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

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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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E

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E

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E

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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?

E

Ground E

TE

RE

Ect if R ≠0

Far from the PV anomaly, one needs to analyse the

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

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No rotation f=0 cases with constant wind shear and constant buoyancy frequency N. Stable case: J=N²/²^>0.25

Taylor Goldstein Eq. In Hydrostatic + Boussinesq:

d²W

dz² J1²

z² W=0

w ' (x , y , z , t)=W (z)eⁱ^(kx⁺^ly⁾

A doppler shift can always place the critical level in z=0:

where =l/k

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No rotation f=0 cases with constant wind shear and constant buoyancy frequency N. Stable case: J=N²/²^>0.25

Taylor Goldstein Eq. In Hydrostatic + Boussinesq:

d²W

dz² J1²

z² W=0

w ' (x , y , z , t)=W (z)eⁱ^(kx⁺^ly⁾

A doppler shift can always place the critical level in z=0:

where =l/k

Upward propagating wave above:

W(z)=z^1/²⁺ⁱ^μ where μ =

√

^J^{(1+ ν}²^)−0.25

Analytic continuation below z=0:

Wz=−i∣z∣¹^/2ⁱ^ e^{ }

R = 0, T = e

^{− }

Exponential factor

z=∣z∣e⁻ⁱ^

This is an upward wave, there is no reflected wave:

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f



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

The critical level =0 is a regular singularity.

The two CLs in are irregular

and are where the wave intrinsic phase speed

k_Λz equal(+-) the Coriolis frequency f

=− 1, 0,1

ξ=±1

The 0-PV condition for a monochromatic disturbance gives

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W ^F₁^{, F}₂^{, F}₃^...

>>1: 1/2+i(upward GW)

: C|^1/2-i+D|^1/2+i

(Downward GW+Upward GW)



α'=(α A+α' ' B) e^{− ν π}

β'=β A+β' ' B

 A− ' ' B= ' ' ' C' D e^{ }

 A−' ' B=' ' ' C' D

Connection in =-1 Connection in =1

Extremely involved

calculations yields:

R = C

D =0, T = 1

D =e

^{− } Exactly as with f=0 ^!

The α^andβ's are products and fractions of Γ^-functions

>1: F₁(1-^-2)

1: A F₂(²)+B F₃(²))

1: CF₄(1-^-2) +D F₅(1-^-2)

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W ^F₁^{, F}₂^{, F}₃^...

>>1: 1/2+i(upward GW)

: C|^1/2-i+D|^1/2+i

(Downward GW+Upward GW)

>1: F₁(1-^-2)

1: A F₂(²)+B F₃(²))

1: CF₄(1-^-2) +D F₅(1-^-2)



R = C

D =0, T = 1

D =e

^{− }

α'=(α A+α' ' B) e^{− ν π}

β'=β A+β' ' B

 A− ' ' B= ' ' ' C' D e^{ }

 A−' ' B=' ' ' C' D

Connection in =-1 Connection in =1

The valve effect is an amplification of the incoming wave for =l/k>0

It is attenuated back at the higher inertial level Exactly as with f=0 ^!

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Mathematical explanation for R and T insensitive to rotation



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.

As between the path near along the real axis and a path at

large distance from the critical levels there is no singularities

the solutions in the far field are the (Jones 1967)same

But how can we explain that attenuations in e^ (so dependent of the orientation  and independent of J) becomes attenuations in e^ (strongly

dependent on J) ?

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WKB expansion for large W =



^W0⁻¹W₁...



^e^^∫ ^

φ =± 1

√

¹^−ξ² ^{, W}⁰^=ξ^∣^ξ−1^∣

−1 4+i ν

2∣_ξ+₁∣⁻

1 4−i ν

2 , W₁=...

=



^J ^^1²^

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

^W0⁻¹W₁...



^e^^∫ ^

φ =± 1

√¹^−ξ² ^{, W}⁰^=ξ^∣^ξ−1^∣

−1 4+iν

2∣_ξ+₁∣⁻

1 4−iν

2 , W₁=...

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^

=



^J ^^1²^

ie^{− }¹^/^2i^

~

∣∣^1/^2i^

~

Upward GW

≈∞

^⁻¹e^{− }

≈0

~

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

^W0⁻¹W₁...



^e^^∫ ^

φ =± 1

√¹^−ξ² ^{, W}⁰^=ξ^∣^ξ−1^∣

−1 4+iν

2∣_ξ+₁∣⁻

1 4−iν

2 , W₁=...

W^III=iW₀ e^{−μ π} eⁱ^μ^log⁽^ξ+^√^ξ²⁻¹⁾

W ^II=1+i

2 W ₀ e^{+ν π} e^−μ(^{asin(ξ)+ π}2)

W ^I=W₀ e^iμ^log⁽^∣^ξ^∣⁺^√^ξ²⁻¹⁾

Valve effect

decaying solution

~Quasi Geostrophic solution

=



^J ^^1²^

ie^{− }¹^/^2i^

~

∣∣^1/^2i^

~

Upward GW

≈∞

≈−∞

^⁻¹e^{− }

≈0

~

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

^W0⁻¹W₁...



^e^^∫ ^

ϕ=± 1

√^1−ξ²^{, W}⁰^=ξ^∣^ξ−1^∣

−1 4+i ν

2∣_ξ+₁∣⁻

1 4−i ν

2, W₁=...

W^III=iW₀ e^{−μ π} eⁱ^μ^log⁽^ξ+^√^ξ²⁻¹⁾

W ^II=1+i

2 W ₀ e^{+ν π} e^−μ(^{asin(ξ)+ π}2)

W ^I=W e^iμ^log⁽^∣^ξ^∣⁺^√^ξ²⁻¹⁾

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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equilibrated by the valve effect when ν>>1:

The solutions between the inertial layers become very large and are no longer

decaying away from the inertial levels(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 pair to sustain momentum fluxes)

e^{−  }≈e^^¹⁻^^J^

20

10

0

-10

-20

ξ ξ

Exact and EP-flux

J^=0.5,⁼⁵

Zoom

Negative EP flux 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

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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 =±1

J=1

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The incoming gravity wave becomes evanescent at the lowest

inertial level, the disturbance amplitude at the upper inertial level is then exponentially small, and this represents the absorptive property of the

shear layer.

This is somehow reminiscent of a tunelling effect, but no reflections are needed at the turning altitudes, because the presence of critical levels allows jumps in

EP fluxes.

In inertially unstable conditions (J<1), the valve effect can yield very large disturbances between the inertial levels.

There, the solution is made of two disturbances that decay moderatly away from the inertial levels. The interaction between these disturbances

yields very large EP fluxes between the inertial levels that will bring back the flow to stable conditions.

Very small effect on GW emission by PV anomalies. One needs to place