1. Harish Dixit and Rama Govindarajan With Anubhab Roy and Ganesh Subramanian Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore September 2008 Instabilities in variable-property flows and the continuous spectrum An aggressive ‘passive’ scalar

2. Re=3000, unstratified Building block for inverse cascade

3. `Perpendicular’ density stratification: baroclinic torque (+ centrifugal + other non-Boussinesq effects) Heavy Light 1 2 ρ(y) ‏ y ρ Brandt and Nomura, JFM (2007): stratification upto Fr=2, Boussinesq Stratification aids merger at Re > 2000 At lower diffusivities, larger stratifications?

4. Re=3000, Pe=30000, Fr (pair) = 1

5. Large scale overturning: a separate story Why does the breakdown happen? Consider one vortex in a (sharp) density gradient In 2D, no gravity

6. Heavy Light point vortex Initial condition: Point vortex at a density jump

7. Homogenised within the yellow patch, if Pe finite A single vortex and a density interface Inviscid The locus seen is not a streamline!

9. Scaling Density is homogenised for e.g. Rhines and Young (1983) Flohr and Vassilicos (1997) (different from Moore & Saffman 1975) When Pe >>> 1, many density jumps between r h and r s Consider one such jump, assume circular

10. Linearly unstable when heavy inside light, Rayleigh-Taylor Vortex sheet of strength Rotates at m times angular velocity of mean flow Point vortex, circular density jump Radial gravity Non-Boussinesq, centrifugal Non-Boussinesq: e.g. Turner, 1957, Sipp et al., Joly et al., JFM 2005

11. m = 2 Vortex sheet at r j In unstratified case: a continuous spectrum of `non-Kelvin’ modes

12. Rankine vortex with density jumps at r j s spaced at r 3 r Kelvin (1880): neutral modes at r=a for a Rankine vortex

13. Vorticity and density: Heaviside functions

14. For j jumps: 2j+2 boundary conditions u r and pressure continuous at jumps and r c Green’s function, integrating across jumps For non-Boussinesq case: For one density jump

15. m = 5 Multiple (7) jumps

17. r j = 2 r c,  =0.1 Single jump Step vs smooth density change

18. Single jump: radial gravity (blue), non-Boussinesq (red) m = 2,  = 0.01

19. (circular jump: pressure balances, but) Lituus spiral Dominant effect, small   non-dimensional) KH instability at positive and negative jump growing faster than exponentially  In the basic flow

20. Simulations: spectral, interfaces thin tanh, up to 1536 2 periodic b.c. Heavy Light Non-Boussinesq equations

21. t=0 3.18 6.4 t=1.59 Boussinesq, g=0, density is a passive scalar

22. 9.5 t=12.7

23. time=0 time=12.7 Vorticity

24. 1.6 3.82 Non-Boussinesq A=0.2

25. t=4.5

26. t=5.1

27. 5.73

28. t=3.2 Notice vorticity contours

29. t=4.5

30. 5.1

31. 5.73

32. A=0.12, t = 7.5Г/r c 2 λ ~ 2.5l d (λ stab ~ 4l d ) Viscous simulations: same instability

33. Re = 8000, Pe = 80000, rho1 = 0.9, rho2 = 1.1 (tanh interface), Circulation=0.8, thickness of the interface = 0.02, rc = 0.1, time = 2.5, N=1024 points Initial condition: Gaussian vortex at a tanh interface

34. Conclusions: Co-existing instabilities: `forward cascade’ unstable wins Beware of Boussinesq, even at small A What does this do to 2D turbulence?

35. Single jump: Boussinesq (blue), non-Boussinesq (red) m = 20,  = 0.1

36. Variation of u r eigenfunction with the jump location: r c = 0.1, m = 2

37. Effect of large density differences m = 2,  = 1

38. Reynolds number: Inertial / Viscous forces For inviscid flow, no diffusion of density, Re, Pe infinite 2D simulations of Harish: Boussinesq approximation Peclet number: Inertial / Diffusive Froude number: Inertial / Buoyancy (1/Fr = T I N)‏

39. Is the flow unstable? Consider radially outward gravity

40. m

41. m

45. Comparison: Boussinesq (blue), non-Boussinesq (red) m = 2,  = 0.1

46. Governing stability PDE’s:

47. Component equations Continuity equations Density evolution equations

48. Background literature:  Studying discontinuities of vorticity / densities or any passive scalar was initiated by Saffman who studies a random distribution of vortices as a model for 2D turbulence and predicted a k -4 spectrum  Bassom and Gilbert (JFM, 1988) studied spiral structures of vorticity and predicted that the spectrum lies between k -3 and k -4  Pullin, Buntine and Saffman (Phys. Fluid, 1994) verify the Lundgren’s model of turbulence based on vorticity spiral

49.  Batchelor (JFM, 1956) argued that at very large Reynolds number, the vorticity field inside closed streamlines evolves towards a constant value.  Rhines and Young (JFM, 1983) showed that any sharp gradients of a passive scalar will be homogenized at Pe 1/3  Bajer et al. (JFM, 2001) showed that the same holds true for the vorticity field, viz. t homo ~ Re 1/3  Flohr and Vassilicos (JFM, 1997) showed that a spiral structure unique among the range of vorticity distribution. Closed spaced spiral lead to an accelerated diffusion where Dk is the Kolmogorov capacity of the spiral

50. Density evolution Continuity Navier-Stokes: Boussinesq approximation, radial gravity

51. Navier-Stokes: Non-Boussinesq equations. For Boussinesq approxmiation,  =  0 Density evolution equation Continuity equation: valid for very high D

52. Linear stability: mean + small perturbation, e.g.

53. First: planar approximation, Rayleigh-Taylor instability When U 1 = U 2, always unstable if ρ heavy > ρ light If D=0, growth rate Using kinematic conditions and continuity of pressure at the interface