Small Magnetic Prandtl Numbers Research Articles

The gap-size influence on the excitation of magnetorotational instability in cylindricTaylor–Couette flows

The excitation conditions of the magnetorotational instability (MRI) are studied for axially unbounded Taylor–Couette (TC) flows of various gap widths between the cylinders. The cylinders are considered as made from both perfect-conducting or insulating material and the conducting fluid with a finite but small magnetic Prandtl number rotates with a quasi-Keplerian velocity profile. The solutions are optimized with respect to the wavenumber and the Reynolds number of the rotation of the inner cylinder. For the axisymmetric modes, we find the critical Lundquist number of the applied axial magnetic field: the lower, the wider the gap between the cylinders. A similar result is obtained for the induced cell structure: the wider the gap, the more spherical the cells are. The marginal rotation rate of the inner cylinder – for a fixed size of the outer cylinder – always possesses a minimum for not too wide and not too narrow gap widths. For perfect-conducting walls the minimum lies at $r_{{\rm in}}\simeq 0.4$ , where $r_{{\rm in}}$ is the ratio of the radii of the two rotating cylinders. The lowest magnetic field amplitudes to excite the instability are required for TC flows between perfect-conducting cylinders with gaps corresponding to $r_{{\rm in}}\simeq ~0.2$ . For even wider and also for very thin gaps the needed magnetic fields and rotation frequencies are shown to become rather huge. Also the non-axisymmetric modes with $|m|=1$ have been considered. Their excitation generally requires stronger magnetic fields and higher magnetic Reynolds numbers in comparison with those for the axisymmetric modes. If TC experiments with too slow rotation for the applied magnetic fields yield unstable modes of any azimuthal symmetry, such as the currently reported Princeton experiment (Wang et al., Phys. Rev. Lett., vol. 129, 115001), then also other players, including axial boundary effects, than the MRI-typical linear combination of current-free fields and differential rotation should be in the game.

Effect of horizontal magnetic field on Küppers–Lortz instability

We investigate the effect of an external horizontal magnetic field on the Küppers–Lortz instability (KLI) in rotating Rayleigh–Bénard convection of Boussinesq fluids using weakly nonlinear theory along with linear theory. By the KLI, we mean the instability where the two-dimensional roll solutions of the system occurring at the onset of convection become unstable against the perturbations by rolls oriented at different angles with the previous one as the rotation rate exceeds a critical value. The governing parameters, namely, the Prandtl number (Pr), the Taylor number (Ta), and the Chandrasekhar number (Q), are varied in the ranges 0.8≤Pr<∞, 0<Ta≤104, and 0≤Q≤104, respectively, by considering the vanishingly small magnetic Prandtl number limit. In the Pr→∞ limit, magnetic field is found to inhibit the KLI by enhancing the critical Taylor number (Tac) for its onset. On the other hand, for finite Prandtl number fluids, the KLI is favored for lower Q, and it is inhibited for higher Q. Interestingly, in the finite Prandtl number range, both KLI and small angle instability are manifested depending on the Prandtl number. No small-angle instability is observed for Pr≥50, and the rotation-induced KLI is inhibited predominantly by the magnetic field, while, for Pr<50, along with the Küppers–Lortz instability, small-angle instability is also observed. However, in this case, the KLI is favored for lower Q, while it is inhibited for higher Q.

From helical to standard magnetorotational instability: Predictions for upcoming liquid sodium experiments

This work is a first preparatory theoretical study for upcoming magnetorotational instability (MRI) experiments with liquid sodium planned within the DRESDYN project. Therefore, we focus on the characteristics of MRI for typical values of the main parameters of the basic magnetized Taylor-Couette (TC) flow achievable in these experiments. In contrast to previous attempts at finding MRI in the laboratory, our results clearly indicate that standard MRI and its helically modified version can be detected for the first time in the DRESDYN-TC device, including the astrophysically important Keplerian rotation, despite the very small magnetic Prandtl number of liquid sodium.

