Energy cascades in strong wave turbulence

R. Rajesh (Institute of Mathematical Sciences [IMSc])

C. Connaughton (Warwick, UK)O. Zaboronski (Warwick, UK)

Outline

• What is wave turbulence?

• Solution for weak wave turbulence

• Strong wave turbulence

• Constant flux relations

• A toy model

Wave turbulence

A collection of weakly interacting waves

Described by a Hamiltonian

H =!

dk [!(k)a!kak + u(k)]

ak = i

!!(k)a(k) +

"

"a!k

"dku(k)

#

H =!

dk [!(k)a!kak + u(k)]

!(k) = ck!

u =!

dk1dk2!(k! k1 ! k2)Tk;k1,k2 [a!kak1ak2 + cc]

u =!

dk1dk2dk3!(k + k1 ! k2 ! k3)Tk,k1;k2,k3

"a!ka!k1

ak2ak3 + cc#

3-wave:

4-wave:

• Drive at intermediate k

• Dissipate at small and large k

• Steady state

• Cascades of conserved quantities

Driving and Dissipation

Conservation Laws

Total energy H

4-wave: wave action N =!!a!kak"

Conservation Laws

N, !

N1, !1 N2, !2

N1 + N2 = N

N1!1 + N2!2 = N!

N1 =N(!2 ! !)!2 ! !1

N2 =N(! ! !1)!2 ! !1

!2 ! !1, then N1 " N and N2!2 " N!

Two cascades

E(

)!

!d(L) ! f !d

(R)

ForcingScale

Large ScaleDissipation

Small ScaleDissipation

Inertialrange

Inertialrange

!

Inverse cascade

(wave action ormomentum)

Direct cascade

(energy)

Weak wave turbulence

Kinetic energy is conserved

u(k)t(k) = !

!n1

!t= "2

!T1234(n2n3n4 + n1n3n4 ! n1n2n4 ! n1n2n3)

#($1 + $2 ! $3 ! $4)d$2$3$4

!n1

!t= Im

!T1234!a!1a!2a3a4"dk2dk3dk4

Weak wave turbulence

!n1

!t= "2

!T1234(n2n3n4 + n1n3n4 ! n1n2n4 ! n1n2n3)

#($1 + $2 ! $3 ! $4)d$2$3$4

Scaling solution (Kolmogorov-Zakharov)

direct cascade :n(k) ! k!(2!+3d)/3

inverse cascade :n(k) ! k!(2!+3d+")/3

Pretty much everything known

!(!k) = !!!(k)

T (!k1,!k2,!k3,!k4) = !!T (k1,k2,k3,k4)

Experiments

Falcon et al, PRL, 2007

Lines have slope 5.5 (gravity, KZ=4.0)and slope 17/6 (capillary KZ)

Strong wave turbulence

• PE greater than KE

• Have to keep higher order terms in Hamiltonian

• Not a relevant limit

Strong wave turbulence A case for studying

u(k)t(k) = !(k)

!(k) ! O(1) when k "# or k " 0

Breakdown of the “weak” limit

(1)

(2) Many problems where theprimitive eq is 3-wave or 4-wave

Example: NLSE

i!"!t + c1!2! + c2|!|2! = 0

Strong wave turbulence A case for studying

(3) The three wave kinetic equation is mathematically identical to a mass aggregation problem

t

0

2

4

6

8

10

12

0 1 2 3 4 5

γ

n

Kolmogoroveps-expansion

Can the ideas be transferred?

Constant Flux Relation (CFR)

Cannot obtain n(k) anymore

More like NS turbulence

Starting point should be conservation laws

Kolmogorov 4/5-th law

S3(r) = ![v||(r) " v||(0)]3# = " 45!r

Constant Flux Relation (CFR)

!Ek

!t=

!JE

!k

!nk

!t=

!Jn

!k

Assume locality

JE , Jn independent of k

Obtain scaling of flux carrying correlation function

Constant Flux Relation (CFR)

!(3)e (k1, k2, k3) =

!d"1d"2d"3!Re[a(k1)a!(k2)a!(k3)]]"

!(4)e (k1, k2, k3, k4) =

!d"1d"2d"3d"4!Re[a(k1)a(k2)a!(k3)a!(k4)]]"

!(4)w (k1, k2, k3, k4) =

!d"1d"2d"3d"4!Im[a(k1)a(k2)a!(k3)a!(k4)]]"

y(3)e = !3d! !

y(4)e = !4d! !

y(4)w = !4d! !

Exact results

Assumption of locality

Check numerically for MMT model

T (k1,k2,k3,k4) = |k1k2k3k4|!/4

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

101 102 103

! 4-p

t "

k

# = 0.25; $ = 0.25exponent = 3.25

A “shell” model of wave turbulence

Would like to study n(k) for strongwave turbulence

Numerically easy to study shell type model

k

-k

k/2

-k/2

k

-k

2k

-2k

Now k = 2m, m!I

Summary of numerical results

Simulated for ! = 0 to 0.75 and " = !2.0 to 0.75

CFR results are validated

Only one cascade!

Direction of cascade fixed by sign of β

n(k) ! k!(5+!)/4

Summary of numerical results

10-5

10-4

10-3

10-2

10-1

100

101

102

103

104

10-4 10-3 10-2 10-1 100 101 102 103 104

n(k)

k

!=0.25; "=0.25exp=21/16

0

0.05

0.1

0.15

0.2

0.25

0.3

0 1000 2000 3000 4000 5000 6000

! w

t

DriveDissipate at large k

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 1000 2000 3000 4000 5000 6000

! ke

t

DriveDissipate at large k

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0 1000 2000 3000 4000 5000 6000

! pe

t

DriveDissipate at large k

Conclusions

• Strong wave turbulence

• Derived equivalent of 4/5-th law

• Toy model

★ Cascade is one direction

★ Exponent different from KZ

• How many of these results carry over

• How can one solve the “strong” limit