Linear and Weakly Nonlinear Energetics of Global Nonhydrostatic Normal Modes

Linear and Weakly Nonlinear Energetics of Global

Nonhydrostatic Normal Modes

Carlos Frederico M. Raupp

*

_{and André Teruya}

Department of Atmospheric Sciences

University of Sao Paulo, Sao Paulo/Brazil

MODES Workshop – Boulder / CO - USA – August 26

th

_{to 28}

th

_{, 2015.}

Financial Support: FAPESP (São Paulo Research Foundation - Grant 09/11643-4)

Introduction



Conventional coarse resolution global circulation atmospheric models (AGCMs)

neglect the vertical acceleration to prevent computational constraints related to

stability of numerical solution in explicit numerical schemes (hydrostatic assumption)

→ Hydrostatic AGCMs;



_{The increase of computer speed and memory, as well as the development of}

massively parallel computing techniques, has allowed the development of global

nonhydrostatic modeling.



_{Thus, with the increasing development of global nonhydrostatic AGCMs, it is}

important to understand the dynamics of these models in a theoretical point of view

→ For this purpose one needs to analyze the normal modes of global nonhydrostatic

atmospheric models;



_{Normal modes → small amplitude oscillations around a background state at rest}

and characterized by a stable stratification → eigensolutions of linearized PDEs;

References on Nonhydrostatic Normal Modes



_{Linear theory of nonhydrostatic normal modes:}

(i) Kasahara and Qian (MWR 2000) ⇒ “Normal modes of a Global Nonhydrostatic

Atmospheric Model”;

(ii) Qian and Kasahara ( Pure Appl. Geophys., 2003) ⇒ “Nonhydrostatic Atmospheric

Normal Modes on Beta Planes”;

(iii) Kasahara (J. Meteor. Soc. Japan, 2003) ⇒ “On the Nonhydrostatic Atmospheric

Models with the Inclusion of the Horizontal Component of the Earth’s Angular

Velocity” (non-traditional Coriolis terms);

(iv) Kasahara (JAS 2003) ⇒ “The Roles of the Horizontal Component of the Earth’s

Angular Velocity in Nonhydrostatic Linear Models”;

(v) Kasahara (NCAR Report 2003) ⇒ “ Free Oscillations of Deep Nonhydrostatic Global

Atmospheres: Theory and a Test of Numerical Schemes”;

Introduction

Goal of this Study

(i) First we further analyze the energetics of the linear eigenmodes of the shallow

global nonhydrostatic model presented by Kasahara and Qian (2000);

(Ii) Then we extend the theory of global nonhydrostatic normal modes by accounting

for the effect of nonlinearity.

Model and Governing Equations

_{Model: shallow nonhydrostatic fluid over a rotating sphere of radius a;}

Traditional Approximation: r = a + z ≈ a, where a = 6370Km (Earth’s radius) and z is the height above the earth’s surface; ∂ / ∂r ≈ ∂ / ∂z

 Governing Equations: λ → longitude;

ϕ→ latitude;

f = 2Ω sinϕ → Coriolis parameter; γ = C_p / C_v ;

g → gravity acceleration;

= (u,v) → horizontal wind field; p → pressure field;

T → Temperature field; ρ→ density field;

w → vertical component of wind field;

; ; λ ϕ ρ ϕ ∂ ∂ − =       + − p a a u f Dt Du cos 1 v tan (1a) ϕ ρ ϕ ∂ ∂ − =       + + p a a u f Dt D 1 u tan v _(1b) z p g D H _∂ ∂ − − = ρ δ 1 Dt w _(1c) 0 z w ₌       ∂ ∂ + • ∇ + V Dt Dρ _ρ  _(1d) Dt D RT Dt Dp _=γ ρ _(1e)

RT

p

=

ρ

(1f)

V



z

V

t

Dt

D

∂

∂

+

∇

•

+

∂

∂

=



w

ϕ

λ

ϕ

∂

∂

+

∂

∂

=

∇

•

a

v

cos

a

u

V



(

)

