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
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