2.1. Thermoelectric effects

Thermoelectric effects enable conversions between thermal and electric energy in electrically conducting materials. There are three different types: the Seebeck, Peltier and Thomson effects (Bhattacharya et al. Reference Bhattacharya, Fornari and Kamimura2011). The Peltier and Thomson effects in our experimental system produce temperature changes of order $\mathrm {\mu }\mathrm {K}$, which are not resolvable with our present thermometric capabilities. Moreover, such small temperature variations will not affect the dynamics of our system. Thus, Peltier and Thomson effects are not considered further.

The Seebeck effect describes the net spatial diffusion of electrons towards or away from a local temperature anomaly (Kasap Reference Kasap2001). As a consequence of this effect, positive and negative charges tend to become sequestered on opposite sides of a regional temperature gradient in the material, leading to the development of a thermoelectric electrical potential. Ohm's law then becomes (Shercliff Reference Shercliff1979)

(2.1)\begin{equation} \boldsymbol{J} = \sigma \left( \boldsymbol{E} + \boldsymbol{u} \times \boldsymbol B - S \boldsymbol{\nabla} T \right), \end{equation}

where $- \sigma S \boldsymbol {\nabla } T$ encapsulates the thermoelectric current. The variables in (2.1) are the electric current density $\boldsymbol {J}$, the electric conductivity $\sigma$ ($\simeq 3.85 \times 10^6\ \textrm {S}\ \textrm {m}^{-1}$ in gallium), the electric field $\boldsymbol {E}$, the fluid velocity $\boldsymbol {u}$, the magnetic flux $\boldsymbol B$, the Seebeck coefficient $S$ and temperature $T$.

Mott & Jones (Reference Mott and Jones1958) derived the following expression for the Seebeck coefficient of a homogeneous and electrically conducting material:

(2.2)\begin{equation} S ={-}\frac{{\rm \pi}^{2} {k_B}^{2}x_0}{3 e E_{F 0}} T. \end{equation}

Here $T$ is measured in Kelvin ($T\approx 300\, \mathrm {K}$ for room temperature), $k_B = 1.38 \times 10^{-23}\, \mathrm {kg}\,\mathrm {m}^2\,\mathrm {s}^{-2}\,\mathrm {K}^{-1}$ is the Boltzmann constant, $x_0$ is an $O(1)$ dimensionless constant that depends on the material properties, $e = 1.60\times 10^{-19}\, \mathrm {C}$ is the elementary charge and $E_{F 0}$ is the material's Fermi energy (${\sim}10\, \mathrm {eV} = 1.6 \times 10^{-18}$ J for metals). In a uniform medium, $S$ is a function only of $T$. In this case, $\boldsymbol {\nabla } S$ is parallel to $\boldsymbol {\nabla } T$ such that $\boldsymbol {\nabla } S \times \boldsymbol {\nabla } T = 0$, which then requires that $S\boldsymbol {\nabla } T$ is irrotational in a uniform medium.

As figure 1 shows, however, a temperature gradient at the interface of two materials with different Seebeck coefficients can generate a net thermoelectric potential. In this case, the Seebeck coefficient $S$ discontinuously varies across the interface of the two materials, $\mathcal {A}$ and $\mathcal {B}$. Near $r_0$ and $r_1$, $\boldsymbol {\nabla } S$ is no longer parallel to $\boldsymbol {\nabla } T$, so a thermoelectric current can form a closed-looped circuit.

Figure 1. A thermoelectric current loop, $\boldsymbol {J}_{T\!E}$, forms across two different conducting materials $\mathcal {A}$ and $\mathcal {B}$ with a horizontal temperature gradient in the $\boldsymbol {\hat e_n}$-direction. The locations where the thermoelectric current flows in and out of the interface are labelled as $r_0$ and $r_1$, respectively. A temperature gradient exists between $r_0$ and $r_1$, where the corresponding temperatures are $T_0 < T_1$. The distance between $r_0$ and $r_1$ is defined as the characteristic length $\mathcal {L} = |r_1-r_0|$. The direction of the current depends on the Seebeck coefficients of both materials, $S_{\mathcal {A}}$ and $S_{\mathcal {B}}$, following (2.6).

The thermoelectric potential, $\varPhi _{T\!E}$, can be calculated via the circuit integral

(2.3)\begin{equation} {\varPhi}_{T\!E}=\oint \frac{\boldsymbol{J}_{T\!E} \boldsymbol{\cdot} \textrm{d} \boldsymbol{r}}{\sigma_{\mathcal{AB}}}, \end{equation}

where $\sigma _{\mathcal {AB}}$ is the effective electric conductivity of the two-material system. The effective electrical resistivity $\tilde {\rho }_{\mathcal {AB}}$ is the sum of the resistivities in each material, i.e.

(2.4)\begin{equation} \tilde{\rho}_{\mathcal{AB}} = \tilde{\rho}_{\mathcal{A}} + \tilde{\rho}_{\mathcal{B}} , \end{equation}

where we assume that the current travels through comparable cross-sectional areas and lengths in each material. Since $\sigma = 1/\tilde {\rho }$, the effective electrical conductivity for the thermoelectric circuit is

(2.5)\begin{equation} \sigma_{\mathcal{AB}} =\frac{1}{\tilde{\rho}_{\mathcal{AB}}} = \frac{1}{\tilde{\rho}_\mathcal{A}+\tilde{\rho}_\mathcal{B}} = \frac{\sigma_\mathcal{A}\, \sigma_\mathcal{B}}{\sigma_\mathcal{A} + \sigma_\mathcal{B}} . \end{equation}

Isolating the thermoelectric current density in the current loop, $\boldsymbol J_{T\!E}$, in (2.1) yields

(2.6)\begin{equation} \boldsymbol{J}_{T\!E} ={-}\sigma_{\mathcal{AB}} \tilde{S} \boldsymbol{\nabla} T \approx{-}\sigma_{\mathcal{AB}} \tilde{S} \left( \frac{T_1-T_0}{\mathcal{L}} \right) , \end{equation}

where $\tilde {S}$ is the net Seebeck coefficient of the two-material system, and the temperature gradient in the $\boldsymbol {\hat e_n}$-direction is approximated by $(T_1-T_0)/{\mathcal {L}}$ (see figure 1). Substituting (2.6) into (2.3), one can show that the net thermoelectric potential ${\varPhi }_{T\!E}$ is the difference between the thermoelectric potentials in each material,

(2.7)\begin{equation} {\varPhi}_{T\!E}=\varPhi_\mathcal{A}-\varPhi_\mathcal{B} ={-}\int_{r_0(T_{0})}^{r_1(T_{1})} S_{\mathcal{A}} \boldsymbol{\nabla} T \boldsymbol{\cdot} \textrm{d} \boldsymbol r+\int_{r_0(T_{0})}^{r_1(T_{1})} S_{\mathcal{B}} \boldsymbol{\nabla} T \boldsymbol{\cdot} \textrm{d} \boldsymbol r, \end{equation}

where $r_0$ and $r_1$ denote the location where the thermoelectric current flows in and out the interface, and $T_0$ and $T_1$ are the temperatures at $r_0$ and $r_1$, respectively. We set $T_0< T_1$, so that the temperature gradient is positive from $r_0$ to $r_1$, following figure 1. Here $S_\mathcal {A}$ and $S_\mathcal {B}$ are the Seebeck coefficients of materials $\mathcal {A}$ and $\mathcal {B}$, respectively.

Substituting (2.2) into (2.7) then yields

(2.8)\begin{equation} \varPhi_{T\!E} = \int_{r_0(T_{0})}^{r_1(T_{1})} (S_{\mathcal{B}}-S_{\mathcal{A}}) \boldsymbol{\nabla} T \boldsymbol{\cdot} \textrm{d} \boldsymbol r = \frac{{\rm \pi}^{2} k_B^{2}}{6 e}\left[\frac{x_{\mathcal{B}}}{E_{F \mathcal{B}}}-\frac{x_{\mathcal{A}}}{E_{F \mathcal{A}}}\right] \left(T_1^{2}-T_{0}^{2}\right), \end{equation}

where $x_\mathcal {A}$, $x_\mathcal {B}$, $E_{F\mathcal {A}}$ and $E_{F\mathcal {B}}$ are numerical constants and Fermi energies of the materials $\mathcal {A}$ and $\mathcal {B}$, respectively.

