Nonlinear Transverse Cascade and Sustenance of MRI Turbulence in Keplerian Disks with an Azimuthal Magnetic Field

Gogichaishvili, D.; Mamatsashvili, G.; Horton, W.; Chagelishvili, G.; Bodo, G.

doi:10.3847/1538-4357/aa7ed1

Nonlinear Transverse Cascade and Sustenance of MRI Turbulence in Keplerian Disks with an Azimuthal Magnetic Field

Journal Article · Thu Aug 10 00:00:00 EDT 2017 · The Astrophysical Journal (Online)

DOI:https://doi.org/10.3847/1538-4357/aa7ed1· OSTI ID:1537191

Gogichaishvili, D. ^[1]; ^[2]; Horton, W. ^[3]; Chagelishvili, G. ^[4]; ^[5]

Univ. of Texas, Austin, TX (United States). Dept. of Physics; DOE/OSTI
Helmholtz-Zentrum Dresden-Rossendorf, Dresden (Germany); Tbilisi State Univ., Tbilisi (Georgia). Dept. of Physics, Faculty of Exact and Natural Sciences; Ilia State Univ., Tbilisi (Georgia). Abastumani Astrophysical Observatory
Univ. of Texas, Austin, TX (United States). Inst. for Fusion Studies and Space and Geophysics Lab.
Ilia State Univ., Tbilisi (Georgia). Abastumani Astrophysical Observatory; Tbilisi State Univ., Tbilisi (Georgia). Inst. of Geophysics
INAF/Osservatorio Astrofisico di Torino, Pino Torinese (Italy). Strada Osservatorio

We investigate magnetohydrodynamic turbulence driven by the magnetorotational instability (MRI) in Keplerian disks with a nonzero net azimuthal magnetic field using shearing box simulations. As distinct from previous studies, we analyze turbulence dynamics in Fourier ($${\boldsymbol{k}}$$-) space to understand its sustenance. The linear growth of the MRI with the azimuthal field has a transient character and is anisotropic in Fourier space, leading to anisotropy of nonlinear processes in Fourier space. As a result, the main nonlinear process appears to be a new type of angular redistribution of modes in Fourier space—the nonlinear transverse cascade—rather than the usual direct/inverse cascade. Here, we demonstrate that the turbulence is sustained by the interplay of the linear transient growth of the MRI (which is the only energy supply for the turbulence) and the transverse cascade. These two processes operate at large length scales, comparable to the box size (disk scale height); the corresponding small wavenumber area, called the vital area in Fourier space, is crucial for the sustenance, while outside the vital area, direct cascade dominates. The interplay of the linear and nonlinear processes in Fourier space is generally too intertwined for a vivid schematization. Nevertheless, we reveal the basic subcycle of the sustenance that clearly shows the synergy of these processes in the self-organization of the magnetized flow system. This synergy is quite robust and persists for the considered different aspect ratios of the simulation boxes. The spectral characteristics of the dynamical processes in these boxes are qualitatively similar, indicating the universality of the sustenance mechanism of the MRI turbulence.

View Accepted Manuscript (DOE)

Research Organization:: Univ. of Texas, Austin, TX (United States)

Sponsoring Organization:: USDOE Office of Science (SC)

Grant/Contract Number:: FG02-04ER54742

OSTI ID:: 1537191

Alternate ID(s):: OSTI ID: 22875923

Journal Information:: The Astrophysical Journal (Online), Journal Name: The Astrophysical Journal (Online) Journal Issue: 1 Vol. 845; ISSN 1538-4357

Publisher:: Institute of Physics (IOP)Copyright Statement

Country of Publication:: United States

Language:: English

References (84)