Unstable spectra of plane Poiseuille flow with a uniform magnetic field

The unstable spectra of plane Poiseuille flow (PF) in the presence of a longitudinal magnetic field are numerically investigated using an eigenvalue solver of incompressible magnetohydrodynamic equations. It is found that the strength of the magnetic field and the dissipative effect of the magnetic perturbation have played different roles in different parameter regions. The magnetic field has a strong suppression effect on the classical plane PF instability with a large Reynolds number in the region with the magnetic Prandtl number or the magnetic Reynolds number . Here, the Reynolds number and the magnetic Reynolds number are defined as and , where a, V 0, ν and η are the typical length, velocity, viscosity and resistivity, respectively. The magnetic Prandtl number is defined as , which is proportional to the ratio of the viscosity and the resistivity of the fluid medium. As the strength of the magnetic field increases, the PF instability can be completely stabilized in the limit of or/and . It is interestingly found that a new instability branch is excited in the small magnetic Prandtl number () or moderate magnetic Reynolds number () and large Reynolds number () regions. This new type of instability is verified to be driven by the magnetic Reynolds stress and modulated by the dissipative effect of the magnetic perturbation. The wavelength of the original PF instability gradually shifts to the long wavelength region, but the wavelength of the new branch is almost unchanged, as increases with fixed . However, the wavelength of the original instability branch is almost unchanged, but the wavelength of the new instability branch shifts to the long wavelength region, as increases with fixed .

Subcritical magnetohydrodynamic instabilities: Chandrasekhar’s theorem revisited

Subcritical instabilities (i.e. finite-amplitude instabilities that occur without any linear instability) in magnetohydrodynamic (MHD) flows are studied by computing finite-amplitude equilibrium solutions of viscous–resistive MHD equations. The plane Couette flow magnetised by a uniform spanwise current is used as a model flow. Solutions are found for broad sub- and super-Alfvenic flow regimes by controlling the magnetic Mach number, but their existence is greatly influenced by the magnetic Prandtl number. When that number is unity, and the walls are perfectly insulating, the solution branch found in the super-Alfvenic regime cannot be continued towards the sub-Alfvenic regime; the boundary between those regimes is called the Chandrasekhar state, where Chandrasekhar (Proc. Natl Acad. Sci. USA, vol. 42, 1956, pp. 273–276) proved the non-existence of a linear ideal instability. Thus, the result may seem to suggest that the Chandrasekhar theorem holds even when diffusivity and nonlinearity are present. This is certainly true, but only when the perturbation magnetic field on the boundary is small. The boundary effects add more complexity to the nonlinear analysis of the Chandrasekhar state. The Chandrasekhar theorem is known to work for flows bounded by perfectly conducting walls. However, somewhat paradoxically, when the walls are perfectly conducting, our large-Reynolds-number computational results show that the nonlinear solutions do exist in the Chandrasekhar state. We give a theoretical reasoning for this curious phenomenon, using a large-Reynolds-number asymptotic analysis. For small magnetic Prandtl numbers, we also show that the solution can be continued for infinitesimally small magnetic Mach number, where the flow is significantly sub-Alfvenic.

Self-consistent single mode investigations of the quasi-geostrophic convection-driven dynamo model

The quasi-geostrophic dynamo model (QGDM) is a multiscale, fully nonlinear Cartesian dynamo model that is valid in the asymptotic limit of low Rossby number. In the additional limit of small magnetic Prandtl number investigated here, the QGDM is a self-consistent, asymptotically exact form of an $\unicode[STIX]{x1D6FC}^{2}$ large-scale dynamo. This article explores methods for simulating the multiscale QGDM and investigates how convection is altered by the magnetic field in the planetary regime of small Rossby number and small magnetic Prandtl number. At present, this combination is beyond the reach of direct numerical simulations. We use a simplified class of solutions whose horizontal structure is restricted to a periodic hexagonal lattice characterized by a single horizontal wavenumber (single mode). In contrast with previous kinematic investigations of the QGDM, the Lorentz force is included to study saturated, self-consistent dynamos. Two methodologies are used to assess handling of the multiple time scales of the QGDM: a stiff, common-in-time approach where all time scales are converted to a single time variable and a heterogeneous multiscale modelling approach employing fast time averaging on the Reynolds, magnetic and buoyancy eddy fluxes that feed back onto the slow scales. These strategies produce consistent results and each illustrates self-similar dynamics as the time-averaging window is increased. The properties of the convection are significantly altered by the dynamo-generated magnetic field. All solutions show a decrease in the overall heat transfer efficiency as compared to non-magnetic convection, suggesting that a change in length scale or flow planform plays a critical role in the enhanced heat transfer efficiency observed in previous dynamo studies. All dynamo solutions show a trend of increasing ohmic dissipation relative to viscous dissipation as the buoyancy forcing is increased.