The system's net Seebeck coefficient is then written as

(2.9)\begin{equation} \tilde{S} = \frac{{\varPhi}_{T\!E}}{T_1-T_0} = \frac{{\rm \pi}^{2} {k_{B}}^{2}}{3 e}\left[\frac{x_{\mathcal{B}}}{E_{F {\mathcal{B}}}}-\frac{x_{\mathcal{A}}}{E_{F \mathcal{A}}}\right] \left( \frac{T_0+T_1}{2} \right), \end{equation}

where $(T_1+T_0)/2$ is the mean temperature of the material interface. Note the structural similarity between the expressions for the Seebeck coefficient for a single material (2.2) and the net Seebeck coefficient across a material interface (2.9).

2.2. Governing equations and non-dimensional parameters

The magnetic Reynolds number, $Rm$, estimates the ratio of magnetic induction and diffusion in a magnetohydrodynamics (MHD) system. In our laboratory experiments, upper bounding values of $Rm$ are estimated by using the convective free-fall velocity (Julien et al. Reference Julien, Legg, McWilliams and Werne1996; Glazier et al. Reference Glazier, Segawa, Naert and Sano1999),

(2.10)\begin{equation} U_{f\!f} =\sqrt{\alpha_T \Delta T g H } , \end{equation}

leading to

(2.11)\begin{equation} Rm = \frac{U_{f\!f} H}{\eta} = Re \, Pm , \end{equation}

where $Re$ is the Reynolds number, which denotes the ratio of inertial and viscous effects,

(2.12)\begin{equation} Re = \frac{U_{f\!f} H}{\nu} , \end{equation}

the magnetic Prandtl number is the ratio of the fluid's magnetic diffusivity $\eta$ and its kinematic viscosity $\nu$,

(2.13)\begin{equation} Pm = \frac{\nu}{\eta} , \end{equation}

and $\alpha _T$ is the thermal expansivity of the fluid, $\Delta T$ is the vertical temperature difference across the fluid layer of depth $H$, $g$ is the gravitational acceleration. In our experiments, $Re \lesssim 9 \times 10^3$ and $Pm \simeq 1.7 \times 10^{-6}$. Thus, $Rm \lesssim 0.015 \ll 1$ for our system, in good agreement with estimates made using ultrasonic velocity measurements in this same set-up by Vogt et al. (Reference Vogt, Horn, Grannan and Aurnou2018a). Further, the free-fall time scale can be defined as

(2.14)\begin{equation} \tau_{f\!f} = H / U_{f\!f} . \end{equation}

The estimates above show that magnetic diffusion dominates induction in our experiments. In this low-$Rm$ regime, the influence of fluid motions on the magnetic field can be neglected and the full magnetic induction equation needs not be solved amongst the governing equations. This results in both $Rm$ and $Pm$ dropping out of the problem (Davidson Reference Davidson2016). This so-called ‘quasistatic approximation’ is commonly applied in low-$Rm$ fluid systems and is valid in most laboratory and industrial liquid metal applications (e.g. Sarris et al. Reference Sarris, Zikos, Grecos and Vlachos2006; Knaepen & Moreau Reference Knaepen and Moreau2008; Davidson Reference Davidson2016).

In addition to quastistaticity, the Boussinesq approximation is applied (Oberbeck Reference Oberbeck1879; Boussinesq Reference Boussinesq1903; Gray & Giorgini Reference Gray and Giorgini1976; Tritton Reference Tritton1977; Chillà & Schumacher Reference Chillà and Schumacher2012) and the governing equations of TEMC are then

(2.15a)\begin{gather} \boldsymbol{J} = \sigma \left( - \boldsymbol{\nabla} \varPhi + \boldsymbol{u} \times \boldsymbol{B} - S \boldsymbol{\nabla} T\right), \end{gather}

(2.15b)\begin{gather} \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol { J } = 0, \end{gather}

(2.15c)\begin{gather} \frac { \partial \boldsymbol{u} } { \partial t } + ( \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nabla} ) \boldsymbol{u} ={-} \frac{1}{\rho } \boldsymbol{\nabla} p + \frac { 1 } { \rho } ( \boldsymbol{J} \times \boldsymbol{B} ) + \nu {\boldsymbol{\nabla}}^{ 2 } \boldsymbol{u} + \alpha _ { T } \Delta T \boldsymbol{g}, \end{gather}

(2.15d)\begin{gather} \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol { u } = 0, \end{gather}

(2.15e)\begin{gather} \frac { \partial T } { \partial t } + (\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nabla} ) T = \kappa {\boldsymbol{\nabla}}^{ 2 } T, \end{gather}

where $\rho$ is fluid density, $p$ is non-hydrostatic pressure, $\boldsymbol {g} = g \boldsymbol {\hat e_z}$ is the gravity vector and $\kappa$ is the thermal diffusivity. The external field is $\boldsymbol {B} = B \boldsymbol {\hat e_b}$. Note that Ohm's law (2.15a) has been simplified via the quasistatic approximation, such that the rotational part of the electric field and perturbative second-order terms from $\boldsymbol {u} \times \boldsymbol {B}$ are not considered. Accordingly, in the bulk fluid, far from material interfaces, where net Seebeck effects are small, the quasistatic Lorentz force is $\boldsymbol {J} \times \boldsymbol {B} \sim -\sigma \boldsymbol {u}_{\perp } B^2$, where $\boldsymbol {u}_{\perp }$ is the velocity perpendicular to the direction of the magnetic field. Therefore, the low-$Rm$ Lorentz force acts as a drag that opposes bulk fluid velocities that are directed perpendicular to $\boldsymbol {B}$ (Sarris et al. Reference Sarris, Zikos, Grecos and Vlachos2006; Davidson Reference Davidson2016). This quasistatic Lorentz drag depends only on $B^2$. In sharp contrast, the thermoelectric component of the Lorentz force, $- \sigma S \boldsymbol {\nabla } T \times \boldsymbol {B}$, varies linearly with $\boldsymbol {B}$. Therefore, the thermoelectric Lorentz force changes sign when the direction of the applied magnetic field is flipped.

The dimensionless form of the TEMC governing equations are given in (A1) and (A2) in Appendix A. The non-dimensional control groups in Appendix A may be decomposed into four parameters: the Prandtl number $Pr$, the Rayleigh number $Ra$, the Chandrasekhar number $Ch$ and the Seebeck number $Se$. The Prandtl number describes the thermo-mechanical properties of the fluid,

(2.16)\begin{equation} Pr = \frac{\nu}{\kappa}; \end{equation}

in liquid gallium, $Pr \approx 0.027$ at $40\,^{\circ }\mathrm {C}$. The Rayleigh number characterizes the buoyancy forcing relative to thermoviscous damping,

(2.17)\begin{equation} Ra = \frac{\alpha_T \Delta T g H^3}{\nu \kappa}. \end{equation}

The Chandrasekhar number describes the ratio of quasistatic Lorentz and viscous forces,

(2.18)\begin{equation} Ch = \frac{\sigma B^2 H^2}{\rho \nu}. \end{equation}

The Seebeck number estimates the ratio of thermoelectric currents in the fluid and currents induced by fluid motions,

(2.19)\begin{equation} Se = \frac{{|\tilde{S}|} \Delta T/H}{U_{f\!f} B } . \end{equation}

Alternatively, $Se$ can be cast as the ratio of the thermoelectrical potential and the motionally induced potential in the fluid. Typical values of $Se$ in our experiments with gallium-copper interfaces range from $O(10^{-2})$ to $O(1)$, implying that the Seebeck effect can generate dynamically significant experimental thermoelectric currents.

Lastly, the aspect ratio acts to describe the geometry of the fluid volume,

(2.20)\begin{equation} \varGamma = \frac{D}{H}, \end{equation}

where $D$ is the inner diameter of the cylindrical container. We focus on $\varGamma = 2$ in this study, similar to Vogt et al. (Reference Vogt, Horn, Grannan and Aurnou2018a), and present only two $\varGamma = 1$ case results for contrast in Appendix C.