Angular Momentum Transport in Accretion Disks: Scaling Laws in MRI-driven Turbulence Pessah, Martin E.; Chan, Chi-kwan; Psaltis, Dimitrios arXiv https://doi.org/10.48550/arxiv.0705.0352	text	January 2007
Stability and Transition in Shear Flows Schmid, Peter J.; Henningson, Dan S. Applied Mathematical Sciences https://doi.org/10.1007/978-1-4613-0185-1	book	January 2001
Spectral Methods in Fluid Dynamics Canuto, Claudio; Hussaini, M. Yousuff; Quarteroni, Alfio Springer Berlin, Heidelberg https://doi.org/10.1007/978-3-642-84108-8	book	January 1988
Statistical theory of magnetohydrodynamic turbulence: recent results Verma, Mahendra K. Physics Reports, Vol. 401, Issue 5-6 https://doi.org/10.1016/j.physrep.2004.07.007	journal	November 2004
Magnetohydrodynamic Turbulence Biskamp, Dieter https://doi.org/10.1017/CBO9780511535222	book	August 2009
Subcritical transition in channel flows Chapman, S. Jonathan Journal of Fluid Mechanics, Vol. 451 https://doi.org/10.1017/S0022112001006255	journal	January 2002
MRI-driven turbulent transport: the role of dissipation, channel modes and their parasites Longaretti, P. -Y.; Lesur, G. Astronomy and Astrophysics, Vol. 516 https://doi.org/10.1051/0004-6361/201014093	journal	June 2010
MHD simulations of the magnetorotational instability in a shearing box with zero net flux: the case Pm = 4 Fromang, S. Astronomy and Astrophysics, Vol. 514 https://doi.org/10.1051/0004-6361/201014284	journal	May 2010
Non-linear energy transfers in accretion discs MRI turbulence: I. Net vertical field case Lesur, G.; Longaretti, P. -Y. Astronomy & Astrophysics, Vol. 528 https://doi.org/10.1051/0004-6361/201015740	journal	February 2011
Dissipative effects on the sustainment of a magnetorotational dynamo in Keplerian shear flow Riols, A.; Rincon, F.; Cossu, C. Astronomy & Astrophysics, Vol. 575 https://doi.org/10.1051/0004-6361/201424324	journal	February 2015
Angular momentum transport and large eddy simulations in magnetorotational turbulence: the small Pm limit Meheut, Heloise; Fromang, Sébastien; Lesur, Geoffroy Astronomy & Astrophysics, Vol. 579 https://doi.org/10.1051/0004-6361/201525688	journal	July 2015
Magnetorotational dynamo chimeras: The missing link to turbulent accretion disk dynamo models?⋆ Riols, A.; Rincon, F.; Cossu, C. Astronomy & Astrophysics, Vol. 598 https://doi.org/10.1051/0004-6361/201629285	journal	February 2017
On hydrodynamic shear turbulence in Keplerian disks: Via transient growth to bypass transition Chagelishvili, G. D.; Zahn, J. -P.; Tevzadze, A. G. Astronomy & Astrophysics, Vol. 402, Issue 2 https://doi.org/10.1051/0004-6361:20030269	journal	April 2003
Nonaxisymmetric stability in the shearing sheet approximation Brandenburg, A.; Dintrans, B. Astronomy & Astrophysics, Vol. 450, Issue 2 https://doi.org/10.1051/0004-6361:20054639	journal	April 2006
Global MHD simulations of stratified and turbulent protoplanetary discs: I. Model properties Fromang, S.; Nelson, R. P. Astronomy & Astrophysics, Vol. 457, Issue 1 https://doi.org/10.1051/0004-6361:20065643	journal	September 2006
MHD simulations of the magnetorotational instability in a shearing box with zero net flux: I. The issue of convergence Fromang, S.; Papaloizou, J. Astronomy & Astrophysics, Vol. 476, Issue 3 https://doi.org/10.1051/0004-6361:20077942	journal	October 2007
MHD simulations of the magnetorotational instability in a shearing box with zero net flux: II. The effect of transport coefficients Fromang, S.; Papaloizou, J.; Lesur, G. Astronomy & Astrophysics, Vol. 476, Issue 3 https://doi.org/10.1051/0004-6361:20077943	journal	October 2007
Aspect ratio dependence in magnetorotational instability shearing box simulations Bodo, G.; Mignone, A.; Cattaneo, F. Astronomy & Astrophysics, Vol. 487, Issue 1 https://doi.org/10.1051/0004-6361:200809730	journal	June 2008