Nested polyhedra model of isotropic magnetohydrodynamic turbulence.

A nested polyhedra model has been developed for magnetohydrodynamic turbulence. Driving only the velocity field at large scales with random, divergence-free forcing results in a clear, stationary k^{-5/3} spectrum for both kinetic and magnetic energies. Since the model naturally effaces disparate scale interactions, does not have a guide field, and avoids injecting any sign of helicity by random forcing, the resulting three-dimensional k spectrum is statistically isotropic. The strengths and weaknesses of the model are demonstrated by considering large or small magnetic Prandtl numbers. It was also observed that the timescale for the equipartition offset with those of the smallest scales shows a k^{-1/2} scaling.

Suppression of magnetorotational instability in viscous resistive magnetized Taylor–Couette flow

Magnetorotational instability (MRI) is an instability that is responsible for accretion, the phenomenon observed in astrophysical disks, for example around black holes. MRI can be modeled using the equations of MHD. A typical linear analysis in the past made use of the small Prandtl number approximation, which results in dropping one term from these equations. Previously, one of the authors (Herron) showed that the small magnetic Prandtl number approximation suppresses MRI in axisymmetric viscous resistive magnetized Taylor–Couette flow when one has no-slip velocity boundary conditions, and insulating magnetic boundary conditions. We follow up here with a proof that MRI is still suppressed with perfectly conducting magnetic boundary conditions on the cylinders.

The Magnetic Effect on Dynamical Tide in Rapidly Rotating Astronomical Objects

Abstract By numerically solving the equations of rotating magnetohydrodynamics, we study the magnetic effect on dynamical tide. We find that a magnetic field has a significant impact not only on the flow structure, i.e., the internal shear layers in a rotating flow can be destroyed in the presence of a moderate or stronger magnetic field (in the sense that the Alfvén velocity is at least of the order of 0.1 of the surface rotational velocity), but also on the dispersion relation of waves excited by tidal force such that the range of tidal resonance is broadened by a magnetic field. A major result is that the total tidal dissipation scales as a square of the field strength, which can be used to estimate the strength of the internal magnetic field in the astronomical object of a binary system. Moreover, with a moderate or stronger field, the ratio of magnetic dissipation to viscous dissipation is almost inversely proportional to the magnetic Prandtl number (i.e., the ratio of viscosity to magnetic diffusivity); thus, in the astrophysical situation at a small magnetic Prandtl number magnetic dissipation dominates over viscous dissipation with a moderate or stronger field.

Small‐scale dynamos in simulations of stratified turbulent convection

Small‐scale dynamo action is often held responsible for the generation of quiet Sun magnetic fields. We aim to determine the excitation conditions and saturation level of small‐scale dynamos in nonrotating turbulent convection at low magnetic Prandtl numbers. We use high‐resolution direct numerical simulations of weakly stratified turbulent convection. We find that the critical magnetic Reynolds number for dynamo excitation increases as the magnetic Prandtl number is decreased, which might suggest that small‐scale dynamo action is not automatically evident in bodies with small magnetic Prandtl numbers, such as the Sun. As a function of the magnetic Reynolds number (Rm), the growth rate of the dynamo is consistent with an Rm1/2 scaling. No evidence for a logarithmic increase of the growth rate with Rm is found.

