cdot\nabla\mathbf{u}+\nabla^{2}\mathbf{b},$
(4) $D_{t}T=\frac{1}{Pr}{\nabla^{2}}T,$ (5) $\nabla\cdot\mathbf{u}=0,$ (6)
$\nabla\cdot\mathbf{b}=0,$ (7)

where the material derivative is denoted by
$D_{t}(\cdot)={\partial_{t}}(\cdot)+\mathbf{u}\cdot\nabla(\cdot)$,
$\mathbf{u}=\left(u,v,w\right)$ is the velocity field, $\mathbf{b}$ is the
induced magnetic field, $T$ is the temperature, and $P$ is the reduced
pressure.

The non-dimensional parameters appearing in the above equations are the
Rayleigh number, the thermal Prandtl number, and the Chandrasekhar number
defined by, respectively

$Ra=\frac{\alpha g\Delta\mathcal{T}H^{3}}{\kappa\nu},\quad
Pr=\frac{\nu}{\kappa},\quad Q=\frac{B_{0}^{2}H^{2}}{\rho\nu\mu\lambda}.$ (8)

In all cases presented here the thermal Prandtl number is fixed at $Pr=1$,
whereas $Ra$ and $Q$ are both varied.

We use the quasi-static magnetohydrodynamic approximation in which the induced
magnetic field is asymptotically smaller than the imposed magnetic field. This
approximation is equivalent to assuming that $Rm=RePm\rightarrow 0$, where
$Rm$ is the magnetic Reynolds number, $Re$ is the Reynolds number and
$Pm=\nu/\lambda$ is the magnetic Prandtl number. The quasi-static
approximation is therefore valid when $Re\ll Pm^{-1}$; this condition is
typically satisfied in laboratory experiments that use liquid metals [e.g. 26,
11, 15].

The mechanical boundary conditions are impenetrable and stress-free, constant
temperature and electrically insulating, which can be written as

$w=\frac{\partial u}{\partial z}=\frac{\partial v}{\partial
z}=0\quad\mbox{at}\quad z=0,1.$ (9)

The thermal boundary conditions are constant temperature

$T=1\quad\mbox{at}\quad z=0,\qquad T=0\quad\mbox{at }\quad z=1.$ (10)

Electrically insulating electromagnetic boundary conditions are used such that
the magnetic field is matched to a potential field at $z=0$ and $z=1$ [e.g.
see 27].

The most unstable eigenmodes consist of longitudinal rolls oriented parallel
to the imposed magnetic field with marginal Rayleigh number

$Ra_{m}=\frac{\left(\pi^{2}+k^{2}\right)}{k^{2}}\left[\left(\pi^{2}+k^{2}\right)^{2}+\pi^{2}\eta_{3}^{2}Q\right].$
(11)

Minimizing the above expression gives

$2{k_{c}}^{6}+3\pi^{2}{k_{c}}^{4}-\pi^{4}Q\eta_{3}^{2}-\pi^{6}=0,$ (12)

where $k_{c}$ is the critical horizontal wavenumber. For the field strengths
used in the present study, $Q=(2\times 10^{3},2\times 10^{5},2\times 10^{6})$,
we find $k_{c}\approx(5.6842,12.8343,18.9823)$ and $Ra_{c}\approx(1.5207\times
10^{4},1.0784\times 10^{6},1.0281\times 10^{7})$. In the asymptotic limit
$Q\rightarrow\infty$, the critical wavenumber and the critical Rayleigh number
are given by, respectively,

$k_{c}^{(a)}\rightarrow\left(\frac{1}{2}\pi^{4}Q\eta_{3}^{2}\right)^{1/6},\qquad
Ra_{c}^{(a)}\rightarrow\pi^{2}\eta_{3}^{2}Q.\qquad\left(Q\rightarrow\infty\right)$
(13)

Both the velocity and magnetic fields are represented in terms of poloidal and
toroidal scalars such that the solenoidal conditions are satisfied exactly.
The resulting equations are solved numerically with a pseudo-spectral
algorithm using Fourier series in the horizontal directions and Chebyshev
polynomials in the vertical direction. A third-order implicit-explicit Runge-
Kutta time stepping method is used. Further details of the code are given in
[28].

We scale the horizontal length of the computational domain, $L$, in integer
multiples of the critical horizontal wavelength, $\lambda_{c}=2\pi/k_{c}$,
such that the aspect ratio is given by

$\Gamma=n\lambda_{c}.$ (14)

We found that $n=10$ is sufficient for convergence of bulk quantities such as
the Nusselt number and the Reynolds number. Thus, for the imposed field
strengths used in the present study, the corresponding aspect ratios are given
by $\Gamma\approx(11.1,4.9,3.3)$.

##  III Results

###  III.1 Parametric overview and flow regimes

Figure 2: Parametric overview of the simulations, as characterized by the
ratio $Ha/Re_{z}$, where $Ha=\sqrt{Q}$ is the Hartmann number and $Re_{z}$ is
the Reynolds number based on the rms of the vertical component of the velocity
field. The critical Rayleigh number is denoted by $Ra_{c}$. The ratio
$Ha/Re_{z}$ represents the relative size of the Lorentz force and inertia:
cases with $Ha/Re_{z}\gtrsim O(1)$ are considered magnetically constrained.

We investigate imposed magnetic field strengths of $Q=(2\times 10^{3},2\times
10^{5},2\times 10^{6})$, which were chosen so that the critical Rayleigh
number, $Ra_{c}$, for each field strength corresponds with those used in
recent VMC simulations [8]. We explore Rayleigh numbers up to
$Ra\approx(132Ra_{c},28Ra_{c},7Ra_{c})$ for each of the three values of $Q$.
Data for the simulations is summarized in Table 1. While spatial resolution
requirements are significant for many of the simulated flows, we find that the
primary limiting factor for the computations done at larger values of $Q$ and
$Ra$ is the presence of slowly evolving relaxation oscillations that require
prohibitively long computation times to obtain converged statistics.

A useful measure for characterizing the relative importance of the imposed
magnetic field is the ratio, $Ha/Re_{z}$, where the Hartmann number is defined
as $Ha=\sqrt{Q}$ and $Re_{z}$ is the Reynolds number based on the rms of the
vertical velocity component. The ratio $Ha/Re_{z}$ represents the relative
influence of the Lorentz force to inertial forces in the quasi-static limit
[e.g. 29, 3]. By restricting the Reynolds number to only include the vertical
component of the velocity field, we are attempting to better characterize the
relative influence of the magnetic field on the small scale convection. As
discussed later, mean flows develop that have magnitudes significantly larger
than the vertical component of the velocity field. Fig. 2 shows this ratio for
all of the simulations. Cases in which $Ha/Re_{z}\gg 1$ are considered
magnetically constrained in the sense that the Lorentz force enters the
leading order force balance [cf. 8]. For values of $Ha/Re_{z}\lesssim 1$,
inertia plays a leading order dynamical role and the resulting motions are
only weakly influenced by the imposed magnetic field. In the present study we
only find cases with $Ha/Re_{z}\lesssim 1$ for the smallest field strength of
$Q=2\times 10^{3}$; for these cases we find that the convective structures are
no longer aligned with the (tilted) magnetic field.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 3: Volumetric renderings of the fluctuating temperature for various
cases. Top row ($Q=2\times 10^{3}$): (a) $Ra/Ra_{c}=1.3$; (b)
$Ra/Ra_{c}=13.2$; (c) $Ra/Ra_{c}=132$. Middle row ($Q=2\times 10^{5}$): (d)
$Ra/Ra_{c}=1.4$; (e) $Ra/Ra_{c}=7.4$; (f) $Ra/Ra_{c}=27.8$. Bottom row
($Q=2\times 10^{6}$): (g) $Ra/Ra_{c}=1.5$; (h) $Ra/Ra_{c}=3.9$; (i)
$Ra/Ra_{c}=6.8$.

