Detailed mathematical and numerical analysis of a dynamo model
TL;DR: In this paper, the role of nonlinear feedback by α-quenching, flux losses, and feedback by differential rotations in dynamo was investigated, and it was shown that the effect of flux loss and α quenching is boosted when combined with toroidal flux loss, indicating that the dynamic balance of dynamo is optimized.
Abstract: We investigate the role of nonlinear feedback by α-quenching, flux losses, and feedback by differential rotations in dynamo. Specifically, by studying the nonlinear dynamo model analytically and numerically, we unfold how frequency p of magnetic field, magnetic field strength |B|, and phase φ are influenced by different types of nonlinear feedback in the limit of a very weak mean and/or fluctuating differential rotation. We find that p and φ are controlled by both flux losses with no influence by α-quenching when there is no back reaction because of fluctuating differential rotation. We find a similar effect of poloidal flux loss and toroidal flux loss on p and |B| in the absence of a back reaction of shear. Their effect becomes totally different with the inclusion of this back reaction. Detailed investigations suggest that toroidal flux loss tends to have more influence than poloidal flux loss (with or without α-quenching) in the presence of fluctuating shear. Furthermore, the effect of α-quenching is boosted when combined with toroidal flux loss, indicating that the dynamic balance of dynamo is optimized in the presence of both α-quenching and flux loss. These results highlight the importance of nonlinear transport coefficients and differential rotation in the regulation of a dynamo.
Citations
More filters
TL;DR: In this article, a spindown model is proposed where loss of angular momentum by magnetic fields is evolved dynamically, instead of being kinematically prescribed, and the authors highlight the importance of self-regulation of magnetic fields and rotation by direct and indirect interactions involving nonlinear feedback in stellar evolution.
Abstract: Since their formation, stars slow down their rotation rates by the removal of angular momentum from their surfaces, e.g. via stellar winds. Despite the complexity of the processes involved, a traditional model, where the removal of angular momentum loss by magnetic fields is prescribed, has provided a useful framework to understand observational relations between stellar rotation and age and magnetic field strength. Here, a spindown model is proposed where loss of angular momentum by magnetic fields is evolved dynamically, instead of being kinematically prescribed. To this end, we evolve the stellar rotation and magnetic field simultaneously over stellar evolution time by extending our previous work on a dynamo model which incorporates the nonlinear feedback mechanisms on rotation and magnetic fields. Our extended model reproduces key observations and explains the presence of the two branches of (fast and slow rotating) stars which have different relations between rotation rate $\Omega$ vs. time (age), magnetic field strength $|B|$ vs. rotation rate, and frequency of magnetic field $\omega_{cyc}$ vs. rotation rate. For fast rotating stars we find: (i) an exponential spindown $\Omega \propto e^{-1.35t}$, with $t$ measured in Gyrs, (ii) $|B|$ saturates for higher rotation rate, (iii) $\omega_{cyc} \propto \Omega^{0.85}$. For slow rotating stars we obtain: (i) a power law spindown $\Omega \propto t^{-0.52}$, (ii) $|B|$ scales almost linearly with rotation rate, (iii) $\omega_{cyc} \propto \Omega^{1.16}$. The results obtained are in good agreement with observations. The Vaughan-Preston gap is consistently explained in our model by the shortest spindown timescale in this transition from fast to slow rotators. Our results highlight the importance of self-regulation of magnetic fields and rotation by direct and indirect interactions involving nonlinear feedback in stellar evolution.
4 citations
Dissertation•
01 Aug 2015
TL;DR: In this paper, a simple parameterized model was proposed to understand the evolution of rotation rate and magnetic fields in the solar dynamo and the spindown of solar-type stars.
Abstract: Stellar rotation plays an important role in maintaining the magnetic fields inside the stellar interior through convection, and starspots are the most visible manifestation of the interplay between stellar rotation rate and magnetic fields. It is revealed through high end observations of evolution of magnetic fields and rotation rate of the Sun and other solar type stars that they exhibit a wide range of variation among their rotation rates yet there are some common ingredients such as rotational shear, turbulent transport and various nonlinear transport mechanisms which contribute towards the evolution and maintenance of the magnetic activity displayed by them. Also, these observations provide us with valuable information about the dependence of differential rotation and magnetic activity on rotation rate of stars with different ages and different rotation rates. Thus, the main challenge in dynamo theory is to explain these observations which is in fact a very strenuous problem and is challenging to do with full MHD simulations due to the various constraints such as expensive computations in terms of time and resolution.
Therefore, it is useful to construct a simple parameterized model in order to understand the evolution of rotation rate and magnetic fields which can provide valuable insight into the various observations.
