1

Tutorial lectures on hydrodynamics instabilities

Lecture notes presented at the Institute of Laser Engineering.

Osaka University (4-1-2006)-(7-1-2006)

Javier Sanz Recio

ETSI Aeronáuticos.

Universidad Politécnica de Madrid

2

Contents

Lecture1: The classical Rayleigh-Taylor (RT) instability. ……………………………3Phenomenology of the instability. Incompressible fluids: Heuristic derivation of the linear instability growth rate (Second Newton Law); analysis method (dispersion relation); Atwood number effects finite thickness effects. Examples.Compressible fluids. Dispersion relation. Introduction to nonlinear classical RT instability. Phenomenology. Layzer models approach.

Lecture 2: Linear ablative RT instability………………………………………………141D ablation front structure. Isobaric model and coronal models.Stabilization mechanisms. Heuristic derivation of the dispersion relation in sharp ablation fronts (Second Newton Law). Linear analysis method. Froude number dependence. The critical surface proximity effects (long wavelength perturbation) and Landau instability.

Lecture 3: Non linear ablative RT instability…………………………………………26Sharp ablation front model: Thermal equation, momentum equation and time evolution equation of the interface. Single mode perturbations: saturation amplitude, inversion of spike bubble asymmetry, non linear cutoff wave number, asymptotic bubble velocity.

References:

• S. Bodner, Phys. Rev. Lett. 33, 761 (1974).

• H. Takabe, K. Mima, L. Monthierth, and R. Morse, Phys Fluids 28, 3676

(1985).

• H. J. Kull, Phys. Fluids B1, 170 (1989).

• J. Sanz, Phys. Rev. Lett. 73, 2700 (1994).

• R. Betti et. al, Phys. Plasmas 2, 3844 (1995).

• A. Piriz, J. Sanz, and L. Ibañez, Phys. Plasmas 4, 1117 (1997).

• J. Sanz, J. Ramirez, R. Ramis, R. Betti, and R. P. J. Twon, Phys. Rev. Lett,

(89, 195002 (2002).

• P. Clavin and L. Masse, Phys. Plasmas 11, 690 (2004).

3

THE CLASSICAL RT INSTABILITY

4

Phenomenology of the instability (*)

- Lord Rayleigh, Proc. London Math. Soc. 14 (1883) 170; G. Taylor, Proc. R. Soc. A 201 (1950) 192;

D. J. Lewis, Proc. R. Soc. A 202 (1950) 81.

*

,kgη η

g12 kλ π −=

η2ρ

1ρ

2 1( )ρ ρ<<

0 ,teγη η kgγ

η3gk

η →logη

log t

0.1λ

20 ,bh g tα 0( 0.065)α ≈

γ

k

Viscosity

Surface tension

ck

Accelerated fluid layers :

g 1P

2 1P P>

g g

g g

5

Rayleigh-Taylor instability

• Geophysics, Astrophysics, ……

• Technological applications:

Inertial Confinement Fusion (ICF)

y

x

g

g

Incompressible fluids and uniform density.

0,ν∇ ⋅ =

( ) ,tv v v p gρ ρ∂ + ⋅∇ = −∇ +

6

Second Newton Law:

3 3a tt k b tt kk kρ ξ ρ ξ− −∂ + ∂ ≈ 2 ,b kk gρ ξ−−

aρg

kξ12 kλ π −=

bρ

2a kk gρ ξ−

ma F=

,a btt k k

a b

kgρ ρξ ξρ ρ−

∂ =+

,tk Ceγξ = 2 0 , ,a b

T Ta b

A kg A ρ ργρ ρ−

− = =+

Atwood number

,a b TA kgρ ρ γ> =± Unstable.RT modes ,a b TA kgiρ ρ γ< =± Stable.

Gravity waves

1k−

12k −

Perturbed solution:

Interfaces:

Method:

0 ,ν∇ ⋅ =

( ) ,tv v v p gρ ρ∂ + ⋅∇ = −∇ +y

1 1

1

1

( ) , ( , )

0,

,,

x y

y y x

x

y y

ikx t ikx tp y v v v

v ikv

v ikpv p

e eγ γ

ργργ

+ +=

∂ + =

= −= −∂

cρdρbρ

aρ

g

Equilibrium solution:

0

0

0, ( ),0,y

v p p yp gρ

= =

−∂ + =

jikx te γξ +

7

Solution for each fluid layer:Method:

y

21 1

1

0,

,

( )

( ),

yy

k y k y

k y k yx

k y k yy

p k p

p Ae Beikv Ae Be

kv Ae Be

γρ

ργ

−

−

−

∂ − =

= +

= − +

= − −

cρdρbρ

aρ

g

Boundary conditions:

jy 1jy −1jy +

1 0 1, ,

,

, variables ,

j j y j

y y j

at y y p p p g continous

v continous and v

at y must bebounded

ξ ρ ξ

γξ

= + ∂ ⇔ +

=

=±∞

Compatibity condition ->Dispersion relation :

( )det ( , ) 0D kγ =( , ) 0 ,

j

ABA

D kB

γ

ξ

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟′⋅ =⎜ ⎟′⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

How the spectrum is?:

For each k value we have different values of .