_











∂

∂

+

∂

∂

=

•

∇

ϕ

ϕ

λ

ϕ

cos

v

cos

1

u

a

V



Model and Governing Equations

 From the governing equations (1), we have considered small (but not infinitesimal) amplitude perturbations around a resting, hydrostatic and isothermal background state:

u = u₀ + u’ ; v = v₀ + v’ ; w = w₀ + w’ , with u₀ = v₀ = w₀ = 0;

p = p₀(z) + p’ ; ρ = ρ₀(z) + ρ’; with dp₀ / dz = -ρ₀ g and T = T₀ + T’, with T₀ = const; (2)

 Inserting (2) into (1) and retaning only the terms until second order in terms of perturbations: ∂_∂ − + _{ρ ϕ} ∂_∂λ = −_ •∇ + ∂_∂ _ + ϕ + _ρ ρ _ϕ ∂_∂λ' cos ' tan a ' v ' ' ' ' ' ' cos 1 v' ' 2 0 0 p a u z u w u V p a f t u 

ϕ

ρ

ρ

ϕ

ϕ

ρ

∂ ∂ + −     ∂ ∂ + •∇ − = ∂ ∂ + + ∂ ∂ ' ' tan ' ' v w' v' ' ' 1 u' v' 2 0 2 0 p a a u z V p a f t  2 0 2 0 2 0 ' ' ' ' w w' w' ' ' ' ' 1 t w'       + ∂ ∂ +     ∂ ∂ + •∇ − =       − + ∂ ∂ + ∂ ∂ ρ ρ ρ ρ θ ρ z g p z V p C g z p S      ₋ ∂ ∂ +       ∂ ∂ + • ∇ −     ∂ ∂ + •∇ − = ∂ ∂ + • ∇ +       ₋ ∂ ∂ w' ' ' 1 w' ' ' p' ' w ' ' 1 w' ' w' ' 1 0 0 2 2 0 2 0 g t p T T C z V z p V C z V g t p C_s ρ _s ρ _s ρ ρ          ₋ ∂ ∂ +     ∂ ∂ + •∇ = + ∂ ∂ w' ' ' z ' w' ' ' V -w' ' 0 0 2 2 0 g z p T T C g N t _S

ρ

θ

θ

ρ

θ

 (3a) (3b) (3c) (3d) (3e)

Model and Governing Equations

'

'

'

₂

ρ

θ

p

g

C

g

S

−

=

g RT H = 0 Where: 0 0 0

'

'

'

ρ

ρ

+

=

T

T

p

p

(3f) ; (3g)

H

g

C

g

dz

d

g

N

S

κ

ρ

ρ



_

=









+

−

=

0 ₂ 0 2

1

κ

γ

−

=

=

1

0 2

gH

RT

C

_S p

C

R

=

κ

 Following Kasahara and Qian (2000) we have rescaled the perturbations according to:













































=





































− − − 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0

p

w

v

'

'

p'

w'

'

v

'

ρρ

θρ

ρ

ρ

ρ

ρ

ρ

θ

u

u

(4)

( )

[

]

      ∂ ∂ + + + ∇ • − = ∂ ∂ + − ∂ ∂ − ₋ λ ϕ ρ ϕ ρ λ ϕ ' cos tan a v cos 1 v 2 1 0 p a u u wL u V p a f t u z  (5a)

( )

[

]

      ∂ ∂ + − + ∇ • − = ∂ ∂ + + ∂ ∂ − − ϕ ρ ϕ ρ ϕ ' tan a v v 1 u v ₂1 2 0 p a u wL V p a f t z  (5b)

( )

[

]

      ₊ ∂ ∂ + + ∇ • − = − Γ + ∂ ∂ + ∂ ∂ − 2 -z 2 1 0 w wL w t w _{θ ρ ρ ρ} g z p V p z p  _(5c)

( )

[

]

(

( )

)

            − ∂ ∂       − + + • ∇ − + ∇ • − = Γ − ∂ ∂ + • ∇ + ∂ ∂ − _{+ −} w 1 w wL 1 w w 1 0 2 z 2 2 1 0 2 g t p RT p C L V p p V C z V t p C_s

ρ

_s

ρ

Z _s

ρ

   (5d)

( )

[

]

            ₋ ∂ ∂       − + + ∇ • = + ∂ ∂ ₊ w RT p wL V 1 -w 1 0 2 2 z 2 2 1 -0 2 g z p C N g N t N

ρ

θ

θ

_S

ρ

θ

 (5e) Where,

( )