To illustrate the structure of the flow field as both $Q$ and $Ra$ are varied
(and therefore also $Ha/Re_{z}$) we show volumetric renderings of the
fluctuating temperature in Fig. 3 for each of the three values of $Q$ and
three particular values of $Ra$. As predicted by linear theory, for $Ra\approx
Ra_{c}$, we observe anisotropic convective rolls that are predominantly
aligned with the $x$-component of the imposed magnetic field – these
structures are evident in Fig. 3(a) and to a lesser degree in panels (d) and
(g). As the Rayleigh number is increased we find that the rolls develop a
large scale, $k_{x}=1$, modulation. This modulation interacts nonlinearly with
the convective rolls and leads to the formation of a large scale mean flow
that is discussed in more detail below. Further increases in $Ra$ lead to the
development of convective columns that are aligned with the imposed magnetic
field. These structures are particularly evident in the $Q=2\times 10^{5}$ and
$Q=2\times 10^{6}$ cases, e.g. Figures 3(e,f,h,i). These cases also show that
the convective structures are elongated in the $x$-direction. For $Q=2\times
10^{3}$ the tilt of the convective structures is less noticeable; in Fig. 3(b)
some tilt is observable, though for sufficiently large Rayleigh number the
tilt is no longer obvious (Fig. 3(c)). Despite the lack of constraint in
simulations with $Ha/Re_{z}\lesssim 1$, we find that the magnetic field still
plays an important dissipative role, as discussed in the next section.

###  III.2 Heat and momentum transport

(a)

(b)

Figure 4:  Nusselt number data: (a) $Nu$ versus $Ra$; (b) compensated Nusselt
number, $Nu(Ra_{c}/Ra)$, versus $Ra/Ra_{c}$. Tilted magnetoconvection cases
are denoted by TMC, and the vertical field cases from Ref. [8] are denoted by
VMC.

Fig. 4 shows the Nusselt number, defined by

$Nu=1+Pr\langle wT^{\prime}\rangle,$ (15)

where the angled brackets, $\langle\cdot\rangle$, denote a volume and time
average. Both the new TMC data and the VMC data from Ref. [8] are shown for
comparison. The slope of the $Nu$-$Ra$ data increases with $Q$, similar to
VMC. However, in VMC there is a steeper increase in $Nu$ near the onset of
convection and a subsequent reduction in the growth rate of $Nu$ with
increasing $Ra$. This difference between VMC and TMC is exhibited in the
corresponding compensated value, $Nu/(Ra/Ra_{c})$, as shown in Fig. 4(b). The
$Nu\sim Ra/Ra_{c}$ scaling behavior is independent of viscous dissipation,
which might be expected in the limit $Q\rightarrow\infty$ [e.g. 30]. However,
we find that viscous dissipation becomes important in all cases just after the
onset of convection, which suggests a possible explanation for why such a
scaling is not observed. For sufficiently large values of $Ra/Ra_{c}$, both
VMC and TMC show similar scaling behavior in $Nu$, at least for $Q=2\times
10^{3}$ and $Q=2\times 10^{5}$.

(a)

(b)

(c)

Figure 5: Vertical profiles of the rms temperature fluctuation: (a) $Q=2\times
10^{3}$; (b) $Q=2\times 10^{5}$; (c) $Q=2\times 10^{6}$. Darker greyscale
lines correspond to larger Rayleigh numbers.

Vertical profiles of the rms temperature fluctuation are shown for all values
of $Q$ in Fig. 5. For a fixed value of $Q$ we find that the temperature
fluctuation within the interior begins to decrease for sufficiently large
$Ra/Ra_{c}$ once robust thermal boundary layers form; the growth of
$T^{\prime}$ with increasing $Ra/Ra_{c}$ is then achieved within the vicinity
of the thermal boundary layer. Comparing the profiles for different values of
$Q$ reveals that the rms temperature is generally a decreasing funtion of $Q$,
as predicted by asymptotic theory [31, 32, 33]. Although not shown, we find
that for all values of $Q$ the horizontally averaged temperature shows a trend
toward a nearly isothermal bulk with well developed thermal boundary layers as
$Ra$ is increased. These results indicate that, like RBC and VMC [8], the heat
transfer is ultimately limited by the thermal boundary layers in TMC. This
data suggests that the initial rapid growth in $Nu$ for small $Ra/Ra_{c}$ is
due to the formation of thermal boundary layers.

(a)

(b)

(c)

Figure 6: Dissipation ratios: (a) $Q=2\times 10^{3}$; (b) $Q=2\times 10^{5}$;
(c) $Q=2\times 10^{6}$. Note the difference in scale on the horizontal axis.

The Nusselt number and the energy dissipation are related via

$\left(Nu-1\right)\frac{Ra}{Pr^{2}}=\varepsilon_{u}+\varepsilon_{b},$ (16)

where the viscous and ohmic dissipation are given by
$\varepsilon_{u}=\langle\boldsymbol{\zeta}^{2}\rangle$ and
$\varepsilon_{b}=Q\langle\mathbf{j}^{2}\rangle$, respectively. The vorticity
vector is denoted by $\boldsymbol{\zeta}=\nabla\times\mathbf{u}$ and the
current density is $\mathbf{j}=\nabla\times\mathbf{b}$.

It is helpful to define the ratios of viscous and ohmic dissipation according
to

$\phi_{u}=\frac{\varepsilon_{u}}{\varepsilon_{b}+\varepsilon_{u}},\qquad\phi_{b}=\frac{\varepsilon_{b}}{\varepsilon_{b}+\varepsilon_{u}},$
(17)

where we note that $\phi_{u}+\phi_{b}=1$. The dissipation ratios are shown in
Fig. 6 for all values of $Q$. For all cases, near the onset of convection we
observe a rapid decrease (increase) in the ohmic (viscous) dissipation ratio
as $Ra/Ra_{c}$ is increased. For $Q=2\times 10^{3}$, there is an approximate
saturation in both dissipation ratios in the range $25\lesssim
Ra/Ra_{c}\lesssim 75$. For the largest value of $Ra/Ra_{c}\approx 132$ we find
a slight increase (decrease) in the ohmic (viscous) dissipation ratio. We note
that within the saturated regime we find significant mean flows, though the
mean flow is negligibly small at the largest value of $Ra/Ra_{c}\approx 132$.
This increase (decrease) in the ohmic (viscous) dissipation ratio may be due
to the strongly 3D flow that occurs, thus leading to significant induced
magnetic field and associated current. The data suggests that viscous
dissipation is important for all $Q$, though it is always smaller than ohmic
dissipation.

(a)

(b)

Figure 7: Normalized dissipation profiles averaged in time for select cases
from $Q=2\times 10^{3}$. (a) normalized viscous dissipation; (b) normalized
ohmic dissipation.

To examine the depth-dependence of the dissipation we compute horizontally
averaged profiles of the squared vorticity and current density, i.e.
$\overline{\boldsymbol{\zeta}^{2}}$ and $Q\overline{\mathbf{j}^{2}}$. Fig. 7
shows these profiles, normalized by the total dissipation, for $Q=2\times
10^{3}$ and three different values of $Ra$. Due to the stress-free mechanical
boundary conditions we find that viscous dissipation is dominant within the
interior of the domain and momentum boundary layers are evident. In contrast,
we find that ohmic dissipation is dominant near the boundaries though we do
not observe obvious boundary layer regions even for the largest values of
$Ra/Ra_{c}$. As a result, it is not relevant to separate the ohmic dissipation
into interior and boundary layer contributions.

