Harish Dixit and Rama Govindarajan With Anubhab Roy and Ganesh Subramanian

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

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

3. y 2 1 Light ρ(y)‏ ρ Heavy `Perpendicular’ density stratification: baroclinic torque (+ centrifugal + other non-Boussinesq effects) 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. Initial condition: Point vortex at a density jump Light point vortex Heavy

7. A single vortex and a density interface Inviscid Homogenised within the yellow patch, if Pe finite 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 rh and rs Consider one such jump, assume circular

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

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

12. r Rankine vortex with density jumps at rjs spaced at r3 Kelvin (1880): neutral modes at r=a for a Rankine vortex

13. Vorticity and density: Heaviside functions

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

15. Multiple (7) jumps m = 5

17. Single jump Step vs smooth density change rj = 2 rc, =0.1

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

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

20. Simulations: spectral, interfaces thin tanh, up to 15362 periodic b.c. Light Heavy Non-Boussinesq equations

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

22. 9.5 t=12.7

23. Vorticity time=0 time=12.7

24. 1.6 Non-Boussinesq A=0.2 3.82

25. t=4.5

26. t=5.1

27. 5.73

28. Notice vorticity contours t=3.2

29. t=4.5

30. 5.1

31. 5.73

32. Viscous simulations: same instability A=0.12, t = 7.5Г/rc2 λ ~ 2.5ld (λstab ~ 4ld)

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 ur eigenfunction with the jump location: rc = 0.1, m = 2

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

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

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 Pe1/3 • Bajer et al. (JFM, 2001) showed that the same holds true for the vorticity field, viz. thomo ~ Re1/3 • 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. Navier-Stokes: Boussinesq approximation , radial gravity Density evolution Continuity