The number of values may be infinite but in any case the spectrum is discrete.

1 2: , , ,γ γ γ …

Advanced comment:Laplace transform in “t “:

Poles of K, (discrete spectrum):

(*)Branche points of K (continuum spectrum):

(*) E. Ott, PRL 1981

( , ) stK s k e ds−∫teγ

tt eα γ−

8

Example 1:

aρbρ

1k yp Ae=

y

,

,

a b

a b

A g B gk k

A B

ρ ξ ρ ξ

γξρ γ ρ γ

+ = +

− = =

1k yp Be−=

2

1 1 ( )

1 0 0

0 1

a b

a

b

b

g AB

k

ρ ρρρ

ξρ γ

⎡ ⎤⎢ ⎥

− −⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎢ ⎥⋅ =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥−⎢ ⎥

⎢ ⎥⎣ ⎦

2 0 , ,a bT T

a b

A kg A ρ ργρ ρ

−− = =

+

g

Example 2:

2

2

1 1 0

0

1 1 0 0,

0

k d k d

a

bk d k d

g

e e g AB

k

e ek

ρ

ρ

γ ρξξγ ρ

−

−

⎡ ⎤⎢ ⎥

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥− ⋅ =⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥−⎢ ⎥⎣ ⎦

4 2 2 0,k gγ − =

1 ,k y k yp Ae Be−= +

0, 0,

( ) , ( ) ,

k d k da b

k d k da b

A B g Ae Be gk k

A B Ae Be

ρ ξ ρ ξ

γξ γξργ ργ

−

−

+ + = + + =

− − = − − =

, ,kg kgiγ γ= ± = ±

aξ

y

ρ

g

bξ

d

2

2

, ,

, ,

kdb a

kda b

kg e RT

kg e GW

γ ξ ξ

γ ξ ξ

−

−

= ⇒ =

= − ⇒ =

9

Compressible fluids:

aξ

0 0 0 0

/( 1) /( 1)0 0

2 20 0

0 , / ,

(1 / ) , (1 / ) ,1/ (1 / ), 1,

ny

n n n na a

aa

a

p g p const

p p y d y ddgnc n p c y d

n p

ρ ρ

ρ ρρρ

− −

= −∂ + =

= + = +

−= = + =

ap

y

ρ

g

bξ

d

Equilibrium solution:

0 1 0 1, ,

( , ) ,

ikx t ikx t

ikx tx y

e p p p e

v v v e

γ γ

γ

ρ ρ ρ + +

+

= + = +

=

1 0 0

0 1 1

0 12

1 0 1

( ) 0,

,

,

,

y y x

y y

x

v ikv

v p g

v ikp

p c

γρ ρ ρ

γρ ρ

γρ

ρ

+ ∂ + =

= −∂ +

= −

=

1 0

1 0

0, ; 0,

0, ; ,a y a

b y b

p g v at y

p g v at y d

ρ ξ γξ

ρ ξ γξ

+ = = =

+ = = = −

Perturbed solution:

0p =

Equations:Boundary conditions:

22

1 1 12 2 20 0 0

2

1 120

( ( )) 0,

( ) 0, 0, ,

yy y y

y

g gp p k pc c c

g p p at y dg c

γ

γ

∂ − ∂ − + +∂ =

+ −∂ = = −

( , , , ) 0D k g dγ =

1 ( , , , ) ( , , , )p A Hyperg k y B Lague k yγ γ= ⋅ + ⋅… …

Dispersion relation:

10

11

2(1 / ) /(1 / )n y d d gp C y d e γ− += +

22

1 1 12 2 20 0 0

2

1 120

( ( )) 0,

( ) 0, 0, ,

yy y y

y

g gp p k pc c c

g p p at y dg c

γ

γ

∂ − ∂ − + +∂ =

+ −∂ = = − 1 0 0 ( )y y y y xv v ikvγρ ρ ρ+ ∂ + ∂ +2

1 120

0, ( ) 0,yg p p

g cγ

= ⇒ + −∂ =

Dispersion relation: We look for incompressible modes!

4 2 20, ( , )kg kg kgγ γ γ− = ⇒ = = −

( )4 ( , , , ) 0,SMkg D k g dγ γ− =

Nonlinear classical RT instability

aρ 0ρ

g p const

• Harmonics generation (starting in the weakly nonlinear phase).

• Subharmonic cascade.

• Bubbles and spikes show different time behavior.

• Bubble competition.

