Free-fall velocities and heat transport enhancement in liquid metal magneto-convection

In geo- and astrophysics, low Prandtl number convective flows often interact with magnetic fields. Although a static magnetic field acts as a stabilizing force on such flow fields, we find that self-organized convective flow structures reach an optimal state where the heat transport significantly increases and convective velocities reach the theoretical free-fall limit, i.e. the maximum possible velocity a fluid parcel can achieve when its potential buoyant energy is fully converted into kinetic energy. Our measurements show that the application of a static magnetic field leads to an anisotropic, highly ordered flow structure and a decrease of the turbulent fluctuations. When the magnetic field strength is increased beyond the optimum, Hartmann braking becomes dominant and leads to a reduction of the heat and momentum transport. The results are relevant for the understanding of magneto-hydrodynamic convective flows in planetary cores and stellar interiors in regions with strong toroidal magnetic fields oriented perpendicular to temperature gradients.


Introduction
Turbulent convective energy coalesces into large coherent flow structures. This is one of the key features that delineates many geo-and astrophysical systems, and also manifests in numerous industrial applications. Often, these systems are subjected to additional, stabilizing forces such as centrifugal and Coriolis forces due to rotation, or Lorentz forces due to magnetic fields. These forces can substantially affect the flow structures and therefore, the heat and momentum transport of convective systems.
The simplest physical model to study thermally driven flows is the so-called Rayleigh-Bénard convection (RBC), where the driving force is a temperature gradient ∇T between a warmer bottom and a cooler top Chilla & Schumacher 2009). An important outcome of RBC studies are the scaling relations for the global heat and momentum transport, expressed non-dimensionally in terms of Nusselt number (N u) and Reynolds number (Re), respectively (Grossmann & Lohse 2002;Stevens 2013).
When stabilizing forces such as rotation (Stellmach 2014; Guervilly 2014), geometrical confinements (Daya & Ecke 2001;Huang 2013) or static magnetic fields are added to highly non-linear systems such as RBC, unexpected features are encountered (Chong 2017;Aurnou 2018). For instance, studies have shown that rotation around a vertical axis has strong influence on the flow structure and consequently the heat transport. Although it is well known that the Coriolis force has a stabilizing effect (Proudman 1916;Taylor 1917;Chandrasekhar 1961), it was found that the application of moderate forces can even enhance scalar transport (Rossby 1969;Zhong 1993;Liu & Ecke 2009;Stevens 2009;Wei 2015;Weiss 2016;Chong 2017).
Similarly, magnetic fields exert an influence on electrically conducting fluids by the induction of eddy currents j = σ(E + u × B) which give rise to a corresponding Lorentz force f l = j×B that acts on the fluid. Here, σ is the electrical conductivity, E is the electric field, u is the velocity and B is the magnetic field. A static magnetic field cannot generate any flow from a quiescent state, however, it can reorganize an electrically conducting flow field so as to minimise the Joule dissipation (Davidson 1995). This is a direct consequence of a reduction of the velocity gradients along the magnetic field direction due to the induced eddy currents (Sommeria & Moreau 1982;Potherat 2017).
In thermal convection in liquid metal, the orientation of the applied magnetic field with respect to the the temperature gradient ∇T, plays a pivotal role on the details of the resulting flow field. Two field orientations are possible: vertical and horizontal. A vertical magnetic field (B ∇T) inhibits the onset of liquid metal convection and the heat transport decreases monotonically with increasing the magnetic field strength, due to a strong suppression of the bulk flow (Cioni 2000;Aurnou & Olsen 2001;Burr & Müller 2001;Liu 2018;Yan 2019;Zürner 2020). Recently, it was shown that convection in fluid with P r = 8, and under the influence of a vertical magnetic field results in increased heat flux with increasing magnetic field strength but accompanied by a decrease in momentum transport (Lim 2019).