This thesis discusses the modelling of solar dynamo and spindown of solar-type stars by using ODE and the effect of shear in kinematic dynamo in full MHD. We propose a simple parameterized model to understand the effect of nonlinear transport coefficients as well as mean/fluctuating differential rotation in the generation and destruction of magnetic fields and their capability in the working of dynamo near marginal stability. This model is then utilised to discuss detailed dynamics to understand the self-regulation of magnetic fields in solar/stellar dynamo. This work is further extended to understand the spindown of solar-type stars where the angular momentum loss is dynamically prescribed via equation of evolution of rotation rate and magnetic fields. The results obtained from this model are consistent with observations.
Furthermore, regulatory behaviour of a kinematic dynamo by shear flow is investigated. Specifically, we study the induction equation by prescribing small scale velocity field to which a large scale radial/latitudinal shear is added in the direction of zonal flow. The results from numerical simulations are analysed and we conclude that the presence of large scale shear suppresses the small scale flows and results in quenching of a kinematic dynamo.
3 citations
Cites background from "Detailed mathematical and numerical..."
..., in parameterisations of transport coefficients in MHD simulations)” (Sood and Kim, 2014)....
[...]
TL;DR: The non-detailed balance observable representation (NOR) provides a useful methodology for understanding the evolution of non-equilibrium complex systems, by mapping out the correlation between two states to a metric space where a small distance represents a strong correlation.
Abstract: Many systems in nature are out of equilibrium and irreversible. The non-detailed balance observable representation (NOR) provides a useful methodology for understanding the evolution of such non-equilibrium complex systems, by mapping out the correlation between two states to a metric space where a small distance represents a strong correlation [1]. In this paper, we present the first application of the NOR to a continuous system and demonstrate its utility in controlling chaos. Specifically, we consider the evolution of a continuous system governed by the Lorenz equation and calculate the NOR by following a sufficient number of trajectories. We then show how to control chaos by converting chaotic orbits to periodic orbits by utilizing the NOR. We further discuss the implications of our method for potential applications given the key advantage that this method makes no assumptions of the underlying equations of motion and is thus extremely general.
3 citations
TL;DR: In this paper, a spindown model is proposed where loss of angular momentum by magnetic fields evolves dynamically, instead of being prescibed kinematically, and the authors show that their extended model reproduces key observations and is capable of explaining the presence of two branches of (fast and slow rotating) stars.
Abstract: After their formation, stars slow down their rotation rates by the removal of angular momentum from their surfaces, e.g., via stellar winds. Explaining how this rotation of solar-type stars evolves in time is currently an interesting but difficult problem in astrophysics. Despite the complexity of the processes involved, a traditional model, where the removal of angular momentum by magnetic fields is prescribed, has provided a useful framework to understand observational relations between stellar rotation, age, and magnetic field strength. Here, for the first time, a spindown model is proposed where loss of angular momentum by magnetic fields evolves dynamically, instead of being prescibed kinematically. To this end, we evolve the stellar rotation and magnetic field simultaneously over stellar evolution time by extending our previous work on a dynamo model which incorporates nonlinear feedback mechanisms on rotation and magnetic fields. We show that our extended model reproduces key observations and is capable of explaining the presence of the two branches of (fast and slow rotating) stars which have different relations between rotation rate Ω versus time (age), magnetic field strength $| B| $ versus rotation rate, and frequency of magnetic field ${\omega }_{\mathrm{cyc}}$ versus rotation rate. For fast rotating stars we find that: (i) there is an exponential spindown ${\rm{\Omega }}\propto {e}^{-1.35t}$, with t measured in Gyr; (ii) magnetic activity saturates for higher rotation rate; (iii) ${\omega }_{\mathrm{cyc}}\propto {{\rm{\Omega }}}^{0.83}$. For slow rotating stars we find: (i) a power-law spindown ${\rm{\Omega }}\propto {t}^{-0.52}$; (ii) that magnetic activity scales roughly linearly with rotation rate; (iii) ${\omega }_{\mathrm{cyc}}\propto {{\rm{\Omega }}}^{1.16}$. The results obtained from our investigations are in good agreement with observations. The Vaughan–Preston gap is consistently explained in our model by the shortest spindown timescale in this transition from fast to slow rotators. Our results highlight the importance of self-regulation of magnetic fields and rotation by direct and indirect interactions involving nonlinear feedback in stellar evolution.
1 citations
Cites background or methods or result from "Detailed mathematical and numerical..."
...We propose a dynamical model for the evolution of rotation rate and magnetic field in spindown by extending a previous nonlinear dynamo model (Sood & Kim, 2013, 2014; Weiss et al. 1984)....
[...]