On self-sustained dynamo cycles in accretion discs Lesur, G.; Ogilvie, G. I. Astronomy & Astrophysics, Vol. 488, Issue 2 https://doi.org/10.1051/0004-6361:200810152	journal	July 2008
Dynamic anisotropy in MHD turbulence induced by mean magnetic field Sundar, Sita; Verma, Mahendra K.; Alexakis, Alexandros Physics of Plasmas, Vol. 24, Issue 2 https://doi.org/10.1063/1.4975609	journal	February 2017
A mostly linear model of transition to turbulence Baggett, Jeffrey S.; Driscoll, Tobin A.; Trefethen, Lloyd N. Physics of Fluids, Vol. 7, Issue 4 https://doi.org/10.1063/1.868606	journal	April 1995
A powerful local shear instability in weakly magnetized disks. I - Linear analysis. II - Nonlinear evolution Balbus, Steven A.; Hawley, John F. The Astrophysical Journal, Vol. 376 https://doi.org/10.1086/170270	journal	July 1991
A Powerful Local Shear Instability in Weakly Magnetized Disks. IV. Nonaxisymmetric Perturbations Balbus, Steven A.; Hawley, John F. The Astrophysical Journal, Vol. 400 https://doi.org/10.1086/172022	journal	November 1992
Parasitic instabilities in magnetized, differentially rotating disks Goodman, Jeremy; Xu, Guohong The Astrophysical Journal, Vol. 432 https://doi.org/10.1086/174562	journal	September 1994
Toward a theory of interstellar turbulence. 2: Strong alfvenic turbulence Goldreich, P.; Sridhar, S. The Astrophysical Journal, Vol. 438 https://doi.org/10.1086/175121	journal	January 1995
Local Three-dimensional Magnetohydrodynamic Simulations of Accretion Disks Hawley, John F.; Gammie, Charles F.; Balbus, Steven A. The Astrophysical Journal, Vol. 440 https://doi.org/10.1086/175311	journal	February 1995
The Effect of Resistivity on the Nonlinear Stage of the Magnetorotational Instability in Accretion Disks Fleming, Timothy P.; Stone, James M.; Hawley, John F. The Astrophysical Journal, Vol. 530, Issue 1 https://doi.org/10.1086/308338	journal	February 2000
Bypass to Turbulence in Hydrodynamic Accretion Disks: An Eigenvalue Approach Mukhopadhyay, Banibrata; Afshordi, Niayesh; Narayan, Ramesh The Astrophysical Journal, Vol. 629, Issue 1 https://doi.org/10.1086/431419	journal	August 2005
Magnetohydrodynamic Shearing Waves Johnson, Bryan M. The Astrophysical Journal, Vol. 660, Issue 2 https://doi.org/10.1086/513867	journal	May 2007
Angular Momentum Transport in Accretion Disks: Scaling Laws in MRI-driven Turbulence Pessah, Martin E.; Chan, Chi-kwan; Psaltis, Dimitrios The Astrophysical Journal, Vol. 668, Issue 1 https://doi.org/10.1086/522585	journal	September 2007
Simulations of Magnetorotational Turbulence with a Higher-Order Godunov Scheme Simon, Jacob B.; Hawley, John F.; Beckwith, Kris The Astrophysical Journal, Vol. 690, Issue 1 https://doi.org/10.1088/0004-637X/690/1/974	journal	December 2008
Locality of mhd Turbulence in Isothermal Disks Guan, Xiaoyue; Gammie, Charles F.; Simon, Jacob B. The Astrophysical Journal, Vol. 694, Issue 2 https://doi.org/10.1088/0004-637X/694/2/1010	journal	March 2009
Zonal Flows and Long-Lived Axisymmetric Pressure Bumps in Magnetorotational Turbulence Johansen, A.; Youdin, A.; Klahr, H. The Astrophysical Journal, Vol. 697, Issue 2 https://doi.org/10.1088/0004-637X/697/2/1269	journal	May 2009
On the Saturation of the Magnetorotational Instability via Parasitic Modes Pessah, Martin E.; Goodman, Jeremy The Astrophysical Journal, Vol. 698, Issue 1 https://doi.org/10.1088/0004-637X/698/1/L72	journal	May 2009
Viscous and Resistive Effects on the Magnetorotational Instability with a net Toroidal Field Simon, Jacob B.; Hawley, John F. The Astrophysical Journal, Vol. 707, Issue 1 https://doi.org/10.1088/0004-637X/707/1/833	journal	November 2009
Sustained Magnetorotational Turbulence in Local Simulations of Stratified Disks with zero net Magnetic flux Davis, Shane W.; Stone, James M.; Pessah, Martin E. The Astrophysical Journal, Vol. 713, Issue 1 https://doi.org/10.1088/0004-637X/713/1/52	journal	March 2010