In this paper, we report about the interaction between liquid metal convection and a static, horizontally imposed magnetic field, which is governed by two non-dimensional parameters. The Rayleigh number is a measure of the thermal forcing that drives the convection. The strength of the stabilizing Lorentz force due to the applied magnetic field is expressed by the Chandrasekhar number Here, H is the distance between the heated and the cooled plates, L is the width of the cell, ∆T is the imposed temperature difference between these plates, B is the strength of the magnetic field, ρ is the density of the liquid metal, α is the isobaric expansion coefficient, ν is the kinematic viscosity, κ is the thermal conductivity, g denotes the gravitational acceleration and Ha is the Hartmann number. The present work is a continuation of the work of Vogt (2018a), where the transition from a three-dimensional to a quasi two-dimensional flow structure in a liquid metal convection under the influence of a horizontal magnetic field was investigated. Vogt (2018a) focused on the qualitative description of large and small-scale flow structures at different parametric combinations of Rayleigh number and Chandrasekhar number. The main goal of this work is to investigate how this transition between flow regimes affects heat and momentum transport. Accordingly, the measuring arrangement at the experiment was extended and the number of measurements was increased significantly to allow a fine increment of the Chandrasekhar number. We find that the reorganization of the convective flow due to the magnetic field results in a significant enhancement of both heat and momentum transport. In the optimum, the convective velocities can even reach the free-fall limit u ff = √ αg∆T H. In classical Rayleigh-Bénard convection in fluids with moderate Prandtl numbers, such as water or air, the flow velocities are well below the free-fall limit and do not exceed u max /u ff 0.2 (Niemela 2003). Therefore, our measurements demonstrate how intense low P r magnetohydrodynamic convective flows can actually become.

Experimental set-up
The experiments were conducted at the Helmholtz-Zentrum Dresden Rossendorf (HZDR). The eutectic liquid metal alloy composed of Gallium, Indium, and Tin (GaInSn, melting point of T = 10.5 • C, P r = ν/κ = 0.03) was used as the working fluid (Plevachuk 2014). The liquid metal is contained in an aspect ratio Γ = L/H = 5 rectangular vessel with a cross section L 2 = 200 × 200 mm 2 and a height H = 40 mm (Fig. 1). The side walls are made of 30 mm thick Polyvinyl chloride and the top and bottom are made of copper. The convection cell is wrapped in 30 mm closed-cell foam to minimize heat loss. The temperature of the top and bottom were adjusted by a constant temperature water bath which flows through channels in the copper plates. The maximum heat flux is 1500 Watts. The applied temperature drop between the plates ranges from 1.1 • C ∆T 11.7 • C whereby the mean fluid temperature was kept constant at T m = 20 • C. The Rayleigh number ranges between 2.3×10 4 Ra 2.6×10 5 . A static, uniform horizontal magnetic field penetrates the liquid metal with a strength 0 B 317 mT, which gives a Chandrasekhar number range 0 Q 6.1 × 10 6 .
The fluid velocities are measured using Ultrasound Doppler Velocimetry (UDV), which provides instantaneous velocity profiles along two horizontal directions, as shown in figure 1. This technique is useful for non-invasively measuring velocities in opaque fluids (Brito 2001;Tsuji 2005;Vogt 2014. The transducers (TR0805SS, Signal Processing SA) detect the velocity component parallel to the ultrasonic beam with resolutions of about 1 mm in beam direction and 1 Hz in time. One UDV transducer measures the flow velocities perpendicular to the magnetic field (u ⊥ ) and is located in the middle of the cell width at L /2 and 10 mm below the upper boundary (3H/4). A second UDV transducer measures the magnetic field parallel velocity component (u ) and is also located in the middle of the cell width at L ⊥ /2, but at a different height, 10 mm above the lower boundary (H/4). Both transducers are in direct contact with the liquid metal which allows a good velocity signal quality even at low velocities.
The difference between the mean temperatures of the heated and the cooled plates were obtained from two sets of nine thermocouples, with each set located in the heated and the cooled plates respectively. The thermocouples are individually calibrated using a high precision thermometer to ensure accuracy better than 0.05 K.
