Transient Dynamics of Perturbations in Astrophysical Discs

Part of the Astrophysics and Space Science Library book series (ASSL, volume 454)


This chapter reviews some aspects of one of the major unsolved problems in understanding astrophysical (in particular, accretion) discs: whether the disc interiors may be effectively viscous in spite of the absence of magnetorotational instability. In this case, a rotational homogeneous inviscid flow with a Keplerian angular velocity profile is spectrally stable, making the transient growth of perturbations a candidate mechanism for energy transfer from regular motion to perturbations. Transient perturbations differ qualitatively from perturbation modes and can grow substantially in shear flows due to the non-normality of their dynamical evolution operator. Since the eigenvectors of this operator, alias perturbation modes, are mutually nonorthogonal, they can mutually interfere, resulting in transient growth of their linear combinations. Physically, a growing transient perturbation is a leading spiral whose branches are shrunk as a result of the differential rotation of the flow. This chapter discusses in detail the transient growth of vortex shear harmonics in the spatially local limit as well as methods for identifying the optimal (fastest growth) perturbations. Special attention is given to obtaining such solutions variationally, by integrating the direct and adjoint equations forwards and backwards in time, respectively. The material is presented in a newcomer-friendly style.


  1. Afshordi N, Mukhopadhyay B, Narayan R (2005) Bypass to turbulence in hydrodynamic accretion: Lagrangian analysis of energy growth. Astrophys J 629:373–382. ADSCrossRefGoogle Scholar
  2. Andersson P, Berggren M, Henningson DS (1999) Optimal disturbances and bypass transition in boundary layers. Phys Fluids 11:134–150. ADSMathSciNetzbMATHCrossRefGoogle Scholar
  3. Armitage PJ (2009) Astrophysics of planet formation. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  4. Avila M (2012) Stability and angular-momentum transport of fluid flows between corotating cylinders. Phys Rev Lett 108(12):124501. ArXiv:1203.4923
  5. Balbus SA (2003) Enhanced angular momentum transport in accretion disks. Annu Rev Astron Astrophys 41:555–597. ArXiv:astro-ph/0306208ADSCrossRefGoogle Scholar
  6. Balbus SA, Hawley JF (1991) A powerful local shear instability in weakly magnetized disks. i - linear analysis. ii - nonlinear evolution. Astrophys J 376:214–233. ADSCrossRefGoogle Scholar
  7. Balbus SA, Hawley JF (1998) Instability, turbulence, and enhanced transport in accretion disks. Rev Mod Phys 70:1–53. ADSCrossRefGoogle Scholar
  8. Balbus SA, Hawley JF, Stone JM (1996) Nonlinear stability, hydrodynamical turbulence, and transport in disks. Astrophys J 467:76ADSCrossRefGoogle Scholar
  9. Binney J, Tremaine S (2008) Galactic dynamics, 2nd edn. Princeton University Press, PrincetonzbMATHGoogle Scholar
  10. Blaes OM, Glatzel W (1986) On the stability of incompressible constant angular momentum cylinders. Mon Not R Astron Soc 220:253–258ADSCrossRefGoogle Scholar
  11. Bodo G, Chagelishvili G, Murante G, Tevzadze A, Rossi P, Ferrari A (2005) Spiral density wave generation by vortices in keplerian flows. Astron Astrophys 437:9–22. ADSCrossRefGoogle Scholar
  12. Brekhovskikh LM, Goncharov V (1985) Mechanics of continua and wave dynamics. Berlin: SpringerzbMATHCrossRefGoogle Scholar
  13. Butler KM, Farrell BF (1992) Three-dimensional optimal perturbations in viscous shear flow. Phys Fluids A 4(8):1637–1650. ADSCrossRefGoogle Scholar
