8 results
Self-similarity of the dipole–multipole transition in rapidly rotating dynamos
- Debarshi Majumder, Binod Sreenivasan, Gaurav Maurya
-
- Journal:
- Journal of Fluid Mechanics / Volume 980 / 10 February 2024
- Published online by Cambridge University Press:
- 05 February 2024, A30
-
- Article
-
- You have access Access
- Open access
- HTML
- Export citation
-
The dipole–multipole transition in rapidly rotating dynamos is investigated through the analysis of forced magnetohydrodynamic waves in an unstably stratified fluid. The focus of this study is on the inertia-free limit applicable to planetary cores, where the Rossby number is small not only on the core depth but also on the length scale of columnar convection. By progressively increasing the buoyant forcing in a linear magnetoconvection model, the slow magnetic–Archimedean–Coriolis (MAC) waves are significantly attenuated so that their kinetic helicity decreases to zero; the fast MAC wave helicity, on the other hand, is practically unaffected. In turn, polarity reversals in low-inertia spherical dynamos are shown to occur when the slow MAC waves disappear under strong forcing. Two dynamically similar regimes are identified – the suppression of slow waves in a strongly forced dynamo and the excitation of slow waves in a moderately forced dynamo starting from a small seed field. While the former regime results in polarity reversals, the latter regime produces the axial dipole from a chaotic multipolar state. For either polarity transition, a local Rayleigh number based on the mean wavenumber of the energy-containing scales bears the same linear relationship with the square of the peak magnetic field measured at the transition. The self-similarity of the dipole–multipole transition can place a constraint on the Rayleigh number for polarity reversals in the Earth.
Evolution of forced magnetohydrodynamic waves in a stratified fluid
- Binod Sreenivasan, Gaurav Maurya
-
- Journal:
- Journal of Fluid Mechanics / Volume 922 / 10 September 2021
- Published online by Cambridge University Press:
- 19 July 2021, A32
-
- Article
-
- You have access Access
- Open access
- HTML
- Export citation
-
The evolution of a buoyancy disturbance in a stratified incompressible fluid permeated by a uniform vertical magnetic field is investigated. Two regimes are considered in the absence of background rotation – that of strong stratification, where the internal gravity wave frequency $\omega _A$ is much higher in magnitude than the magnetic (Alfvén) wave frequency $\omega _M$, and that of strong magnetic field, where $\omega _M$ is dominant. For small but finite magnetic diffusion, perturbations that initially lie in the strong-field regime are shown to cross over to the regime of strong stratification, so that small-scale motions may exist as damped internal gravity waves at large times. The induced magnetic field propagates as damped Alfvén waves for a much longer time than the velocity before undergoing the above transition. With strong rotation, the unstably stratified system that satisfies the inequality $|\omega _C| > |\omega _M| \gg |\omega _A| \gg |\omega _\eta |$, where $\omega _C$ is the inertial wave frequency and $\omega _\eta$ is the diffusion frequency, is of relevance to convection-driven dynamos. Here, a parameter space with $|\omega _M/\omega _C| \sim 0.1$ is found wherein the flow intensity of the slow magnetic-Archimedean-Coriolis (MAC) waves is of the same order of magnitude as that of the fast MAC waves. Slow wave motions at horizontal length scales much smaller than the width of the fluid layer can therefore generate substantial helicity in rapidly rotating dynamos. The excitation of slow MAC waves at scales of $\sim$10 km in the Earth's core may play a crucial role in the generation of the axial dipole field.