...In particular, Sood & Kim (2013, 2014) incorporated various nonlinear transport coefficients such as α-quenching and flux losses and took the control parameter D known as the dynamo number to scale with rotation rate as D ∝ Ω2....
[...]
...The parameters ν, ν0, κ and λ1,2 are much the same as in our previous work (Sood & Kim 2013, 2014)....
[...]
...We note that Sood & Kim (2013, 2014) have demonstrated that nonlinear feedback plays a vital role in the generation and destruction of magnetic fields as well as self-regulation of the dynamo....
[...]
...To this end, we evolve the stellar rotation and magnetic field simultaneously over the stellar evolution time by extending our previous work (Sood & Kim 2013, 2014) which incorporates the nonlinear feedback mechanisms on rotation and magnetic fields via α-quenching and magnetic flux losses as well…...
[...]
References
More filters
1,882 citations
"Detailed mathematical and numerical..." refers background in this paper
...Based upon the traditional α − Ω dynamo (Parker 1955; Moffat 1978; Parker 1979), the dynamo number in this work, which controls the efficiency of dynamo action, is proportional to the product of α and Ω effects, which generate a toroidal field from a poloidal field by cyclonic convection (α effect)…...
[...]
TL;DR: Magnetic Field Generation in Electrically-Conducting Fluids by H. K. Moffatt as mentioned in this paper is a seminal work in the field of magnetic field generation in electrically conducting fluids.
Abstract: Magnetic Field Generation in Electrically-Conducting Fluids. By H. K. Moffatt. Pp. 343. (Cambridge University Press: Cambridge and London, 1978.) £15.50.
1,529 citations
Book•
01 Jan 1980
1,486 citations
"Detailed mathematical and numerical..." refers methods in this paper
...It will also be of great interest to utilize our model to understand the spin down of solar type stars (Leprovost & Kim 2010; Kepens et al. 1995) by using a magnetic field that is consistently obtained from a dynamo model (work in progress)....
[...]
TL;DR: In this paper, a nonlinear theory is developed for three-dimensional, homogeneous, isotropic, incompressible MHD turbulence with helicity, i.e. not statistically invariant under plane reflexions.
Abstract: To understand the turbulent generation of large-scale magnetic fields and to advance beyond purely kinematic approaches to the dynamo effect like that introduced by Steenbeck, Krause & Radler (1966)’ a new nonlinear theory is developed for three-dimensional, homogeneous, isotropic, incompressible MHD turbulence with helicity, i.e. not statistically invariant under plane reflexions. For this, techniques introduced for ordinary turbulence in recent years by Kraichnan (1971~~)’ Orszag (1970, 1976) and others are generalized to MHD; in particular we make use of the eddy-damped quasi-normal Markovian approximation. The resulting closed equations for the evolution of the kinetic and magnetic energy and helicity spectra are studied both theoretically and numerically in situations with high Reynolds number and unit magnetic Prandtl number. Interactions between widely separated scales are much more important than for non-magnetic turbulence. Large-scale magnetic energy brings to equipartition small-scale kinetic and magnetic excitation (energy or helicity) by the ‘AlfvBn effect ’; the small-scale ‘residual’ helicity, which is the difference between a purely kinetic and a purely magnetic helical term, induces growth of largescale magnetic energy and helicity by the ‘helicity effect’. In the absence of helicity an inertial range occurs with a cascade of energy to small scales; to lowest order it is a - power law with equipartition of kinetic and magnetic energy spectra as in Kraichnan (1965) but there are - 2 corrections (and possibly higher ones) leading to a slight excess of magnetic energy. When kinetic energy is continuously injected, an initial seed of magnetic field willgrow to approximate equipartition, at least in the small scales. If in addition kinetic helicity is injected, an inverse cascade of magnetic helicity is obtained leading to the appearance of magnetic energy and helicity in ever-increasing scales (in fact, limited by the size of the system). This inverse cascade, predicted by Frisch et aZ. (1975), results from a competition between the helicity and Alfvh effects and yields an inertial range with approximately - 1 and - 2 power laws for magnetic energy and helicity. When kinetic helicity is injected at the scale Zinj and the rate k (per unit mass), the time of build-up of magnetic energy with scale L 9 Zinl is t % L( prp;nj)-k 21 FLM 77
982 citations
"Detailed mathematical and numerical..." refers background in this paper
...…+ κ1|B|2, F2 = 1 + λ1|B|2, F3 = 1 + λ2|B|2, where κ1, λ1, and λ2 are constant parameters; F1 represents the α source term (i.e., helicity by the magnetic field) due to the back reaction by magnetic field that was studied in previous works including Pouquet et al. (1976) and Kleeorin et al. (1982)....
[...]