H

z

dz

d

z

L

_z

2

1

2

1

₀ 0

−

∂

∂

=

+

∂

∂

=

+

ρ

ρ

( )

dz

z

H

d

z

L

_z

2

1

2

1

₀ 0

+

∂

∂

=

−

∂

∂

=

−

ρ

ρ

H

C

g

dz

d

S

2

2

1

2

1

2 0 0

κ

ρ

ρ

−

=

+

=

Γ

Model and Governing Equations

 Substituting (4) into (3) we get:

Eigenmodes of the Linear Problem (Normal Modes)

 If the second-order nonlinear terms are disregarded, equations (5) become:

₀ cos 1 v = ∂ ∂ + − ∂ ∂ λ ϕ p a f t u _(6a) 0 1 u v ₌ ∂ ∂ + + ∂ ∂ ϕ p a f t (6b) 0 t w _{+Γ − =} ∂ ∂ + ∂ ∂ _θ p z p _(6c) 0 w w 1 2 ∂ −Γ = ∂ + • ∇ + ∂ ∂ z V t p C_s  (6d) 0 w 1 2 ∂ + = ∂ t N

θ

(6e)

 _{The eigensolutions of (6) were determined by Kasahara and Qian (2000) for the following} boundary conditions:

(i) w = 0 at z = 0 and at z = z_T (7a) (ii) Periodic solutions in longitude (7b) (iii) Regularity at the poles (7c)

Eigenmodes of the Linear Problem

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

t i is e z P z iP z P z iV z U w p u σ λ

ϕ

η

ϕ

ξ

ϕ

ξ

ϕ

ξ

ϕ

θ

−                 Θ =                 v (8)

The eigensolutions of (6) with boundary conditions (7) are given by:

where: 0 1 1 2  =      _−Γ +       − η η σξ dz d C gH_{e s}

0

2

=

+

Θ

−

σ

N

η

0

=

Θ

−













₊

_Γ

+

ξ

ξ

ση

dz

d

Vertical structure equations

0 cos = + − −

ϕ

σ

a sP fV U

0

1

₌

+

+

ϕ

σ

d

dP

a

fU

V

(

)

e gH P V d d sU a σ ϕ ϕ ϕ _ =     ₊ cos cos 1

horizontal structure equations (Laplace’s tidal equations)

Eigenmodes of the Linear Problem

(

2

)

0 2 2 = Γ − +

λ

η

η

dz d

 The vertical structure equations can be written in terms of η as follows: ; with BCs: η = 0 at z = 0 and at z = z_T ;       = z z k A z T k

π

η

( ) sin , k = 1, 2, 3 ...

(

2 2

)

2

1

1

_δ

_σ

λ

_H s e

N

C

gH



_

−









−

=

 _{The solution is given by: , provided that}

2 2

Γ

+













=

T k

z

k

π

λ

Eigenvalues of the vertical structure _equations

1 2 2 2 2 1 −       − + = σ δ λ H s k s e N C g C H  _{Separation constant:}

(

s

l

H

_e

)

F

,

,

=

Eigenmodes of the Linear Problem

 Different oscillation regimes: (I) → → inertio-acoustic modes;

(ii) → → inertio-gravity modes;

g C H_e S 2 > 2 2 N > σ g C H_e S 2 < 2 2 N < σ

Dispersion curves for acoustic and gravity modes for l = 0, k = 1, and symmetric about the equator.

Eigenmodes of the Linear Problem

 Different oscillation regimes: (I) → → inertio-acoustic modes;

(ii) → → inertio-gravity modes;

g C H_e S 2 > 2 2 N > σ g C H_e S 2 < 2 2 N < σ

Equivalent heights H_e for acoustic and gravity modes for l = 0, k = 1, and symmetric about the equator.