(a)

(b)

Figure 8: Reynolds number data for both TMC and VMC. $Re_{z}$ is shown for TMC
and $Re$ is shown for VMC, where we note that the Reynolds number for VMC is
dominated by the vertical component of velocity. (a) Reynolds number $Re_{z}$
versus $Ra/Ra_{c}$; (b) Rescaled Reynolds number, $Re_{z}\,Q^{-1/4}$, versus
$Ra/Ra_{c}$.

The convective flow speeds, as characterized by the Reynolds number based on
the vertical component of the velocity, $Re_{z}$, are shown in Fig. 8(a). For
the VMC data we show the total Reynolds number, as derived from Ref. [8],
though the vertical component of the velocity is larger than the corresponding
horizontal components so long as the Lorentz force remains dominant [e.g. 31].
We find that the scaling of the convective flow speeds are qualitatively
similar in both TMC and VMC where there is a rapid rise in amplitude near the
onset of convection, and a slower growth for larger $Ra/Ra_{c}$. There is a
general trend of increasing $Re_{z}$ with increasing $Q$ for both data sets –
we find a good collapse of the data by rescaling the convective flow speeds
according to $Re_{z}\,Q^{-1/4}$, as shown in Fig. 8(b). This $Q^{1/4}$ scaling
was used in the asymptotic models of [32, 33], though only single mode (i.e.
single wavenumber) solutions were analyzed. Asymptotic behavior is only
expected in the magnetically constrained regime in which $Ha/Re_{z}\gg 1$; as
previously mentioned, for $Q=2\times 10^{3}$ the cases with $Ra/Ra_{c}\gtrsim
10$ are characterized by $Ha/Re_{z}=O(1)$.

(a)

(b)

(c)

Figure 9: Time-averaged velocity ratios: (a) $Q=2\times 10^{3}$; (b)
$Q=2\times 10^{5}$; (c) $Q=2\times 10^{6}$. All quantities are volumetric rms
values and the error bars show the standard deviation.

The anisotropy in the velocity field can be characterized by computing the
ratio of the volumetric rms of each velocity component to the total volumetric
rms velocity  which we denote $u_{tot}$; the resulting data is shown in Fig.
9. Near the onset of convection, in which the flow consists of two-dimensional
rolls, we find that the vertical component of the velocity dominates. However,
we find a rapid decrease in the relative size of $w$ as the large scale flow
forms and the $y$-component of the velocity field, $v$, dominates. All cases
show a saturation in $v/u_{tot}\approx 0.9$. Moreover, this saturation is
observed to occur at smaller values of $Ra/Ra_{c}$ as $Q$ increases. The
relative size of $u/u_{tot}$ exhibits a slow but steady increase with
$Ra/Ra_{c}$ and we find that the standard deviation also increases. For
$Q=2\times 10^{5}$ we find relaxation oscillations (discussed more below) for
supercriticalities $Ra/Ra_{c}\gtrsim 7.5$ – in this regime both $u$ and $v$
become of comparable magnitude and exhibit large amplitude variations with
time.

###  III.3 Horizontally averaged mean fields

As previously noted, mean flows form in TMC for all values of $Q$ investigated
here. These mean flows take various forms and are associated with
corresponding mean magnetic fields. In the present section we analyze mean
fields, both velocity and magnetic, that are defined by averages over the
entire horizontal plane. In the next subsection we examine the dynamics of
what we refer to as zonal flows, which are $y$-directed flows averaged only in
the $y$-direction.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 10: Time and horizontally averaged x-component of the velocity field
and magnetic field: (a,d) $Q=2\times 10^{3}$; (b,e) $Q=2\times 10^{5}$; (c,f)
$Q=2\times 10^{6}$. Note that the volume averaged x-velocity was removed from
the horizontally averaged x-velocity. Darker lines correspond to higher
Rayleigh numbers.

We restrict the present analysis to the $x$-component of the mean velocity
field, $\overline{u}$, and the associated magnetic field $\overline{b}_{x}$,
since the $y$-components are observed to be significantly smaller in magnitude
and approach zero for sufficiently long time averages. The mean momentum
equation in the $x$-direction is given by

${\partial_{t}}\overline{u}+{\partial_{z}}\overline{\left(u^{\prime}w^{\prime}\right)}=Q\eta_{3}{\partial_{z}}\overline{b}_{x}+\partial^{2}_{z}\overline{u},$
(18)

where $u^{\prime}=u-\overline{u}$, etc. The corresponding mean magnetic field
is governed by

$0=\eta_{3}{\partial_{z}}\overline{u}+\partial^{2}_{z}\overline{b}_{x},$ (19)

so that vertical shear in the mean flow is associated with a mean magnetic
field.

The computed $x$-components of the means fields, $\overline{u}$ and
$\overline{b}_{x}$, are shown in Fig. 10 for a range of $Ra$ and all three
values of $Q$. We find that the magnitude of the mean fields increases with
$Ra/Ra_{c}$, with the singular exception of the case ($Q=2\times 10^{3}$,
$Ra/Ra_{c}=132$), which shows a sudden decrease; profiles for this case are
indicated by the darkest shade in panels (a) and (d). This latter case has
reached sufficiently large Rayleigh number such that the magnetic field is no
longer constraining the motion. We find that the $x$-components of the mean
fields are robust in the sense that they maintain a similar structure with
varying $Q$ and $Ra$, and evolve on a timescale much longer than the
underlying convection. The mean velocity flows in the same direction as the
imposed magnetic field; i.e. it points in the negative (positive)
$x$-direction in the lower (upper) half of the flow domain. We find that the
magnitude of $\overline{u}$ is small relative to the convective fluctuations,
$u^{\prime}$ and $w^{\prime}$, though it remains non-zero for all cases
investigated. This non-zero vertical flux of $x$-momentum is associated with
the tendency for the convection to align with the direction of the magnetic
field.

Estimates for the sizes of the various terms in the above equations can be
made, and help to explain some of the numerical findings shown in Fig. 10. We
scale vertical derivatives of mean quantities as
${\partial_{z}}\rightarrow\overline{\ell}_{z}^{-1}$, where
$\overline{\ell}_{z}$ is some characteristic length scale in the vertical
dimension. In what follows we assume $\eta_{3}=O(1)$. Equation (19) then gives

$\overline{u}\sim\frac{\overline{b}_{x}}{\overline{\ell}_{z}}.$ (20)

If, for the purpose of the present scale analysis, we interpret the overline
as also including a time average, then we have three terms in the mean
momentum equations to consider, which we scale as

${\partial_{z}}\overline{\left(u^{\prime}w^{\prime}\right)}\rightarrow\frac{\overline{u^{\prime}w^{\prime}}}{\overline{\ell}_{z}},\qquad
Q\eta_{3}{\partial_{z}}\overline{b}_{x}\rightarrow\frac{Q\overline{b}_{x}}{\overline{\ell}_{z}},\qquad\partial^{2}_{z}\overline{u}\rightarrow\frac{\overline{u}}{\overline{\ell}_{z}^{2}}.$
(21)

Using relationship (20) in the above Lorentz force term we then have

$\frac{Q\overline{b}_{x}}{\overline{\ell}_{z}}\rightarrow Q\overline{u}.$ (22)