Another five thermocouples measure the temperature inside the liquid metal at a distance of 3 mm below the cold plate (c.f. figure 1 The convective heat transport is expressed dimensionless by means of the Nusselt number, N u =Φ/Φ cond . Here,Φ cond = λL 2 ∆T /H is the conductive heat flux, with λ being the thermal conductivity of the liquid metal.Φ = ρc pV (T in − T out ) is the total heat flux injected at the bottom and removed from the top wall heat exchanger, whereby c p is the isobaric heat capacity of water. The total heat flux is determined by the flow rateV and the temperature change (T in − T out ) of the circulating water inside the hot and cold wall heat exchangers.

Non-dimensional quantities and characteristic length scale
The length, velocity and time are made non-dimensional throughout this work using the width L of the cell, the free-fall velocity u ff , and the free-fall time t ff = H/u ff , respectively.
In contrast to the work of Vogt (2018) the Chandrasekhar number was not determined with the cell height H but with the distance L of the horizontal walls in magnetic field direction. The definition of Q using the height H goes back to the studies of Burr & Müller (2002), who used the same definition of the Hartmann number Ha with H as characteristic length for their investigations both in the vertical (Burr & Müller 2001) and the horizontal magnetic field (Burr & Müller 2002). In our opinion, the use of the horizontal dimension of the cell is better suited for the case of a horizontal magnetic field, since the effect of Hartmann braking scales with the dimension of the flow domain in magnetic field direction (Müller & Bühler 2001;Knaepen & Moreau 2008).

Results
All results presented here were recorded after the temperature difference between the hot and the cold plates reached a constant value, and the system had attained thermal equilibrium. At low magnetic field strength, the convection at sufficiently high Ra forms a large scale circulation with a three-dimensional cellular structure that fills the entire cell ( Fig. 2a,b,g), whereby upwelling takes place in the center and all four corners of the vessel. A detailed description of this structure can be found in (Akashi 2019). The large amplitude oscillation that can be seen in Fig. 2(a,b) is a typical feature of inertia-dominated liquid metal flows due to their low viscosity and high density (Vogt 2018b). Applying a horizontal magnetic field to such a three-dimensional flow promotes the formation of quasi two-dimensional convection rolls that are aligned parallel to the magnetic field lines. The number of rolls formed depends on the ratio between the driving and the stabilizing force Ra/Q and the aspect ratio Γ of the vessel (Tasaka 2016). Fig.  2(c,d) shows the example of a convective flow field under the influence of an intermediate magnetic field strength and exhibits an unstable roll configuration. In this range the magnetic field is not yet intense enough to produce a stable quasi-two-dimensional flow in perfection. The character of the global flow is still three-dimensional, but the magnetic field has caused a breaking of the symmetry, which characterizes the cell structure. The flow structure is dominated by the convection rolls, but their shape and orientation is still transient and subject to strong three-dimensional disturbances. Four rolls are formed in this transitional range, but these are irregular, and temporary changes to three or five roll configurations can occur. At higher field strength, the flow develops five counter rotating convection rolls which are very stable in time (Fig.2e,f,h). At this stage, convection has restructured into a quasi two-dimensional flow field oriented parallel to the magnetic field direction. The symmetric but weak flow that appears along L is evoked by the Ekman-pumping that originates in the Bödewadt boundary layers where the convection rolls meet the sidewalls (Vogt 2018a). A weak but regular oscillation is visible in figure  2(e,f), which is due to inertial waves within the convection rolls Yang (2020). However, apart from these regular oscillations, the flow appears to be laminar. Based on the flow fields shown in Fig. 2, we distinguish here mainly between three characteristic regimes, the "cell structure" where the influence of the magnetic field on the flow is negligible, the "unstable 3,4,5-roll" state where the field starts to reorganize the flow but is not strong enough to form stable roll configurations, and finally the "stable 5-roll" state at higher Chandrasekhar numbers that results in the formation of stable, quasi two-dimensional convection rolls. with T avg =< T (t) > t the average over the whole measurement and N the number of measurement points. The vertical dashed lines in Figure 3(b) show the boundaries between the three different flow regimes. However, this is only indicative since the