The stability of non-dissipative couette flow in hydromagnetics

doi:10.1073/PNAS.46.2.253

Journal Article•DOI•

The stability of non-dissipative couette flow in hydromagnetics

Subrahmanyan Chandrasekhar¹•Institutions (1)

01 Feb 1960-Proceedings of the National Academy of Sciences of the United States of America (National Academy of Sciences)-Vol. 46, Iss: 2, pp 253-257

About: This article is published in Proceedings of the National Academy of Sciences of the United States of America.The article was published on 1960-02-01 and is currently open access. It has received 586 citations till now. The article focuses on the topics: Couette flow & Dissipative system.

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Book•

An Introduction to Magnetohydrodynamics

[...]

Peter Davidson¹•Institutions (1)

University of Cambridge¹

05 Mar 2001

TL;DR: An introductory text on magnetohydrodynamics (MHD) is presented in this paper, which is intended to serve as an introductory text for advanced undergraduates and postgraduate students in physics, applied mathematics and engineering.

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Abstract: Magnetic fields influence many natural and man-made flows. They are routinely used in industry to heat, pump, stir and levitate liquid metals. There is the terrestrial magnetic field which is maintained by fluid motion in the earth's core, the solar magnetic field, which generates sunspots and solar flares, and the galactic field which influences the formation of stars. This is an introductory text on magnetohydrodynamics (MHD) - the study of the interaction of magnetic fields and conducting fluids. This book is intended to serve as an introductory text for advanced undergraduates and postgraduate students in physics, applied mathematics and engineering. The material in the text is heavily weighted towards incompressible flows and to terrestrial (as distinct from astrophysical) applications. The final sections of the text also contain an outline of the latest advances in the metallurgical applications of MHD and so are relevant to professional researchers in applied mathematics, engineering and metallurgy.

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1,060 citations

Journal Article•DOI•

Dynamo action by differential rotation in a stably stratified stellar interior

[...]

Hendrik C. Spruit¹•Institutions (1)

Max Planck Society¹

01 Jan 2002-Astronomy and Astrophysics

TL;DR: In this article, a dynamo model is developed from these ingredients, and applied to the problem of angular momentum transport in stellar interiors, which is found to be more effective in transporting angular momentum than the known hydrodynamic mechanisms.

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Abstract: Magnetic fields can be created in stably stratified (non-convective) layers in a differentially rotating star. A magnetic instability in the toroidal field (wound up by differential rotation) replaces the role of convection in closing the field amplification loop. Tayler instability is likely to be the most relevant magnetic instability. A dynamo model is developed from these ingredients, and applied to the problem of angular momentum transport in stellar interiors. It produces a predominantly horizontal field. This dynamo process is found to be more effective in transporting angular momentum than the known hydrodynamic mechanisms. It might account for the observed pattern of rotation in the solar core.

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955 citations

Cites background from "The stability of non-dissipative co..."

...The instability can either be of the VelikhovChandrasekhar-Balbus-Hawley type (hereafter BH instability, see Velikhov 1959; Chandrasekhar 1960; Balbus & Hawley 1991), or a buoyancy-driven instability (Parker 1966), though the properties of the resulting field may depend on which of these…...
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Journal Article•DOI•

Dynamo-generated Turbulence and Large-Scale Magnetic Fields in a Keplerian Shear Flow

[...]

Axel Brandenburg, Åke Nordlund, Robert Stein, Ulf Torkelsson

01 Jun 1995-The Astrophysical Journal

TL;DR: In this article, the nonlinear evolution of magnetized Keplerian shear fields is simulated in a local, three-dimensional model, including the eeects of compressibility and stratiication.