  14. Canuto VM, Goldman I, Hubickyj O (1984) A formula for the shakura–sunyaev turbulent viscosity parameter. Astrophys J Lett 280:L55ADSCrossRefGoogle Scholar
  15. Chagelishvili GD, Rogava AD, Segal IN (1994) Hydrodynamic stability of compressible plane couette flow. Phys Rev E 50:4283. ADSCrossRefGoogle Scholar
  16. Chagelishvili GD, Chanishvili RG, Lominadze DG (1996) Physics of the amplification of vortex disturbances in shear flows. JETP Lett 7:543–549. ADSCrossRefGoogle Scholar
  17. Chagelishvili GD, Tevzadze AG, Bodo G, Moiseev SS (1997) Linear mechanism of wave emergence from vortices in smooth shear flows. Phys Rev Lett 79:3178–3181. ADSCrossRefGoogle Scholar
  18. Chagelishvili GD, Zahn JP, Tevzadze AG, Lominadze JG (2003) On hydrodynamic shear turbulence in keplerian disks: via transient growth to bypass transition. Astron Astrophys 402:401–407. ADSCrossRefGoogle Scholar
  19. Charney JG, Fjörtoft R, Von Neuman J (1950) Numerical integration of the barotropic vorticity equation. Tellus 2:237–254ADSMathSciNetCrossRefGoogle Scholar
  20. Cherubini S, De Palma P (2013) Nonlinear optimal perturbations in a couette flow: bursting and transition. J Fluid Mech 716:251–279. ADSMathSciNetzbMATHCrossRefGoogle Scholar
  21. Cherubini S, Robinet JC, Bottaro A, de Palma P (2010) Optimal wave packets in a boundary layer and initial phases of a turbulent spot. J Fluid Mech 656:231–259. ADSMathSciNetzbMATHCrossRefGoogle Scholar
  22. Cherubini S, de Palma P, Robinet JC, Bottaro A (2011) The minimal seed of turbulent transition in the boundary layer. J Fluid Mech 689:221–253. ADSzbMATHCrossRefGoogle Scholar
  23. Churilov SM, Shuhman IG (1981) On the relation between volume and surface adiabatic indices for gaseous subsystems of flat galaxies. Astron Tsirk 1157:1–2ADSGoogle Scholar
  24. Coles D (1965) Transition in circular couette flow. J Fluid Mech 21:385–425. ADSzbMATHCrossRefGoogle Scholar
  25. Corbett P, Bottaro A (2001) Optimal linear growth in swept boundary layers. J Fluid Mech 435:1–23ADSzbMATHCrossRefGoogle Scholar
  26. Davis SW, Stone JM, Pessah ME (2010) Sustained magnetorotational turbulence in local simulations of stratified disks with zero net magnetic flux. Astrophys J 713:52–65ADSCrossRefGoogle Scholar
  27. Drazin PG, Reid WH (1981) Hydrodynamic stability. NASA STI/Recon Technical Report A 82:17950ADSGoogle Scholar
  28. Drury LO (1985) Acoustic amplification in discs and tori. Mon Not R Astron Soc 217:821–829ADSMathSciNetzbMATHCrossRefGoogle Scholar
  29. Edlund EM, Ji H (2014) Nonlinear stability of laboratory quasi-keplerian flows. Phys Rev Lett 89(2):021004. ArXiv:1401.6183
  30. Farrell BF (1988) Optimal excitation of perturbations in viscous shear flow. Phys Fluids 31:2093–2102. ADSCrossRefGoogle Scholar
  31. Farrell BF, Ioannou PJ (1996a) Generalized stability theory. Part i: autonomous operators. J Atmos Sci 53:2025–2040.<2025:GSTPIA>2.0.CO;2 ADSMathSciNetGoogle Scholar
  32. Farrell BF, Ioannou PJ (1996b) Generalized stability theory. Part ii: nonautonomous operators. J Atmos Sci 53:2041–2053.<2041:GSTPIN>2.0.CO;2 MathSciNetGoogle Scholar
  33. Frank J, Robertson JA (1988) Numerical studies of the dynamical stability of differentially rotating tori. Mon Not R Astron Soc 232:1–33ADSzbMATHCrossRefGoogle Scholar
  34. Fridman AM (1989) On the dynamics of a viscous differentially rotating gravitating medium. Sov Astron Lett 15:487ADSGoogle Scholar
  35. Fridman AM, Bisikalo DV (2008) The at of accretion disks of close binary stars: overreflection instability and developed turbulence. Phys Usp 51:551–576ADSCrossRefGoogle Scholar
  36. Friedman JL, Schutz BF (1978) Lagrangian perturbation theory of nonrelativistic fluids. Astrophys J 221:937–957. ADSMathSciNetCrossRefGoogle Scholar