Azimuthal magnetorotational instability with super-rotation

It is demonstrated that the azimuthal magnetorotational instability (AMRI) also works with radially increasing rotation rates contrary to the standard magnetorotational instability for axial fields which requires negative shear. The stability against non-axisymmetric perturbations of a conducting Taylor–Couette flow with positive shear under the influence of a toroidal magnetic field is considered if the background field between the cylinders is current free. For small magnetic Prandtl number $Pm\rightarrow 0$ the curves of neutral stability converge in the (Hartmann number,Reynolds number) plane approximating the stability curve obtained in the inductionless limit $Pm=0$. The numerical solutions for $Pm=0$ indicate the existence of a lower limit of the shear rate. For large $Pm$ the curves scale with the magnetic Reynolds number of the outer cylinder but the flow is always stable for magnetic Prandtl number unity as is typical for double-diffusive instabilities. We are particularly interested to know the minimum Hartmann number for neutral stability. For models with resting or almost resting inner cylinder and with perfectly conducting cylinder material the minimum Hartmann number occurs for a radius ratio of $r_{\text{in}}=0.9$. The corresponding critical Reynolds numbers are smaller than $10^{4}$.

Quasi-two-dimensional nonlinear evolution of helical magnetorotational instability in a magnetized Taylor–Couette flow

Magnetorotational instability (MRI) is one of the fundamental processes in astrophysics, driving angular momentum transport and mass accretion in a wide variety of cosmic objects. Despite much theoretical/numerical and experimental efforts over the last decades, its saturation mechanism and amplitude, which sets the angular momentum transport rate, remains not well understood, especially in the limit of high resistivity, or small magnetic Prandtl numbers typical to interiors (dead zones) of protoplanetary disks, liquid cores of planets and liquid metals in laboratory. Using direct numerical simulations, in this paper we investigate the nonlinear development and saturation properties of the helical magnetorotational instability (HMRI)—a relative of the standard MRI—in a magnetized Taylor–Couette flow at very low magnetic Prandtl number (correspondingly at low magnetic Reynolds number) relevant to liquid metals. For simplicity, the ratio of azimuthal field to axial field is kept fixed. From the linear theory of HMRI, it is known that the Elsasser number, or interaction parameter determines its growth rate and plays a special role in the dynamics. We show that this parameter is also important in the nonlinear problem. By increasing its value, a sudden transition from weakly nonlinear, where the system is slightly above the linear stability threshold, to strongly nonlinear, or turbulent regime occurs. We calculate the azimuthal and axial energy spectra corresponding to these two regimes and show that they differ qualitatively. Remarkably, the nonlinear state remains in all cases nearly axisymmetric suggesting that this HMRI-driven turbulence is quasi two-dimensional in nature. Although the contribution of non-axisymmetric modes increases moderately with the Elsasser number, their total energy remains much smaller than that of the axisymmetric ones.

Magnetic and velocity fields in a dynamo operating at extremely small Ekman and magnetic Prandtl numbers

AbstractNumerical simulations of the geodynamo are becoming more realistic because of advances in computer technology. Here, the geodynamo model is investigated numerically at the extremely low Ekman and magnetic Prandtl numbers using the PARODY dynamo code. These parameters are more realistic than those used in previous numerical studies of the geodynamo. Our model is based on the Boussinesq approximation and the temperature gradient between upper and lower boundaries is a source of convection. This study attempts to answer the question how realistic the geodynamo models are. Numerical results show that our dynamo belongs to the strong-field dynamos. The generated magnetic field is dipolar and large-scale while convection is small-scale and sheet-like flows (plumes) are preferred to a columnar convection. Scales of magnetic and velocity fields are separated, which enables hydromagnetic dynamos to maintain the magnetic field at the low magnetic Prandtl numbers. The inner core rotation rate is lower than that in previous geodynamo models. On the other hand, dimensional magnitudes of velocity and magnetic fields and those of the magnetic and viscous dissipation are larger than those expected in the Earth’s core due to our parameter range chosen.

