100 years to the Orr mechanism of shear instability

Nili Harnik & Eyal Heifetz Tel Aviv University

The Orr mechanism (Kelvin, 1887)

shear

growth decay

• 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)

Inviscid 2D barotropic shear flow:

Energy growth via Reynolds stress

Enstrophy growth and wave action

The linearized vorticity eq’ :

The wave action :

(Eliassen-Palm flux divergence)

For energy growth constant shear is enough

For enstrophy growth the vorticity gradient should change sign

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 ?

+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

+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

Inviscid 2D barotropic shear flow:

The Kernel Rossby Wave approach

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

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:

KRW practical representation

Energy versus Enstrophy growth –

the Orr mechanism

++ --

++ --

-- ++

-- ++

++ --

++ --

2y

++ ++

++++

---- -- --

++ ++ ++ ++

y

time

0 ++ --

++--

--++ -- ++

++--

++ --

y

The Orr mechanism – CRW description

0--++ ++

-- ++ -- 0U

-- ++ --

-- ++ -- 0

00U

C.R.W

C.R.W

Later…

---- ++

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

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

++ -- ++

The Orr mechanism – KRW description

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

vorticity growth generalized Orr

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)

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.

• 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

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

Non-normal growth (Farrell, 1982)

eigenvectors are orthogonal

However:

is obtained

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 ?

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)

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

SVD and optimal growth :

Generalized Stability Theory (Farrell) :

&

Larger growth in energy than in enstrophy:

Larger growth in enstrophy than in energy:

Conclusions

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

(with some minor modifications)

What’s next ?

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

Thank you !