Energetics of Normal modes

 Kasahara and Qian (2000) have demonstrated the orthogonality condition for the eigenmodes:

(

)

_{∫ ∫ ∫}

(

)

−

=

















+

+

+

+

−

T z H k j k j k j

i

u

u

a

d

d

dz

i

0 2 0 2 2 2 2 * k j 2 S * k j * k j * *

0

cos

N

C

p

p

w

w

v

v

π π π

λ

ϕ

ϕ

θ

θ

δ

σ

σ

 For the case j = k we have the total energy of the j-th eigenmode of the system:

[

]

∫ ∫ ∫

−

>

+

+

=

T z j j j j

K

E

A

a

d

d

dz

ET

0 2 0 2 2 2

0

cos

π π π

λ

ϕ

ϕ

Where:

∫ ∫ ∫

[

(

)

]

− + + = T z H j j j u a d d dz K 0 2 0 2 2 2 2 j 2 2 cos w v 2 1 π π π

λ

ϕ

ϕ

δ

Kinetic energy

∫ ∫ ∫

− = T z j a d d dz E 0 2 0 2 2 2 2 S 2 j cos C p 2 1 π π π λ ϕ ϕ elastic energy

∫ ∫ ∫

− = T z j a d d dz A 0 2 0 2 2 2 2 2 j cos N 2 1 π π π λ ϕ ϕ θ Termobaric energy

Energetics of Normal modes

Energetics of Normal modes

Energetics of eastward inertio-acoustic modes with meridional index l = 0 and k = 1. Each energy type is normalized by total energy.

Energetics of Normal modes

Energetics of Normal modes

Energetics of eastward inertio-gravity modes with meridional index l = 0 and k = 1. Each energy type is normalized by total energy.

( )

[

]

      ∂ ∂ + + + ∇ • − = ∂ ∂ + − ∂ ∂ − ₋ λ ϕ ρ ϕ ρ λ ϕ ' cos tan a v cos 1 v 2 1 0 p a u u wL u V p a f t u z  (5a)

( )

[

]

      ∂ ∂ + − + ∇ • − = ∂ ∂ + + ∂ ∂ − − ϕ ρ ϕ ρ ϕ ' tan a v v 1 u v ₂1 2 0 p a u wL V p a f t z  (5b)

( )

[

]

      ₊ ∂ ∂ + + ∇ • − = − Γ + ∂ ∂ + ∂ ∂ − 2 -z 2 1 0 w wL w t w _{θ ρ ρ ρ} g z p V p z p  _(5c) (5d) (5e) Where,

( )

H

z

dz

d

z

L

_z

2

1

2

1

₀ 0

−

∂

∂

=

+

∂

∂

=

+

ρ

ρ

( )

dz

z

H

d

z

L

_z

2

1

2

1

₀ 0

+

∂

∂

=

−

∂

∂

=

−

ρ

ρ

H

C

g

dz

d

S

2

2

1

2

1

2 0 0

κ

ρ

ρ

−

=

+

=

Γ

Resonant Nonlinear Interactions of Global Nonhydrostatic Modes

( )

[

]

(

( )

)

            − ∂ ∂       − + + • ∇ − + ∇ • − = Γ − ∂ ∂ + • ∇ + ∂ ∂ − + − w 1 w L p wL 1 w w 1 0 2 z z 2 2 1 0 2 g t p RT p C V p V C z V t p C_s ρ _S ρ _S ρ   

( )

[

]

            ₋ ∂ ∂       − + + ∇ • − = + ∂ ∂ − + w wL 1 w 1 0 2 2 z 2 2 1 0 2 g t p RT p C N g V N t N ρ θ θ _S ρ θ 

Ansatz ⇒ Solution with three modes:

With the following resonance relations satisfied:

k₃ = k₁ + k₂ (not excluding) s₁ = s₂ + s₃

σ1 = σ2 + σ3

condition for meridional structures satisfied

Nonlinear resonant triad interaction conditions

Resonant Interactions of nonhydrostaic nonrmal modes: general case

(

)

( )

( )

( )

e CC z z z z z z U t A e z z z z z z U t A e z z z z z z U t A t z u t i is t i is t i is . ) , ( ) , ( ) , ( P ) , ( iW ) , ( iV ) , ( ) , ( ) , ( ) , ( P ) , ( iW ) , ( iV ) , ( ) , ( ) , ( ) , ( P ) , ( iW ) , ( iV ) , ( , , , p w v 3 3 2 2 1 1 3 3 3 3 3 3 3 2 2 2 2 2 2 2 1 1 1 1 1 1 1 +                   +                   +                   =                   − − −σ λ σ λ σ λ ϕ ρ ϕ θ ϕ ϕ ϕ ϕ ϕ ρ ϕ θ ϕ ϕ ϕ ϕ ϕ ρ ϕ θ ϕ ϕ ϕ ϕ ϕ λ ρ θ

(

)

