Top Banner
100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University
49

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

Dec 20, 2015

Download

Documents

Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Page 1: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.

100 years to the Orr mechanism of shear instability

Nili Harnik & Eyal Heifetz Tel Aviv University

Page 2: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.
Page 3: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.

The Orr mechanism (Kelvin, 1887)

shear

growth decay

Page 4: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.

• O-M is non-modal, applied to normal modes ?

• O-M conducted originally for shear 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)

Page 5: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.

Inviscid 2D barotropic shear flow:

Page 6: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.

Energy growth via Reynolds stress

Page 7: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.

Enstrophy growth and wave action

The linearized vorticity eq’ :

The wave action :

(Eliassen-Palm flux divergence)

Page 8: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.

For energy growth constant shear is enough

For enstrophy growth the vorticity gradient should change sign

Page 9: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.

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 ?

Page 10: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.

+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

qy>0

-q+q

- Waves can be maintained

q

qy>0

C

Potential vorticity

Page 11: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.

+q

The generation of Rossby waves

q

qy<0

+q-q

- Waves can be maintained

q

qy<0

C

If you flip the direction of PV gradient, the wave phase speed changes as well

Page 12: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.

Inviscid 2D barotropic shear flow:

Page 13: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.
Page 14: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.
Page 15: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.
Page 16: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.
Page 17: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.
Page 18: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.
Page 19: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.
Page 20: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.

The Kernel Rossby Wave approach

Page 21: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.

1

2

3

Divide the PV field into infinitesimal kernelsEach 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

Page 22: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.

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:

Page 23: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.

KRW practical representation

Page 24: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.

Energy versus Enstrophy growth –

the Orr mechanism

++ --

++ --

-- ++

-- ++

++ --

++ --

2y

++ ++

++++

---- -- --

++ ++ ++ ++

y

time

0 ++ --

++--

--++ -- ++

++--

++ --

y

Page 25: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.

The Orr mechanism – CRW description

0--++ ++

-- ++ -- 0U

-- ++ --

-- ++ -- 0

00U

C.R.W

C.R.W

Later…

---- ++

-- ++ ----++ ++

-- ++ --++++ --

++ -- ++

Page 26: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.

The Orr mechanism – KRW description

For the 2 CRW paradigm (Heifetz & Methven 2005) :

vorticity growth generalized Orr

Page 27: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.

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)

Page 28: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.

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.

Page 29: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.
Page 30: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.

• 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

Page 31: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.

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

Page 32: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.

Non-normal growth (Farrell, 1982)

eigenvectors are orthogonal

However:

is obtained

Page 33: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.
Page 34: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.

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 ?

Page 35: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.

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)

Page 36: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.

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

Page 37: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.

SVD and optimal growth :

Page 38: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.

Generalized Stability Theory (Farrell) :

Page 39: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.
Page 40: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.
Page 41: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.
Page 42: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.
Page 43: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.
Page 44: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.

&

Page 45: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.

Larger growth in energy than in enstrophy:

Page 46: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.

Larger growth in enstrophy than in energy:

Page 47: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.

Conclusions

Orr’s outstanding insight on the fundamental mechanism of shear instability is still valid !

(with some minor modifications)

Page 48: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.

What’s next ?

Establishing a “CRW-KRW” analogous description to gravity wave type.

Page 49: 100 years to the Orr mechanism of shear instability Nili Harnik & Eyal Heifetz Tel Aviv University.

Thank you !