11

aρg ρ

0φΔ =

ξx

y

( , )ar tα n

v φ=∇

n nt a ar φ⋅ ∂ = ⋅ ∇

212 ( )t a a

g con stφ ξ φ∂ = − ∇ +

1 2 3 . . .

amplitude

•Weakly non-linear results

2 33

38 Lkη ξ≈

22

12 Lkη ξ≈

2 31

14L Lkη ξ ξ≈ − ( ),L

t kgeγξ γ≈ =

1 2 3( , ) cos cos 2 cos3x t kx kx kxξ η η η+ + +

12

•Spike bubble asymmetry112s L La kξ ξ⎛ ⎞≈ +⎜ ⎟

⎝ ⎠spike amplitude:

bubble amplitude: 112b L La kξ ξ⎛ ⎞≈ −⎜ ⎟

⎝ ⎠

log t

log a

linear theory

ba

Sa

gba

sa

• Long wavelength modes generation

1212 1 2

2 1 22 2

1 2 12

1 2( )( ) ( )2

( )( )

tkt t e γ γξ ξ ξ

γ γ γγ γ γ

+∝

+= −

+ −

2k1,k 12 1 2

1 2

,k k kk k

= −+

13

• Bubble competition. Acceleration of bubble front.

bh

2 4 6 8 10 12 14

0.2

0.4

0.6

0.8

1

bh

2t

20 ,bh h g tα− = ( 0.065 )α ≈

aρg ρ

0φΔ =

ξx

y

( , )ar tα n

v φ=∇

n nt a ar φ⋅ ∂ = ⋅ ∇

212 ( )t a a

g con stφ ξ φ∂ = − ∇ +

14

LINEAR ABLATIVE RT INSTABILITY

15

1D ablation front structure:

Lasertarget

dT

ρ

Lc

La

v

y

1D ablation front structure: Isobaric model

52

,,

,

( ) ,

a a

y y

a a

na y

v v mv v p g

T T

m T T KT T

ρ ρρ ρ

ρ ρ

= =∂ = −∂ +

− = ∂

52

2 22

2, , (0.01 0.3 )5

, (0.05 10) 1,

nn a a

a a aa

a ar a

a a

T KTmT KT L mL m

v vF MgL T

μ= ⇒ = −

= − = <<

1/

1/

/

/

( / ) ,( )

( / ) ,

(1 ),( )

(1 ),

na a

ana a

a

aa

a

a

y L

y L

T T ny Ly

Lny L

T Ty

L

e

e

ρ ρ

ρ ρ

−

⎫⎪

→ ∞⎬⎪⎭

⎫+⎪

→ −∞⎬⎪− ⎭

Minimum density gradient scale length1(1 ) /n n

m aL L n n+= +

aρ

yρ

v

T

avaT

16

Cold compressed target:d

T

ρ

Lc

La

v

y

2

/ / ,

0, ( ( )),

( ) 0,

:( ( )) 0, ( 0) ,( ( )) ( ),( 0) ,

a a

y a

t y

a

t

a

P P const

P g O M

v

BCP y d t P y Pv y d t d tv y v

α αρ ρ

ρ

ρ ρ

= =

−∂ +

∂ + ∂ =

= − = = == − = −∂= =

∼

1/ / (1 / ( )) ,a ap p y d tααρ ρ −= +

11 ,1 ( )a

yv vd tα

⎛ ⎞= −⎜ ⎟−⎝ ⎠

0 ,1 ad d v tα

α= −

−

2( )( 1) ( )

aa

a

pg O Md t

αα ρ

= +−

Linear ablative RTI: Stabilization mechanisms. Scaling Laws

aρ

0mΔ <

0mΔ >

ablation surface

g

flow

0pΔ >

0pΔ <

heat

kξ

20 ,t kk m ξ−− ∂

12 kπ −

2dk p−− Δ

20

d kk

mp kξρ

Δ1k −

kρ1( )k ay kρ ρ ρ− <<

1/0 nk a

k

mV k Vρ

−= >>∼

3 2( ) ( )a k tt k a k kk k gρ ρ ξ ρ ρ ξ− −+ ∂ ≈ −

Hydrostatic pressure

Dynamical pressure

Ablation:

•Fire polishising.