Convection in a rapidly rotating cylindrical annulus with laterally varying boundary heat flux
- Swarandeep Sahoo, Binod Sreenivasan
-
- Journal:
- Journal of Fluid Mechanics / Volume 883 / 25 January 2020
- Published online by Cambridge University Press:
- 19 November 2019, A1
-
- Article
- Export citation
-
Convection in a rapidly rotating cylindrical annulus subject to azimuthal variations in outer boundary heat flux is investigated experimentally. The motivation for this problem stems from the influence of the laterally inhomogeneous lower mantle on the geodynamo. The absence of axial ($z$) gradients of boundary temperature ensures that the condition of quasi-geostrophy, often used to model convection outside the tangent cylinder in spherical shells, is realized in a cylindrical annulus even in strongly driven convection. Experiments are performed with water from below onset of convection to highly supercritical states (measured by the flux Rayleigh number, $Ra\sim 10^{10}$) and for boundary heat flux heterogeneity $q^{\ast }$ (defined by the ratio of the azimuthal variation to the mean boundary heat flux) in the range 0–2. The power requirement for onset of convection reduces substantially with increasing $q^{\ast }$, in line with earlier studies of the onset in rotating spherical shells. For strongly driven convection at $q^{\ast }>1$, the long-time structure is that of localized coherent cyclone–anticyclone vortex pairs, which produce narrow downwellings between them. However, shorter-time averages of the flow reveal the presence of small-scale motions, which may have an important role in magnetic field generation. For a twofold heat flux heterogeneity of $q^{\ast }\approx 2$, convection within the annulus fully homogenizes at ${\sim}30$ times the onset Rayleigh number, and no coherent vortices remain. Finally, the measured heat flux variation on the inner boundary is considerably larger compared with that on the outer boundary, which provides a plausible mechanism for inner-core heterogeneity in the Earth.
Experimental study of the convection in a rotating tangent cylinder
- Kélig Aujogue, Alban Pothérat, Binod Sreenivasan, François Debray
-
- Journal:
- Journal of Fluid Mechanics / Volume 843 / 25 May 2018
- Published online by Cambridge University Press:
- 21 March 2018, pp. 355-381
-
- Article
- Export citation
-
This paper experimentally investigates the convection in a rapidly rotating tangent cylinder (TC), for Ekman numbers down to $E=3.36\times 10^{-6}$. The apparatus consists of a hemispherical fluid vessel heated in its centre by a protruding heating element of cylindrical shape. The resulting convection that develops above the heater, i.e. within the TC, is shown to set in for critical Rayleigh numbers and wavenumbers respectively scaling as $Ra_{c}\sim E^{-4/3}$ and $a_{c}\sim E^{-1/3}$ with the Ekman number $E$. Although exhibiting the same exponents as for plane rotating convection, these laws reflect much larger convective plumes at onset. The structure and dynamics of supercritical plumes are in fact closer to those found in solid rotating cylinders heated from below, suggesting that the confinement within the TC induced by the Taylor–Proudman constraint influences convection in a similar way as solid walls would do. There is a further similarity in that the critical modes in the TC all exhibit a slow retrograde precession at onset. In supercritical regimes, the precession evolves into a thermal wind with a complex structure featuring retrograde rotation at high latitude and either prograde or retrograde rotation at low latitude (close to the heater), depending on the criticality and the Ekman number. The intensity of the thermal wind measured by the Rossby number $Ro$ scales as $Ro\simeq 5.33(Ra_{q}^{\ast })^{0.51}$ with the Rayleigh number based on the heat flux $Ra_{q}^{\ast }\in [10^{-9},10^{-6}]$. This scaling is in agreement with heuristic predictions and previous experiments where the thermal wind is determined by the azimuthal curl of the balance between the Coriolis force and buoyancy. Within the range $Ra\in [2\times 10^{7},10^{9}]$ which we explored, we also observe a transition in the heat transfer through the TC from a diffusivity-free regime where $Nu\simeq 0.38E^{2}Ra^{1.58}$ to a rotation-independent regime where $Nu\simeq 0.2Ra^{0.33}$.
