1. P. Meunier Institut de Recherche sur les Phénomènes Hors-Equilibre, Marseille, France Collaborators: X. Riedinger, N. Boulanger, S. Le Dizès, P. Billant (LadhyX) Instabilities of a columnar vortex in a stratified fluid Workshop on “Rotating stratified turbulence and turbulence in the atmosphere and oceans”, Newton Institute, Dec. 8-12 2008

2. Motivations Shedding of tilted vortices in oceans Pawlak et al. (2003) Schecter and Montgomery (2006)

3. Outline Motivations Parameters and experimental set-up Dynamics of a tilted vortex: - 2D base flow - Tilt-induced instability Instability of a vertical vortex: - Radiating modes - Instability via resonance of modes Conclusion

4. Presentation of the problem Lamb-Oseen vortex:-Gaussian distribution of vorticity - Circulation  - Core size a - Infinite length … a stable stratification of density with Brunt-Väisälä frequency z  A stable vortex tilted of an angle  with respect to …  3 parameters: - Reynolds number Re=  /2  - Froude number F=  /2  Na 2 =0.2-5 - Angle of tilt  lengths dimensionalised by a time dimensionalised by  a 2 

5. Experimental set-up - Linear density profile created by the two tank method Variation of 20% of the density (salted water)  Buoyancy frequency: N=1.5-3 rad s -1 - Vortex generated by a flap rotated impulsively: Empirical law for the motion of the flap  Gaussian profile of vorticity  No stopping vortex Appropriate width of the plate: 10 or 30 cm - Measurement techniques:  50cm Axial vorticity by PIV Dye visualisationShadowgraph

6. Outline Motivations Parameters and experimental set-up Dynamics of a tilted vortex: - 2D base flow - Tilt-induced instability Instability of a vertical vortex: - Radiating modes - Instability via resonance of modes Conclusion

7. First part: stratified tilted vortex

8. Three stages A critical layerA 3D instabilityA turbulent mixing

9. The critical layer 1/F rcrc r c /a o : experiment —: theory  (r c )=N Appearance of a critical layer where the angular velocity  =u/r equals the buoyancy frequency (Cariteau and Flor 2001)

10. Inviscid base flow Euler equations + Boussinesq approx. Change of variables: r’ = r – z.sin(  ).e x U = U 0 +  U 1 with  small O(  ) : U 0 = r  (r) =  /2  r (1-exp(-r 2 /a 2 )) O(  ) : u 1 = v 1 = p 1 = 0 w 1 = r  . sin(  )  (  2 – N 2 )  1 = r   .cos(  )  (  2 – N 2 )  This solution breaks down at r c where  (r c ) = N ’ ’

11. Viscous critical layer Axial velocity in a transverse plane Theory Experiment Strong jet in the plane of the tilt Strong shear in the plane perpendicular to the tilt w ~  Re 1/3 and  ~  Re 2/3 Nonlinear terms vanish when  Re 1/3 <<1 - If F>1, a critical layer appears at r c where  (r c ) = 1/F - At small angles, inside the critical layer: U = u 0 +  Re 1/3 (u 1 e i  + c.c.) where: ~

12. Exp.(o) and theory (-) Viscous critical layer Axial velocity in a transverse plane Theory Strong jet in the plane of the tilt Strong shear in the plane perpendicular to the tilt w ~  Re 1/3 and  ~  Re 2/3 Nonlinear terms vanish when  Re 1/3 <<1 - If F>1, a critical layer appears at r c where  (r c ) = 1/F - At small angles, inside the critical layer: U = u 0 +  Re 1/3 (u 1 e i  + c.c.) where: ~

13. Viscous critical layer Axial velocity in a transverse plane Theory Strong jet in the plane of the tilt Strong shear in the plane perpendicular to the tilt w ~  Re 1/3 and  ~  Re 2/3 Nonlinear terms vanish when  Re 1/3 <<1 - If F>1, a critical layer appears at r c where  (r c ) = 1/F - At small angles, inside the critical layer: U = u 0 +  Re 1/3 (u 1 e i  + c.c.) where: ~ Exp.(o) and theory (-)

14. Instability of the tilted vortex in the plane  = 0.  t=1s, F= 3, Re = 720,  = 0.07 rad - sinusoïdal perturbation confined in a thin layer - corotating structures - no helical mode - identical to Cariteau and Flor instability Shadowgraph visualisations of the instability

15. Azimuthal vorticity in the plane  =  PIV measurements of the instability - periodic modulation of the vorticity sheet - corotating structures - looks like Kelvin- Hemholtz billows - strong growth rate compared to the advection

16. PIV measurements of the instability - periodic modulation of the vorticity sheets - van Karman alley - looks like the sinuous mode of the jet instability Azimuthal vorticity in the plane  =  Local instability: U = u 0 +  Re 1/3 u 1 + u vortex tilt perturbation