•Vorticity

17

Linear ablative RTI: stabilization mechanisms

aρ

0mΔ <

0mΔ >

ablation surface

g

flow

0pΔ >

0pΔ <

heat

kξ

Thermal pressure,Rocket effect,

Fire polishing+

Vorticity

12 kπ −

22 24 0,

1a a

Tb b

kV Vk A kgr r

γ γ+ + − ≈+

/( 1) 1n nc a rk L F − − <∼cutoff

1k −

kρ

1/(2 / ) ,nb ar kL n=

1 ,1

bT

b

rAr

−=

+,a

kb

VVr

=

,kteγξ ⇒∼

222 2

1 1a a

T a kb b

kV kV A kg k V Vr r

γ⎛ ⎞

= − + + −⎜ ⎟+ +⎝ ⎠

0 0 0

0 0 0 0 0

50 0 02

,,

,

( ) ,

a a

y y

a a

na y

v v mv v p g

T T

m T T KT T

ρ ρρ ρ

ρ ρ

= =

∂ = −∂ +

− = ∂

Equilibrium solutionaρ

y0ρ

0v

0T

avaT

Linear analysis method: Isobaric model ( )2 2 11, 1a a rM M F −<< <<

t ikxeγ +∝

Perturbed solutionPerturbed quantities are expanded as

g

5th ODS

1 1 0 0 1 0 1( ) 0,y y xv v ikvγρ ρ ρ ρ+ ∂ + + =

0 1 1 1 ,y yv p gγρ ρ+ = −∂ +

0 1 1 0 1 1 0, ( 0)xv ikp T Tγρ ρ ρ+ = − + =

50 0 1 0 12 ( ) ( ) 0,n

y y yv T KT Tρ∂ − ∂ ∂ =

,at y = −∞ 2 bounded modes

,at y = +∞ 3 bounded modes

Numerical eigenvalueproblem for γ

( , , ) 0aa r

a

LF kL Fvγ

=

18

Analytical model: Isobaric model ( )1akL <<

y

aρ1/

0 ( / ) na any Lρ ρ −=

1

1 1

0

0, ,

( ) ,

0,

a y

a t a y

y a

v v V e

V v p

p g

φ

ρ

ρ

∇ ⋅ = = +∇

∂ + ∂ = −∇

∂ + =

1k y

xv ikAe=

1k y

yv k Ae=

1 ( ) k ya ap V k Aeρ γ= − +

g

aV

at ikxeγξ +

y

1 1 1(0 ) 2 ,a a aQ p g V mρ ξ−= + +

1ikx tm e γ+

1ikx tQ e γ+

Mass ablation

Momentum

2 22

1/(1 ) 0,( / )

aa n

a

k Vf kV q kgk L n

γ γ+ + + − =

Cold region Hot

region

1/1

2

( / ) ,n

a

a a a

Q kL nqV kρ ξ

≡

1 ,a a a

mfV kρ ξ

≡

( , ),( , ),

q kf kγγ

?

Analytical model: Isobaric model ( )1akL <<

at ikxeγξ +

y1

ikx tm e γ+

1ikx tQ e γ+

Momentum

Mass ablation rate

Hot region

5th ODS

1 1 0 0 1 0 1( ) 0,y y xv v ikvγρ ρ ρ ρ+ ∂ + + =

0 1 1 1 ,y yv p gγρ ρ+ = −∂ +

0 1 1 0 1 1 0, ( 0)xv ikp T Tγρ ρ ρ+ = − + =

50 0 1 0 12 ( ) ( ) 0,n

y y yv T KT Tρ∂ − ∂ ∂ =

Normalized variables:

1 2 ˆ( ) ( ) ; ,F F F kyη η γ η= + + ≡

1 2 ˆq q q γ∗ ∗= + +

1 2 ˆf f f γ∗ ∗= + +

1/ˆ ( / ) /( ) 1,na akL n kVγ γ≡ <<

1 1/ 11 1 1 1, , , ,nv T p k y kρ− − −∼ ∼Scaling:

( )t yv∂ << ∂

Eigenvalue problem for: 1 1 2 2( , ), ( , ),q f q f∗ ∗ ∗ ∗

1/1

1

2

2 (1 1/ ),

1,

2,

nq n

f

q

∗ −

∗

∗

Γ +

……

11( 1)2

q∗< <

19

Analytical model: Dispersion relation ( )1akL <<

2 22

1/(1 ) 0,( / )

aa n

a

k Vf kV q kgk L n

γ γ+ + + − =

1 2 ˆ ,q q q γ∗ ∗= + + 1f f ∗= +

2 22

1 2 1/1

(1 ) 0,( /( ) )

aa n n

a

k Vf q kV kgkL q n

γ γ∗ ∗∗+ + + + − =

1 2

1/1( 2 (1 1/ ))nq n∗ − Γ +

1 1( ) ,a a am V k fρ ξ ∗ 1/ 21 1 1 2 ˆ( / ) ( )( ),n

a a a a aQ p kL n V k q qρ ξ γ− ∗ ∗+

1 ( ) ,x a a aQ Vρ ω∂ ⋅ 1/1 1 2 ˆ( / ) ( )( ),na a a aikV kL n k q qω ξ γ− ∗ ∗+

Blow-off to ablation density ratio

br =

11 1/

1/1

( ) , ( 1)n rc a rn

Fk L Fq n

−−

∗= >cutoff:

Extension of the analytical model: (for every value)

•Atwood number effects

0 1( ( ) / )( ) ,n

b aa

y q kr kL ρρ

∗==

0 0 0

11 ,n

a a ay

aLρ ρ ρρ ρ ρ

⎛ ⎞ ⎛ ⎞− = ∂⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

Coronal model 1 ,1

bT

b

rAr

−=

+

•Lateral heat transport at the ablation front: (based in a SBM)