actual regime boundaries depend not only on Q but also on Ra. The fluctuations are strongest in the cell structure regime and the unstable 3,4,5-roll regime. The larger the Ra number, the stronger the fluctuations. At the transition to the stable 5-roll regime, the fluctuations decrease significantly and are close to zero, which indicates that from this point on the position and orientation of the rolls within the convection cell is arrested by the applied magnetic field. The slight but systematic increase of the fluctuations in the stable regime is surprising at first sight, but can be explained by the occurence of inertial waves inside the convection rolls (Yang 2020). Finally, at very high magnetic field strengths, these oscillations are also damped and the temperature fluctuations decrease again and approach zero. Figure 3(b) shows the cross correlation of the different temperature sensors within the liquid metal calculated as whereby R i,j is the Pearson's correlation coefficient. In the cell structure regime all cross correlation coefficients are scattered around zero and indicate a negligible correlation between the different measurement points due to a complex and turbulent flow field. In the unstable roll regime, first rolls form along the magnetic field and the cross correlation coefficient between the corresponding adjacent sensors in the magnetic field direction increases and approaches R i,j ≈ 1.
In the stable 5-roll regime, sensors T1, T2 and T3 are located along the same roll. The sensor T5 is located centrally above the neighboring convection roll with opposite rotation direction. Sensor T4 is located centrally between two neighboring convection rolls. During transition to the stable 5-rolle regime, the values for R T 1−T 2 and R T 1−T 3 decrease initially, and then approache a value of 1 again. Correlation of about 1 is expected as these thermocouples (T1, T2 and T3) are located along the same roll. The reason for the initial decrease of the cross correlation coefficient is that the oscillations at the beginning of the stable roll regime are still weak and three-dimensional in nature (Yang 2020). With increasing magnetic field strength the three-dimensional character of the oscillations is suppressed, and from Q > 10 6 onwards only quasi-two-dimensional oscillations take place. The cross correlation coefficient of R T 1−T 2 and R T 1−T 3 then reaches its maximum. The oscillations of neighboring rolls take place with a phase shift of π. For this reason, T5 which is located centrally above the neighbouring roll with opposite direction of rotation, registers a cross correlation coefficient R T 1−T 5 ≈ −1. At the measuring position T4 between two adjacent rolls, remaining oscillations vanish with increasing Q and the correlation coefficient R T 1−T 4 approaches zero for high magnetic field strengths.
For the case of RBC without magnetic field, our measurements show that the heat transport properties scale as: N u 0 = 0.166Ra 0.250 as shown in figure 4(a). This is in reasonable agreement to N u 0 = 0.147Ra 0.257 measured in Mercury (Rossby 1969) and N u 0 = 0.19Ra 0.249 measured in Gallium (King 2013). Note, that Mercury, Gallium and GaInSn have comparable Prandtl numbers ranging from P r = 0.025 − 0.033. Fig.  4(b) presents the relative deviation (N u − N u 0 )/N u 0 for convection with imposed magnetic field where N u 0 is the corresponding Nusselt number for RBC (without magnetic field). In the case of cellular flow structures, which is the prevailing structure for 0 < Q < 1 × 10 4 , the heat transfer does not vary remarkably with increasing Q. This behavior changes in the range 1 × 10 4 < Q < 1.6 × 10 5 where the formation of unstable convection rolls proceeds with significant increase of heat transfer. Finally, for Q > 1.6 × 10 5 , N u reaches a maximum before the heat transfer decreases for even higher Q. The investigation of a wider Ra range is not possible with our current experimental setup. This is due to limited power range of the thermostats, and the cell height which limits Rayleigh number, Ra max ≈ 3 × 10 5 . On the other hand, lowering Ra 10 5 triggers a transition from a stable 5-roll to a 4-roll structure regime which is beyond the scope of this paper, as we focus only on the Ra range where the stable 5-roll configuration fits well into the aspect ratio Γ = 5 of the cell. Enhancement of heat transfer in a liquid metal layer due to the application of a horizontal magnetic field was also investigated by Burr & Müller (2002). Temperature measurements revealed an increase of N u in a certain range of Q. The correlation of temperature signals suggests that the enhancement of the convective heat transfer is accompanied by the existence of non-isotropic time-dependent flows. However, there are still no direct flow measurements of this phenomenon to explain the increase in convective heat transport.
