The Instability of Laminar Axisymmetric Flows .

1. The Instability of Laminar Axisymmetric Flows. V. Zhuravlev, Moscow State University. • The search of hydrodynamical instabilities of stationary flows is classical way to predict theoretically the onset of turbulence in laboratory experiments (since the beginning of the last century) and in astrophysical disks (since early 1980’s). • Inspite of the evident difference between laboratory and accretion flows the basic theoretical ideas can be applied both to earth and space. • The well-known physical example that slightly resembles an astrophysical shear flow is Couette flow between coaxial cylinders (Taylor,1923). • For rotating perfect fluid Rayleigh (1916) discovered his famous criterion: is necessary for the stability.

2. An astrophysical disks seem to have rotational profiles with specific angular momentum increasing outwards – what sorts of instabilities can act in this case? • The global non-axisymmetric perturbationsin perfect fluidwith free boundaries have been studied by several authors mainly in 1980’s: 1. Papaloizou & Pringle (1984) – compressible torus with constant specific angular momentum in unstable. 2.Goldreich et al.(1986)– slender torus with a rotation law is unstable for and stable for independent on the polytropic index. 3. Blaes & Glatzel (1986) – differentially rotating incompressible cylinder with constant angular momentum is unstable. • Sekiya & Miyama (1988) – differentially rotating thin incompressible shell is unstable for 1.5<q<2.0. • Glatzel (1987) – differentially rotating compressible cylinder is unstable in wide range of parameters.

3. In present work the above sort of perturbations is reexamined in one-dimentional approach (rotating shear flow without vertical structure) for both incompressible and compressible fluid. • For incompressible fluid both rigid and free boundaries were considered. • Two kinds of rotation angular velocity profile were involved: 1. usual power law profile 2. keplerian law with sine deviation profile The difference is that the second one implies the absence of pressure (enthalpy) gradient at the boundaries.

4. Incompressible fluid.

5. Basic Equation. Equations for perfect incompressible fluid: Put Eulerian perturbations: Obtain the equations for perturbations: Cylindrical coordinates +

6. Basic Equation. It’s convenient to define the flow function: And take non-axisymmetric modes in the form:

7. Basic Equation. - Angular velocity profile that is determined by stationarity relation: where

8. Basic Equation. (Lord Rayleigh) At rigid boundaries: - necessary Rayleigh condition.

9. Basic Equation. - Complex quantities. - Need boundary conditions. Looking for

10. Boundary Conditions. Rigid boundaries: Free boundaries: Lagrangian perturbation: - Lagrangian displacement. Then:- ? (Tassoul,1978)

11. Boundary Conditions. The projection onto r:

12. Boundary conditions. Using the above relations between Fourier amplitudes we finally get at the boundaries: Rigid - Free -

13. The way of solution. 1.May exclude 1st derivative from the basic equation: , 2.Must solve equation: In real variables. 3. According to Lin’s rule (1945 ye.) the growing mode solution can be integrated along the real axis. Lin’s rule chooses the integration path bypass corotation point in complex plane so that inviscid solutions approximate viscid ones in the limit of low viscosity.

14. The way of solution. 3.Common way to solve linear boundary problems: to find general solution of differential system and substitute it into boundary conditions. 4. System: 5. General solution: . - FSS. 6. Apply the condition on in : 7. Then - is just a solution in 7. In: 8. In :where .

15. The way of solution. Finally, construct the algebraic system: Secular equation:

16. The dimensionless units.

17. Types of angular velocity profile considered. • Keplerian law with sinusoidal deviation. • Power law.

18. Types of angular velocity profile considered.

19. Verification: axisymmertic perturbations obeying the Rayleigh criterion. Rigid boundaries. Free boundaries.

20. The determinant curve for non-axisymmetric perturbations. Rigid boundaries. Free boundaries.

21. The radial profiles of flow function for growing non-axisymmetric modes. Rigid boundaries. Free boundaries.

22. 1. The dispersion curves for growing modes: rotation profile is keplerian law with sine deviation.

23. Increment depending on m: rigid boundaries: Increment depending on m: free boundaries:

24. Increment depending on m: rigid boundaries: Increment depending on m: free boundaries:

25. Growing modes maximal m as function of K & w. Dashed lines – rigid boundaries. Solid lines – free boundaries.

26. The instability limits.

27. 2. The dispersion curves for growing modes: rotation profile is a power law. No rigid boundaries here: necessary Rayleigh condition.

28. Increment depending on m. Increment depending on m.

29. Increment as a function of q. m=1. Increment for changing radial extent of the basic flow. m=1.

30. Compressible fluid

31. The compressibility obeys a polytropic equation of state: It’s convenient to introduce enthalpy: , So that: (Eulerian pert.) & , - the sound velocity.

32. Basic Equation. Equations for perfect compressible fluid: Put Eulerian perturbations: Obtain the equations for perturbations: Cylindrical coordinates +

33. Basic Equation. Take non-axisymmetric modes in the same form as it was previously:

34. Basic Equation. - Angular velocity profile that is determined by stationarity relation: where again

35. Basic Equation. Extra definitions: - epicyclic frequency - shifted perturbation frequency

36. Basic Equation. (Balbus, 2003) Still need boundary conditions.

37. Boundary Conditions. Consider only free boundaries. Again: - for keplerian law with sine deviation. - for power law.

38. The way of solution. Must solve equation with singularities at the boundaries. The solution close to the boundary points should be found in the form of generalized series expansion:

39. The way of solution. The recursive relation on the expansion coefficients: is to be determined from quadratic equation: Two values of correspond to lineary independent solutions of differential equation.

40. The way of solution. The values of corresponding to different rotation profiles: 1. power law: 2. keplerian law with sine deviation: Theregularity conditionat the boundaries:

41. The way of solution. The regularity condition and boundary condition. 1. power law: just choose 2. keplerian law with sine deviation: the numerical test shows that in wide range of parameteres at least one of ‘s doesn’t satisfy the regularity condition. If the another one does, corresponding solution automatically satisfies the boundary condition. The computation:

42. The way of solution. 1. Choose in each boundary point. 2. Produce an expansion in some points close to the boundaries (leave the finite number of series terms). 3. Integrate differential equation in real variables from these points up to some point in the middle of radial interval with two values of : 4. Finally get two lineary independent solution vectors from each side of middle point.

43. The way of solution. 5. The coupling of enthalpy and its derivative in the middle point: 6. The condition of non-trivial solution:

44. The results of compressible growing modes investigation.1. The power law rotation profile. (Glatzel,1987)

45. Increment for changing radial extent of the basic flow. m=1. Pattern speed for changing radial extent of the basic flow. m=1.

46. Increment for changing radial extent of the basic flow. m=10. Pattern speed for changing radial extent of the basic flow. m=10.

47. Increment for changing radial extent of the basic flow. m=10: two branches. Pattern speed for changing radial extent of the basic flow. m=10: two branches.

48. The radial profile of enthalpy for growing non-axisymmetric modes. For peak w: For w close to the peak:

49. The radial profile of radial velocity component.

50. Minimal radial width of unstable flow depending on m.