Alternatively, the groups of the above parameters that exist in (A1) and (A2) are the Péclet number $Pe$, the Reynolds number $Re$, the convective interaction parameter $N_{\mathcal {C}}$ and the thermoelectric interaction parameter $N_\mathcal {C}Se$. The Péclet number,

(2.21)\begin{equation} Pe = \frac{U_{f\!f} H}{\kappa} = \sqrt{Ra Pr}, \end{equation}

estimates the ratio of thermal advection and thermal diffusion in the thermal energy equation. The convective interaction parameter $N_{\mathcal {C}}$ is the ratio of quasistatic Lorentz drag and fluid inertia. It is defined as

(2.22)\begin{equation} N_{\mathcal{C}} = \frac{\sigma B^2H}{\rho U_{f\!f}} = Ch\sqrt{\frac{Pr}{Ra}} = \frac{Ch}{Re}. \end{equation}

When $N_\mathcal {C}\gtrsim 1$, the Lorentz force will tend to strongly damp buoyancy-driven convective turbulence. Lastly, the thermoelectric interaction parameter, $N_{T\!E}$, is the product of the convective interaction parameter $N_{\mathcal {C}}$ and the Seebeck number $Se$. This parameter approximates the ratio between the thermoelectric Lorentz force and the fluid inertia, and is given by

(2.23)\begin{equation} N_{T\!E} = \frac{\sigma B {|\tilde{S}|} \Delta T}{\rho U_{f\!f}^2} = Se \, N_{\mathcal{C}} . \end{equation}

Thus, when $Se \sim 1$, the thermoelectric forces can become comparable to the MHD drag, at least in the vicinity of the material interfaces where the thermoelectric currents are maximal.

All the non-dimensional parameters and their estimated values for our study are summarized in table 1.

Table 1. Non-dimensional parameters and parameter groups in TEMC. The low values of the top two parameters show that the current experiments fall within the quasistatic approximation. The next five are the base parameters used to describe most of the experimental cases. The next four parameters are alternative groupings that arise in the non-dimensional version of (2.15) given in Appendix A.

2.3. Previous studies of turbulent MC

Despite its broad relevance to natural and industrial systems, MC has not been studied in great detail relative to non-magnetic RBC (e.g. Ahlers, Grossmann & Lohse Reference Ahlers, Grossmann and Lohse2009) and rotating convection (e.g. Aurnou et al. Reference Aurnou, Calkins, Cheng, Julien, King, Nieves, Soderlund and Stellmach2015). Further, laboratory and numerical studies of turbulent MC have largely neglected thermoelectric effects to date (cf. Zhang et al. Reference Zhang, Cramer, Lange and Gerbeth2009). Thus, in reviewing the current state of turbulent MC studies, thermoelectric effects will not be considered.

In the limit of weak magnetic fields, such that $N_{\mathcal {C}} \rightarrow 0$, turbulent MC behaves similarly to RBC (Cioni et al. Reference Cioni, Chaumat and Sommeria2000; Zürner et al. Reference Zürner, Liu, Krasnov and Schumacher2016), with the flow self-organizing into a LSC. Thus, the LSC is the base flow structure in turbulent MC when the dynamical effects of the magnetic field are subdominant (Zürner et al. Reference Zürner, Schindler, Vogt, Eckert and Schumacher2020). Large-scale circulations, the largest turbulent overturning structure in the bulk fluid, have been studied extensively in RBC systems (e.g. Xia, Sun & Zhou Reference Xia, Sun and Zhou2003; Xi, Lam & Xia Reference Xi, Lam and Xia2004; Sun, Xia & Tong Reference Sun, Xia and Tong2005; Von Hardenberg et al. Reference Von Hardenberg, Parodi, Passoni, Provenzale and Spiegel2008; Ahlers et al. Reference Ahlers, Grossmann and Lohse2009; Brown & Ahlers Reference Brown and Ahlers2009; Chillà & Schumacher Reference Chillà and Schumacher2012; Pandey, Scheel & Schumacher Reference Pandey, Scheel and Schumacher2018; Stevens et al. Reference Stevens, Blass, Zhu, Verzicco and Lohse2018; Vogt et al. Reference Vogt, Horn, Grannan and Aurnou2018a; Zürner et al. Reference Zürner, Schindler, Vogt, Eckert and Schumacher2020).

Vogt et al. (Reference Vogt, Horn, Grannan and Aurnou2018a) carried out turbulent RBC laboratory (and associated numerical) experiments in a $\varGamma = 2$ liquid gallium cell using the same laboratory device as we employ in this study. Coupling the DNS outputs to laboratory thermo-velocimetric data, Vogt et al. (Reference Vogt, Horn, Grannan and Aurnou2018a) found that the turbulent liquid metal convection was dominated by a so-called JRV LSC mode, instead of the sloshing and torsional modes found in the majority of $\varGamma = 1$ experiments (e.g. Funfschilling & Ahlers Reference Funfschilling and Ahlers2004; Funfschilling, Brown & Ahlers Reference Funfschilling, Brown and Ahlers2008; Brown & Ahlers Reference Brown and Ahlers2009; Xi et al. Reference Xi, Zhou, Zhou, Chan and Xia2009; Zhou et al. Reference Zhou, Xi, Zhou, Sun and Xia2009). The JRV had a characteristic oscillation frequency $\widetilde {f}_{JRV}$ of

(2.24)\begin{equation} \widetilde{f}_{JRV} = \,f_{JRV}/\,f_\kappa = 0.027Ra^{0.419}, \end{equation}

where $f_\kappa$ is the inverse of the thermal diffusion time scale

(2.25)\begin{equation} \tau_\kappa = H^2 / \kappa. \end{equation}

Ultrasonic measurements yielded an LSC velocity scaling corresponding to

(2.26)\begin{equation} Re_{JRV} = 0.99\left({Ra}/{Pr} \right)^{0.483}, \end{equation}

formulated using their mean Prandtl number value, $Pr \simeq 0.027$. These velocity measurements approach the free-fall velocity scaling in which $Re = U_{f\!f} H / \nu = (Ra/Pr)^{1/2}$. Thus, we will use $U_{f\!f}$ as the characteristic velocity scale when non-dimensionalizing our equations in Appendix A and in the model of thermoelectric LSC precession developed in § 6.

The quasistatic Lorentz force does, however, impede the convective motions in finite $N_{\mathcal {C}}$ cases. Zürner et al. (Reference Zürner, Schindler, Vogt, Eckert and Schumacher2020) used ultrasonic velocimetry measurements to develop an empirical scaling law for the global characteristic velocity, $U_{M\text{-}C}$, in GaInSn MC experiments,

(2.27)\begin{equation} U_{M\text{-}C} = \left( \frac{1}{1+0.68 \, N_{\mathcal{C}}^{0.87}} \right) \, U_{f\!f}. \end{equation}

In § 6 we will test both $U_{M\text{-}C}$ and $U_{f\!f}$ in our model for thermoelectrical precession of the LSC, and show that the $U_{M\text{-}C}$-based predictions better fit our precessional frequency measurements.

The turbulent LSC mode breaks down in MC when $N_{\mathcal {C}} \gtrsim 1$ (Cioni et al. Reference Cioni, Chaumat and Sommeria2000; Zürner et al. Reference Zürner, Schindler, Vogt, Eckert and Schumacher2019, Reference Zürner, Schindler, Vogt, Eckert and Schumacher2020). This is roughly analogous to the loss of the LSC in rotating convection when the Rossby number is decreased below unity (Kunnen, Clercx & Geurts Reference Kunnen, Clercx and Geurts2008; Horn & Shishkina Reference Horn and Shishkina2015). In the supercritical $N_{\mathcal {C}} \gtrsim 1$ regime, the convection in the fluid bulk should then become multi-cellular, akin to the flows shown in Yan et al. (Reference Yan, Calkins, Maffei, Julien, Tobias and Marti2019).

Near the onset of the MC, wall modes appear near the vertical boundaries and will become unstable before bulk convection in many geometrically confined MC systems (Busse Reference Busse2008).

It is important to stress that MC wall modes do not drift along the wall, in contrast to rotating convection (Ecke, Zhong & Knobloch Reference Ecke, Zhong and Knobloch1992), since the quasistatic Lorentz force does not break azimuthal reflection symmetry (e.g. Houchens, Witkowski & Walker Reference Houchens, Witkowski and Walker2002). The multi-cellular and magneto-wall mode regimes were both investigated in the numerical MC simulations of Liu, Krasnov & Schumacher (Reference Liu, Krasnov and Schumacher2018). The wall modes were found not to drift in their large-aspect ratio simulations, similar to the experimental findings of Zürner et al. (Reference Zürner, Schindler, Vogt, Eckert and Schumacher2020). Furthermore, Liu et al. (Reference Liu, Krasnov and Schumacher2018) showed that the wall modes could become unstable and inject nearly axially invariant jets into the fluid bulk.