Based on velocity measurements, we analyze the Q dependance of the amplitude of the velocity components perpendicularû ⊥ and parallelû to the magnetic field as shown in Fig. 5. The maximum velocity valuesû i for figure 5 were determined as follows: For each measurement, a velocity threshold was defined, such that 95% of the velocity values of a measurement are below the threshold value. This approach provides very reliable values for the vast majority of measurements. Only at the largest Q and the associated very low velocitiesû , the signal-to-noise ratio of the velocity measurements is not sufficient for applying this method. For these measurements,û was determined from the timeaveraged quasi stationary velocity profile.
The flow velocities, and as such the Re number increases with Ra in all three regimes. As for the heat transfer, the velocity components of the cell structure do not significantly change for Q < 1 × 10 4 . Both velocity components,û ⊥ andû are at the same order of magnitude and reach an amplitude of aboutû ⊥ /u ff ≈ 0.7, which is an expected velocity value for low P r thermal convection at this Ra number range (Vogt 2018b;Zürner 2019). For Q > 1 × 10 4 , the development of the unstable 3,4,5-roll state goes along with a separation of the velocity components. The increase ofû ⊥ and the decrease of u indicates that the flow field starts to become quasi two-dimensional. The relatively large scatter of the velocity data in this regime is caused by the transient flow behavior with frequent reversals of the flow direction. At Q > 1.6 × 10 5 the flow structure changes into the stable 5-roll state, which remains the dominant flow structure for at least one decade of Q numbers. The small scattering of the measured velocity amplitudes in this regime reflects the stable characteristic of this flow configuration. The velocity component parallel to the magnetic fieldû decreases monotonically for higher Q whileû ⊥ reaches maximum arround Q ≈ 2.5 × 10 5 , where the velocity amplitudes reach the theoretical free-fall limit u ff . The normalization of the velocity amplitude with the free-fall velocity yields good conformity for the different Ra numbers.

Discussion
We have demonstrated that the rearrangement of a three-dimensional thermal convection flow into a quasi two-dimensional flow field, due to an applied static magnetic field, results in significantly increased heat and momentum transport. The convection forms five counter rotating rolls, whereby the diameter of the rolls corresponds to the height of the fluid layer and the number of rolls results from the aspect ratio Γ = 5 of the vessel. The preferred orientation of the rolls implies that momentum oriented parallel to the magnetic field is redirected in the direction perpendicular to the field. Therefore,û decreases whileû ⊥ increases. In addition, the intensity of fluctuations in the temperature and velocity field decreases and the stabilized convection rolls appear laminar and quasi-stationary. Q dependence of the normalized horizontal velocity amplitudes for three different Ra numbers. The field-normal velocity amplitude increases from u ⊥ /u ff ≈ 0.7 at cellular flow structures (Q < 1 × 10 4 ) to O(u ff ) in the stable 5-roll regime (Q ≈ 2.5 × 10 5 ). The diverging branches ofû andû ⊥ start with the transition from cellular flow structure to magnetic field aligned convection rolls at Q > 1 × 10 4 .
