J ( J ) A re-examination of the inertial levels in verticallysheared rotating stratified flows

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A re-examination of the inertial levels in vertically sheared rotating stratified flows

F. Lott

LMD/CNRS, Ecole Normale Supérieure , Paris

C. Millet

Laboratoire de détection et de Géophysique, CEA-DAM Ile de France, Arpajon

J. Vanneste

Maxwell Institute, University of Edimburg

I Motivation: Emission of GWs from balanced flows

II Object : R and T of an IGWs propagating through critical levels III Interpretation: for large Richardson number

( J )

IV Triggering of very large disturbances via the valve effect at small

J

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Lott, Plougonven and Vanneste, JAS 2010 and 2013.

Orr mechanism' in rotating vertically stratified and shear flow : SET Up : a 3D (x,y,z) PV anomaly is advected, Coriolis f =cte,

BV freq N=cte, wind vertical shear ^=cte).

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A re-examination of the inertial levels in vertically sheared rotating stratified flows

I Motivation: Emission of GWs from balanced flows

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Question : Does the presence of the ground affect the results in terms of GWs emission, triggering of Sub-Synoptic instabilities?

Near the PV anomaly, can trigger non-geostrophic modes of baroclinic instabilities (next time!)

E

Ground

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Question : Does the presence of the ground affect the results in terms of GWs emission, or triggering of Sub-Synoptic instabilities?

Far from the PV anomaly, one needs to analyse the

Transmission (T) and Reflection ( R ) of the gravity waves Emitted (E).

E

Ground E

TE

RE

Ect if R

≠ 0

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The Booker and Bretherton (1967) result with constant wind shear



and constant Brunt Vaisala frequency

N .

Stable case:

J=N

²

/ 

²

^>0.25

Taylor Goldstein Eq. In Hydrostatic + Boussinesq:

d²W

dz² J1²

z² W=0

w 'x , y , z , t=Wze^i^kx^ly^ Critical level in z=0:

où =l/k

Solution above, upward propagating wave :

W  z=z^1/^2i^ where=



^J^¹^²^−^0.25

Analytic continuation below z=0:

Wz=−i∣z∣¹^/2ⁱ^e^{ }

This is again an upward propagating waves, there is no reflected wave:

− 

Exponential factor

z=∣z∣e⁻ⁱ^

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Case with rotation and

f

=cte (Jones 1967, Yamanaka et Tanaka 1984, YT84)

The 0-PV condition for a monochromatic disturbance gives



¹^−² ²



^dd²^W² −



²³−2i

²



^W⁻



^J^1² ²^2i

³



^W⁼⁰

Where=k z f

Three critical levels, in

Fortunately, the BB critical level



=0 corresponds to a regular singularity.

This not the case of the other two « Jones » Cls where the wave intrinsic phase speed

k Λ z

^equal

(+-) the Coriolis frequency

f

=−1, 0,1

II Object: R & T of an IGWs propagating through critical levels

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W ^F

₁

^{, F}

₂

^{, F}

₃

^...

>>1: 1/2+i(upward GW)

: C|^1/2-i+D|^1/2+i

(Downward GW+Upward GW)

>1: (1+^-i^{ii}F1(1-^-2)

1: (1+^-i(A F2(²)+B ^F2(²))

1: (|^-i

[

^C^{ii}^F4^(1-^-2⁾⁺

D^{ii}F5(1-^-2)

]



Extremely involved

calculations yields:

R = C

=0, T = 1

=e

^{− } Exactly as in BB67!

'= A' ' Be^{− }

'= A' ' B

 A−' ' B= ' ' ' C' D e^{ }

 A−' ' B=' ' ' C' D Connection in =1

Connection in =1

The

α

^and

β

's are products and fractions of

Γ

^-functions

which complex arguments are combinations of

ν

^and

µ

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The exact solution

W

involves Hypergeometric functions (

F

₁

, F

₂

, F

₃

...