Strong wall mode injections are also found in the numerical MC simulations of Akhmedagaev et al. (Reference Akhmedagaev, Zikanov, Krasnov and Schumacher2020). These injected axially invariant jets are accompanied by a net azimuthal drift of the flow field, whose drift direction appears to be randomly set. We interpret these drifting flows as being controlled by the collisional interaction of the jets, qualitatively similar in nature to the onset of the shearing flows in the plane layer simulations of Goluskin et al. (Reference Goluskin, Johnston, Flierl and Spiegel2014). Therefore, we argue that the drifting effect found in the $N_{\mathcal {C}} > 1$ near-onset numerical simulation by Akhmedagaev et al. (Reference Akhmedagaev, Zikanov, Krasnov and Schumacher2020) fundamentally differs from the LSC precession found in the thermoelectrically active $N_{\mathcal {C}} \lesssim 1$ experiments reported herein.

5.1. Fixed $Ra \approx 2\times 10^6$ TEMC survey

To characterize the system's behavioural regimes, we have conducted a survey of turbulent TEMC with the Rayleigh number fixed at approximately $2\times 10^6$ and the Chandrasekhar number varying from $0$ to $8\times 10^5$ all with a vertically downward applied magnetic field ($\boldsymbol {\hat e_b} = -\boldsymbol {\hat e_z}$). Three regimes are found: (i) the JRV regime, (ii) the MP regime, and (iii) the MCMC regime.

Figure 9(a) shows a thermal spectrogram made using the $T_{ij}^{mid}$ data, plotted as functions of $f /f_\kappa$ on the vertical axis and the Chandrasekhar number $Ch$ on the horizontal axis. Here, we use $Pr = 0.027$, and the RBC case's $Ra_0 = 2.12 \times 10^6$ value to calculate $N_\mathcal {C}$ for all the cases. The peak frequency at each $Ch$-value is marked by an open black circle. Starting from the left of the figure, the predicted peak RBC frequency, $\widetilde {f}_{JRV}$ derived from equation (2.24), is marked by the white, horizontal dashed line. In the JRV regime, the $0 < N_\mathcal {C} \lesssim 10^{-1}$ experimental data are in good agreement with $\widetilde {f}_{JRV}$, with sidewall thermal fields that correspond to that of an LSC-like flow (e.g. figure 9b). In this regime, buoyancy-driven inertia is the dominant forcing in the system. As $N_\mathcal {C}$ increases and exceeds unity, it is expected that the LSC will weaken and eventually disappear (e.g. Cioni et al. Reference Cioni, Chaumat and Sommeria2000; Zürner et al. Reference Zürner, Schindler, Vogt, Eckert and Schumacher2020). However, the MP regime exists in the intermediate $N_\mathcal {C}$ TEMC system. In this regime, the spectral peak switches from near to $\widetilde {f}_{JRV}$ to the slow MP frequency above $N_\mathcal {C} \approx 0.1$, corresponding to the MP sidewall thermal signal shown in figure 9(c). The MP frequency grows with $N_\mathcal {C}$, reaching a value of $0.60\ f_\kappa$ near $N_\mathcal {C} \approx 0.4$. At higher $N_\mathcal {C}$, the peak frequency decays, becomes unstable, and mixes with other complex modes at $N_\mathcal {C} \gtrsim 1$ (e.g. figure 9d). The single, turbulent LSC likely gives way to multi-cellular bulk flow in this $N_\mathcal {C} \gtrsim 1$ MCMC regime.

Figure 9. (a) Power spectral density (PSD) of the $T_{ij}^{mid}$ temperature data vs $Ch$ and $N_{\mathcal {C}}$ for the $Ra \approx 2\times 10^6$ cases shown in table 5. The vertical axis is thermal diffusion frequency $\widetilde{f} = f/f_\kappa$. The peak frequency for each case is marked with black open circles. The interaction parameter is calculated here as $N_\mathcal {C} = \sqrt {Ch^2 Pr/Ra_0}$, where $Ra_0 = 2.12 \times 10^6$ corresponds to the case with no magnetic field. The JRV, the MP and the MCMC regimes are separated by the two black vertical dot-dashed lines. The JRV frequency (Vogt et al. Reference Vogt, Horn, Grannan and Aurnou2018a), $\widetilde {f}_{JRV}$, is shown as the white horizontal dotted line near $\widetilde{f} \approx 12.1$. The black square marks the peak frequency of the $Conducting\ MC^-$ case, and the white cross-marks the peak frequency of the $Insulating\ MC^-$ case. The black dashed curve denotes the second-order fit to the experimental data in the MP regime. The white stars are magnetoprecession frequency estimates calculated using (6.20) for each MP case, and the white dotted curve is the second-order fit of these theoretical estimates developed in § 6.5. The lower panels show sidewall midplane temperature contour maps in (b) JRV, (c) MP and (d) MCMC regimes. The blue downwards and red upwards triangles in the lower panels denote the heat exchanger inlet and outlet azimuth locations, respectively.

We contend that it is the existence of coherent thermoelectric current loops existing across the top and bottom horizontal interfaces of the fluid layer that drive the MP mode observed in the MP regime. Following the arguments of § 2.1, this requires horizontal temperature gradients to exist along these bounding interfaces as shown schematically in figure 1. To quantify this, the horizontal temperature difference at height $z_k$ and time $t_j$ is estimated using the best fit of the data to (4.1) as

(5.1)\begin{equation} \delta T_j^k= \max \left( T_{fit}^k(t_j)\right) - \min \left(T_{fit}^k(t_j)\right) = 2 A_j^k . \end{equation}

Its time-mean value is denoted by

(5.2)\begin{equation} \delta T^k = \frac{2}{N} \sum_{j=1}^{N} A_j^k , \end{equation}

where $j$ is the $j$th step in the discrete temperature time series and $N$ is the total number of time steps. Thus, $\delta T^{top}$ estimates the time-averaged, maximum horizontal temperature difference in the top block thermistors located at $z = 100.6\, \mathrm {mm} = 1.022 H$ and $r {=0.71} R$. Similarly, $\delta T^{bot}$ estimates this value using the bottom block thermistors located at $z = -2.0 \, \mathrm {mm} = -0.020 H$ and $r {=0.71} R$.

Figure 10(a) shows time series of the maximum horizontal temperature variations in the top and bottom boundaries in the reference frame of the fitted LSC plane, $\delta T_j^k$, calculated using (5.1). This data is from our canonical $Conducting\ MC^-$ case at $Ra = 1.83\times 10^6$, $Ch = 2.59\times 10^3$ and $N_\mathcal {C}= 0.31$. The temperature time series for this case has been shown in figure 6. In the top block, the time-averaged maximum horizontal temperature variation, plotted in blue, is $2.24\, \mathrm {K}$, which is $1.2\, \mathrm {K}$ smaller than that of the bottom block (in red), $3.44\, \mathrm {K}$. Moreover, the top has a larger magnitude of fluctuation of ${\sim}2\, \mathrm {K}$, while the bottom remains relatively stable with a fluctuation of ${\sim}1\, \mathrm {K}$. The difference in the top and bottom $\delta T_j^k$ fluctuation amplitudes is likely due to the structure of the heat exchanger, in which the inlet and the outlet of the cooling water are antipodal to one another. This imposes a small (${\sim}1$ K), spatially fixed temperature gradient along the line connecting these two points. This causes the horizontal temperature variation at the top boundary to fluctuate as the LSC precesses across the top block's spatially fixed temperature gradient. We hypothesize that this fluctuation propagates to the bottom boundary, generating a smaller fluctuation there than on the top block and lagging the top fluctuation by approximately $0.6$ thermal diffusion times.