The vertical velocity component u z is responsible for convective heat flux, but this component was not directly measured in the experiment. However, our measurements show a fully three-dimensional flow in the cell structure regime wherein the three velocity components are of similar amplitude. It can therefore be assumed that the velocity components in this regime are as follows:: u z ≈ u ⊥ ≈ u ≈ 0.7u ff . By contrast, in the quasi two-dimensional, stable 5-roll regime, the flow component parallel to the magnetic field was measured to be significantly weaker compared to the flow component perpendicular to the magnetic field. These experimental results, in conjunction with the prevailing topology of the flow structure, and the requirement imposed by continuity indicate that the following relation would hold for the velocity components at the optimal state of the stable 5-roll regime: u z ≈ u ⊥ ≈ u ff .
In classical Rayleigh-Bénard convection in fluids with moderate Prandtl numbers, such as water or air, the flow velocities are well below the free-fall limit and never exceed u max /u ff 0.2 (Niemela 2003). Our results show unequivocally that the flow has a predisposition to reorient itself perpendicular to the magnetic field, which allows it to attain the optimal state wherein the fluid parcel traverses with the maximum possible velocity, the free-fall velocity. Consequently, the vigour of the convective flow in such state is intense, leading to an enhancement of the heat flux. Further increase of Lorentz force or Q, beyond the optimum state yields reduced convective transport due to the increasing dominance of the Hartmann braking in the lateral boundary layers perpendicular to the magnetic field (Vogt 2018a;Yang 2020). In an infinite fluid layer, the Hartmann braking for an ideal two-dimensional flow structure aligned with the magnetic field direction would disappear since the characteristic Hartmann damping time scale τ HB = ρL 2 /σνB 2 shows a linear dependence on the distance L between the Hartmann walls (Sommeria & Moreau 1982). In previous works (Chong 2017;Lim 2019), the increase in Nusselt number was explained as a result of increased coherency of the flow structures, which acts as the main carrier for the heat transport. Moreover, the authors concluded that the maximum heat flux is achieved when the thermal and viscous boundary layers reach the same thickness. Our results differ in several respects from the studies mentioned above. First, the low P r 1 implies that the viscous boundary layer is always nested well inside the much thicker thermal boundary layer. A crossover of the boundary layer thicknesses is therefore, not expected in very low Prandtl number fluids such as liquid metals. Second, in our case, not only the heat flux, but also the momentum transport perpendicular to the magnetic field is increased. And finally, the low P r of liquid metals implies that the Peclet number P e = Re P r remains low when compared to moderate P r flows at a comparable turbulence level (Vogt 2018b). The consequence is a pronounced coherence in the flow field even without the influence of the magnetic field. The application of a horizontal magnetic field supports an increase in the coherence of the flow pattern in a particularly striking way by transforming unsteady threedimensional flows into stable two-dimensional structures. In this context, it is very interesting to point out that the application of small magnetic fields in the range of the three-dimensional flow (Q < 10 4 ) does not show any measurable effects for the heat and momentum transport. From MHD turbulence it is known that the transition from isotropic to anisotropic turbulence starts at values of the magnetic interaction parameter N = Q/Re ≈ 1 (Davidson 1995;Eckert 2001). When crossing this threshold the effect of the Lorentz force sets in, which prevents three-dimensional structures from absorbing the energy supplied by the thermal driving. Instead the development of quasi-two-dimensional structures are promoted. The interaction parameter reaches values of about 5 at the transition from the cell structure to the unstable roll regimes. The cell structure is completely three-dimensional and an amplification of the flow by the Lorentz force is not plausible in view of the described mechanism. Only with the emergence of the convection rolls are the quasi-two-dimensional structures available into which energy can be transferred. Accordingly, our measurements show a simultaneous increase of both momentum and heat transport in the regime of unstable roll structures.
In conclusion, we have shown how a stabilising, static magnetic field can significantly alter the flow dynamics such that the free-fall velocity is attained by the flow structure, resulting in enhanced heat and momentum transport in thermal convection. These trends remained a consistent feature for all the Rayleigh numbers investigated. It is likely that the optimum state for the heat and momentum transport does not solely depend on Q, but also on a combination of Q, Ra and Γ . Further investigation of this phenomenon with other combinations of parameters would therefore be desirable.