∫

      ± ± = T − z T z dz z z k k k I 0 3 2 1 2 1 0 cos

π

ρ

 _{Substituting the ansatz into the PDEs (5) we get:}

Resonant Interactions between Acoustic and Gravity Modes

3 2 23 1 1 1

i

A

A

dt

dA

ET

=

α

* 3 1 13 2 2 2

i

A

A

dt

dA

ET

=

α

* 2 1 12 3 3 3

i

A

A

dt

dA

ET

=

α

α₁23_{, α}

213 , α312 ⇒ Nonlinear coupling constants;

Resonant Interactions of nonhydrostaic nonrmal modes: general case

[

N

u

N

N

N

u

N

N

]

a

d

dz

T z u u 2 1 0 2 0 2 2 * 1 ) 3 , 2 ( * 1 ) 3 , 2 ( p * 1 ) 3 , 2 ( * 1 ) 3 , 2 ( w * 1 ) 3 , 2 ( v * 1 ) 3 , 2 ( 23 1

v

w

p

cos

− −

∫ ∫

+

+

+

+

+

=

θ

ϕ

ϕρ

α

π π θ

[

N

u

N

N

N

u

N

N

]

a

d

dz

T z u u 2 1 0 2 0 2 2 * 2 ) 3 , 1 ( * 2 ) 3 , 1 ( p * 2 ) 3 , 1 ( * 2 ) 3 , 1 ( w * 2 ) 3 , 1 ( v * 2 ) 3 , 1 ( 13 2

v

w

p

cos

− −

∫ ∫

+

+

+

+

+

=

θ

ϕ

ϕρ

α

π π θ

[

N

u

N

N

N

u

N

N

]

a

d

dz

T z u u 2 1 0 2 0 2 2 * 3 ) 2 , 1 ( * 3 ) 2 , 1 ( p * 3 ) 2 , 1 ( * 3 ) 2 , 1 ( w * 3 ) 2 , 1 ( v * 3 ) 2 , 1 ( 12 3

v

w

p

cos

− −

∫ ∫

+

+

+

+

+

=

θ

ϕ

ϕρ

α

π π θ

Resonant Interactions of nonhydrostaic nonrmal modes: general case

( )

is p CP a u u u a a u is u N_{u }+ + +      ₊ ∂ ∂ + − = 2 3 2 _{3 3} 3 -z 2 3 2 3 3 2 ) 3 , 2 ( cos tan a v L w v cos

ϕ

ρ

ϕ

ϕ

ϕ

( )

p CP a u u a a is u N + ∂ ∂ + −       + ∂ ∂ + − = ϕ ρ ϕ ϕ ϕ 3 2 3 2 3 -z 2 3 2 3 3 2 ) 3 , 2 ( v tan a v L w v v cos v

( )

L

( )

p

g

CP

L

a

N

_{H z}

_

+

_z

+

+











+

∂

∂

+

−

=

− + 3 2 3 2 3 2 3 2 3 3 2 ) 3 , 2 ( w

w

w

w

v

w

is

cos

a

u

_ρ

_ρ

_ρ

ϕ

ϕ

δ

( )

(

)

( )

(

i p g

)

CP RT p C L u is a a p is a u C N S z S p + − −       − +       +       ∂ ∂ + −       ₊ ∂ ∂ + − = + − 3 3 3 2 0 2 2 3 3 3 3 2 3 z 2 3 2 3 3 2 2 ) 3 , 2 ( w 1 w cos v cos 1 p L w p v cos 1 σ ρ ϕ ϕ ϕ ρ ϕ ϕ

( )

(

i

p

g

)

CP

RT

p

N

C

g

a

is

a

u

N

N

S

+

−

−













−

+













+

∂

∂

+

−

=

+ 3 3 3 2 0 2 2 2 3 z 2 3 2 3 3 2 2 ) 3 , 2 (

w

L

w

v

cos

1

_θ

_ρ

_σ

ϕ

θ

θ

ϕ

θ

 _{From the complex amplitude equations it is easy to get the energy equations:}

Resonant Interactions of nonhydrostaic nonrmal modes

(

* *

)

1 23 1 2 1 1

Im

A

A

₂

A

₃

dt

A

d

ET

=

−

α

(

* *

)

1 13 2 2 2 2

Im

A

A

₂

A

₃

dt

A

d

ET

=

α

(

* *

)

1 12 3 2 3 3

Im

A

A

₂

A

₃

dt

A

d

ET

=

α

 _{Condition for total energy to be conserved within a resonant triad interaction is:}

0

12 3 13 2 23 1

+

+

=

−

α

α

α

Mode 1 ⇒ unstable mode of the triad.