Figure 10. (a) Time series of the horizontal temperature difference at different heights $\delta T_j^k$, defined in (5.1), from the $Conducting\ MC^-$ case at $Ra = 1.83\times 10^6$, $Ch = 2.59\times 10^3$ and $N_\mathcal {C}= 0.31$. The horizontal axis is normalized time $t_j/\tau _\kappa$. The black dotted line denotes the mean values of $\delta T^{top} = 2.24\, \mathrm {K}$, and the black dashed line denotes $\delta T^{bot} = 3.44\, \mathrm {K}$. (b) Time-averaged horizontal temperature difference estimates $\delta T^{k}$ on the top and bottom boundaries for the fixed-$Ra$ cases ($Ra\approx 2\times 10^6$) at $N_{\mathcal {C}} < 1$. The fixed-$Ra$ cases are marked by triangles; blue (red) colour represents top (bottom) boundary measurements. The MP regime lies between the two vertical dot-dashed lines. The right-hand $y$-axis denotes $\delta T^{k}$ normalized by the averaged vertical temperature difference $\Delta T = 7.68\, \mathrm {K}$ of the fixed-$Ra$ cases shown here. Values of $\delta T_j^k$ for the $Conducting\ MC^-$ case are marked by the square symbols.

Figure 10(b) shows $\delta T^k$, the time-mean horizontal temperature differences calculated via (5.2) on the top block (blue triangles) and on the bottom block (red triangles) for the $N_{\mathcal {C}} < 1$ experiments in the fixed $Ra$ survey ($Ra\approx 2\times 10^6$). The $Conducting\ MC^-$ case in panel (a) corresponds to the square markers on the right in panel (b). The horizontal dashed and dotted lines show the mean values of $\delta T^k$ for the $Conducting\ MC^-$ case. The two vertical dot-dashed lines denote the boundaries between the JRV, MP and MCMC regimes. In the lowest $N_{\mathcal {C}}$ case shown in figure 10(b), it is found that $\delta T^{top} \approx \delta T^{bot} \approx 2.6$ K. This value is nearly $40\,\%$ of the vertical temperature gradient across the tank, and is similar to values found for comparable RBC cases. We argue that this 2.6 K value is predominantly generated by the JRV imprinting its thermal anomalies onto the top and bottom boundary thermistors. The values of $\delta T^{bot}$ exceed $\delta T^{top}$ in the MP regime ($0.1 \lesssim N_{\mathcal {C}} \lesssim 1$). For $N_{\mathcal {C}} \gtrsim 1$, the jump rope-style LSC breaks down into multi-cellular flow (e.g. Zürner et al. Reference Zürner, Schindler, Vogt, Eckert and Schumacher2020) and it is not possible to fit a sinusoidal function of the form (4.1) to the thermistor data in the top and bottom blocks.

The right-hand vertical axis in figure 10(b) shows thermal block temperature differences normalized by the vertical temperature difference, $\delta T^{k}/ \Delta T$. The fixed $Ra$ survey cases have $\delta T^{k}/ \Delta T$ values ranging from roughly 0.3 to 0.5, demonstrating that low $Pr$ convective heat transfer occurs via large-scale, large amplitude thermal anomalies that may alter the thermal boundary conditions in finite $Bi$ experiments. Such conditions differ from those typically assumed in theoretical models of low Prandtl number convection (e.g. Clever & Busse Reference Clever and Busse1981; Thual Reference Thual1992).

The slow MP modes only appear in MC experiments with electrically conducting boundaries for $0.1\lesssim {N_\mathcal {C}} \lesssim 1$. Strong, coherent horizontal temperature gradients exist along the top and bottom boundaries in these cases, as shown in figure 10. This suggests that magnetoprecession is controlled by the material properties of the boundaries and the horizontal temperature gradients on the liquid–solid interfaces. Based on these arguments, we develop a simple model for thermoelectrically driven magnetoprecession of the LSC in the following section.

6.1. Angular momentum of the LSC flywheel

A Cartesian coordinate frame is used in our model of thermoelectrically driven magnetoprecession of the LSC. This Cartesian frame is fixed in the LSC plane such that $\boldsymbol {\hat e_y}$ always points along the $\xi _j = 0$ direction. The thermal gradient is also aligned in the same direction, yielding $\boldsymbol {\hat e_n} = \boldsymbol {\hat e_y}$. The magnetic field direction is oriented in $\pm \boldsymbol {\hat e_z}$, and the right-handed normal to the LSC plane is oriented in the $\boldsymbol {\hat e_x}$-direction.

We treat the LSC as a solid cylindrical flywheel that spins around a midplane $\boldsymbol {\hat e_x}$-axis, as shown in figure 11. The LSC is taken to have the same cross-sectional area as the LSC plane, $A_{LSC} = \varGamma H^2$. The corresponding radius of the solid LSC cylinder is then

(6.1)\begin{equation} R^* = \sqrt{A_{LSC}/{\rm \pi}} = \sqrt{\varGamma H^2/{\rm \pi}}, \end{equation}

which corresponds to $R^* \approx 0.8 H$ for the $\varGamma = 2$ experiments carried out here. The volume of the turbulent LSC is $V_{LSC}$. For convenience, we take the depth of the LSC in $\boldsymbol {\hat e_x}$ to be $R^*$ so that $V_{LSC} = {\rm \pi}R^{*3}$, noting that the assumed depth and $V_{LSC}$ both drop out of our eventual prediction for the LSC's magnetoprecession rate $\omega _{M\!P}$. The LSC flywheel, as constructed, does not physically fit within the tank since $R^* > H/2$, as shown in figure 11. (It is not shown to scale in figure 13b.)

Figure 11. Cross-section view of the precessional flywheel model (pink) in the LSC plane (yellow). The precessional flywheel is assumed to have the same cross-sectional area as the LSC plane, ${\rm \pi} (R^*)^2 = 2H^2$. The angular velocity of the overturning LSC flywheel is estimated by assuming it rotates at the free-fall speed $\omega _{LSC} \approx U_{f\!f}/R^*$.

The angular momentum of the flywheel is taken to be that of a uniform-density solid cylinder with mass $M_{LSC} = \rho V_{LSC}$ and radius $R^*$, rotating around $\boldsymbol {\hat e_x}$. Its moment of inertia with respect to the $\boldsymbol {\hat e_x}$-axis is

(6.2)\begin{equation} I = \tfrac{1}{2} M_{LSC} R^{*2} = \tfrac{1}{2} \rho V_{LSC} R^{*2} . \end{equation}

We use the upper bounding free-fall velocity as an estimate of the angular velocity vector for the LSC flywheel,

(6.3)\begin{equation} \boldsymbol{\omega}_{LSC} \approx U_{f\!f}/R^* \boldsymbol{\hat e_x} . \end{equation}

Thus, the angular momentum due to the overturning of the flywheel, $\boldsymbol {L}_{LSC}$, is oriented along $\boldsymbol {\hat e_x}$ and is estimated as

(6.4)\begin{equation} \boldsymbol{L}_{LSC} = I \boldsymbol{\omega}_{LSC} \approx \left( \frac{1}{2} \rho V_{LSC} R^{*2} \right) \left( \frac{U_{f\!f}}{R^*}\right) \boldsymbol{\hat e_x} = \frac{1}{2} \rho V_{LSC} U_{f\!f} R^{*}\ \boldsymbol{\hat e_x}. \end{equation}

6.2. Thermoelectric currents at the electrically conducting boundaries

Figure 12(a) shows a schematicized vertical slice through our experimental tank in the low $N_{\mathcal {C}}$ regime. The LSC generates horizontal thermal gradients on both horizontal boundaries. Thus, the end blocks have a higher temperature near the upwelling branch of the LSC, which carries warmer fluid upwards, and the end blocks are cooler near the downwelling branch of the LSC, which carries cooler fluid downwards. These temperature gradients on the top and bottom fluid–solid interfaces generate thermoelectric current loops. Equation (2.7) is used to calculate the net Seebeck coefficient of such a thermoelectric current loop in our Cu–Ga system,

(6.5)\begin{equation} \varPhi_{T\!E} = \int_{r_0(T_{0})}^{r_1(T_{1})} (S_{Cu}-S_{Ga}) \boldsymbol{\nabla} T \boldsymbol{\cdot} \textrm{d} \boldsymbol r = \frac{{\rm \pi}^{2} k_B^{2}}{6 e}\left[\frac{x_{Cu}}{E_{F Cu}}-\frac{x_{Ga}}{E_{F Ga}}\right] \left(T_1^{2}-T_{0}^{2}\right), \end{equation}

where $S_{Ga}$ and $S_{Cu}$ are Seebeck coefficients for gallium and copper, respectively, calculated via (2.2). For gallium, the numeric coefficient $x_0$ is $x_{Ga} = 0.7$ (Cusack Reference Cusack1963) and the Fermi energy is $E_{FGa} = 10.37\, \mathrm {eV}$ (Kasap Reference Kasap2001). For copper, $x_{Cu} = -1.79$ and $E_{FCu} = 7.01\, \mathrm {eV}$. The temperatures $T_0$ and $T_1$ represent the minimum and maximum temperatures, respectively, in a given horizontal plane (e.g. figure 1).