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Abstract: The nonlinear evolution of magnetized Keplerian shear ows is simulated in a local, three-dimensional model, including the eeects of compressibility and stratiication. Supersonic ows are initially generated by the Balbus-Hawley magnetic shear instability. The resulting ows regenerate a turbulent magnetic eld which, in turn, reinforces the turbulence. Thus, the system acts like a dynamo that generates its own turbulence. However, unlike usual dynamos, the magnetic energy exceeds the kinetic energy of the turbulence by a factor of 3{10. By assuming the eld to be vertical on the outer (upper and lower) surfaces we do not constrain the horizontal magnetic ux. Indeed, a large scale toroidal magnetic eld is generated, mostly in the form of toroidal ux tubes with lengths comparable to the toroidal extent of the box. This large scale eld is mainly of 1 even (i.e. quadrupolar) parity with respect to the midplane and changes direction on a timescale of about 30 orbits, in a possibly cyclic manner. The eeective Shakura-Sunyaev alpha viscosity parameter is between 0.001 and 0.005, and the contribution from the Maxwell stress is about 3-7 times larger than the contribution from the Reynolds stress.

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863 citations

Cites background or methods or result from "The stability of non-dissipative co..."

...This instability has been known for more than three decades (Velikhov 1959, Chandrasekhar 1960, 1961), but its importance for accretion discs was rst pointed out by Balbus & Hawley (1991), and it is therefore often referred to as the Balbus-Hawley instability. The instability requires a magnetic eld that can be generated either outside the disc (e.g. in the central object) or in the disc itself. Here we consider the latter possibility where the motions resulting from the instability act as a dynamo to sustain the magnetic eld. In a recent paper, Hawley, Gammie, & Balbus (1994) have studied the Balbus-Hawley instability in the presence of an irregular magnetic eld, and have shown that this eld can sustain self-excited turbulence. In their model there is no vertical gravity, and therefore the Parker instability (Parker 1979) does not operate. This is in contrast to the dynamo model of Tout & Pringle (1992), which invokes the Parker instability to transform an azimuthal magnetic eld into a vertical eld. Meanwhile, Stone & Hawley (1994) included vertical gravity with substantial density strati cation and found qualitatively similar results as Hawley, Gammie, & Balbus (1994). They also showed that the saturated state is essentially independent of the initial magnetic eld geometry. The purpose of the present paper is twofold. First of all, simulating dynamo action which generates its own turbulence is a formidable problem in dynamo theory, and is central to the understanding of accretion discs which would not exist if there was no enhanced (e.g., turbulent) angular momentum transport. The papers by Stone & Hawley (1994) and Hawley, Gammie, & Balbus (1994) are based on the same algorithm, and it is desirable to verify their results using an independent method....
[...]
...This instability has been known for more than three decades (Velikhov 1959, Chandrasekhar 1960, 1961), but its importance for accretion discs was rst pointed out by Balbus & Hawley (1991), and it is therefore often referred to as the Balbus-Hawley instability. The instability requires a magnetic eld that can be generated either outside the disc (e.g. in the central object) or in the disc itself. Here we consider the latter possibility where the motions resulting from the instability act as a dynamo to sustain the magnetic eld. In a recent paper, Hawley, Gammie, & Balbus (1994) have studied the Balbus-Hawley instability in the presence of an irregular magnetic eld, and have shown that this eld can sustain self-excited turbulence. In their model there is no vertical gravity, and therefore the Parker instability (Parker 1979) does not operate. This is in contrast to the dynamo model of Tout & Pringle (1992), which invokes the Parker instability to transform an azimuthal magnetic eld into a vertical eld....
[...]
...This instability has been known for more than three decades (Velikhov1959, Chandrasekhar 1960, 1961), but its importance for accretion discs was rst pointedout by Balbus & Hawley (1991), and it is therefore often referred to as the Balbus-Hawleyinstability....
[...]
...This instability has been known for more than three decades (Velikhov 1959, Chandrasekhar 1960, 1961), but its importance for accretion discs was rst pointed out by Balbus & Hawley (1991), and it is therefore often referred to as the Balbus-Hawley instability. The instability requires a magnetic eld that can be generated either outside the disc (e.g. in the central object) or in the disc itself. Here we consider the latter possibility where the motions resulting from the instability act as a dynamo to sustain the magnetic eld. In a recent paper, Hawley, Gammie, & Balbus (1994) have studied the Balbus-Hawley instability in the presence of an irregular magnetic eld, and have shown that this eld can sustain self-excited turbulence. In their model there is no vertical gravity, and therefore the Parker instability (Parker 1979) does not operate. This is in contrast to the dynamo model of Tout & Pringle (1992), which invokes the Parker instability to transform an azimuthal magnetic eld into a vertical eld. Meanwhile, Stone & Hawley (1994) included vertical gravity with substantial density strati cation and found qualitatively similar results as Hawley, Gammie, & Balbus (1994). They also showed that the saturated state is essentially independent of the initial magnetic eld geometry....
[...]
...This instability has been known for more than three decades (Velikhov 1959, Chandrasekhar 1960, 1961), but its importance for accretion discs was rst pointed out by Balbus & Hawley (1991), and it is therefore often referred to as the Balbus-Hawley instability. The instability requires a magnetic eld that can be generated either outside the disc (e.g. in the central object) or in the disc itself. Here we consider the latter possibility where the motions resulting from the instability act as a dynamo to sustain the magnetic eld. In a recent paper, Hawley, Gammie, & Balbus (1994) have studied the Balbus-Hawley instability in the presence of an irregular magnetic eld, and have shown that this eld can sustain self-excited turbulence....
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Journal Article•DOI•