  37. Glatzel W (1987a) On the stability of compressible differentially rotating cylinders. Mon Not R Astron Soc 225:227–255ADSMathSciNetzbMATHCrossRefGoogle Scholar
  38. Glatzel W (1987b) On the stability of compressible differentially rotating cylinders. ii. Mon Not R Astron Soc 228:77–100ADSMathSciNetzbMATHCrossRefGoogle Scholar
  39. Glatzel W (1988) Sonic instabilities in supersonic shear flows. Mon Not R Astron Soc 231:795–821ADSCrossRefGoogle Scholar
  40. Goldreich P, Lynden-Bell D (1965) II. Spiral arms as sheared gravitational instabilities. Mon Not R Astron Soc 130:125ADSCrossRefGoogle Scholar
  41. Goldreich P, Narayan R (1985) Non-axisymmetric instability in thin discs. Mon Not R Astron Soc 213:7P–10PADSCrossRefGoogle Scholar
  42. Goldreich P, Goodman J, Narayan R (1986) The stability of accretion tori. i - long-wavelength modes of slender tori. Mon Not R Astron Soc 221:339–364ADSzbMATHCrossRefGoogle Scholar
  43. Golub GH, Reinsch C (1970) Singular value decomposition and least squares solutions. Numer Math 14:403–420. MathSciNetzbMATHCrossRefGoogle Scholar
  44. Golub GH, Van Loan CF (1996) Matrix computations. Johns Hopkins University Press, BaltimorezbMATHGoogle Scholar
  45. Guégan A, Schmid PJ, Huerre P (2006) Optimal energy growth and optimal control in swept hiemenz flow. J Fluid Mech 566:11–45. ADSMathSciNetzbMATHCrossRefGoogle Scholar
  46. Gunzburger MD (2003) Perspectives in flow control and optimization. J Soc Ind Appl Math.
  47. Hanifi A, Schmid PJ, Henningson DS (1996) Transient growth in compressible boundary layer flow. Phys Fluids 8:826–837. ADSMathSciNetzbMATHCrossRefGoogle Scholar
  48. Hawley JF, Gammie CF, Balbus SA (1995) Local three-dimensional magnetohydrodynamic simulations of accretion disks. Astrophys J 440:742. ADSCrossRefGoogle Scholar
  49. Hawley JF, Balbus SA, Winters WF (1999) Local hydrodynamic stability of accretion disks. Astrophys J 518:394–404. ArXiv:astro-ph/9811057ADSCrossRefGoogle Scholar
  50. Heading JA (2013) Introduction to phase-integral methods. Dover Publications, MineolazbMATHGoogle Scholar
  51. Heinemann T, Papaloizou JCB (2009a) The excitation of spiral density waves through turbulent fluctuations in accretion discs - I. WKBJ theory. Mon Not R Astron Soc 397:52–63. ArXiv:0812.2068ADSCrossRefGoogle Scholar
  52. Heinemann T, Papaloizou JCB (2009b) The excitation of spiral density waves through turbulent fluctuations in accretion discs - ii. numerical simulations with MRI-driven turbulence. Mon Not R Astron Soc 397:64–74. ArXiv:0812.2471ADSCrossRefGoogle Scholar
  53. Henningson DS, Reddy SC (1994) On the role of linear mechanisms in transition to turbulence. Phys Fluids 6:1396–1398. ADSMathSciNetzbMATHCrossRefGoogle Scholar
  54. Hill GW (1878) Researches in the lunar theory. Am J Math 1:5–26MathSciNetzbMATHCrossRefGoogle Scholar
  55. Horton W, Kim JH, Chagelishvili GD, Bowman JC, Lominadze JG (2010) Angular redistribution of nonlinear perturbations: a universal feature of nonuniform flows. Phys Rev E 81(6):066304. ADSCrossRefGoogle Scholar
  56. Ioannou PJ, Kakouris A (2001) Stochastic dynamics of keplerian accretion disks. Astrophys J 550:931–943. ADSCrossRefGoogle Scholar
  57. Ji H, Burin M, Schartman E, Goodman J (2006) Hydrodynamic turbulence cannot transport angular momentum effectively in astrophysical disks. Nature 444:343–346,, ArXiv:astro-ph/0611481ADSCrossRefGoogle Scholar
  58. Johnson BM, Gammie CF (2005) Linear theory of thin, radially stratified disks. Astrophys J 626:978–990. ADSCrossRefGoogle Scholar