1 1 ,af kL∗ +

2(2 ) (1 2 ) (2 )

1 1 1a a b a r a

T ab b b b

kL kL r kL F kLkg A kVr r r r

γ⎛ ⎞ ⎛ ⎞+ + + +

= − − −⎜ ⎟ ⎜ ⎟+ + +⎝ ⎠ ⎝ ⎠

rF

210 rF− < < ∞

20

Cutoff wavenumber versus Froude number

0.01 0.1 1 10 1000.01

0.02

0.05

0.1

0.2

0.5

1

rF

2c ak Vg

2.5n =

1.5

2

0.01 0.1 1 10 1000.001

0.01

0.1

1

10

2

2.5n =

1.5

rF

c ak L

•Rocket effect-Vor.conv.

Fire polishing-Vor.conv

At number corrections

•Lateral heat transp.

•Scale length density-

At number corrections

0.01 0.1 1 10 1000.01

0.02

0.05

0.1

0.2

0.5

1

Cutoff wave number versus Froude number

rF

2c ak Vg

2.5

21.5

,1emp a

m

kg kVkL

γ β= −+

1.75β =

4β =

21

Landau-Darrieus instability

aρ

c aρ ρ<

0pΔ <

0pΔ >

aV

g

2 2 aLDRT a

c

k V kgργρ

+

aLD a

c

kV ργρ

0( 1)cky <<

1/ 2 21

0 1/( 1, 1) ,( )

na

c r na

q n k Vky F kgkL

γ∗

>> >> ⇒ −

Landau-RT instability in ICF :

Va

T

ρ

aρ

aT

Critical surface

0→M

aL

I II III

Isobaric approximation with , and every length much larger than .

•Region I incompressible and potential flow ( ). …etc.……

•Region II :

where .

•Region III :

/ 0c aρ ρ → aL

0a aV mρ =

1/0( / ) n

c cT T y y= 1/0( / ) n

c cy yρ ρ −= 1/0( / ) n

c cu u y y=

0 ( / ) /nc a c ay L nρ ρ=

, ,c c cT T u uρ ρ= = =

y

0cy

22

Landau-RT instability in ICF : Critical surface

0→M

I II IIIAblation surface

y

x

g

ikx tae

γξ + ikx tce

γξ +

2 22

1/(1 ) 0,( / )

aa n

a

k Vf kV q kgkL n

γ γ+ + + − = 1( )x a a aaa

ik mv k Vk

ξ γξρ−

+ = − +

1/ 1/0

2 2

( ) ( / ) ,( ) ( )

n nc a

c c a a a a

Q ky Q kL nqu k V kρ ξ ρ ξ

≡ =

1 ,( )a a a

mfV kρ ξ

≡

( , ) ?q k γ

( , ) ?f k γ

Landau-RT instability in ICF : Critical surface

0→M

I II IIIAblation surface

y

x

g

ikx tae

γξ + ikx tce

γξ +

( ) 0t vρ ρ∂ +∇⋅ =

t yv v v p geρ ρ ρ∂ + ⋅∇ = −∇ +

( )52 ( )n

a c cP v KT T I y yδ∇ ⋅ − ∇ = −

2 2( ) ( ),5

ikx ct av r a y c c

a

Im v e e y yP

ρ θ γξ ρ ρ δ∂ + ∇ + + ⋅∇ = − −

,t yv v v p geρ ρ ρ∂ + ⋅∇ = −∇ +

( )ikx tavr a y

mv v e eγθ γ ξρ

+= ∇ + +

0,rv∇ ⋅ =

1/( .)n constρθ =

0

0(1 ( ) / ) ,

ac

c aikx ikx

c a c a

y ys yy y

y e y eξ ξ ξ

−⎛ ⎞≡ ≈⎜ ⎟−⎜ ⎟⎜ ⎟− − −⎝ ⎠

( )nTθ ∝

23

Landau-RT instability in ICF : 0→M

• Perturbed equations :

( )0

2 200 1 1 1 0(( ) / / ) ( ( ) / ) 0,ry

c a c ss c a c a

v my s k k s yn nsγ ξ ξ θ θ θ ξ ξ ξ

ρ− − − + ∂ − + − + =

0 0 0

2 21 0 0 0

2 2 2 20 0 1 0 0 0 0 0 0 1

( ( ) / ) ( )