Balancing the mean Lorentz force with the mean viscous force would imply
$\overline{\ell}_{z}\sim Q^{-1/2}$, which is the well-known Hartmann boundary
layer scaling. We assume that such a strong dependence on $Q$ is only relevant
within the Hartmann layer, and not in the bulk of the domain. We do observe a
Hartmann boundary layer in the horizontally averaged fields, though its
effects appear to be negligibly small. Outside of the Hartmann layer it is
unclear that there should be any $Q$-dependence on the vertical length scale
with regards to mean quantities. Therefore, as a first approximation we
neglect any $Q$-dependence on this length scale and the mean Lorentz force can
then only be balanced by the divergence of the Reynolds stress in the bulk. By
directly integrating (in $z$) the resulting balance, the (time-averaged) mean
momentum equation gives

$\overline{b}_{x}\approx\frac{1}{\eta_{3}Q}\overline{u^{\prime}w^{\prime}},$
(23)

where we have used the impenetrable and electrically insulating boundary
conditions. In terms of the asymptotic dependence only, the above balance
suggests

$\overline{b}_{x}=O\left(\frac{\overline{u^{\prime}w^{\prime}}}{Q}\right),$
(24)

This result shows that Reynolds stresses are directly responsible for the
generation of a mean magnetic field. Using the mean induction equation then
gives a relationship between the mean flow and the Reynolds stresses,

${\partial_{z}}\overline{u}\approx-\frac{1}{Q\eta_{3}^{2}}\partial_{z}^{2}\overline{\left(u^{\prime}w^{\prime}\right)}.$
(25)

Assuming $\overline{\ell}_{z}=O(1)$, the above balance leads to

$\overline{u}=O\left(\frac{\overline{u^{\prime}w^{\prime}}}{Q}\right),$ (26)

so that both the mean velocity and mean magnetic field scale similarly with
$Q$. These relationships suggest that we require knowledge on the asymptotic
size of the fluctuating velocity components in order to estimate the
asymptotic size of both $\overline{u}$ and $\overline{b}_{x}$. Fig. 8 suggests
that $w^{\prime}\sim Q^{1/4}$ and, though not shown, we also find that the
$u^{\prime}\sim Q^{1/4}$, which suggests that

$\overline{u}=O\left(Q^{-1/2}\right),\qquad\overline{b}_{x}=O\left(Q^{-1/2}\right).$
(27)

(a)

(b)

(c)

(d)

Figure 11: Scalings of the rms values for the horizontally averaged
quantities: (a) $\overline{u}$; (b) $\overline{u}\,Q^{1/2}$; (c)
$\overline{b}_{x}$; (d) $\overline{b}_{x}\,Q^{1/2}$. The rms is taken in the
vertical direction, then a time-average is computed. Note that for
$\overline{u}$, the volume-averaged x-velocity was subtracted off before the
rms is calculated.

Figs. 11(a) and (c) show rms values of the mean velocity and magnetic field,
respectively. The corresponding rescaled components are given in Figs. 11(b)
and (d). The collapse of the rescaled quantities suggests that the scaling
arguments given above lead to good estimates for the asymptotic dependence of
these mean fields. These results suggest that the mean flow (and associated
mean magnetic field) becomes less significant dynamically in the limit
$Q\rightarrow\infty$.

###  III.4 Zonal flow dynamics

The horizontally averaged fields analyzed in the previous section represent
one particular component of the mean flows that are observed in TMC. However,
the strongest mean flows observed in the simulations are dominated by
horizontal wavenumber $k_{x}=1$, and such motions are eliminated by a
horizontal average over the entire horizontal plane. These mean flows are a
combination of ‘zonal’ ($y$-directed) flows and large scale vortices that
result in a meandering jet structure. For a fixed value of $Q$, we find that
the $y$-component of the flow dominates, though for sufficiently strong
forcing for $Q=2\times 10^{5}$ the non-zonal ($x$-directed) flow becomes
comparable to the zonal component and relaxation oscillations occur. Given the
directional dependence of the flow it is helpful to average in only a single
horizontal direction, either $x$ or $y$. We denote such averages with an
overline and a superscript that specifies the direction or directions of the
average. Here we focus only on the zonal, or $y$-directed, mean flows since
they are observed for all values of $Q$ and tend to dominate over much of the
parameter space investigated here. A brief description of the relaxation
oscillations is given in the next section.

(a)

(b)

(c)

Figure 12: Volumetric renderings of the $y$-component of the velocity, $v$,
illustrating the development of the zonal flow for $Q=2\times 10^{6}$ and
increasing $Ra$: (a) $Ra=1.04\times 10^{7}$; (b) $Ra=1.1\times 10^{7}$; (c)
$Ra=1.3\times 10^{7}$. The view is approximately along the imposed magnetic
field.

As $Q$ is increased we find that the zonal flow quickly becomes the dominant
component of the velocity. To illustrate this behavior, Fig. 12 shows
instantaneous volumetric renderings of the $y$-component of the velocity, $v$,
for $Q=2\times 10^{6}$ and three different values of $Ra$, increasing from
left to right. Just beyond the onset of convection we find a mode with
$k_{x}=1$ appears, as shown in panel (a), which suggests the formation of the
zonal flow. As $Ra$ increases in panels (b) and (c), this zonal flow becomes
stronger and eventually becomes energetically dominant relative to the small
scale convection by which it is driven.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 13: Structure of the zonal flow from representative cases. (a,d)
$Q=2\times 10^{3}$, $Ra=1\times 10^{5}$; (b,e) $Q=2\times 10^{5}$, $Ra=8\times
10^{6}$; (c,f) $Q=2\times 10^{6}$, $Ra=7\times 10^{7}$. The top row shows
instantaneous views of the zonal velocity, $\overline{v}^{y}$, and the bottom
row shows instantaneous views of the $y$-component of the velocity, $v$, in
the horizontal plane at depth $z\approx 0.5$.

Figs. 13(a)-(c) show instantaneous views of $\overline{v}^{y}(x,z,t)$ in the
$x$-$z$ plane for all three values of $Q$ with values of $Ra$ chosen such that
the zonal flow is energetically dominant (relative to the convection). For all
values of $Q$ we find that the zonal flow is aligned with the imposed magnetic
field, and dominated by a $k_{x}=1$ structure at all depths. Corresponding
top-down views of $v$ at depth $z\approx 0.5$ are shown in panels (d)-(f),
where the meandering structure of this mean flow can be seen. Comparing the
$y$-averaged flows shown in (a)-(c) with the full field in (d)-(f) shows how
the action of $y$-averaging removes much of the small scale features present
in (d)-(f).

(a)

(b)

Figure 14: Scaling behavior of the rms value for the zonal flow: (a) raw data;
(b) rescaled data.

The scaling behavior of the zonal velocity is shown in Fig. 14(a) where rms
values are plotted for all values of $Q$. We observe a trend of increasing
magnitude with both increasing $Ra/Ra_{c}$ and increasing $Q$. As shown in
Fig. 14(b), the data can be collapsed by scaling the zonal velocity as
$\overline{v}^{y}\sim Q^{1/3}$. As done previously for the horizontally
averaged mean flows, the cause of this zonal flow scaling can be found by
analyzing dominant balances in the zonal momentum equation.