Damping of magnetohydrodynamic waves in a rotating fluid
- Binod Sreenivasan, Ghanesh Narasimhan
-
- Journal:
- Journal of Fluid Mechanics / Volume 828 / 10 October 2017
- Published online by Cambridge University Press:
- 12 September 2017, pp. 867-905
-
- Article
- Export citation
-
The long-time evolution of a flow structure subject to background rotation and a coaxial uniform magnetic field is investigated in this paper. The conditions of magnetic Reynolds number $Rm\ll 1$ and Rossby number $Ro\ll 1$ apply, while the condition of magnetic interaction parameter $N\gg 1$ ensures that nonlinear inertial forces are small in the system. Cylindrical polar coordinates $(s,\unicode[STIX]{x1D719},z)$ are used, where the velocity and the induced magnetic field are axisymmetric. Two regimes are analysed in the inviscid limit, that of strong rotation, where the inertial wave frequency is much higher than the Alfvén wave frequency, and that of weak rotation, where the Alfvén wave frequency is dominant. In either regime, the evolution consists of a damped wave-dominated phase followed by a diffusion-dominated phase. For strong rotation, the laws of energy decay in the damped wave phase are obtained by considering the decay of the fast and slow magneto-Coriolis (MC) waves individually. The diffusion-dominated phase obeys the decay laws in the well-known quasistatic approximation. The wave–diffusion transition time scale indicates that the wave phase of decay is very long, so that small-scale turbulence is characterized by damped wave motions. The ratio of kinetic to magnetic energies of the slow MC wave in the early stages of evolution is $O(Le^{2})$, where $Le$ is the initial ratio of the inertial wave to Alfvén wave time scales. The induced magnetic field is hence far more efficient than the velocity in supporting slow MC waves for $Le\ll 1$. In the regime of weak rotation, the fast and slow MC wave solutions merge and tend to the classical damped Alfvén wave solution. Here, the decay laws in non-rotating magnetohydrodynamic turbulence (Moffatt, J. Fluid Mech., vol. 28, 1967, pp. 571–592) are recovered. Computations of the general solution for the long-time decay of an isolated vortex confirm the theoretical energy scalings as well as the wave–diffusion transition time scale of the kinetic energy. It is shown that a magnetically damped system that initially generates Alfvén waves because of relatively weak rotation can subsequently give rise to MC waves. Small-scale motions of $Rm\sim 1$ in the Earth’s core probably generate slow magnetostrophic waves only for $Le>0.1$, which suggests that a mean intensity of ${\sim}10~\text{mT}$ or higher is plausible for the toroidal magnetic field within the core.
Confinement of rotating convection by a laterally varying magnetic field
- Binod Sreenivasan, Venkatesh Gopinath
-
- Journal:
- Journal of Fluid Mechanics / Volume 822 / 10 July 2017
- Published online by Cambridge University Press:
- 07 June 2017, pp. 590-616
-
- Article
- Export citation
-
Spherical shell dynamo models based on rotating convection show that the flow within the tangent cylinder is dominated by an off-axis plume that extends from the inner core boundary to high latitudes and drifts westward. Earlier studies explained the formation of such a plume in terms of the effect of a uniform axial magnetic field that significantly increases the length scale of convection in a rotating plane layer. However, rapidly rotating dynamo simulations show that the magnetic field within the tangent cylinder has severe lateral inhomogeneities that may influence the onset of an isolated plume. Increasing the rotation rate in our dynamo simulations (by decreasing the Ekman number $E$) produces progressively thinner plumes that appear to seek out the location where the field is strongest. Motivated by this result, we examine the linear onset of convection in a rapidly rotating fluid layer subject to a laterally varying axial magnetic field. A Cartesian geometry is chosen where the finite dimensions $(x,z)$ mimic $(\unicode[STIX]{x1D719},z)$ in cylindrical coordinates. The lateral inhomogeneity of the field gives rise to a unique mode of instability where convection is entirely confined to the peak-field region. The localization of the flow by the magnetic field occurs even when the field strength (measured by the Elsasser number $\unicode[STIX]{x1D6EC}$) is small and viscosity controls the smallest length scale of convection. The lowest Rayleigh number at which an isolated plume appears within the tangent cylinder in spherical shell dynamo simulations agrees closely with the viscous-mode Rayleigh number in the plane layer linear magnetoconvection model. The lowest Elsasser number for plume formation in the simulations is significantly higher than the onset values in linear magnetoconvection, which indicates that the viscous–magnetic mode transition point with spatially varying fields is displaced to much higher Elsasser numbers. The localized excitation of viscous-mode convection by a laterally varying magnetic field provides a mechanism for the formation of isolated plumes within the Earth’s tangent cylinder. The polar vortices in the Earth’s core can therefore be non-axisymmetric. More generally, this study shows that a spatially varying magnetic field strongly controls the structure of rotating convection at a Rayleigh number not much different from its non-magnetic value.