  59. Joseph DD (1976) Stability of fluid motions. i, ii. NASA STI/Recon Technical Report A 77:287Google Scholar
  60. Julian WH, Toomre A (1966) Non-axisymmetric responses of differentially rotating disks of stars. Astrophys J 146:810. ADSCrossRefGoogle Scholar
  61. Kato S (1987) Instability of isentropic geometrically thin disks due to corotation resonance. Astron Soc Jpn 39(4):645–666ADSGoogle Scholar
  62. Kato S (2001) Basic properties of thin-disk oscillations. Publ Astron Soc Jpn 53(1):1–24ADSMathSciNetCrossRefGoogle Scholar
  63. Kelvin L (1887) On the stability of steady and of periodic fluid motion. Philos Mag 23:459–539CrossRefGoogle Scholar
  64. Klahr H, Hubbard A (2014) Convective overstability in radially stratified accretion disks under thermal relaxation. Astrophys J 788:8. ADSCrossRefGoogle Scholar
  65. Kojima Y (1986) The dynamical stability of a fat disk with constant specific angular momentum. Prog Theor Phys 75:251–261. ADSCrossRefGoogle Scholar
  66. Kojima Y (1989) Non-axisymmetric unstable modes of a differentially rotating torus. Mon Not R Astron Soc 236:589–602ADSCrossRefGoogle Scholar
  67. Kojima Y, Miyama SM, Kubotani H (1989) Effects of entropy distributions on non-axisymmetric unstable modes in differentially rotating tori and cylinders. Mon Not R Astron Soc 238:753–768ADSCrossRefGoogle Scholar
  68. Korn GA, Korn TM (1968) Mathematical Handbook for Scientists and Engineers. McGraw-Hill, New YorkzbMATHGoogle Scholar
  69. Kurbatov EP, Bisikalo DV, Kaygorodov PV (2014) On the possible turbulence mechanism in accretion disks in nonmagnetic binary stars. Phys Usp 57:787–198ADSCrossRefGoogle Scholar
  70. Landau LD, Lifshitz EM (1980) Statistical physics, vol 1. Pegramon Press, OxfordzbMATHGoogle Scholar
  71. Landau LD, Lifshitz EM (1987) Fluid mechanics. Pergamon Press, OxfordGoogle Scholar
  72. Lesur G, Longaretti PY (2005) On the relevance of subcritical hydrodynamic turbulence to accretion disk transport. Astron Astrophys 444:25–44. ADSzbMATHCrossRefGoogle Scholar
  73. Lin C (1955) The theory of hydrodynamic stability. Cambridge University Press, CambridgezbMATHGoogle Scholar
  74. Lominadze JG (2011) Development of the theory of instabilities of differentially rotating plasma with astrophysical applications. In: Bonanno A, de Gouveia Dal Pino E, Kosovichev AG (eds) Proceedings of the International Astronomical Union, IAU symposium: advances in plasma astrophysics, IAU symposium, vol 274, pp 318–324. CrossRefGoogle Scholar
  75. Lominadze DG, Chagelishvili GD, Chanishvili RG (1988) The evolution of nonaxisymmetric shear perturbations in accretion disks. Sov Astron Lett 14(5):364ADSGoogle Scholar
  76. Longaretti PY (2002) On the phenomenology of hydrodynamic shear turbulence. Astrophys J 576:587–598. ArXiv:astro-ph/0205430ADSCrossRefGoogle Scholar
  77. Lovelace RVE, Li H, Colgate SA, Nelson AF (1999) Rossby wave instability of keplerian accretion disks. Astrophys J 513:805–810. ArXiv:astro-ph/9809321ADSCrossRefGoogle Scholar
  78. Luchini P (2000) Reynolds-number-independent instability of the boundary layer over a flat surface:optimal perturbations. J Fluid Mech 404:289–309. ADSMathSciNetzbMATHCrossRefGoogle Scholar
  79. Malik M, Alam M, Dey J (2006) Nonmodal energy growth and optimal perturbations in compressible plane couette flow. Phys Fluids 18(3):034103–034103-14. ArXiv:0804.0065ADSMathSciNetzbMATHCrossRefGoogle Scholar
  80. Mamatsashvili GR, Gogichaishvili DZ, Chagelishvili GD, Horton W (2014) Nonlinear transverse cascade and two-dimensional magnetohydrodynamic subcritical turbulence in plane shear flows. Phys Rev E 89(4):043101. ArXiv:1409.8543