17. + terms in  Re 1/3 We neglect: viscous terms, non-Boussinesq terms, advection terms due to the change of variable, advection terms due to the tilt, terms due to radial variation of density and pressure… AND angular advection, Coriolis terms, vertical stratification Local stability analysis Rayleigh equation: with Growth rate: Wavenumber:

18. shear layer jet Growth rate for various Re Exp. (symbols) and theory (lines) Comparison theory-experiment Stability diagram Stable - Unstable for F>1 and  Re 2/3 large - Theory overestimates growth rate: competition between wavelengths, advection effects  Theory ~

19. Late stages Vortices are robust to the instability The core size increases if the c.l. is close to the vortex core, i.e. for F~1 Circulation of the vortex Square of the core size C.L.   =0 ○  =0.12, Re=2000, F~1.5 □  =0.12, Re=4200, F~3 Stable Unstable Unstable, F~1.5 Unstable, F~3

20. Outline Motivations Parameters and experimental set-up Dynamics of a tilted vortex: - 2D base flow - Tilt-induced instability Instability of a vertical vortex: - Radiating modes - Instability via resonance of modes Conclusion

21. Linear stability analysis of a vertical vortex - Base flow: The « frozen » stratified Gaussian vortex. -Perturbation equations: Linearisation of Navier-Stokes with Boussinesq approximation. Five unknowns: (u,v,w,p,ρ) - Normal modes decomposition: (u’,v’,w’,p’,ρ’) = (u,v,w,p,ρ)(r) e i(kz+mθ-ωt) k axial wavenumber m azimuthal wavenumber - Eigenvalue problem for ω with (k, m, Re, F) chosen Re(ω)=frequency Im(ω)=growth rate - Pseudo-spectral Chebyshev collocation code Dispersion relation and mode structures.

22. Boundary conditions Boundary conditions Trick: Resolution of the problem on a complex path r  r’ = r e i  e i  r  e i  r’ =e i  r cos(  ) e -  r sin(  )  Re(r’) Im(r’)  Numerical problem: Radiative modes extend to infinity The radiative modes become localized on the complex path They can be resolved with the pseudo spectral code  not described by Chebyshev polynomials

23. - most unstable mode: m=1 - unstable for all Reynolds numbers - Stabilisation at large F (non stratified) - For F<1 and large Re, k max ~1/F viscous stabilisation at small F Maximum growth rate contours F Re ii Stability results Stability results

24. - most unstable mode: m=1 - unstable for all Reynolds numbers - Stabilisation at large F (non stratified) - For F<1 and large Re, k max ~1/F viscous stabilisation at small F Maximum growth rate contours F Re x Density perturbation Structure of the most unstable mode F=0.9 Re=10 5 k=2.5  =0.055+i0.00206 y 2a - A displacement mode at the center - Radiative structures far from the center ii Stability results Stability results

25. F>1: Appearance of a critical layer x y Critical radius r c for:  -mV 0 (r c )/r c =  1/F Density perturbation (F=1.5, Re=10 5, k=1.4)

26. F>1: Appearance of a critical layer x y Critical radius r c for:  -mV 0 (r c )/r c =  1/F Density perturbation (F=1.5, Re=10 5, k=1.4) Decrease of the growth rate (Re=3000) - Radius of critical layer r c increases with F - Stabilisation increases with F F ii

27. F>2.58: A different instability mechanism Growth rate versus axial wavenumber (m=1) k - At moderate F, appearance of oscillations - At high F, bands of instability F=2.5 F=3 F=4 F=5 F=7 ii

28. F>2.58: A new instability Resonances with unstratified Kelvin modes k Radiative modes (low F)Unstratified Kelvin modes (large F) rr rr

29. m=1, F=8 Crossing of branches between modes leads to bands of instability F>2.58: A new instability Resonances with unstratified Kelvin modes k ii Radiative modes (low F)Mixed modes (intermediate F)Unstratified Kelvin modes (large F) rr k rr

30. Dye visualisation (F=1.15, Re=610) Accelerated 5 times 20 cm Experimental difficulties: - Small values of the growth rate - Creation of negative vorticity (stopping vortex)  risk of zigzag instability - Viscous effects:  variation of the core size - Finite size effects

31. Dye visualisation (F=1.15, Re=610) Accelerated 5 times 20 cm t = 120s /a = 7 th /a =3.7

32. Conclusions Conclusions Stratified tilted vortices have a critical layer when F>1, at  (r) = N, leading to strong jets and shears with velocity amplitude scaling as  Re 1/3 The critical layer leads to a 3D instability due to a Kelvin-Helmholtz or a jet instability with a growth rate  ~ 0.1  Re 2/3 N Tilted vortices remain but their core size is increased if F~1 until F<1 Vertical vortices have unstable radiative modes for F~1 For large Froude (F>2.58), radiative and non-stratified modes resonate and lead to an instability on narrow bands of wavenumbers

33. We neglect: angular advection, Coriolis effects, vertical stratification Local stability analysis Rayleigh equation