Angular Momentum Transport in Protoplanetary and Black hole Accretion Disks: the role of Parasitic Modes in the Saturation of mhd Turbulence Pessah, Martin E. The Astrophysical Journal, Vol. 716, Issue 2 https://doi.org/10.1088/0004-637X/716/2/1012	journal	May 2010
Radially Extended, Stratified, Local Models of Isothermal Disks Guan, Xiaoyue; Gammie, Charles F. The Astrophysical Journal, Vol. 728, Issue 2 https://doi.org/10.1088/0004-637X/728/2/130	journal	January 2011
Turbulence and Steady Flows in Three-Dimensional Global Stratified Magnetohydrodynamic Simulations of Accretion Disks Flock, M.; Dzyurkevich, N.; Klahr, H. The Astrophysical Journal, Vol. 735, Issue 2 https://doi.org/10.1088/0004-637X/735/2/122	journal	June 2011
Symmetries, Scaling Laws, and Convergence in Shearing-Box Simulations of Magneto-Rotational Instability Driven Turbulence Bodo, G.; Cattaneo, F.; Ferrari, A. The Astrophysical Journal, Vol. 739, Issue 2 https://doi.org/10.1088/0004-637X/739/2/82	journal	September 2011
Large-Scale Azimuthal Structures of Turbulence in Accretion Disks: Dynamo Triggered Variability of Accretion Flock, M.; Dzyurkevich, N.; Klahr, H. The Astrophysical Journal, Vol. 744, Issue 2 https://doi.org/10.1088/0004-637X/744/2/144	journal	December 2011
Global Simulations of Accretion Disks. i. Convergence and Comparisons with Local Models Sorathia, Kareem A.; Reynolds, Christopher S.; Stone, James M. The Astrophysical Journal, Vol. 749, Issue 2 https://doi.org/10.1088/0004-637X/749/2/189	journal	April 2012
On Hydromagnetic Stresses in Accretion disk Boundary Layers Pessah, Martin E.; Chan, Chi-kwan The Astrophysical Journal, Vol. 751, Issue 1 https://doi.org/10.1088/0004-637X/751/1/48	journal	May 2012
Magnetorotational Turbulence in Stratified Shearing Boxes with Perfect gas Equation of State and Finite Thermal Diffusivity Bodo, G.; Cattaneo, F.; Mignone, A. The Astrophysical Journal, Vol. 761, Issue 2 https://doi.org/10.1088/0004-637X/761/2/116	journal	December 2012
Turbulence in Weakly Ionized Protoplanetary Disks Flock, M.; Henning, Th.; Klahr, H. The Astrophysical Journal, Vol. 761, Issue 2 https://doi.org/10.1088/0004-637X/761/2/95	journal	November 2012
Testing Convergence for Global Accretion Disks Hawley, John F.; Richers, Sherwood A.; Guan, Xiaoyue The Astrophysical Journal, Vol. 772, Issue 2 https://doi.org/10.1088/0004-637X/772/2/102	journal	July 2013
Convection Causes Enhanced Magnetic Turbulence in Accretion Disks in Outburst Hirose, Shigenobu; Blaes, Omer; Krolik, Julian H. The Astrophysical Journal, Vol. 787, Issue 1 https://doi.org/10.1088/0004-637X/787/1/1	journal	April 2014
Magnetic flux Concentration and Zonal Flows in Magnetorotational Instability Turbulence Bai, Xue-Ning; Stone, James M. The Astrophysical Journal, Vol. 796, Issue 1 https://doi.org/10.1088/0004-637X/796/1/31	journal	November 2014
Magnetorotational Instability: Nonmodal Growth and the Relationship of Global Modes to the Shearing box Squire, J.; Bhattacharjee, A. The Astrophysical Journal, Vol. 797, Issue 1 https://doi.org/10.1088/0004-637X/797/1/67	journal	November 2014
On the Anisotropic Nature of Mri-Driven Turbulence in Astrophysical Disks Murphy, Gareth C.; Pessah, Martin E. The Astrophysical Journal, Vol. 802, Issue 2 https://doi.org/10.1088/0004-637X/802/2/139	journal	April 2015
Turbulent cascades, transfer, and scale interactions in magnetohydrodynamics Alexakis, A.; Mininni, P. D.; Pouquet, A. New Journal of Physics, Vol. 9, Issue 8 https://doi.org/10.1088/1367-2630/9/8/298	journal	August 2007
Fully Convective Magnetorotational Turbulence in Stratified Shearing Boxes Bodo, G.; Cattaneo, F.; Mignone, A. The Astrophysical Journal, Vol. 771, Issue 2 https://doi.org/10.1088/2041-8205/771/2/L23	journal	June 2013