Helicity generation and subcritical behaviour in rapidly rotating dynamos
- Binod Sreenivasan, Chris A. Jones
-
- Journal:
- Journal of Fluid Mechanics / Volume 688 / 10 December 2011
- Published online by Cambridge University Press:
- 19 August 2011, pp. 5-30
-
- Article
- Export citation
-
Numerical dynamo models based on convection-driven flow in a rapidly rotating spherical shell frequently give rise to strong, stable, dipolar magnetic fields. Dipolar dynamos can be subcritical in the sense that strong magnetic fields are sustained at a Rayleigh number lower than that required for a dynamo to grow from a small seed field. In this paper we find subcritical behaviour in dynamos in line with previous studies. We explore the action of Lorentz force in a rotating dynamo which gives rise to a strong preference for dipolar modes over quadrupolar modes, and also makes subcritical behaviour more likely to occur. The coherent structures that arise in rapidly rotating convection are affected by the magnetic field in ways which strongly increase their helicity, particularly if the magnetic field is dipolar. As helicity enhances dynamo action, an existing magnetic field can hold itself up, which leads to subcritical behaviour in the dynamo. We investigate this mechanism by means of the asymptotic small Ekman number theory of rapidly rotating magnetoconvection, and compare our results with fully nonlinear dynamo simulations. There are also other mechanisms which can promote subcritical behaviour. When Reynolds stresses are significant, zonal flows can lower the helicity and disrupt the onset of dynamo action, but an established dipole field can suppress the zonal flow, and hence boost the helicity. Subcriticality means that a slow gradual reduction in Rayleigh number can lead to a catastrophic collapse of the dynamo once a critical Rayleigh number is reached. While there is little evidence that the Earth is currently in a subcritical regime, this may have implications for the long-term evolution of the geodynamo.
Experimental study of a vortex in a magnetic field
- BINOD SREENIVASAN, THIERRY ALBOUSSIÈRE
-
- Journal:
- Journal of Fluid Mechanics / Volume 464 / 10 August 2002
- Published online by Cambridge University Press:
- 21 August 2002, pp. 287-309
-
- Article
- Export citation
-
It is well-known that magnetohydrodynamic (MHD) flows behave differently from conventional fluid flows in two ways: the magnetic field makes the flow field anisotropic in the sense that it becomes independent of the coordinate parallel to the field; and the flow of liquid across the field lines induces an electric current, leading to ohmic damping. In this paper, an experimental study is presented of the long-time decay of an initially three-dimensional flow structure subject to a steady magnetic field, when the ratio of the electromagnetic Lorentz forces to the nonlinear inertial forces, quantified by the magnetic interaction parameter, N0, takes large as well as moderate values. This investigation is markedly different from previous studies on quasi-two-dimensional MHD flows in thin layers of conducting fluids, where only Hartmann layer friction held the key to the dissipation of the flow.
The initial ‘linear’ phase of decay of an MHD flow, characterized by dominant Lorentz forces and modelled extensively in the literature, has been observed for the first time in a laboratory experiment. Further, when N0 is large compared to unity, a distinct regime of decay of a vortex follows this linear phase. This interesting trend can be explained in terms of the behaviour of the ratio of the actual magnitudes of the Lorentz to the nonlinear inertial forces – the true interaction parameter – which decreases to a constant of order unity towards the end of the linear phase of decay, and remains invariant during a subsequent ‘nonlinear’ phase.