  81. Marchuk GI (1998) Construction of adjoint operators in non-linear problems of mathematical physics. Sbornik Math 189:1505ADSMathSciNetzbMATHCrossRefGoogle Scholar
  82. Marcus P, Pei S, Jiang CH, Barranco J, Hassanzadeh P, Lecoanet D (2014) Zombie vortex instability i: a purely hydrodynamic instability to resurrect the dead zones of protoplanetary disks. ArXiv e-prints1410.8143Google Scholar
  83. Maretzke S, Hof B, Avila M (2014) Transient growth in linearly stable Taylor–Couette flows. J Fluid Mech 742:254–290. ArXiv:1304.7032ADSMathSciNetCrossRefGoogle Scholar
  84. Menou K (2000) Viscosity mechanisms in accretion disks. Science 288:2022–2024. ArXiv:astro-ph/0009022ADSCrossRefGoogle Scholar
  85. Meseguer Á (2002) Energy transient growth in the taylor-couette problem. Phys Fluids 14:1655–1660. ADSMathSciNetzbMATHCrossRefGoogle Scholar
  86. Monin AS, Yaglom AM (1971) Statistical fluid mechanics, vol 1. MIT Press, CambridgeGoogle Scholar
  87. Mukhopadhyay B, Afshordi N, Narayan R (2005) Bypass to turbulence in hydrodynamic accretion disks: an eigenvalue approach. Astrophys J 629:383–396. ADSCrossRefGoogle Scholar
  88. Narayan R (1991) Instabilities in thick disks. In: Bertout C, Collin-Souffrin S, Lasota JP (eds) Proceedings of IAU Colloq. 129, the 6th Institute d’Astrophysique de Paris (IAP) meeting: structure and emission properties of accretion disks, p 153. Editions Frontieres, Gif-sur-YvetteGoogle Scholar
  89. Narayan R, Goodman J (1989) Non-axisymmetric shear instabilities in thick accretion disks. In: Meyer F (ed) Proceedings of a NATO advanced research workshop: theory of accretion disks, vol 290. NATO Advanced Science Institutes (ASI), Series C, Garching, p 231CrossRefGoogle Scholar
  90. Narayan R, Goldreich P, Goodman J (1987) Physics of modes in a differentially rotating system - analysis of the shearing sheet. Mon Not R Astron Soc 228:1–41ADSzbMATHCrossRefGoogle Scholar
  91. Okazaki AT, Kato S, Fukue J (1987) Global trapped oscillations of relativistic accretion disks. Astron Soc Jpn 39:457–473ADSGoogle Scholar
  92. Orr WM (1907a) The stability or instability of the steady motions of a liquid i. Proc R Ir Acad A 27:9–68Google Scholar
  93. Orr WM (1907b) The stability or instability of the steady motions of a perfect liquid and of a viscous liquid ii. Proc R Ir Acad A 27:69–138Google Scholar
  94. Paoletti MS, Lathrop DP (2011) Angular momentum transport in turbulent flow between independently rotating cylinders. Phys Rev Lett 106(2):024501. ArXiv:1011.3475
  95. Paoletti MS, van Gils DPM, Dubrulle B, Sun C, Lohse D, Lathrop DP (2012) Angular momentum transport and turbulence in laboratory models of keplerian flows. Astron Astrophys 547:A64. ArXiv:1111.6915ADSCrossRefGoogle Scholar
  96. Papaloizou JCB, Pringle JE (1984) The dynamical stability of differentially rotating discs with constant specific angular momentum. Mon Not R Astron Soc 208:721–750ADSzbMATHCrossRefGoogle Scholar
  97. Papaloizou JCB, Pringle JE (1985) The dynamical stability of differentially rotating discs. ii. Mon Not R Astron Soc 213:799–820ADSzbMATHCrossRefGoogle Scholar
  98. Papaloizou JCB, Pringle JE (1987) The dynamical stability of differentially rotating discs. iii. Mon Not R Astron Soc 225:267–283ADSzbMATHCrossRefGoogle Scholar
  99. Pringle JE, King A (2007) Astrophysical flows. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
  100. Rayleigh L (1880) On the stability or instability of certain fluid motions. Sci Papers 1:474–484zbMATHGoogle Scholar
  101. Rayleigh L (1916) On the dynamics of revolving fluids. Sci Papers 6:447–453Google Scholar
  102. Razdoburdin DN, Zhuravlev VV (2012) Optimal growth of small perturbations in thin gaseous disks. Astron Lett 38:117–127. ADSCrossRefGoogle Scholar