⁾

>>1: 1/2+i(upward GW)

: C|^1/2-i+D|^1/2+i

(Downward GW+Upward GW)

>1: (1+^-i^{ii}F1(1-^-2)

1: (1+^-i(A F2(²)+B ^F2(²))

1: (|^-i

[

^C^{ii}^F4^(1-^-2⁾⁺

D^{ii}F5(1-^-2)

]



II Object: R & T of an IGWs propagating through critical levels

Extremely involved

calculations yields:

R = C

D =0, T = 1

D =e

^{− } Exactly as in BB67!

'= A' ' Be^{− }

'= A' ' B

 A−' ' B= ' ' ' C' D e^{ }

 A−' ' B=' ' ' C' D Connection in =1

Connection in =1

The valve effect (Grimshaw, 1977) is an amplification of the incoming wave for =l/k>0

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Mathematical explanation for R and T as in BB67 (Jones 1967)



¹^−^² ²



^dd²^W² −



²³−2i

²



^W⁻



^J^^1^² ²^^²^ⁱ³^



^W⁼⁰

The 0-PV equation

For large

 ^d

2W

d² J1²

² W=0

The Taylor Goldstein Eq.

Explanation :

as between the path near along the real axis (vertical) and a path at

large distance from the critical levels there is no singularities

the solutions in the far field are the same.

Physicallly : How explain that

attenuations in e^ (so dependant of the orientation  and independant of J) becomes attenuations in e^ (strongly dependant on J) ?

To answer, we propose an

approximate (WKB) solution along a path

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III Interpretation: for large Richardson number

(J)

WKB expansion for large _W_=



^W0⁻¹W₁...



^e^^∫^^ ^=±

1

^1−²

W₀=∣−1∣⁻

1 4i

2

∣1∣

1 4i

2

W^III=iW₀ e^{− } eⁱ^log^^^^²⁻¹^

W ^II=1i

2 W₀ e^{ } e^−^asin^

W ^I=W ₀ eⁱ^^log^^∣∣^^²⁻¹^

Valve effect

decaying solution

~Quasi Geostrophic solution

=



^J ^^1²^

The value of the evanescent solution in =+1 is T,the transmission coefficient

ie^{− }¹^/^2i^

~

Upward GW

∣∣^1/^2i^

~

Upward GW

≈∞

≈−∞

^⁻¹e^{− }

≈0

~

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Verification: Exact versus WKB solution,

J

^=5,



^=0.1

Real (solid) and Imag (dashed) of W

ξ

2 0 -2 -4 -6 -8 -10

0 -0.2 -0.4 -0.6 -0.8 -1

ξ

Zoom

Lower Inertial level

Decaying solution Propagating

solution

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IV Triggereing of very large disturbances via the valve effect when

(J<1)

When J<1, WKB theory tells that the decay of the evanescent solution can largely be equilibrated by the valve effect when ν>>1:

The solution between the inertial layers becomes very large and is no longer decaying (the critical levels are no longer near turning points)

The two solutions between the inertial layers have now comparable amplitudes and can sustained substantial EP fluxes (non-propagating solutions need to be in pairs to sustain momentum fluxes)

e^{−  }≈e^^¹⁻^^J^

20

10

0

-10

-20

ξ ξ

Exact and EP-flux

J

^=0.5,



⁼⁵

Zoom

-1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5

Negative EP flux due to the incident wave

Very large positive EP flux due to the

interaction

between two solutions Very small negative EP flux due to the transmitted wave

1.5

0

-1.5

Real (solid) and Imag (dashed) of W EP Flux in blue

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inertial layers. This will inevitably restore the mean wind shear to returns to J>1 conditions (e.g. inertially stable conditions).

Very small incoming disturbances will induce very large in ∂_z F^z

J ( J ) A re-examination of the inertial levels in verticallysheared rotating stratified flows

A re-examination of the inertial levels in vertically sheared rotating stratified flows

( J )

J

≠ 0



N .

J=N

/ 

>0.25



f















k Λ z

f

=−1, 0,1

W F

, F

, F

...

[

]



R = C

=0, T = 1

=e

α

β

Γ

ν

µ

W

F

, F

, F

...

[

]



R = C

D =0, T = 1

D =e













(J)







J



ξ

ξ

(J<1)

ξ ξ

J



=±1

^>0.25

W ^F

^{, F}

^{, F}

^...