Figure 12. (a) Cross-sectional schematics of the experimental MC system with electrically conducting boundaries. In the plane of LSC, the turbulent LSC imprints large-scale thermal anomalies onto the boundaries: the top boundary has a minimum temperature $T_{0}^{top}$ and a maximum temperature $T_{1}^{top}$; the bottom boundary has a minimum temperature $T_{0}^{bot}$ and a maximum temperature $T_{1}^{bot}$. Thermoelectric potentials are generated at the Cu–Ga interfaces and form current density loops across the boundaries, $\boldsymbol {J}_{T\!E}^{top}$ and $\boldsymbol {J}_{T\!E}^{bot}$, with a width of $\mathcal {L} \approx \varGamma H$. (b) Circuit diagram of the Cu–Ga system at the bottom boundary. The thermoelectric potential in gallium is denoted as $\varPhi _{Ga}$, which is smaller in magnitude and has an opposite sign to the thermoelectic potential in copper, $\varPhi _{Cu}$. Thus, the thermoelectric current flows from cold to hot in liquid gallium (in $+\boldsymbol {\hat e_n}$), and from hot to cold in copper (in $-\boldsymbol {\hat e_n}$).

Figure 13. (a) Free-body diagram of thermoelectric LSC precession. The red arrows enclosed inside the tank represents the LSC. The thermoelectric potentials generate current in the liquid gallium at the top and bottom boundaries: $\boldsymbol {J}_{T\!E}^{top}$ and $\boldsymbol {J}_{T\!E}^{bot}$ with $J_{T\!E}^{top}< J_{T\!E}^{bot}$. The thermoelectric Lorentz forces, $\boldsymbol {F}_{T\!E}^{top}$ and $\boldsymbol {F}_{T\!E}^{bot}$, create a net torque $\boldsymbol {\tau }_{net}$ perpendicular to the LSC's angular momentum vector, $\boldsymbol {L}_{LSC}$, which drives the LSC to precess around the tank's vertical $\boldsymbol {\hat e_z}$-axis. (b) Precessional flywheel schematic (not to scale). The LSC is simplified into a flywheel-like cylinder of radius $R^*$, with angular velocity $\omega _{LSC} \lesssim U_{f\!f}/R^*$ and angular momentum $\boldsymbol {L}_{LSC} = \omega _{LSC} MR^{*2} \boldsymbol {\hat e_x}/ 2$. The LSC is assumed to respond to $\boldsymbol {\tau }_{net}$ in a solid-body manner.

Following (2.9), the net Seebeck coefficient on the $k$-level Cu–Ga interface is

(6.6)\begin{equation} \tilde{S}^k = \frac{{\rm \pi}^{2} k_B^{2}}{3 e}\left[\frac{x_{C u}}{E_{F C u}}-\frac{x_{G a}}{E_{F G a}}\right] T^k \equiv X_0 \, T^k \, (\mathrm{V}\,\mathrm{K}^{{-}1}), \end{equation}

where $T^k$ is the time-azimuthal mean temperature on the $k$-interface. The Cu–Ga Seebeck prefactor $X_0$ collects all the constant and material properties in (6.6). Its value in our system is

(6.7)\begin{equation} X_0 \approx{-}7.89 \times 10^{{-}9}\, \mathrm{V}\,\mathrm{K}^{{-}2}. \end{equation}

Unlike the net Seebeck coefficient, $X_0$ does not depend on temperature.

The thermoelectric current density vector in liquid gallium is approximated via (2.6),

(6.8)\begin{equation} \boldsymbol J_{T\!E}^k \approx \sigma_0 X_0 T^k \left( \frac{\delta T^k}{\mathcal{L}} \right) \, \boldsymbol{\hat e_y}, \end{equation}

where $\sigma _0 = 3.63\times 10^6\, \mathrm {S}\,\mathrm {m}^{-1}$ is the effective electric conductivity for the Cu–Ga system, calculated by substituting the conductivity of gallium ($\sigma _{Ga}\approx 3.88\times 10^6\, \mathrm {S}\,\mathrm {m}^{-1}$) and copper ($\sigma _{Cu}\approx 5.94\times 10^7\, \mathrm {S}\,\mathrm {m}^{-1}$) into (2.5). The horizontal temperature gradient is approximated by the maximum temperature difference across the $k$-interface, $\delta T^k$, divided by a characteristic width of the current loop, $\mathcal {L}$. We assume this width is the same as the diameter $D$ of the tank, $\mathcal {L} \approx \varGamma H = 2H = 197.2\, \mathrm {mm}$. (Effects of possible thermoelectric currents in the stainless steel sidewall ($\sigma _{St.Stl.}\approx 1.4\times 10^6\, \mathrm {S}\,\mathrm {m}^{-1}$) are not accounted for here.)

Figure 12(b) shows a circuit diagram for the thermoelectric current loop near the experiment's bottom liquid–solid interface at $\approx 40\,^{\circ }\mathrm {C}$. The thermoelectric potential in gallium has a negative sign, so the currents within the fluid are always aligned in the direction of the thermal gradient $\boldsymbol {\hat e_n} = \boldsymbol {\hat e_y}$. In contrast, the thermoelectric potential in copper has a positive sign, so the current flows from the hot to the cold region in $-\boldsymbol {\hat e_n}$.

6.3. Thermoelectric forces and torques

The thermoelectric component of the mean Lorentz forces density on the index $k$ horizontal interface can be calculated, using (6.8), as

(6.9)\begin{equation} \boldsymbol{f}_{T\!E}^k = \boldsymbol J_{T\!E}^k \times \boldsymbol B = \sigma_0 X_0 T^k \left(\frac{\delta T^k }{{\varGamma}H} \boldsymbol{\hat e_y}\right) \times B \boldsymbol{\hat e_b} = \frac{\sigma_0 X_0 B \, T^k \delta T^k }{{\varGamma} H} \, (\boldsymbol{\hat e_y} \times \boldsymbol{\hat e_b}) , \end{equation}

where we have taken $\mathcal {L} \approx \varGamma H$. The definitions of the experimental parameters and their values from the Conducting $MC^-$ subcase is shown in table 2. Since the thermoelectric currents are predominantly in the LSC plane and are aligned parallel to the thermal gradient, the thermoelectric Lorentz forces point in the $+\boldsymbol {\hat e_x}$-direction for upward directed $\boldsymbol {B}$, and in the $-\boldsymbol {\hat e_x}$-direction for downward directed $\boldsymbol {B}$ (corresponding to figure 13a).

The Lorentz force is the volume integral of the force density

(6.10)\begin{equation} \boldsymbol{F}_{T\!E}^k = \int{\boldsymbol{f}_{T\!E}^k \, \mathrm{d}V} = \int \boldsymbol J_{T\!E}^k \times \boldsymbol{B}\, \mathrm{d}V. \end{equation}

We coarsely assume that each thermoelectric current loop exists in the top half or bottom half of the LSC, and generates uniform Lorentz forces that act, respectively, on the upper or lower half of the LSC. With these assumptions, we take each hemicylindrical integration volume to be $V_{LSC}/2$. The Lorentz forces due to thermoelectric currents generated across the $k$-level Cu–Ga interface are then estimated to be

(6.11)\begin{equation} \boldsymbol F_{T\!E}^k = \tfrac{1}{2} V_{LSC} \boldsymbol{f}_{T\!E}^k . \end{equation}

Table 2. Experimental parameter values from the $Conducting\ MC^-$ subcase. These values are characteristic of those used in calculating $\omega _{M\!P}$ in figure 14.