On the Convergence of Magnetorotational Turbulence in Stratified Isothermal Shearing Boxes Bodo, G.; Cattaneo, F.; Mignone, A. The Astrophysical Journal, Vol. 787, Issue 1 https://doi.org/10.1088/2041-8205/787/1/L13	journal	May 2014
The non-axisymmetric instability of a cylindrical shear flow containing an azimuthal magnetic field Ogilvie, G. I.; Pringle, J. E. Monthly Notices of the Royal Astronomical Society, Vol. 279, Issue 1 https://doi.org/10.1093/mnras/279.1.152	journal	March 1996
On the stability of an accretion disc containing a toroidal magnetic field Terquem, C.; Papaloizou, J. C. B. Monthly Notices of the Royal Astronomical Society, Vol. 279, Issue 3 https://doi.org/10.1093/mnras/279.3.767	journal	April 1996
On the stability of an accretion disc containing a toroidal magnetic field: the effect of resistivity Papaloizou, J. C. B.; Terquem, C. Monthly Notices of the Royal Astronomical Society, Vol. 287, Issue 4 https://doi.org/10.1093/mnras/287.4.771	journal	June 1997
Global simulations of magnetorotational turbulence – I. Convergence and the quasi-steady state Parkin, E. R.; Bicknell, G. V. Monthly Notices of the Royal Astronomical Society, Vol. 435, Issue 3 https://doi.org/10.1093/mnras/stt1450	journal	August 2013
Revisiting linear dynamics of non-axisymmetric perturbations in weakly magnetized accretion discs Mamatsashvili, G. R.; Chagelishvili, G. D.; Bodo, G. Monthly Notices of the Royal Astronomical Society, Vol. 435, Issue 3 https://doi.org/10.1093/mnras/stt1470	journal	September 2013
On characterizing non-locality and anisotropy for the magnetorotational instability Nauman, F.; Blackman, E. G. Monthly Notices of the Royal Astronomical Society, Vol. 441, Issue 3 https://doi.org/10.1093/mnras/stu706	journal	May 2014
A study of the transient dynamics of perturbations in Keplerian discs using a variational approach Zhuravlev, V. V.; Razdoburdin, D. N. Monthly Notices of the Royal Astronomical Society, Vol. 442, Issue 1 https://doi.org/10.1093/mnras/stu848	journal	June 2014
The stress–pressure relationship in simulations of MRI-induced turbulence Ross, Johnathan; Latter, Henrik N.; Guilet, Jerome Monthly Notices of the Royal Astronomical Society, Vol. 455, Issue 1 https://doi.org/10.1093/mnras/stv2286	journal	November 2015
Saturation of the magnetorotational instability in the unstratified shearing box with zero net flux: convergence in taller boxes Shi, Ji-Ming; Stone, James M.; Huang, Chelsea X. Monthly Notices of the Royal Astronomical Society, Vol. 456, Issue 3 https://doi.org/10.1093/mnras/stv2815	journal	December 2015
Transient growth of perturbations on scales beyond the accretion disc thickness Razdoburdin, D. N.; Zhuravlev, V. V. Monthly Notices of the Royal Astronomical Society, Vol. 467, Issue 1 https://doi.org/10.1093/mnras/stx050	journal	February 2017
Universal small-scale structure in turbulence driven by magnetorotational instability Zhdankin, Vladimir; Walker, Justin; Boldyrev, Stanislav Monthly Notices of the Royal Astronomical Society, Vol. 467, Issue 3 https://doi.org/10.1093/mnras/stx372	journal	February 2017
On the nature of magnetic turbulence in rotating, shearing flows Walker, Justin; Lesur, Geoffroy; Boldyrev, Stanislav Monthly Notices of the Royal Astronomical Society: Letters, Vol. 457, Issue 1 https://doi.org/10.1093/mnrasl/slv200	journal	January 2016
Chaos transition despite linear stability Gebhardt, Thomas; Grossmann, Siegfried Physical Review E, Vol. 50, Issue 5 https://doi.org/10.1103/PhysRevE.50.3705	journal	November 1994
Energy transfer in anisotropic magnetohydrodynamic turbulence Teaca, B.; Verma, M. K.; Knaepen, B. Physical Review E, Vol. 79, Issue 4 https://doi.org/10.1103/PhysRevE.79.046312	journal	April 2009
Angular redistribution of nonlinear perturbations: A universal feature of nonuniform flows Horton, W.; Kim, J. -H.; Chagelishvili, G. D. Physical Review E, Vol. 81, Issue 6 https://doi.org/10.1103/PhysRevE.81.066304	journal	June 2010