( ) / ( ) ,

rx s rx

c a c a

c a c s

ikv v ikv

k p k s y g u

k u u k y k u u

ρ γ ρ

ξ ξ ξ ρ γ

γ ρ θ ρ ξ ξ ρ θ

+ ∂ =

− − + + +

− − + ∂

0 0 0 0 0 1

2 0 1 0 10 0 1 0 0 0 0 0 1 0 0 1 0

20 0 0 0 0 1 0

( ) 2

(( 2 ) )( ) / ,

ry ry s s ry s

ss s s s s

s c a c a

v v u u v p

u uu u u u u

ns nss u u g u y g

ρ γ ρ ρ

θ θρ θ ρ ρ θ ρ γ θ ρ γ

γ γ ξ ξ ρ ρ ρ γ ξ

+ ∂ + ∂ = −∂ −

∂ − ∂ − ∂ ∂ − ∂ − +

+ ∂ + + − + −

0,s ry rxv ikv∂ + =

0(0 )cs y< <

Landau-RT instability in ICF : 0→M

• Boundary conditions:

1 0,θ = 1 1 0 00/ ( ) / ,s c a cm m yθ ξ ξ+∂ = + −

0,ryv = 1( ),rx aa

ik mvk

γξρ

= − +

1 ,p Q=

1 0θ =

1 0( ) / ,s c a cyθ ξ ξ∂ = −

21( ) ( ) ,c c cry a c c rx

c

p g k uv u k ikvξ ργξ ξγρ γ

−+ = + +

at 0s +→

at 0cs y−→

( , ),Q kγ 1( , ),m kγ ( , ),c kξ γ

24

Landau-RT instability in ICF : 0→M

• Resolution method:0

ˆ (1),ck ky O≡ = )1 1 (1)nra F O− − =( 1 / 0,c aa ρ ρ− ≡ →

11 /( )cku aγ −>> >>

1/

1 22

ˆˆ

( )

n

c c a

Qk q q qu k

γρ ξ

≡ = + +

11 2 ˆ

( )a a a

m f f fV k

γρ ξ

≡ = + +

1/ˆˆ ,

n

c

kkuγγ

⎛ ⎞=⎜ ⎟⎜ ⎟

⎝ ⎠

…… and all perturbed quantities are perturbed in the same way

The system of ODE is iteratively solved ………

1ˆ( )q k

1ˆ( )f k

2ˆ( )q k

2ˆ( )f k

Landau-RT instability in ICF : 0→M

• Dispersion relation:2 2

21/(1 ) 0,

( / )a

a na

k Vf kV q kgkL n

γ γ⎛ ⎞

+ + + − =⎜ ⎟⎝ ⎠

y rx a ryv ik ikvω γξ≡ −∂ + +Vorticity:

1/

1 2

ˆˆ ˆ( 0 )

( )

n

a c

i ks q qk kuωω γξ

+= ≡ = − − +

1 0 ( 0 )x ap m sω +∂ = =

1 2

2 22 1

1/(1 )/ )

0 ,( a

aa nf q

nq k VkV kgkL

γ γ+ ++ + − =

1 2 ˆq q q γ= + +

1 2 ˆf f f γ= + +

25

Landau-RT instability in ICF :

1 2

2 22 1

1/(1 )/ )

0 ,( a

aa nf q

nq k VkV kgkL

γ γ+ ++ + − =

0.5 1 1.5 2 2.5 3 3.5

-0.5

0

0.5

1

1.5

22q

1q

1f

0cky

StabilizationLandau

instability 0.01 0.1 1 10 100

0.5

1

5

10

50

100

nc ak a L

1 1nra F− −

a = ∞ 105/ 2n =

ˆ ˆ1/

1 01 1/

1 1ˆ ˆ ˆ( ) ( ,2 ) , ( )ˆ ˆcosh 2 cosh

n k k

cn

n n

n n

k e eq k k kyk k

−

+

+ +⎛ ⎞= − + Γ − Γ =⎜ ⎟⎝ ⎠

1 0 2 00

1tanh( ), 1 tanh( ), ,cosh( )

cc c

a c

f ky q kyky

ξξ

= + =

Landau instability in ICF : ( 1, 0)rF g>> →

11/

(/ )