In order to estimate the torques due to each thermoelectric current loop, we assume that the thermoelectric Lorentz forces act on the LSC via a moment arm $\boldsymbol {\ell }$ of approximate length $H/2$. Thus, $\boldsymbol {\ell }^{top} = H/2 \, \boldsymbol {\hat {e}_z}$ and $\boldsymbol {\ell }^{bot} = - H/2 \, \boldsymbol {\hat {e}_z}$. The net thermoelectric torque on the LSC then becomes

(6.12a)\begin{align} \boldsymbol{\tau}^{net} &= \boldsymbol{\tau}^{top}+ \boldsymbol{\tau}^{bot} \end{align}

(6.12b)\begin{align} &= \left( \boldsymbol{\ell}^{top} \times \boldsymbol F_{T\!E}^{top} \right) + \left( \boldsymbol{\ell}^{bot} \times \boldsymbol F_{T\!E}^{bot} \right) \end{align}

(6.12c)\begin{align} &= (H/2) \boldsymbol{\hat e_z} \times \left( \boldsymbol F_{T\!E}^{top}- \boldsymbol F_{T\!E}^{bot} \right) \end{align}

(6.12d)\begin{align} &= (H V_{LSC}/4) \boldsymbol{\hat e_z} \times \left( \boldsymbol{f}_{T\!E}^{top}- \boldsymbol{f}_{T\!E}^{bot} \right) . \end{align}

Substituting (6.9) into (6.12d) yields the net thermoelectric torque on the LSC to be

(6.13)\begin{align} \boldsymbol{\tau}^{net} &= \frac{\sigma_0 V_{LSC} X_0 B \, \left( T^{bot} \delta T^{bot} - T^{top} \delta T^{top} \right)}{4 \varGamma} \ \left(\boldsymbol{\hat e_x} \times \boldsymbol{\hat e_b} \right)\nonumber\\ &= \frac{\sigma_0 V_{LSC} X_0 B \, \mathcal{T}}{4 \varGamma} \, \left(\boldsymbol{\hat e_x} \times \boldsymbol{\hat e_b} \right) , \end{align}

where

(6.14)\begin{equation} \mathcal{T} \equiv \left( T^{bot} \delta T^{bot} - T^{top} \delta T^{top} \right) \end{equation}

describes the difference in thermal conditions on the bottom relative to top horizontal Cu–Ga interfaces. Since $T^{bot} > T^{top}$ in all our convection experiments, the data in figure 10(b) implies that $\mathcal {T} > 0$ in the MP regime. Since all the other parameters in (6.13) are positive, the net thermoelectric torque is directed in $-\boldsymbol {\hat e_y}$ for upwards directed magnetic fields ($\boldsymbol {\hat e_b} = + \boldsymbol {\hat e_z}$) and, as shown in figure 13, the net torque is directed in $+\boldsymbol {\hat e_y}$ for downwards directed magnetic fields ($\boldsymbol {\hat e_b} = - \boldsymbol {\hat e_z}$). Equation (6.13) also shows that the bottom torque will tend to dominate even when $\delta T^{bot} \approx \delta T^{top}$ since $T^{bot} > T^{top}$ in all convectively unstable cases.

6.4. Thermoelectrically driven LSC precession

The net thermoelectric torque on the LSC acts in the direction perpendicular to $\boldsymbol {L}_{LSC}$. The LSC must then undergo a precessional motion in order to conserve angular momentum. This precession can be quantified via Euler's equation (Landau & Lifshitz Reference Landau and Lifshitz1976), in which the net torque is the time derivative of the angular momentum,

(6.15)\begin{equation} \boldsymbol \tau^{net} = \frac{\textrm{d}{\boldsymbol{L}}_{LSC}}{\textrm{d} t} = I\ \frac{\textrm{d} {\boldsymbol{\omega}}}{\textrm{d} t}+\boldsymbol{\omega} \times \boldsymbol{L}_{LSC}. \end{equation}

The angular velocity vector $\boldsymbol {\omega }$ is comprised of two components here,

(6.16)\begin{equation} \boldsymbol{\omega} = \omega_{LSC} \boldsymbol{\hat e_x} + \boldsymbol \omega_{M\!P} , \end{equation}

where $\omega _{LSC}$ is the angular velocity component of the flywheel in $\boldsymbol {\hat e_x}$ and $\boldsymbol \omega _{M\!P}$ is the angular velocity vector of the LSC's magnetoprecession. We assume that the precession frequency and the angular speed of the flywheel are nearly time invariant, $\textrm {d} {\boldsymbol {\omega }}/{\textrm {d} t} \approx 0$. Then (6.15) reduces to

(6.17)\begin{equation} \boldsymbol{\tau}^{net} = \left( \omega_{LSC} \boldsymbol{\hat e_x} + \boldsymbol \omega_{M\!P} \right) \times L_{LSC} \boldsymbol{\hat e_x} = \boldsymbol{\omega}_{M\!P} \times L_{LSC} \boldsymbol{\hat e_x} , \end{equation}

where $\boldsymbol {\omega }_{LSC} \times \boldsymbol {L}_{LSC} = 0$ since these vectors are parallel. Expression (6.17) requires that $\boldsymbol \omega _{M\!P}$ remain orthogonal to both $\boldsymbol L_{LSC}$ and $\boldsymbol \tau _{net}$ such that

(6.18)\begin{equation} {\tau}^{net} \left(\boldsymbol{\hat e_x} \times \boldsymbol{\hat e_b} \right) = \boldsymbol{\omega}_{M\!P} \times L_{LSC} \boldsymbol{\hat e_x} = {\omega}_{M\!P} L_{LSC} \, \left( \boldsymbol{\hat{\omega}}_{M\!P} \times \boldsymbol{\hat e_x} \right) . \end{equation}

Therefore, $\boldsymbol {\hat {\omega }}_{M\!P} = - \boldsymbol {\hat e_b}$ and the precessional angular velocity vector is

(6.19)\begin{equation} \boldsymbol{\omega}_{M\!P} = \frac{ \tau_{net}}{L_{LSC}} \left( \! - \boldsymbol{\hat e_b} \right) . \end{equation}

Substituting (6.4) and (6.13) into (6.19) yields our analytical estimate for the thermoelectrically driven angular velocity of LSC precession in the $N_{\mathcal {C}} \lesssim 1$ TEMC experiments,

(6.20a)\begin{align} \boldsymbol \omega_{M\!P} &= \frac{\sigma_0 X_0 B}{2 \rho U_{f\!f}\varGamma R^{*} } \left( T^{bot} \delta T^{bot} - T^{top} \delta T^{top} \right) \left( \! - \boldsymbol{\hat e_b} \right) \end{align}

(6.20b)\begin{align} &= \frac{{\rm \pi}^{1/2}}{\varGamma^{3/2}} \frac{\sigma_0 X_0 B \mathcal{T} }{2 \rho U_{f\!f} H } \left( \! - \boldsymbol{\hat e_b} \right)\, . \end{align}

Expressions (6.19) and (6.20) predict that the LSC's MP angular velocity vector, $\boldsymbol {\omega }_{M\!P}$, will always be antiparallel to the imposed magnetic field vector in our Cu–Ga TEMC experiments in which $\mathcal {T} > 0$. Further, the magnetoprecession should flip direction such that $\boldsymbol {\omega }_{M\!P}$ would be parallel to $\boldsymbol {B}$ in a comparable Cu–Ga TEMC system with $\mathcal {T} < 0$. These predictions agree with the precession directions found in our MP regime experiments, as shown in figure 7(a).

6.5. Experimental verification

The direction of magnetoprecession is sensitive to the sign of $\mathcal {T}$. The value and sign of $\mathcal {T}$ are, however, both likely related to the details of the experimental set-up. For instance, we have a thermostated bath controlling the top thermal block temperature, whereas a fixed thermal flux is input below the bottom thermal block. Further, the top and bottom thermal blocks have different thicknesses. It is possible that $\mathcal {T}$ could behave differently with much thinner or thicker end-blocks, and for differing thermal boundary conditions. Thus, further modelling efforts are required before $\mathcal {T}$ can be predicted a priori.

Figure 14(a) provides a test of our magnetoprecession model using data from the $Conducting\ MC^-$ case. The blue line shows the experimentally determined LSC angular velocity via measurements of the azimuthal drift velocity of the LSC plane, $\textrm {d} \xi ^{mid}_j / \textrm {d}t$, made with the best fits to (4.1). The magenta line shows the instantaneous angular velocity of LSC precession, $(\omega _{M\!P})_j$, calculated by feeding instantaneous thermal data from the $Conducting\ MC^-$ case into (6.20). The green line is a modified theoretical estimate, in which the $U_{M\!C}$ velocity prediction (2.27) is used in (6.20) instead of the upper bounding $U_{f\!f}$ estimate.