Periodic magnetorotational dynamo action as a prototype of nonlinear magnetic-field generation in shear flows Herault, J.; Rincon, F.; Cossu, C. Physical Review E, Vol. 84, Issue 3 https://doi.org/10.1103/PhysRevE.84.036321	journal	September 2011
Magnetized stratified rotating shear waves Salhi, A.; Lehner, T.; Godeferd, F. Physical Review E, Vol. 85, Issue 2 https://doi.org/10.1103/PhysRevE.85.026301	journal	February 2012
Nonlinear transverse cascade and two-dimensional magnetohydrodynamic subcritical turbulence in plane shear flows Mamatsashvili, G. R.; Gogichaishvili, D. Z.; Chagelishvili, G. D. Physical Review E, Vol. 89, Issue 4 https://doi.org/10.1103/PhysRevE.89.043101	journal	April 2014
Dynamics of homogeneous shear turbulence: A key role of the nonlinear transverse cascade in the bypass concept Mamatsashvili, G.; Khujadze, G.; Chagelishvili, G. Physical Review E, Vol. 94, Issue 2 https://doi.org/10.1103/PhysRevE.94.023111	journal	August 2016
The onset of shear flow turbulence Grossmann, Siegfried Reviews of Modern Physics, Vol. 72, Issue 2 https://doi.org/10.1103/RevModPhys.72.603	journal	April 2000
Impact of dimensionless numbers on the efficiency of magnetorotational instability induced turbulent transport Lesur, G.; Longaretti, P. -Y. Monthly Notices of the Royal Astronomical Society, Vol. 378, Issue 4 https://doi.org/10.1111/j.1365-2966.2007.11888.x	journal	June 2007
MRI channel flows and their parasites Latter, Henrik N.; Lesaffre, Pierre; Balbus, Steven A. Monthly Notices of the Royal Astronomical Society, Vol. 394, Issue 2 https://doi.org/10.1111/j.1365-2966.2009.14395.x	journal	April 2009
Turbulence in global simulations of magnetized thin accretion discs: Turbulence in magnetized thin discs Beckwith, Kris; Armitage, Philip J.; Simon, Jacob B. Monthly Notices of the Royal Astronomical Society https://doi.org/10.1111/j.1365-2966.2011.19043.x	journal	July 2011
Regimes of the non-exponential temporal growth in thin Keplerian discs under toroidally dominated magnetic fields: Keplerian discs in toroidal magnetic fields Shtemler, Yuri M.; Mond, Michael; Liverts, Edward Monthly Notices of the Royal Astronomical Society https://doi.org/10.1111/j.1365-2966.2011.20346.x	journal	January 2012
On the angular momentum transport due to vertical convection in accretion discs Lesur, Geoffroy; Ogilvie, Gordon I. Monthly Notices of the Royal Astronomical Society: Letters, Vol. 404, Issue 1 https://doi.org/10.1111/j.1745-3933.2010.00836.x	journal	May 2010
The nuclear stellar disc in Andromeda: a fossil from the era of black hole growth: Origin of the eccentric disc and BH in M31 Hopkins, Philip F.; Quataert, Eliot Monthly Notices of the Royal Astronomical Society: Letters, Vol. 405, Issue 1 https://doi.org/10.1111/j.1745-3933.2010.00855.x	journal	April 2010
Hydrodynamic Stability Without Eigenvalues Trefethen, L. N.; Trefethen, A. E.; Reddy, S. C. Science, Vol. 261, Issue 5121 https://doi.org/10.1126/science.261.5121.578	journal	July 1993
Pseudospectra of the Orr–Sommerfeld Operator Reddy, Satish C.; Schmid, Peter J.; Henningson, Dan S. SIAM Journal on Applied Mathematics, Vol. 53, Issue 1 https://doi.org/10.1137/0153002	journal	February 1993
Dynamics of Protoplanetary Disks Armitage, Philip J. Annual Review of Astronomy and Astrophysics, Vol. 49, Issue 1 https://doi.org/10.1146/annurev-astro-081710-102521	journal	September 2011
Enhanced Angular Momentum Transport in Accretion Disks Balbus, Steven A. Annual Review of Astronomy and Astrophysics, Vol. 41, Issue 1 https://doi.org/10.1146/annurev.astro.41.081401.155207	journal	September 2003
Nonmodal Stability Theory Schmid, Peter J. Annual Review of Fluid Mechanics, Vol. 39, Issue 1 https://doi.org/10.1146/annurev.fluid.38.050304.092139	journal	January 2007