  103. Razdoburdin DN, Zhuravlev VV (2017) Transient growth of perturbations on scales beyond the accretion disc thickness. Mon Not R Astron Soc 467:849–872. ArXiv:1701.02535ADSCrossRefGoogle Scholar
  104. Rebusco P, Umurhan OM, Klúzniak W, Regev O (2009) Global transient dynamics of three-dimensional hydrodynamical disturbances in a thin viscous accretion disk. Phys Fluids 21:076601. ADSzbMATHCrossRefGoogle Scholar
  105. Reddy SC, Henningson DS (1993) Energy growth in viscous channel flows. J Fluid Mech 252:209–238ADSMathSciNetzbMATHCrossRefGoogle Scholar
  106. Reddy SC, Schmid PJ, Henningson DS (1993) Pseudospectra of the Orr–sommerfeld operator. SIAM J Appl Math 53(01):15–47. MathSciNetzbMATHCrossRefGoogle Scholar
  107. Reddy SC, Schmid PJ, Baggett JS, Henningson DS (1998) On stability of streamwise streaks and transition thresholds in plane channel flows. J Fluid Mech 365:269–303ADSMathSciNetzbMATHCrossRefGoogle Scholar
  108. Regev O, Umurhan OM (2008) On the viability of the shearing box approximation for numerical studies of MHD turbulence in accretion disks. Astron Astrophys 481:21–32. ArXiv:0711.0794ADSzbMATHCrossRefGoogle Scholar
  109. Remillard RA, McClintock JE (2006) X-ray properties of black-hole binaries. Annu Rev Astron Astrophys 44:49–92. ArXiv:astro-ph/0606352ADSCrossRefGoogle Scholar
  110. Richard D, Zahn JP (1999) Turbulence in differentially rotating flows. what can be learned from the Couette–Taylor experiment. Astron Astrophys 347:734–738. ArXiv:astro-ph/9903374Google Scholar
  111. Richtmyer RD (1981) Principles of advanced mathematical physics, vol 2. Springer, New YorkzbMATHCrossRefGoogle Scholar
  112. Salhi A, Pieri AB (2014) Wave-vortex mode coupling in neutrally stable baroclinic flows. Phys Rev E 90(4):043003. ADSCrossRefGoogle Scholar
  113. Savonije GJ, Heemskerk MHM (1990) Non-axisymmetric unstable modes in a thin differentialy rotating gaseous disk. Astron Astrophys 240(1):191–202ADSzbMATHGoogle Scholar
  114. Schartman E, Ji H, Burin MJ (2009) Development of a Couette–Taylor flow device with active minimization of secondary circulation. Rev Sci Instrum 80(2):024501. ADSCrossRefGoogle Scholar
  115. Schartman E, Ji H, Burin MJ, Goodman J (2012) Stability of Quasi–Keplerian shear flow in a laboratory experiment. Astron Astrophys 543:A94. ArXiv:1102.3725ADSCrossRefGoogle Scholar
  116. Schmid PJ (2007) Nonmodal stability theory. Annu Rev Fluid Mech 39:129–162. ADSMathSciNetzbMATHCrossRefGoogle Scholar
  117. Schmid PJ, Henningson DS (2001) Stability and transition in shear flows. Springer, New YorkzbMATHCrossRefGoogle Scholar
  118. Schmid PJ, Reddy SC, Henningson DS (1996) Transition thresholds in boundary layer and channel flows. In: Gavrilakis S, Machiels L, Monkewitz PA (eds) Advances in turbulence VI. Kluwer Academic Publishers, Dordrecht, pp 381–384CrossRefGoogle Scholar
  119. Sekiya M, Miyama SM (1988) The stability of a differentially rotating cylinder of an incompressible perfect fluid. Mon Not R Astron Soc 234:107–114ADSzbMATHCrossRefGoogle Scholar
  120. Shen Y, Stone JM, Gardiner TA (2006) Three-dimensional compressible hydrodynamic simulations of vortices in disks. Astrophys J 653:513–524. ADSCrossRefGoogle Scholar
  121. Simon JB, Hawley JF, Beckwith K (2009) Nonlinear stability, hydrodynamical turbulence, and transport in disks. Astrophys J 690:974–997ADSCrossRefGoogle Scholar
  122. Squire J, Bhattacharjee A (2014a) Magnetorotational instability: nonmodal growth and the relationship of global modes to the shearing box. Astrophys J 797:15. CrossRefGoogle Scholar