Figure 14. (a) Time series of LSC precessional rate for the $Conducting\ MC^-$ case at $Ra = 1.83\times 10^6$, $Ch = 2.59\times 10^3$ and $N_\mathcal {C}= 0.31$. The horizontal axis shows time $t$ normalized by the diffusion time scale $\tau _\kappa$. The vertical axis is the LSC's instantaneous angular precession speed $\omega$. The blue line shows angular velocity of the LSC plane, $\textrm {d} \xi ^{mid}_j / \textrm {d}t$, measured via (4.1) using temperature data on the midplane sidewall. The magenta line marks $\omega _{M\!P}$ model predictions made using (6.20) and instantaneous temperature data. The green line shows the alternative MP prediction made using (6.20) with Zürner et al. (Reference Zürner, Schindler, Vogt, Eckert and Schumacher2020)'s $U_{M\!C}$ velocity scaling (2.27). See table 4 for $Conducting\ MC^-$ case details. The horizontal dashed lines are mean values for their corresponding angular speeds. The black horizontal dashed line represents the peak frequency from the FFT converted into the angular speed. (b) Scatter plot of $\textrm {d} \xi ^{mid}_j / \textrm {d}t$ vs $\mathcal {T}$. The red dashed line is a linear fit for this particular case. The best fit slope, $3.42\times 10^{-6}\, \mathrm {K}^2\,\mathrm {s}^{-1}$, is $55.3\,\%$ of the theoretical prediction from (6.20), where the prefactor $\sigma _0 X_0 B / ( 4\rho U_{f\!f} R^{*} ) \approx 6.18\times 10^{-6}\, \mathrm {K}^2\,\mathrm {s}^{-1}$.

The three time series in figure 14(a) show that the predicted MP angular speeds compare well with the LSC azimuthal drift speed, especially when accounting for the simplifications that are underlying (6.20). The time series all have similar gross shapes, but with the peaks in $\textrm {d} \xi ^{mid}_j /\textrm {d}t$ slightly delayed in time relative to those in the $\omega _{M\!P}$ curves. The dashed lines in figure 14 show the time-mean angular velocity values. The time-mean LSC plane velocity is $\textrm {d} \xi ^{mid} /\textrm {d}t = 5.27\times 10^{-3}\, \mathrm {rad}\,\mathrm {s}^{-1}$. The mean value of the MP flywheel (magenta) is $\omega _{M\!P} = 2.48\times 10^{-3}\, \mathrm {rad}\,\mathrm {s}^{-1} = 0.47 \, \textrm {d} \xi ^{mid} /\textrm {d}t$. The mean value of the modified estimate (green) is $\omega _{M\!P}\, U_{f\!f}/ U_{M\!C}= 3.10\times 10^{-3}\, \mathrm {rad}\,\mathrm {s}^{-1} = 0.59 \, \textrm {d} \xi ^{mid} /\textrm {d}t$. Moreover, there is a good agreement between the converted angular speed from the peak frequency $2{\rm \pi} \,f_{peak}$ of the FFTs (black dashed lines) and the mean LSC azimuthal drift speed by direct thermal measurement $\textrm {d} \xi ^{mid} /\textrm {d}t$ (blue dashed lines). The direct measurement is approximately $2.5\,\%$ higher than $2{\rm \pi} \,f_{peak}$.

Figure 14(b) shows the correlation between the measured angular velocity $\textrm {d} \xi ^{mid}_j /\textrm {d}t$ and $\mathcal {T}$. The angular velocity values correspond to the blue curve in panel (a). The dashed red line is the best linear fit to the data, and shows that there is a net positive correlation between $\textrm {d} \xi ^{mid}_j /\textrm {d}t$ and the asymmetry of the thermal condition in end blocks, represented by $\mathcal {T}$. The best fit slope is $(\textrm {d} \xi ^{mid}_j /\textrm {d}t) / \mathcal {T} = 3.42\times 10^{-6}\, \mathrm {K}^2\ \mathrm {s}^{-1}$. Theoretical prediction (6.20) gives $\omega _{M\!P}/ \mathcal {T} = \sigma _0 X_0 B / ( 4\rho U_{f\!f} R^{*} ) = 6.18\times 10^{-6}\, \mathrm {K}^2\ \mathrm {s}^{-1}$. Thus, the best fit slope agrees with theory to within approximately a factor of two.

The thermoelectric LSC precession model is tested further in figure 9(a) and figure 15 using the fixed $Ra \approx 2 \times 10^6$ case results. In figure 9(a) the peak of each experiment's thermal FFT spectrum is marked by an open black circle. The circles in the MP regime are connected by a best fit parabola (black dashed curve). The white stars show the predicted MP frequency, $f_{M\!P} = \omega _{M\!P}/2 {\rm \pi}$, calculated using (6.20). The white dotted line is the best parabolic fit to the $f_{M\!P}$ values. Adequate agreement is found between the spectral peaks and the $f_{M\!P}$ values. The low frequency tail of the best fits appears to correlate with secondary peaks in the thermal spectra. This suggests that MP modes exist down to $Ch \simeq 10^3$ in the JRV regime, but with weaker spectral signatures that do not dominate those of the jump rope LSC flow.

Figure 15. (a) Normalized precessional frequency, $\widetilde{f} = f/\,f_{\kappa}$, vs convective interaction parameter, $N_{\mathcal {C}}$, where the thermal diffusion frequency is $f_{\kappa} \equiv 1/\tau _{\kappa} \approx 1.34\times 10^{-3}$ Hz. Black circles denote the peak frequencies of FFT spectra for the fixed $Ra \approx 2 \times 10^6$ experimental survey. Magenta stars are frequencies predicted by the TEMC precession model, $f_{M\!P}/\,f_{\kappa} = \omega _{M\!P}/(2 {\rm \pi}\,f_{\kappa} )$. Green stars correspond to $\omega _{M\!P} U_{f\!f} / (2 {\rm \pi}U_{M\!C}\,f_{\kappa} )$, the frequency of the precession model using the $U_{M\!C}$ scaling velocity. Dashed curves represent the second-order best fit curves. (b) Plot of $\mathcal{T}$ vs $N_{\mathcal {C}}$. (c) The product $B \mathcal{T}$ vs $N_{\mathcal {C}}$. The magneta dashed curve is the second degree best fit curve from panel (a) normalized by the factor $\sigma _{0} X_{0}/ (8{\rm \pi}\,f_{\kappa} \rho U_{f\!f} R^{\ast}) \approx 0.0612\, (\textrm{T}^{-1}\ \textrm{K}^{-2}\ \textrm{s}^{-1})$.

Figure 15(a) shows a linear plot of FFT spectral peaks (black circles), $f_{M\!P}$ predictions (magenta stars), and the modified predictions $f_{M\!P} (U_{f\!f}/U_{M\!C})$ (green stars) plotted vs $N_{\mathcal {C}}$ for the fixed $Ra \approx 2 \times 10^6$ MC experiments with $N_\mathcal {C} < 1$. The dashed lines show best parabolic fits. The vertical dot-dashed lines are the regime boundaries that separate the JRV, MP and MCMC regimes in figure 9(a). The $f_{M\!P}$ estimates differ by $\lesssim 37\,\%$ from the experimental spectral data, while the improved model that makes use of $U_{M\!C}$ differs from the spectral peak frequencies by $\lesssim 16\,\%$. Further, the intersections of the best fit parabolas with $f/f_\kappa = 0$, near $N_\mathcal {C} \approx 0.1$ and $N_\mathcal {C}\approx 1$, agree well with the empirically located regime boundaries.

Figure 15(b) shows $\mathcal {T} = ( T^{bot} \delta T^{bot} - T^{top} \delta T^{top} )$ vs $N_{\mathcal {C}}$ for the experimental cases shown in panel (a). Although it resembles the curves in panel (a), the shape of the $\mathcal {T}$ curve is steeper on its $N_{\mathcal {C}} < 0.4$ branch and its peak is shifted to slightly lower $N_{\mathcal {C}}$. Figure 15(c) shows the product $B \mathcal {T}$ plotted in orange as a function of $N_{\mathcal {C}}$. The magenta dashed curve in figure 15(c) is the best fit parabolic curve from panel (a) normalized by $8\rho U_{f\!f}R^*/(\sigma _0 X_0)$, which, according to (6.20), separates these values. By comparing figures 15(b) and 15(c), we argue that the quasi-parabolic structure of the MP frequency data in the MP regime is controlled by trade-offs between the $B$ and $\mathcal {T}$ trends.