Cited By (3)

Turbulence and order in magnetized flowing plasmas Ebrahimi, Fatima; Blackman, Eric; Horton, Wendell arXiv https://doi.org/10.48550/arxiv.2011.09633	preprint	January 2020
Active Modes and Dynamical Balances in MRI Turbulence of Keplerian Disks with a Net Vertical Magnetic Field Gogichaishvili, D.; Mamatsashvili, G.; Horton, W. The Astrophysical Journal, Vol. 866, Issue 2 https://doi.org/10.3847/1538-4357/aadbad	journal	October 2018
Active modes and dynamical balances in MRI-turbulence of Keplerian disks with a net vertical magnetic field Gogichaishvili, D.; Mamatsashvili, G.; Horton, W. arXiv https://doi.org/10.48550/arxiv.1808.06147	text	January 2018

Similar Records

Nonlinear Transverse Cascade and Sustenance of MRI Turbulence in Keplerian Disks with an Azimuthal Magnetic Field

Journal Article · Thu Aug 10 00:00:00 EDT 2017 · Astrophysical Journal · OSTI ID:22875923

Active Modes and Dynamical Balances in MRI Turbulence of Keplerian Disks with a Net Vertical Magnetic Field

Journal Article · Sun Oct 21 20:00:00 EDT 2018 · The Astrophysical Journal (Online) · OSTI ID:1609572

Self-Sustaining Nonlinear Dynamo Process in Keplerian Shear Flows

Journal Article · Fri Jun 22 00:00:00 EDT 2007 · Physical Review Letters · OSTI ID:20953310

Related Subjects

79 ASTRONOMY AND ASTROPHYSICS
accretion
accretion disks
astronomy & astrophysics
instabilities
magnetohydrodynamics (MHD)
turbulence

Nonlinear Transverse Cascade and Sustenance of MRI Turbulence in Keplerian Disks with an Azimuthal Magnetic Field

Citation Formats

References (84)

Cited By (3)

Similar Records

Related Subjects