Resonant Interactions between Acoustic and Gravity Modes

 _{Analytical solutions of the conservative triad equations, assuming that}

( ) and the amplitude of mode 1 is zero initially:

( )

              = m u sn A t A ET ₁₃ 2 2 23 1 2 2 2 1 1 ( ) 0 α α

( )













=

m

u

cn

A

t

A

ET

_{2 2}

(

)

2 ₂

0

2 2

( )













=

m

u

dn

A

t

A

ET

_{3 3}

(

)

2 ₃

0

2 2

t

A

u

=

₃

(

0

)

α

₁23

α

13₂ 2 3 2 13 2 12 3

)

0

(

)

0

(

















=

A

A

m

α

α

Where sn, cn and dn are the Jacobian Elliptic functions, with argument u and parameter m given by 23 1 13 2 12 3

α

α

α

< <

Resonant Interactions between Acoustic and Gravity Modes

 _{Numerical results for a representative example of resonant triad containing two} acoustic-inertia modes and one gravity-acoustic-inertia mode:

Determination of a resonant triad involving a long inertio-acoustic, a short acoustic mode and a short gravity mode. The acoustic modes have k = 1 vertical structure, while the gravity mode has a k = 2 vertical structure.

 _{Numerical results for a representative example of resonant triad containing two} acoustic-inertia modes and one gravity-acoustic-inertia mode:

Mode 1: unstable (pump) mode

Acoustic mode with k = 1; s = 476, l = 0 (first symmetric mode)

Mode 2:

Acoustic mode with k = 1, s = 1, l = 0 (first symmetric mode)

Mode 3:

Gravity mode with k = 2, s = 475, l = 0 (first symmetric mode)

Mode 1 Mode 2 Mode 3 σ₁

(cHz) σ₂ (cHz) σ₃ (cHz) ET1 (J) ET2 (J) ET3 (J) α₁23 _α 213 α312 (1,476,0,A) (1,1,0,A) (2,475,0, G) 6.293 5.888 0.407 1.3x 1010 1.3 x 1020 7x 106 4x1011 _3.7x1011 _2.6x1010

Resonant Interactions between Acoustic and Gravity Modes

 Numerical results for a representative example of resonant triad containing two acoustic-inertia modes and one gravity-acoustic-inertia mode:

Resonant Interactions between Acoustic and Gravity Modes

Numerical results for a representative example of resonant triad containing two acoustic-inertia modes and one gravity-acoustic-inertia mode:

Resonant Interactions between Acoustic and Gravity Modes

 Numerical results for a representative example of resonant triad containing two acoustic-inertia modes and one gravity-acoustic-inertia mode:

0<< m <1

Vertical velocity at ϕ = 10ºS and z = 9Km;

zonal velocity at ϕ = 0o

and z = 4.5Km

Short acoustic mode activity.

Short gravity mode activity.

Summary and Remarks



_{Here we have investigated the possibility of resonant interactions involving}

inertio-acoustic and inertio-gravity modes in a shallow-nonhydrostatic global atmospheric

model (weakly nonlinear extension of Kasahara and Qian (2000))



_{For the internal modes (rigid lid boundary condition), we found that the only}

possibility for such resonances is that one gravity mode interacts with two acoustic

modes (similar to Rossby-gravity-gravity interaction in the hydrostatic dynamics);



_{This kind of resonant interaction can potentially yield vacillations in the dynamical}

fields with periods varying from a daily (and intra-diurnal) time-scale up to almost a

month long, depending on the way in which the initial energy is distributed on the

triad components;



_{Acoustic modes are usually filtered out from numerical models to avoid}

computational constraints associated with explicit numerical schemes, even in

Next Steps of the Project



_{To investigate the possibility of resonant interactions for the limiting case of}

vertical modes where z

_T

→ ∞ .

 To study the possibility of long-short wave interactions.

 _{To investigate the dynamics of these resonant interactions with the inclusion of diabatic} effects;