  123. Squire J, Bhattacharjee A (2014b) Nonmodal growth of the magnetorotational instability. Phys Rev Lett 113(2):025006. ArXiv:1406.6582
  124. Stepanyants YA, Fabrikant AL (1989) Sov Phys Usp 32:783 (1989)Google Scholar
  125. Stone JM, Hawley JF, Gammie CF, Balbus SA (1996) Three-dimensional magnetohydrodynamical simulations of vertically stratified accretion disks. Astrophys J 463:656. ADSCrossRefGoogle Scholar
  126. Tassoul JL (1978) Theory of rotating stars. Princeton University Press, PrincetonGoogle Scholar
  127. Taylor GI (1936) Fluid friction between rotating cylinders. i. Torque measurements. R Soc Lond Proc Ser A 157:546–564. ADSCrossRefGoogle Scholar
  128. Tevzadze AG, Chagelishvili GD, Zahn JP, Chanishvili RG, Lominadze JG (2003) On hydrodynamic shear turbulence in stratified keplerian disks: transient growth of small-scale 3d vortex mode perturbations. Astron Astrophys 407:779–786. ADSCrossRefGoogle Scholar
  129. Tevzadze AG, Chagelishvili GD, Zahn JP (2008) Hydrodynamic stability and mode coupling in keplerian flows: local strato-rotational analysis. Astron Astrophys 478:9–15. ADSzbMATHCrossRefGoogle Scholar
  130. Tevzadze AG, Chagelishvili GD, Bodo G, Rossi P (2010) Linear coupling of modes in two-dimensional radially stratified astrophysical discs. Mon Not R Astron Soc 401:901–912. ADSCrossRefGoogle Scholar
  131. Trefethen LN, Trefethen AE, Reddy SC, Driscoll TA (1993) Hydrodynamic stability without eigenvalues. Science 261(5121):578–584. ADSMathSciNetzbMATHCrossRefGoogle Scholar
  132. Umurhan OM, Nemirovsky A, Regev O, Shaviv G (2006) Global axisymmetric dynamics of thin viscous accretion disks. Astron Astrophys 446:1–18. ADSzbMATHCrossRefGoogle Scholar
  133. Vilenkin NY, Gorin EA, Kostyuchenko AG, Krasnosel’skii MA, Maslov VP, Mityagin BS, Petunin YI, Rutitskii YB, Sobolev VI, Ya SV, Fadeev LD, Tsitlanadze ES (1972) Functional analysis. Wolters-Noordhoff, GroningenzbMATHGoogle Scholar
  134. Volponi F (2010) Linear transport in fully stratified discs. Mon Not R Astron Soc 406:551–667. ADSCrossRefGoogle Scholar
  135. Waleffe F (1995) Transition in shear flows. nonlinear normality versus non-normal linearity. Phys Fluids 7:3060–3066ADSMathSciNetzbMATHCrossRefGoogle Scholar
  136. Wendt G (1933) Potentialtheoretische behandlung des wehneltzylinders. Ann Phys 409:445–459. zbMATHCrossRefGoogle Scholar
  137. Yecko PA (2004) Accretion disk instability revisited. transient dynamics of rotating shear flow. Astron Astrophys 425:385–393. ADSzbMATHCrossRefGoogle Scholar
  138. Youdin AN, Kenyon SJ (2013) From disks to planets. Springer Science, Dordrecht. Google Scholar
  139. Zeldovich YB (1981) On the friction of fluids between rotating cylinders. R Soc Lond Proc Ser A 374:299–312. ADSMathSciNetCrossRefGoogle Scholar
  140. Zhuravlev VV, Razdoburdin DN (2014) A study of the transient dynamics of perturbations in keplerian discs using a variational approach. Mon Not R Astron Soc 442:870–890. ADSCrossRefGoogle Scholar
  141. Zhuravlev VV, Shakura NI (2007a) Dynamical instability of laminar axisymmetric flows of ideal incompressible fluid. Astron Lett 33:536ADSCrossRefGoogle Scholar
  142. Zhuravlev VV, Shakura NI (2007b) Dynamical instability of laminar axisymmetric flows of ideal fluid with stratification. Astron Lett 33:673. ADSCrossRefGoogle Scholar
  143. Zhuravlev VV, Shakura NI (2009) Temporal behavior of global perturbations in compressible axisymmetric flows with free boundaries. Astron Nachr 330(88):84–91. ADSzbMATHCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Sternberg Astronomical InstituteLomonosov Moscow State UniversityMoscowRussia

Personalised recommendations