) ,(a

an

kn

qkVkL

γ −0 10.7, ( 0)cky q< < Landau instability

0 10.7, ( 0)cky q> > Oscillations

ˆ ˆ1/

1 01 1/

1 1ˆ ˆ ˆ( ) ( ) ( ,2 ) , ( )ˆ ˆcosh 2 cosh

n k k

cn

n n

n n

k e eq k k k kyk k

−

+

+ +⎛ ⎞= − + Γ − Γ =⎜ ⎟⎝ ⎠

1/0 1 11 2 (1 1/ ), ( 0.67, 5 / 2)n

cky q q n n∗ −>> ⇒ = = Γ + =

1/0 1 01 ( ) ,n

c cky q ky<< ⇒ −

26

NONLINEAR ABLATIVE RAYLEIGH-TAYLOR INSTABILITY

27

Linear ablative RTI: stabilization mechanisms

aρ

0mΔ <

0mΔ >

ablation surface

g

flow

0pΔ >

0pΔ <

heat

kξ

Thermal pressure,Rocket effect,

Fire polishing+

Vorticity

12 kπ −

22 2

2 4k a kk a k a

k

d dkg k V kVdt dtξ ρ ξξ ξ

ρ≈ − −

/( 1) 1n nc a rk L F − − <∼cutoff

1k −

kρ

Nonlinear model

28

Cold region

aρg ρ

0φΔ =

ξ

x

y

( , )ar tα n

a yV e φ+∇

n n n /t a a y aar V e mφ ρ⋅ ∂ = ⋅∇ + ⋅ −

212 ( )t a ya aa

a a

pg Vφ ξ φ φρ −

∂ = − ∇ − ∂ −

Mass ablation rate

2 / aap q m ρ− = − q≈

m

Hot region

gaT Pρ =

52( ) 0n

aP v KT T∇ ⋅ − ∇ =

( ) 0t vρ ρ∂ + ∇ ⋅ =

( )tv v v pρ ∂ + ⋅∇ −∇

1(2 /5 ) ,nrn K Tv vρ−= ∇ +

2 0 ,θ∇ = ( , ) 0 ,ar tθ =

ξ

x

y

0 na

m m θ= ∇ ⋅• Mass ablation rate :

( , , ) 1y x y tθ∂ = ∞ =

n

0( 2 / 5 )nKT nmθ =

20 1 ( ln ln )t rm v

nθ θ θ

ρ∇ = ∂ + ⋅∇

29

Hot region

g

10 ,rmv vρ θ−= ∇ +

ξ

x

y

n

0( 2 / 5 )nKT nmθ =

( )z re vω = ∇×

( ) ,y x x ya ae eφ ψ ψ∇ = ∂ − ∂

( ) ,ψ ω χΔ = −

10( ) 0,t rm vω ρ θ ω−∂ + ∇ + ⋅∇ =

0,rv∇⋅ =

,r a av φ+ −∇a yV eφ∇ +

( . . . )H C ofχ θ=( ) ,ω χ

( , ) ( , )Conformal map x y and k FT onχ θ χ→ −

20

( n )k ikikid dd

k kk

e eθ χ

χφ τθ χχωθθ

∞ ∞ ∞

−∞ −∞

− +∇ ⋅ +

=∇∇∫ ∫ ∫

τ

Linear theory :2 2

02 2 ( , )x y x ty

x tω φ ξ=

= ∂ ≈ ∂

aρ

ghρ

0φΔ =

• Non-linear rocket effect:ξ

x

y

n

• Momentum flux : 2 /h hq p m ρ= +

2 210 02

1( )h

q m m m dω χρ

= − + ∫

flow

0θΔ =

q

?

30

• Restoring force

aρ

gρ

ξx

y

nflow

2 2 110 02 ( ) ,bla

q p m m m dρ ω χ−− ∗ + ∫

2 21 12 2( )v p v vρ ρω ρ∇ + − × ∇

221/ 01

02 1 1/0

1( ) n

naa a

mnq p d m dL n

θθ ω χ

ρ θ

∞

−

∇ −+∫ ∫

1 1( ( ) )bl kIFT onρ ρ χ− −=

10 ,rmv vρ θ−= ∇ +

2 210

0( )

2k

k kk

m mq m k ωρ

−−+

•Ablating surface equations:aρ ρ

0φΔ =n flow

0θΔ =

g ξx

y

a yV e φ+∇

0θ = 1yθ∂ =

n n / n( ),t a a a yr m V eφ ρ⋅ ∂ = ⋅∇ − − ⋅

212

2 2 10

( )

1 ( )2

,

t a ya aa

bl aa

g V

m m V d

φ ξ φ φ

ρ ωρ

χ−

∂ = − ∇ − ∂

⎡ ⎤− − ∗ −⎣ ⎦ ∫

0 n ,a

m m θ= ∇ ⋅

20

( n ),

k iki kid dd

k kk

e eθ χ

χφ τθ χχωθθ

∞ ∞ ∞

−∞ −∞

− +∇ ⋅ +

=∇∇∫ ∫ ∫

31

Single mode results

1 2 3 . . .

amplitude

1c

kk

< <

•Weakly non-linear results

2 33

3 (1 4 )(1 4 )8

/3 Lk k kη ξ≈ − −

22

1 (1 2 )2 Lk kη ξ≈ −

2 31

(1 )(1 2 )4(1 )/ 2

L Lkk kk

η ξ ξ− −≈ −

−

amplitude

1 / 4 1 / 2k< <1 / 2 3 / 4k< <

amplitudeamplitude

3 / 4 1k< <

( )Lteγξ ≈

( )1 1 /( / ) nck k k −≈

( )ck k<

1 2 3( , ) cos cos 2 cos3x t kx kx kxξ η η η+ + +

2 31

14L Lkη ξ ξ≈ −

22

12 Lkη ξ≈

2 33

38 Lkη ξ≈

Classical

RTI

32

• Inversion of spike bubble asymmetry

( )1 (1 / 2 )s L La k kξ ξ≈ + −spike amplitude:

bubble amplitude: ( )1 (1/ 2 )b L La k kξ ξ≈ − −

g

12

k <

g

1 12

k< <

( )1 1 /( / ) nck k k −≈

ba

t

linear theory

1/ 2k <

1/ 2k >

• Nonlinear exponential growth. Saturation amplitude.