The alignment of the zonal flow with the imposed magnetic field suggests that
the use of a non-orthogonal (i.e. skewed) coordinate system is helpful. Such a
coordinate system was used, for example, in Ref. [34] for studying the
asymptotic behavior of rapidly rotating convection with a tilted rotation
axis. We follow Ref. [34] and define the non-orthogonal coordinate system with
unit vectors ${\bf\widehat{x}}$, ${\bf\widehat{y}}$, and
$\widehat{{\boldsymbol{\eta}}}$ and coordinates
$(\widetilde{x},\widetilde{y},\eta)$. The relevant transformations between the
orthogonal and non-orthogonal variables are given by

$\widetilde{x}=x-\frac{\eta_{1}}{\eta_{3}}z,\qquad\widetilde{y}=y,\qquad\eta=\frac{1}{\eta_{3}}z$
(28)
$\widetilde{u}=u-\frac{\eta_{1}}{\eta_{3}}w,\qquad\widetilde{v}=v,\qquad\widetilde{w}=\frac{1}{\eta_{3}}w.$
(29)

Similarly we have $\widetilde{b}_{y}=b_{y}$. The $y$-component of the momentum
equation can now be written as

$\frac{\partial v}{\partial
t}+\partial_{\widetilde{x}}\left(\widetilde{u}v\right)+\partial_{y}(v^{2})+\partial_{\eta}\left(v\widetilde{w}\right)=-\partial_{y}P+\left(\frac{1}{\eta_{3}^{2}}\partial_{\widetilde{x}}^{2}+\partial_{y}^{2}+\frac{1}{\eta_{3}^{2}}\partial^{2}_{\eta}-2\frac{\eta_{1}}{\eta_{3}}\partial^{2}_{\widetilde{x}\eta}\right)v+Q\partial_{\eta}b_{y}.$
(30)

Averaging in the $y$-direction, $\eta$-direction, and in time yields

$\overline{\overline{\widetilde{u}}^{y\eta}\partial_{\widetilde{x}}\overline{v}^{y\eta}}^{t}+\partial_{\widetilde{x}}\overline{\left(\widetilde{u}^{\prime}v^{\prime}\right)}^{y\eta
t}=\left(\frac{1}{\eta_{3}}-2\right)\partial_{\eta}\overline{v}^{yt}\bigg{|}^{1/\eta_{3}}_{0}+\frac{1}{\eta_{3}^{2}}\partial_{\widetilde{x}}^{2}\overline{v}^{y\eta
t}.$ (31)

Here the fluctuating terms are given by $v^{\prime}=v-\overline{v}^{y\eta}$,
etc. We note the absence of the Lorentz force in the above equation,
indicating that it plays an indirect role in the zonal dynamics. The first
term on the left side represents advection by the mean whereas the first term
on the right side contains boundary terms associated with averaging the
viscous force along $\eta$; the numerical simulations show that both of these
terms are small in comparison to the two other terms present in the above
equation. Advection by the mean is small because
$\overline{\widetilde{u}}^{y}$ is not generally aligned with
$\widehat{{\boldsymbol{\eta}}}$. In addition, because $\overline{v}^{y}$ is
aligned with $\widehat{{\boldsymbol{\eta}}}$, the $\eta$-averaged value of
$\partial_{\eta}\overline{v}^{y}$ is expected to be small. Therefore, the
largest terms are the Reynolds stress term and the viscous force,

$\partial_{\widetilde{x}}\overline{\left(\widetilde{u}^{\prime}v^{\prime}\right)}^{y\eta
t}\sim\frac{1}{\eta_{3}^{2}}\partial_{\widetilde{x}}^{2}\overline{v}^{y\eta
t}.$ (32)

Furthermore, if we let $\overline{\ell}^{y\eta}$ denote a length scale
associated with $y$ and $\eta$-averaged quantities, we find

$\overline{v}^{y\eta
t}\sim\overline{\ell}^{y\eta}\,\overline{\left(\widetilde{u}^{\prime}v^{\prime}\right)}^{y\eta
t},$ (33)

where we have dropped the factor of $\eta_{3}^{2}$ since it is of order unity.
In all simulations in which a zonal flow is observed we find that the zonal
flow grows to fill the domain and is dominated by a $k_{x}=1$ structure. As
mentioned previously, as $Q$ is varied we fix the total number of horizontal
critical wavelengths in the domain to be $n=10$. Since the critical wavelength
changes with $Q$, i.e. $\lambda_{c}\sim Q^{-1/6}$ as $Q$ becomes large, this
implies that the horizontal (and vertical) scale of the zonal flow is also
changing as we vary $Q$ in our simulations. Therefore, we can scale the
characteristic zonal length scale as

$\overline{\ell}^{y\eta}\sim
n\lambda_{c}\qquad\Rightarrow\qquad\overline{\ell}^{y\eta}\sim nQ^{-1/6}.$
(34)

Combining this scaling with relationship (33) then yields

$\overline{v}^{y\eta t}=O\left(Q^{1/3}\right),$ (35)

where we have again used the numerical data that indicates
$(\widetilde{u}^{\prime},v^{\prime})=O\left(Q^{1/4}\right)$.

The zonal length scaling relation (34) implies that for a fixed value of $Q$
the zonal flow magnitude increases linearly with the horizontal dimension (as
quantified by $n$) of the simulation domain. Although not shown here, this
scaling was confirmed in a set of numerical simulations in which $Q$ was fixed
and $n$ was varied. This finding is in agreement with a previous investigation
of the inverse kinetic energy cascade in rotating convection where the same
scaling was observed [35].

####  III.4.1 Relaxation Oscillations

(a)

(b)

Figure 15: Times series data illustrating the temporal behavior of relaxation
oscillations for $Q=2\times 10^{5}$ and $Ra=1.5\times 10^{7}$: (a) Reynolds
number, $Re(t)$; (b) Nusselt number, $Nu(t)$.

Relaxation oscillations were observed in simulations with $Q=2\times 10^{5}$
and $10Ra_{c}\lesssim Ra\leq 30Ra_{c}$, where $30Ra_{c}$ was the highest
supercritical Rayleigh number achieved for this value of $Q$. Relaxation
oscillations were also found for a case with $Q=2\times 10^{6}$ but no
statistics were collected due to the long time integration required. The
relaxation oscillations are characterized by a zonal flow magnitude that
exhibits large oscillations in time. Figs. 15(a) and (b) show the Reynolds
number and Nusselt number, respectively, as a function of time for $Q=2\times
10^{5}$ and $Ra=1.5\times 10^{7}$. During times when the mean flow is strong
(weak), the Nusselt number is generally smaller (larger) than the
corresponding time-averaged value. Fig. 16 shows how the horizontal components
of velocity change when the Reynolds number goes from increasing to
decreasing. The $y$-component of the velocity decays rapidly and the
$x$-component of velocity grows to be the dominant velocity component.

(a)

(b)

(c)

(d)

Figure 16: Instantaneous plots of the horizontal velocity components
illustrating relaxation oscillations for $Q=2\times 10^{5}$ and $Ra=1.5\times
10^{7}$. The two plots on the left (a,c) are taken at time $t=5.200$, just
prior to the peak in Reynolds number. The two plots on the right (b,d) are
taken at time $t=5.228$, just after the Reynolds number reaches a peak. (a,b)
$y$-velocity component; (c,d) $x$-velocity component.

##  IV Conclusion

The direct numerical simulations reported here have shown that
magnetoconvection with a tilted magnetic field (TMC) has both dynamic
similarities, and key differences with magnetoconvection with a vertical field
(VMC). For a meaningful comparison between these two cases we have chosen
imposed field strengths that yield critical parameters that are identical to
those used in an analogous VMC study [8]; i.e. the strength of the vertical
component of the imposed magnetic fields are identical for our chosen values
of $Q$. The simulations span dynamical regimes that are characterized as
magnetically constrained, in which $Ha/Re_{z}\gg 1$, and those that might be
considered magnetically influenced when $Ha/Re_{z}=O(1)$. Like VMC, we find
that when $Ha/Re_{z}\gg 1$ and for sufficiently strong buoyancy forcing, the
convective structures consist of anisotropic ‘columns’ that align with the
direction of the magnetic field. This alignment is no longer obvious when
$Ha/Re_{z}\lesssim O(1)$.