Rossby Wave Instability of Keplerian Accretion Disks

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Richard V. E. Lovelace¹, Hui Li², Stirling A. Colgate², Andrew F. Nelson³•Institutions (3)

Cornell University¹, Los Alamos National Laboratory², University of Arizona³

10 Mar 1999-The Astrophysical Journal

TL;DR: In this paper, a linear instability of nonaxisymmetric Rossby waves in a thin nonmagnetized Keplerian disk was found when there is a local maximum in the radial profile of a key function (r)≡(r)S2/Γ(r), where −1=(∇×v)/Σ is the potential vorticity, S=P/ΣΓ is the entropy, Σ is surface mass density, P is the vertically integrated pressure, and Γ are the adiabatic index.

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Abstract: We find a linear instability of nonaxisymmetric Rossby waves in a thin nonmagnetized Keplerian disk when there is a local maximum in the radial profile of a key function (r)≡(r)S2/Γ(r), where −1=(∇×v)/Σ is the potential vorticity, S=P/ΣΓ is the entropy, Σ is the surface mass density, P is the vertically integrated pressure, and Γ is the adiabatic index. We consider in detail the special case where there is a local maximum in the disk entropy profile S(r). This maximum acts to trap the waves in its vicinity if its height-to-width ratio max(S)/Δr is larger than a threshold value. The pressure gradient derived from this entropy variation provides the restoring force for the wave growth. We show that the trapped waves act to transport angular momentum outward. A plausible way to produce an entropy variation is when an accretion disk is starting from negligible mass and temperature, therefore, negligible entropy. As mass accumulates by either tidal torquing, magnetic torquing, or Roche-lobe overflow, confinement of heat will lead to an entropy maximum at the outer boundary of the disk. Possible nonlinear developments from this instability include the formation of Rossby vortices and the formation of spiral shocks. What remains to be determined from hydrodynamic simulations is whether or not Rossby wave packets (or vortices) "hold together" as they propagate radially inward.

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585 citations

Cites background from "The stability of non-dissipative co..."

...The widely cited magneto-rotational instability (Velikhov 1959; Chandrasekhar 1960) occurs only when the B field is weak....
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Book•

The Exoplanet Handbook

[...]

Michael Perryman¹•Institutions (1)

University of Bristol¹

01 May 2011

TL;DR: In this paper, the authors present an overview of the solar system and its evolution, including the formation and evolution of stars, asteroids, and free-floating planets, as well as their internal and external structures.

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Abstract: 1. Introduction 2. Radial velocities 3. Astrometry 4. Timing 5. Microlensing 6. Transits 7. Imaging 8. Host stars 9. Brown dwarfs and free-floating planets 10. Formation and evolution 11. Interiors and atmospheres 12. The Solar System Appendixes References Index.

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527 citations

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The stability of non-dissipative couette flow in hydromagnetics

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