J ( J ) « Gain » of Balance and Critical Level Absorption for Inertio Gravity Waves

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« Gain » of Balance and Critical Level Absorption for Inertio Gravity Waves

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.

Gain of balance and critical level absorption for inertio gravity waves

I Motivation: Emission of GWs from balanced flows

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

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

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

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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^ where =



^J ^1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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Case with rotation and f=cte (Jones 1967, Yamanaka et Tanaka 1984, YT84)



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 regular.

The two inertial levels (ILs) in are irregular

(intrinsic phase speed k_Λz equal (+-) f )

=− 1, 0,1

=±1

The 0-PV condition for a monochromatic disturbance gives

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The exact solution

W

involves Hypergeometric functions (F₁, F₂, F₃...⁾

>>1: 1/2+i(upward GW)



Connection in =-1 Connection in =1

C 1

>1: F₁(1-^-2)

1: A F₂(²)+B F₃(²))

1: CF₄(1-^-2) +D F₅(1-^-2)

: C|^1/2-i+D|^1/2+i

(Downward GW+Upward GW) Two (often) non-propagating

solutions

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The exact solution

W

involves Hypergeometric functions (F₁, F₂, F₃...⁾

>>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)



'=  A' ' B e^{− }

 '= A ' ' 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

Extremely involved

R C

0, T 1

=e

^{− }

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III Interpretation: for large Richardson number (J)

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 Becomes

for large



d²W

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 same (Jones 1967)

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

ν

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WKB expansion for large W =



^W0⁻¹W₁...



^e^^∫^ ^

=± 1



^1−² ^{, W}⁰^=^∣^−¹^∣

−1 4i

2∣_1∣⁻

1 4−i

2 , W₁=...

=



^J ^^1²^

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

^W0⁻¹W₁...



^e^^∫^ ^

=± 1

^1−² ^{, W}⁰^=^∣^−¹^∣

−1 4i

2∣_1∣⁻

1 4−i

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^^∫^ ^

=± 1

^1−² ^{, W}⁰^=^∣^−¹^∣

−1 4i

2∣_1∣⁻

1 4−i

2, W₁=...

W^III=i W₀ e^− eⁱ^^log^^^^²⁻¹^

W^II=1i

2 W₀ e^{ /}² e^−



^asin^{ }^2



W^I=W₀ eⁱ^^log^^∣∣^^²⁻¹^

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^^∫^ ^

=± 1

^1−² ^{, W}⁰^=^∣^−¹^∣

−1 4i

2∣_1∣⁻

1 4−i

2, W₁=...

W^III=i W₀ e^− eⁱ^^log^^^^²⁻¹^

W^II=1i

2 W₀ e^{ /}² e^−



^asin^{ }^2



W^I=W 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, The decay of the solution can largely be dominanted by the valve effect when ν>>1:

The solutions between the inertial layers become very large

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 solution and EP-flux

J^=0.5,⁼²

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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IV Triggereing of very large disturbances via the valve effect when (J<1)

When J<1,Very large EP flux between the inertial layers

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

EP flux=100 J=1 Flux = 1

ming)

ν =l/k

Log 10 of EP-Flux between the ILs

ν=0

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Gain of Balance and Critical Level Absorption For Inertio Gravity Waves

Exact results for all J : R and T as in the non-rotating case Inertially stable case (J>1), interpretation of R & T:

The incoming gravity wave becomes evanescent (« gain of balance ») 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.

Inertially unstable case ((J<1) and valve effect:

the valve effect can yield very large disturbances between the inertial levels.

There, the disturbance is made of two solutions that interact and