The heat transport for TMC is qualitatively similar to VMC in which there is a
rapid increase in the Nusselt number near the onset of convection that is
associated with the formation of the thermal boundary layers, and the growth
of $Nu$ with increasing $Ra/Ra_{c}$ slows beyond this regime. In contrast to
VMC, we do not observe a clear power law scaling of the Nusselt number for TMC
over our investigated range of parameter space. Ohmic dissipation dominates
viscous dissipation in all simulations, including those in which
$Ha/Re_{z}=O(1)$. However, viscous dissipation remains important and
represents approximately 40$\%$ of the total dissipation as the Rayleigh
number is increased.

Convective flow speeds, as characterized by the Reynolds number based on the
vertical component of the velocity, $Re_{z}$, show behavior that is similar to
VMC. However, it is unclear whether $Re_{z}$ exhibits power law behavior due
to the limited range of parameter space accessible in the present simulations.
We find that an asymptotic scaling of $Re_{z}\sim Q^{1/4}$ describes the
$Q$-dependence of the convective flow speeds. This scaling is the same that
was used in the asymptotic models of Refs. [32, 33]. Further investigation,
particularly simulations with larger values of $Q$, is necessary to confirm
the robustness of this scaling.

Mean flows, and associated mean magnetic fields, form for all values of $Q$
investigated. We find and study two distinct forms of mean flows, as
characterized by the horizontal wavevector $\bf k$. Those mean flows in which
$\mathbf{k}=(0,0)$ (i.e. averaged over the horizontal plane) are dominated by
the $x$-components, $\overline{u}$ and $\overline{b}_{x}$. The mean velocity
field ($\overline{u}$) is generally aligned with the imposed external field in
the sense that it flows in the positive direction above the mid-plane (i.e.
$z>0$), and flows in the negative direction below the midplane ($z<0$). The
simulations show that the rms value of $\overline{u}$ is a decreasing function
of $Q$; a balance analysis suggests that
$\overline{u}=O\left(Q^{-1/2}\right)$, which is in agreement with the
numerical findings. An identical scaling holds for $\overline{b}_{x}$,
suggesting that this particular component of the mean fields becomes less
significant dynamically as $Q$ is increased.

The energetically dominant mean flows are characterized by a meandering jet
structure that tends to be aligned with the tilt of the imposed magnetic
field. These mean flows are dominated by either $\mathbf{k}=(1,0)$ at lower
values of $Ra/Ra_{c}$, or exhibit relaxation oscillations that shift between
the $\mathbf{k}=(0,1)$ and $\mathbf{k}=(1,0)$ modes for sufficiently large
$Ra/Ra_{c}$ and $Q>2\times 10^{3}$. We find that the ‘zonal’ flows scale
strongly with $Q$, i.e. $\overline{v}^{y}\sim Q^{1/3}$; this scaling is the
result of the $Q$-dependent Reynolds stresses and the fact that the zonal flow
magnitude scales linearly with the horizontal dimensions of the simulation
domain. The zonal flows become energetically negligible for sufficiently large
$Ra/Ra_{c}$, since then $Ha/Re_{z}\lesssim 1$ and the convection is no longer
aligned with the imposed magnetic field. The relaxation oscillations evolve on
the timescale of a large scale viscous diffusion unit and therefore require
substantial computational resources to study; for this reason we were unable
to study their characteristics in significant detail.

It is interesting to note that several phenomena observed here for convection
in a tilted magnetic field are also observed for rotating convection [24, 25].
In convection with a tilted rotation vector, the fluid structures tend to
align along the tilt axis of the rotation, much like how the fluid structures
in this paper were aligned along the magnetic field. In addition, a strong
shear flow arises in convection with a tilted rotation vector that closely
resembles the mean flow found for convection in a tilted magnetic field. We
anticipate that scalings for the strength of the mean flow and convective flow
speeds can be found in tilted rotating convection, as was done in the present
investigation for magnetoconvection.

The structure of the observed mean flows in the present work is a
manifestation of the plane periodic geometry that was used. Obviously,
confined geometries will yield different mean flows. However, the present work
has shown that a robust inverse kinetic energy cascade is nevertheless present
in TMC, and there is no a priori reason why such a cascade would not also be
present in confined geometries such as cylinders or cubes. We might expect
that the inverse cascade would manifest itself in the form of only large scale
vortices in confined geometries, but future work is necessary to confirm this
assertion.

## Acknowledgements

The authors gratefully acknowledge funding from the National Science
Foundation through grants EAR-1620649, EAR-1945270 and SPG-1743852. This work
utilized the RMACC Summit supercomputer, which is supported by the National
Science Foundation (awards ACI-1532235 and ACI-1532236), the University of
Colorado Boulder, and Colorado State University. The Summit supercomputer is a
joint effort of the University of Colorado Boulder and Colorado State
University. This work used the Extreme Science and Engineering Discovery
Environment (XSEDE) Stampede2 supercomputer at the Texas Advanced Computing
Center (TACC) through allocation PHY180013. Volumetric renderings of the flow
field were performed with VAPOR [36].