Turbulent 2.5-dimensional dynamos

We study the linear stage of the dynamo instability of a turbulent two-dimensional flow with three components $(u(x,y,t),v(x,y,t),w(x,y,t))$ that is sometimes referred to as a 2.5-dimensional (2.5-D) flow. The flow evolves based on the two-dimensional Navier–Stokes equations in the presence of a large-scale drag force that leads to the steady state of a turbulent inverse cascade. These flows provide an approximation to very fast rotating flows often observed in nature. The low dimensionality of the system allows for the realization of a large number of numerical simulations and thus the investigation of a wide range of fluid Reynolds numbers $Re$, magnetic Reynolds numbers $Rm$ and forcing length scales. This allows for the examination of dynamo properties at different limits that cannot be achieved with three-dimensional simulations. We examine dynamos for both large and small magnetic Prandtl-number turbulent flows $Pm=Rm/Re$, close to and away from the dynamo onset, as well as dynamos in the presence of scale separation. In particular, we determine the properties of the dynamo onset as a function of $Re$ and the asymptotic behaviour in the large $Rm$ limit. We are thus able to give a complete description of the dynamo properties of these turbulent 2.5-D flows.

Convection-driven kinematic dynamos at low Rossby and magnetic Prandtl numbers: Single mode solutions.

The onset of dynamo action is investigated within the context of a newly developed low Rossby, low magnetic Prandtl number, convection-driven dynamo model. This multiscale model represents an asymptotically exact form of an α^{2} mean field dynamo model in which the small-scale convection is represented explicitly by finite amplitude, single mode solutions. Both steady and oscillatory convection are considered for a variety of horizontal planforms. The kinetic helicity is observed to be a monotonically increasing function of the Rayleigh number. As a result, very small magnetic Prandtl number dynamos can be found for sufficiently large Rayleigh numbers. All dynamos are found to be oscillatory with an oscillation frequency that increases as the strength of the convection is increased and the magnetic Prandtl number is reduced. Kinematic dynamo action is strongly controlled by the profile of the helicity; single mode solutions which exhibit boundary layer behavior in the helicity show a decrease in the efficiency of dynamo action due to the enhancement of magnetic diffusion in the boundary layer regions. For a given value of the Rayleigh number, lower magnetic Prandtl number dynamos are excited for the case of oscillatory convection in comparison to steady convection. With regard to planetary dynamos, these results suggest that the low magnetic Prandtl number dynamos typical of liquid metals are more easily driven by thermal convection than by compositional convection.

Saturation of the turbulent dynamo.

The origin of strong magnetic fields in the Universe can be explained by amplifying weak seed fields via turbulent motions on small spatial scales and subsequently transporting the magnetic energy to larger scales. This process is known as the turbulent dynamo and depends on the properties of turbulence, i.e., on the hydrodynamical Reynolds number and the compressibility of the gas, and on the magnetic diffusivity. While we know the growth rate of the magnetic energy in the linear regime, the saturation level, i.e., the ratio of magnetic energy to turbulent kinetic energy that can be reached, is not known from analytical calculations. In this paper we present a scale-dependent saturation model based on an effective turbulent resistivity which is determined by the turnover time scale of turbulent eddies and the magnetic energy density. The magnetic resistivity increases compared to the Spitzer value and the effective scale on which the magnetic energy spectrum is at its maximum moves to larger spatial scales. This process ends when the peak reaches a characteristic wave number k☆ which is determined by the critical magnetic Reynolds number. The saturation level of the dynamo also depends on the type of turbulence and differs for the limits of large and small magnetic Prandtl numbers Pm. With our model we find saturation levels between 43.8% and 1.3% for Pm≫1 and between 2.43% and 0.135% for Pm≪1, where the higher values refer to incompressible turbulence and the lower ones to highly compressible turbulence.