25 50 75 100 125 150 175

0.005

0.01

0.05

0.1

0.5

1

baλ

t k g

classical50 mμ

20 mμ14 mμ

12 mμ 11 mμ 10.5 mμ

0.025λ

( )S k

0.18 kgγ =

4.5 , 10r cF mλ μ= =

33

2 4 6 8 10 12 14

0.001

0.01

0.1

1

2 4 6 8 10 12 14

0.001

0.01

0.1

1

• Non linear exp. growth ( ). Simulations ART 2D

2.5 5 7.5 10 12.5 15

0.001

0.01

0.1

1

1ξλ

( )t ns

0.86

1ξλ

( )t ns

1ξλ

( )t ns

ck k<I

aV II Ia aV V>

III IIa aV V>

0.02λ

0.02λ

• Non linear instability for . Bifurcation diagram

1( ) cos( )e ej

jx jk xξ ξ

=

=∑

k

e jξ

ck

1( ) cos( )e e cx k xξ ξ≈Linear theory shows:

31 1 1 1( )tt kξ ξ ξ ξ∂ ≈ − − +

ck k>

34

• Non linear instability for . Bifurcation diagram

1( ) cos( )e ej

jx jk xξ ξ

=

=∑

ck k>

0.8 1 1.2 1.4 1.6

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 π 2π

−0.2

0

0.2

ejξ

/ ck k

1

2

3

4

5

• Full nonlinear instability ( )ck k>

0.2 0.4 0.6 0.8 1 1.2 1.4

-0.2

-0.1

0.1

0.2

aa

1 .5 ck k=

a bubble

35

• Non linear instability ( ). Simulations ART 2D

2 4 6 8

0.001

0.01

0.1

1

2.5 5 7.5 10 12.5 15 17.5

0.001

0.01

0.1

1

1ξλ

0.86

1ξλ

( )t ns

ck k>

0.02λ

0.02λ

• Asymptotic bubble velocity

23sc ca

gk kV

= >

0 / 3 ,b aV g k V≈ −

2

010,3

sc ab

k VVg

= → =

super cutoff

36

• Stability regions:

1.5 2 2.5 3 3.5 4 4.5

0.05

0.1

0.15

0.2

0.25/thsa λ

/ ck k

5rF = 10 20

Multi mode results

37

• Long wavelength modes generation

1212 1 2

1/2 1 2 1 2

2 212 2 1 21 2 12

1 2( )( ) ( )2

( ) (1 )1( )( )

nc

c

tkt t

n k k gkk nk

e γ γξ ξ ξγ γ γ

γ γ γγ γ γ+⎛ ⎞⎛ ⎞⎜ ⎟ ∝⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

+ += − +

++ −

k

γ

2k 1k12 1 2k k k= −

0.2 0.4 0.6 0.8 1

0.5

1

5

10

50

Classical:

• Long wavelength modes generation

1212 1 2( ) ( )

4k t t Gξ ξ ξ ×= −

12 1/k k

G

1 optk k=

1 /10optk k=

1 optk k= 1 2 optk k=

ART SIMUL.

MULTI 2D SIMUL.

Full nonlinear theory

Weakly nonlinear

theory

38

Bubble competition

Acceleration of bubble front

20 ,bh h g tα− = ( 0.06 )α ≈

2 4 6 8 10 12 14

0.2

0.4

0.6

0.8

1

1.2

1.4

bh

2t

1 0 cλ λ=

bh

39

Acceleration of bubble front

20 ,bh h g tα− = ( 0.06 )α ≈

2 4 6 8 10 12 14

0.2

0.4

0.6

0.8

1

bh

2t

1 0 cλ λ=

bh

bh

Acceleration of bubble front

20 ,bh h g tα− = ( 0.03 )α ≈

5 10 15 20 25 30 35

0.25

0.5

0.75

1

1.25

1.5

2 cλ λ= bh

2t

40

Acceleration of bubble front

20bh h g tα− =

0.2 0.4 0.6

0.01

0.02

0.03

0.04

0.05

0.06

Fr=3

Fr=20

Fr=1

Fr=5α

/ SCk k

Acceleration of bubble front :

( )2

0 1 / SCk kα α= −

0.2 0.4 0.6 0.8 1

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Fr=3

Fr=5

Fr=10

Fr=20α

/ sck k

2

20 ,

3b agh h C k V tk

⎛ ⎞− = −⎜ ⎟⎜ ⎟

⎝ ⎠03C α=

20( )bh h g tα− =

41