## References

  * Weiss and Proctor [2014] N. O. Weiss and M. Proctor,  _Magnetoconvection_ (Cambridge University Press, 2014).
  * Chandrasekhar [1961] S. Chandrasekhar,  _Hydrodynamic and Hydromagnetic Stability_ (Oxford University Press, U.K., 1961).
  * Knaepen and Moreau [2008] B. Knaepen and R. Moreau, Magnetohydrodynamic turbulence at low magnetic Reynolds number, Annu. Rev. Fluid Mech. 40, 25 (2008).
  * Cattaneo  _et al._ [2003] F. Cattaneo, T. Emonet, and N. Weiss, On the interaction between convection and magnetic fields, Astrophys. J. 588, 1183 (2003).
  * Liu  _et al._ [2018] W. Liu, D. Krasnov, and J. Schumacher, Wall modes in magnetoconvection at high Hartmann numbers, J. Fluid Mech. 849 (2018).
  * Zürner  _et al._ [2020] T. Zürner, F. Schindler, T. Vogt, S. Eckert, and J. Schumacher, Flow regimes of Rayleigh–Bénard convection in a vertical magnetic field, J. Fluid Mech. 894 (2020).
  * Akhmedagaev  _et al._ [2020] R. Akhmedagaev, O. Zikanov, D. Krasnov, and J. Schumacher, Turbulent Rayleigh–Bénard convection in a strong vertical magnetic field, J. Fluid Mech. 895 (2020).
  * Yan  _et al._ [2019] M. Yan, M. A. Calkins, S. Maffei, K. Julien, S. M. Tobias, and P. Marti, Heat transfer and flow regimes in quasi-static magnetoconvection with a vertical magnetic field, J. Fluid Mech. 877, 1186 (2019).
  * Cioni  _et al._ [2000] S. Cioni, S. Chaumat, and J. Sommeria, Effect of a vertical magnetic field on turbulent Rayleigh-Bénard convection, Phys. Rev. E 62 (2000).
  * Fauve  _et al._ [1981] S. Fauve, C. Laroche, and A. Libchaber, Effect of a horizontal magnetic field on convective instabilities in mercury, Journal de Physique Lettres 42, 455 (1981).
  * Burr and Müller [2002] U. Burr and U. Müller, Rayleigh–Bénard convection in liquid metal layers under the influence of a horizontal magnetic field, J. Fluid Mech. 453, 345 (2002).
  * Andreev  _et al._ [2003] O. Andreev, A. Thess, and C. Haberstroh, Visualization of magnetoconvection, Phys. Fluids 15, 3886 (2003).
  * Vogt  _et al._ [2021] T. Vogt, J.-C. Yang, F. Schindler, and S. Eckert, Free-fall velocities and heat transport enhancement in liquid metal magneto-convection, J. Fluid Mech. 915 (2021).
  * Tasaka  _et al._ [2016] Y. Tasaka, K. Igaki, T. Yanagisawa, T. Vogt, T. Zuerner, and S. Eckert, Regular flow reversals in Rayleigh-Bénard convection in a horizontal magnetic field, Phys. Rev. E 93, 043109 (2016).
  * Vogt  _et al._ [2018] T. Vogt, W. Ishimi, T. Yanagisawa, Y. Tasaka, A. Sakuraba, and S. Eckert, Transition between quasi-two-dimensional and three-dimensional Rayleigh-Bénard convection in a horizontal magnetic field, Phys. Rev. Fluids 3, 013503 (2018).
  * Yang  _et al._ [2021] J. Yang, T. Vogt, and S. Eckert, Transition from steady to oscillating convection rolls in rayleigh-bénard convection under the influence of a horizontal magnetic field, Phys. Rev. Fluids 6, 023502 (2021).
  * HurlBurt  _et al._ [1996] N. HurlBurt, P. Matthews, and M. Proctor, Nonlinear compressible convection in oblique magnetic fields, Ap. J. 457, 933 (1996).
  * Aurnou  _et al._ [2015] J. M. Aurnou, M. A. Calkins, J. S. Cheng, K. Julien, E. M. King, D. Nieves, K. M. Soderlund, and S. Stellmach, Rotating convective turbulence in Earth and planetary cores, Phys. Earth Planet. Int. 246, 52 (2015).
  * Julien  _et al._ [2012] K. Julien, A. M. Rubio, I. Grooms, and E. Knobloch, Statistical and physical balances in low Rossby number Rayleigh-Bénard convection, Geophys. Astrophys. Fluid Dyn. 106, 392 (2012).
  * Rubio  _et al._ [2014] A. M. Rubio, K. Julien, E. Knobloch, and J. B. Weiss, Upscale energy transfer in three-dimensional rapidly rotating turbulent convection, Phys. Rev. Lett. 112, 144501 (2014).
  * Guervilly  _et al._ [2014] C. Guervilly, D. W. Hughes, and C. A. Jones, Large-scale vortices in rapidly rotating Rayleigh-Bénard Convection, J. Fluid Mech. 758, 407 (2014).
  * Favier  _et al._ [2014] B. Favier, L. J. Silvers, and M. R. E. Proctor, Inverse cascade and symmetry breaking in rapidly rotating Boussinesq convection, Phys. Fluids 26 (2014).
  * Julien  _et al._ [2018] K. Julien, E. Knobloch, and M. Plumley, Impact of domain anisotropy on the inverse cascade in geostrophic turbulent convection, J. Fluid Mech. 837 (2018).
  * Novi  _et al._ [2019] L. Novi, J. von Hardenberg, D. Hughes, A. Provenzale, and E. Spiegel, Rapidly Rotating Rayleigh-Bénard convection with a tilted axis, Phys. Rev. E 99 (2019).
  * Currie  _et al._ [2020] L. K. Currie, A. J. Barker, Y. Lithwick, and M. K. Browning, Convection with misaligned gravity and rotation: simulations and rotating mixing length theory, Mon. Not. Roy. Astron. Soc. 493, 5233 (2020).
  * Aurnou and Olson [2001] J. M. Aurnou and P. Olson, Experiments on Rayleigh-Bénard convection, magnetoconvection, and rotating magnetoconvection in liquid gallium, J. Fluid Mech. 430, 283 (2001).
  * Jones and Roberts [2000] C. A. Jones and P. H. Roberts, Convection-driven dynamos in a rotating plane layer, J. Fluid Mech. 404, 311 (2000).
  * Marti  _et al._ [2016] P. Marti, M. A. Calkins, and K. Julien, A computationally efficent spectral method for modeling core dynamics, Geochem. Geophys. Geosys. 17, 3031 (2016).
  * Hunt and Shercliff [1971] J. Hunt and J. Shercliff, Magnetohydrodynamics at high Hartmann number, Annu. Rev. Fluid Mech. 3, 37 (1971).
  * Bhattacharyya [2006] S. Bhattacharyya, Scaling in magnetohydrodynamic convection at high Rayleigh number, Phys. Rev. E 74, 035301 (2006).
  * Matthews [1999] P. C. Matthews, Asymptotic solutions for nonlinear magnetoconvection, J. Fluid Mech. 387, 397 (1999).
  * Julien  _et al._ [1999] K. Julien, E. Knobloch, and S. M. Tobias, Strongly nonlinear magnetoconvection in three dimensions, Physica D 128, 105 (1999).
  * Julien  _et al._ [2000] K. Julien, E. Knobloch, and S. M. Tobias, Nonlinear magnetoconvection in the presence of strong oblique fields, J. Fluid Mech. 410, 285 (2000).
  * Julien  _et al._ [2006] K. Julien, E. Knobloch, R. Milliff, and J. Werne, Generalized quasi-geostrophy for spatially anistropic rotationally constrained flows, J. Fluid Mech. 555, 233 (2006).
  * Maffei  _et al._ [2021] S. Maffei, M. J. Krouss, K. Julien, and M. A. Calkins, On the inverse cascade and flow speed scaling behaviour in rapidly rotating rayleigh–bénard convection, J. Fluid Mech. 913 (2021).
  * Li  _et al._ [2019] S. Li, S. Jaroszynski, S. Pearse, L. Orf, and J. Clyne, Vapor: A visualization package tailored to analyze simulation data in earth system science, Atmosphere 10, 488 (2019).