Angular momentum transport and large eddy simulations in magnetorotational turbulence: the small Pm limit

Angular momentum transport in accretion discs is often believed to be due to magnetohydrodynamic turbulence mediated by the magnetorotational instability. Despite an abundant literature on the MRI, the parameters governing the saturation amplitude of the turbulence are poorly understood and the existence of an asymptotic behavior in the Ohmic diffusion regime is not clearly established. We investigate the properties of the turbulent state in the small magnetic Prandtl number limit. Since this is extremely computationally expensive, we also study the relevance and range of applicability of the most common subgrid scale models for this problem. Unstratified shearing boxes simulations are performed both in the compressible and incompressible limits, with a resolution up to 800 cells per disc scale height. The latter constitutes the largest resolution ever attained for a simulation of MRI turbulence. In the presence of a mean magnetic field threading the domain, angular momentum transport converges to a finite value in the small Pm limit. When the mean vertical field amplitude is such that {\beta}, the ratio between the thermal and magnetic pressure, equals 1000, we find {\alpha}~0.032 when Pm approaches zero. In the case of a mean toroidal field for which {\beta}=100, we find {\alpha}~0.018 in the same limit. Both implicit LES and Chollet-Lesieur closure model reproduces these results for the {\alpha} parameter and the power spectra. A reduction in computational cost of a factor at least 16 (and up to 256) is achieved when using such methods. MRI turbulence operates efficiently in the small Pm limit provided there is a mean magnetic field. Implicit LES offers a practical and efficient mean of investigation of this regime but should be used with care, particularly in the case of a vertical field. Chollet-Lesieur closure model is perfectly suited for simulations done with a spectral code.

Energy transfers in dynamos with small magnetic Prandtl numbers

We perform numerical simulation of dynamo with magnetic Prandtl number Pm = 0.2 on 10243 grid, and compute the energy fluxes and the shell-to-shell energy transfers. These computations indicate that the magnetic energy growth takes place mainly due to the energy transfers from large-scale velocity field to large-scale magnetic field and that the magnetic energy flux is forward. The steady-state magnetic energy is much smaller than the kinetic energy, rather than equipartition; this is because the magnetic Reynolds number is near the dynamo transition regime. We also contrast our results with those for dynamo with Pm = 20 and decaying dynamo.

Hydromagnetic oscillatory Couette flow in rotating system with induced magnetic field

This paper presents a study of hydromagnetic Couette flow of an incompressible and electrically conducting fluid between two parallel rotating plates, one of which is oscillating in its own plane. A uniform transverse magnetic field is used, and the induced magnetic field is taken into account. The exact solution to the governing equations is obtained in a closed form. The solution to the problem in the case of vanishing and small finite magnetic Prandtl numbers is also derived from the general solution. The asymptotic behavior of the solution for large values of the frequency parameter is analyzed to gain some physical insights into the flow pattern. Expressions for the shear stress at both the oscillatory and stationary plates due to primary and secondary flows and mass flow rate in the primary and secondary flow directions are also obtained. The results of the fluid velocity and the induced magnetic field are presented. The shear stresses on the plates due to the primary and secondary flows and the corresponding mass flow rates are presented in a tabular form.

α effect in a turbulent liquid-metal plane Couette flow.

We calculate the mean electromotive force in plane Couette flows of a nonrotating conducting fluid under the influence of a large-scale magnetic field for driven turbulence. A vertical stratification of the turbulence intensity results in an α effect owing to the presence of horizontal shear. Here we discuss the possibility of an experimental determination of the components of the α tensor using both quasilinear theory and nonlinear numerical simulations. For magnetic Prandtl numbers of the order of unity, we find that in the high-conductivity limit the α effect in the direction of the flow clearly exceeds the component in spanwise direction. In this limit, α runs linearly with the magnetic Reynolds number Rm, while in the low-conductivity limit it runs with the product Rm·Re, where Re is the kinetic Reynolds number, so that for a given Rm the α effect grows with decreasing magnetic Prandtl number. For the small magnetic Prandtl numbers of liquid metals, a common value for the horizontal elements of the α tensor appears, which makes it unimportant whether the α effect is measured in the spanwise or the streamwise directions. The resulting effect should lead to an observable voltage of about 0.5 mV in both directions for magnetic fields of 1 kG and velocity fluctuations of about 1m/s in a channel of 50-cm height (independent of its width).