1. 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University

3. The Orr mechanism (Kelvin, 1887) shear growthdecay

4. O-M is non-modal, applied to normal modes ? O-M conducted originally for sh ear flow with zero mean vorticity gradient, relevant to non zero gradients? How O-M is related to the necessary conditions for instability? : 1) Rayleigh inflection point (1880) - mean vorticity gradient should change sign 2) The Fjortoft condition (1953) – Positive correlation between the mean wind and the mean vorticity gradient Revisiting the Orr mechanism (O-M)

5. Inviscid 2D barotropic shear flow:

6. Energy growth via Reynolds stress

7. Enstrophy growth and wave action The linearized vorticity eq’ : The wave action : (Eliassen-Palm flux divergence)

8. For energy growth constant shear is enough For enstrophy growth the vorticity gradient should change sign

9. well… mathematically : 0 without shear (U= const) no energy growth but why the mean vorticity gradient does not play any direct role in energy growth ?

10. +q Essence of action at a distance : Basic PV action at a distance: positive PV - cyclonic flow If we have a background PV gradient: -Creation of new PV anomalies by advection q y >0 -q +q - Waves can be maintained q q y >0 C Potential vorticity

11. +q The generation of Rossby waves q q y <0 +q -q - Waves can be maintained q q y <0 C If you flip the direction of PV gradient, the wave phase speed changes as well

12. Inviscid 2D barotropic shear flow:

20. The Kernel Rossby Wave approach

21. 1 2 3 Divide the PV field into infinitesimal kernels Each has an associated velocity field The velocity is the sum over contributions from all kernels Look at how the kernel amplitude and phase positions change with time

22. And the math looks something like this: Each KRW induces a meridional wind everywhere – a Green Function approach The total velocity field is therefore: From the PV equation, obtain the KRW evolution equations for the amplitude and phase of the PV kernels the basic evolution dynamics is the same as the 2-CRWs… Growth: Propagation:

23. KRW practical representation

24. Energy versus Enstrophy growth – the Orr mechanism + - - + - + + - + + - + - - + - + + - + - + + + + - - - - + + + + time + - + - - + - + + - + -

25. The Orr mechanism – CRW description-++ -+-- +-- + - C.R.W Later… - -+ -+- -++ -+- + +- +-+

26. The Orr mechanism – KRW description For the 2 CRW paradigm (Heifetz & Methven 2005) : vorticity growth generalized Orr

27. The Orr mechanism – CRW description vorticity growth: generalized Orr: classic Orr (shear) counter prop’ CRW inter’ The shear is the only source for instantaneous energy growth !!! (whether or not a mean PV gradient exists)

28. The Orr mechanism – KRW description For a continuous set of KRWs : which is equivalent to the common expression : Reynolds stress Orr is a non-modal (transient) mechanism, but acts as the only energy source in NMs as well.

30. Could be interpreted in 2 equivalent forms: a)All KRWs are phase-locked, i.e. the shear never succeeds to form relative KRW motion. Energy growth is proportional to the shear, however resulted from the KRW amplitude growth due to mean PV advection. b) The Orr mechanism operates but continuously re-stoked by the KRW interaction – Lindzen view The Orr mechanism in normal modes

31. If the matrix A is non-normal (AA = AA) Growth can be found even if all eigenvalues are negative & Rapid transient growth can be much larger than the largest exopnential eigenvalue The Orr mechanism and optimal non- normal transient growth For a given linearized system : TT

32. Non-normal growth (Farrell, 1982) eigenvectors are orthogonal However: is obtained

34. Singular Value Decomposition (SVD) Shear flows are generally highly non- normal systems Can we identify for a given target time t : a) what is the initial optimal perturbation? b) by how much it will grow ? c) what would be its final structure ?

35. The SVD recipe : Let’s seek for a matrix M : 2 sets of vectors u and v & one set of scalars which satisfy : Since both and Hermitian both and are orthonormal sets and are real (where if M is normal then U = V)

36. The SVD recipe (cont) : - the eigenvalues of both and unitarian matrices - the (single) singular values of M Taking real and positive, so that & - the two set of singular vectors of M - SVD

37. SVD and optimal growth :

38. Generalized Stability Theory (Farrell) :

44. &

45. Larger growth in energy than in enstrophy:

46. Larger growth in enstrophy than in energy:

47. Conclusions Orr’s outstanding insight on the fundamental mechanism of shear instability is still valid ! (with some minor modifications)

48. What’s next ? Establishing a “CRW-KRW” analogous description to gravity wave type.

49. Thank you !