$Q$ | $Ra$ | $Nu$ | $Re$ | $Re_{z}$ | $\Delta t$ | $N_{x}\times N_{y}\times N_{z}$
---|---|---|---|---|---|---
2 $\times 10^{3}$ | 1.53 $\times 10^{4}$ | 1.012 $\pm$ 0.000 | 0.822 $\pm$ 0.000 | 0.719 $\pm$ 0.000 | $10^{-3}$ | $96\times 96\times 48$
2 $\times 10^{3}$ | 2 $\times 10^{4}$ | 1.30 $\pm$ 0.03 | 5.05 $\pm$ 0.19 | 4.16 $\pm$ 0.18 | $10^{-3}$ | $96\times 144\times 48$
2 $\times 10^{3}$ | 2.5 $\times 10^{4}$ | 1.54 $\pm$ 0.04 | 8.29 $\pm$ 0.24 | 6.61 $\pm$ 0.21 | $10^{-3}$ | $144\times 192\times 48$
2 $\times 10^{3}$ | 4 $\times 10^{4}$ | 2.19 $\pm$ 0.04 | 17.86 $\pm$ 0.24 | 11.34 $\pm$ 0.30 | $5\times 10^{-4}$ | $144\times 192\times 48$
2 $\times 10^{3}$ | 6 $\times 10^{4}$ | 2.83 $\pm$ 0.05 | 30.98 $\pm$ 0.35 | 16.47 $\pm$ 0.36 | $2\times 10^{-4}$ | $192\times 288\times 72$
2 $\times 10^{3}$ | 1 $\times 10^{5}$ | 3.75 $\pm$ 0.05 | 54.80 $\pm$ 0.37 | 25.12 $\pm$ 0.43 | $1\times 10^{-4}$ | $288\times 288\times 144$
2 $\times 10^{3}$ | 2 $\times 10^{5}$ | 5.22 $\pm$ 0.07 | 103.43 $\pm$ 0.71 | 40.81 $\pm$ 0.66 | $4\times 10^{-5}$ | $288\times 384\times 144$
2 $\times 10^{3}$ | 4 $\times 10^{5}$ | 7.06 $\pm$ 0.09 | 179.71 $\pm$ 0.79 | 63.67 $\pm$ 1.03 | $4\times 10^{-5}$ | $384\times 576\times 144$
2 $\times 10^{3}$ | 6 $\times 10^{5}$ | 8.38 $\pm$ 0.09 | 229.58 $\pm$ 1.90 | 81.39 $\pm$ 1.14 | $2\times 10^{-5}$ | $576\times 576\times 144$
2 $\times 10^{3}$ | 1 $\times 10^{6}$ | 10.35 $\pm$ 0.11 | 285.17 $\pm$ 2.17 | 110.31 $\pm$ 1.25 | $1\times 10^{-5}$ | $768\times 768\times 192$
2 $\times 10^{3}$ | 2 $\times 10^{6}$ | 13.42 $\pm$ 0.12 | 214.17 $\pm$ 2.16 | 154.75 $\pm$ 1.39 | $5\times 10^{-6}$ | $768\times 768\times 288$
2 $\times 10^{5}$ | 1.1 $\times 10^{6}$ | 1.024 $\pm$ 0.006 | 2.11 $\pm$ 0.29 | 2.01 $\pm$ 0.28 | $10^{-5}$ | $96\times 144\times 96$
2 $\times 10^{5}$ | 1.3 $\times 10^{6}$ | 1.17 $\pm$ 0.02 | 8.13 $\pm$ 0.52 | 7.16 $\pm$ 0.55 | $10^{-5}$ | $96\times 144\times 96$
2 $\times 10^{5}$ | 1.5 $\times 10^{6}$ | 1.31 $\pm$ 0.02 | 13.84 $\pm$ 0.54 | 10.73 $\pm$ 0.67 | $10^{-5}$ | $144\times 192\times 96$
2 $\times 10^{5}$ | 1.7 $\times 10^{6}$ | 1.46 $\pm$ 0.03 | 20.69 $\pm$ 0.62 | 14.33 $\pm$ 0.79 | $10^{-5}$ | $144\times 192\times 96$
2 $\times 10^{5}$ | 2 $\times 10^{6}$ | 1.72 $\pm$ 0.03 | 32.44 $\pm$ 0.63 | 19.53 $\pm$ 0.92 | $10^{-5}$ | $192\times 192\times 96$
2 $\times 10^{5}$ | 2.2 $\times 10^{6}$ | 1.87 $\pm$ 0.04 | 41.26 $\pm$ 0.64 | 22.65 $\pm$ 0.96 | $10^{-5}$ | $192\times 192\times 96$
2 $\times 10^{5}$ | 2.5 $\times 10^{6}$ | 2.10 $\pm$ 0.04 | 55.79 $\pm$ 0.70 | 27.05 $\pm$ 1.02 | $10^{-5}$ | $192\times 192\times 96$
2 $\times 10^{5}$ | 3 $\times 10^{6}$ | 2.45 $\pm$ 0.04 | 81.53 $\pm$ 0.76 | 34.23 $\pm$ 1.11 | $10^{-5}$ | $192\times 288\times 96$
2 $\times 10^{5}$ | 4 $\times 10^{6}$ | 3.11 $\pm$ 0.07 | 134.82 $\pm$ 1.32 | 47.60 $\pm$ 1.70 | $10^{-5}$ | $288\times 384\times 96$
2 $\times 10^{5}$ | 6 $\times 10^{6}$ | 4.24 $\pm$ 0.13 | 240.66 $\pm$ 1.88 | 70.43 $\pm$ 3.00 | $10^{-5}$ | $288\times 384\times 144$
2 $\times 10^{5}$ | 8 $\times 10^{6}$ | 5.18 $\pm$ 0.16 | 344.60 $\pm$ 2.37 | 90.74 $\pm$ 3.77 | $5\times 10^{-6}$ | $576\times 576\times 192$
2 $\times 10^{5}$ | 1 $\times 10^{7}$ | 6.55 $\pm$ 0.27 | 323.43 $\pm$ 72.48 | 112.83 $\pm$ 3.82 | $5\times 10^{-6}$ | $576\times 576\times 192$
2 $\times 10^{5}$ | 1.5 $\times 10^{7}$ | 8.30 $\pm$ 0.38 | 548.14 $\pm$ 114.34 | 154.27 $\pm$ 6.46 | $2\times 10^{-6}$ | $576\times 576\times 288$
2 $\times 10^{5}$ | 3 $\times 10^{7}$ | 12.67 $\pm$ 0.40 | 921.73 $\pm$ 195.51 | 251.40 $\pm$ 6.44 | $2\times 10^{-6}$ | $576\times 768\times 288$
2 $\times 10^{6}$ | 1.04 $\times 10^{7}$ | 1.015 $\pm$ 0.000 | 2.651 $\pm$ 0.000 | 2.575 $\pm$ 0.000 | $10^{-6}$ | $96\times 144\times 144$
2 $\times 10^{6}$ | 1.1 $\times 10^{7}$ | 1.07 $\pm$ 0.01 | 6.75 $\pm$ 0.37 | 6.54 $\pm$ 0.35 | $10^{-6}$ | $96\times 144\times 144$
2 $\times 10^{6}$ | 1.3 $\times 10^{7}$ | 1.21 $\pm$ 0.01 | 16.17 $\pm$ 0.75 | 13.33 $\pm$ 0.76 | $10^{-6}$ | $144\times 192\times 144$
2 $\times 10^{6}$ | 1.5 $\times 10^{7}$ | 1.37 $\pm$ 0.02 | 29.24 $\pm$ 0.91 | 20.16 $\pm$ 1.20 | $10^{-6}$ | $144\times 192\times 144$
2 $\times 10^{6}$ | 2 $\times 10^{7}$ | 1.81 $\pm$ 0.03 | 73.14 $\pm$ 1.03 | 36.13 $\pm$ 1.59 | $10^{-6}$ | $144\times 288\times 144$
2 $\times 10^{6}$ | 2.5 $\times 10^{7}$ | 2.25 $\pm$ 0.04 | 125.88 $\pm$ 1.29 | 51.13 $\pm$ 2.28 | $10^{-6}$ | $192\times 288\times 144$
2 $\times 10^{6}$ | 3 $\times 10^{7}$ | 2.64 $\pm$ .06 | 184.21 $\pm$ 1.91 | 63.98 $\pm$ 3.06 | $10^{-6}$ | $288\times 288\times 144$
2 $\times 10^{6}$ | 4 $\times 10^{7}$ | 3.41 $\pm$ .07 | 305.61 $\pm$ 2.88 | 89.59 $\pm$ 4.62 | $10^{-6}$ | $384\times 384\times 192$
2 $\times 10^{6}$ | 5 $\times 10^{7}$ | 4.14 $\pm$ .10 | 422.72 $\pm$ 5.20 | 113.64 $\pm$ 5.24 | $10^{-6}$ | $384\times 576\times 192$
2 $\times 10^{6}$ | 7 $\times 10^{7}$ | 5.46 $\pm$ .14 | 663.70 $\pm$ 3.09 | 156.39 $\pm$ 8.06 | $10^{-6}$ | $576\times 576\times 288$
Table 1: Details of the simulations. $Q$ is the Chandraskehar number, $Ra$ is
the Rayleigh number, $Nu$ is the Nusselt number, $Re$ is the Reynolds number,
$Re_{z}$ is the Reynolds number based only on the vertical component of the
velocity, $\Delta t$ is the timestep size and $N_{x}\times N_{y}\times N_{z}$
denotes the physical space resolution. The thermal Prandtl number is fixed at
$Pr=1$ for all simulations.