numerical studies of collapses in ... -...

Collapses inNLS Turbulence

N. Vladimirova andP. Lushnikov

Turbulence

Collapses

ExtraNumerical Studies of

Collapses in Turbulencein Nonlinear Schrodinger Equation

N. Vladimirova and P. Lushnikov

University of New Mexico

May 20, 2012

Turbulence

Collapses

Nonlinear Schrodinger equation (NLS)

iψt +∇2ψ + |ψ|2ψ = 0

iψt + (1− iεa)∇2ψ + (1 + iεc)|ψ|2ψ = iεbψ

application: propagation of light through nonlinear media;ψ(x , y , t) — envelope of electric field

Turbulence

pdf of |ψ|... side note

pdf of collapses

collapse similarity

F (hmax) → pdf (ψ)

Collapses

Distribution of |ψ| in the field

Notations: h ≡ |ψ|, havg = 〈|ψ|〉 ∝√

N, N =∫|ψ|2d2r .

0 1 2 3

h / havg

b = 20, ε=0.01

c=0, a= 1 c=1, a=1/4c=1, a= 1 c=1, a= 2 c=2, a= 1 2x exp(-x2)

1 10 100

h / havg

c=0, a= 1 c=1, a=1/4c=1, a= 1 c=1, a= 2 c=2, a= 1

Turbulent background:

I pdf scales with havg

I universal shape for h ∼ havg

Tails:

I power-law (?) for h�havg

I depend on a, c , not on b

Turbulence

pdf of collapses

collapse similarity

Collapses

Other field quantities scale with havg

Spatial correlation length (∼ havg ) ←Correlation time (∼ h2

avg ) ← depend on a, c , not on b

Spectra of |ψ|2k (∼ havg ) ←

When rescaled to havg ,

The shape of spatial correlation function is universal.The shapes of temporal correlation function and spectra

(slightly) depend on a, c .

Turbulence

pdf of collapses

collapse similarity

Collapses

Frequency of collapses with h > hmax.

0 100 200 300 400 500

c = 1, b = 20a= 1/4a= 1/2a= 1 a= 2

0 100 200 300 400 500F

a = 1, b = 20c= 0 c= 1/4c= 1/2c= 1 c= 2

I Frequency of collapses with h > hmax per unit area: F (hmax).

I The dependence is exponential.

I The exponent depends on a, c , and b.

I No obvious scaling with havg .

Turbulence

pdf of collapses

collapse similarity

Collapses

Universality of collapses in turbulence: a=1, c=0

Turbulence

pdf of collapses

collapse similarity

Collapses

Turbulence

pdf of collapses

collapse similarity

Collapses

Turbulence

pdf of collapses

collapse similarity

Collapses

Turbulence

pdf of collapses

collapse similarity

Collapses

Connecting PDF of |ψ| to frequency of collapses

0 50 100 150

c = 1, b = 20 a = 1/4a = 1/2a = 1 a = 2

0 50 100 150F

a = 1, b = 20 c = 1/4c = 1/2c = 1 c = 2

I assume similarity among collapses, h ∼ (tc − t)−12

I assume known frequency of collapses above hmax: Fmax(hmax)

I conclude that PDF in the field P(h) ∼ h−5Fmax(h)

I Solid lines: F (hmax). Points: 0.012 h5P(h).

Turbulence

Collapses

diagnostics

growth and decay

rescaled h(t)

maximum height

growth rate

ODE model

ODE terms

...side note

ODE test

Diagnostics of collapses

-1.5 -1 -0.5 0 0.5 1

I In slow variables ρ = rL and τ =

∫L−2(t ′)dt ′

ψ(r , t) = 1LV (ρ, τ)e iτ+iγ(τ)ρ2/4 (1)

critical NLS becomes

∇2V + V 3 − V + 14βρ

2V = 0. (2)

I Solve (2) for each β, tabulate V0(β) at the center.

I Taylor expansion of (2) at the center and (1) give

L = L ( |ψ|(0)|, |ψ|′′rr (0) ) ⇒ V0 ⇒ β; γ = 2L2φ′′rr (0)

Turbulence

Collapses

diagnostics

growth and decay

rescaled h(t)

maximum height

growth rate

ODE model

ODE terms

...side note

ODE test

Diagnostics of collapses

-1.5 -1 -0.5 0 0.5 1

I In slow variables ρ = rL and τ =

∫L−2(t ′)dt ′

ψ(r , t) = 1LV (ρ, τ)e iτ+iγ(τ)ρ2/4 (1)

critical NLS becomes

∇2V + V 3 − V + 14βρ

2V = 0. (2)

I Solve (2) for each β, tabulate V0(β) at the center.

I Taylor expansion of (2) at the center and (1) give

L = L ( |ψ|(0)|, |ψ|′′rr (0) ) ⇒ V0 ⇒ β; γ = 2L2φ′′rr (0)

Turbulence

Collapses

diagnostics

growth and decay

rescaled h(t)

maximum height

growth rate

ODE model

ODE terms

...side note

ODE test

Collapse growth and decay

-0.02 -0.01 0 0.01

t - tc

εa = 1e-4εa = 8e-5εa = 6e-5εa = 0

1e-05 1e-04 1e-03 1e-02 1e-01h

|t - tc|

|t-tc|-1/2

|t-tc|-1

Evolution of collapses with h0 = 2.8, r0 = 1 with c = 0, b = 0.Growth rate depends on a and c , decay rate does not.Notation: h ≡ |ψ|r=0.

Turbulence

Collapses

diagnostics

growth and decay

rescaled h(t)

maximum height

growth rate

ODE model

ODE terms

...side note

ODE test

Rescaled evolution: (a + 2c) similarity

-0.004 -0.002 0 0.002

t - tc

a = 2, c = 0c = 1, a = 0

-200 -100 0 100h

(t - tc) h2max

a = 2, c = 0c = 1, a = 0a + 2c = 0.2

Left:NLS coordinatescollapses with ε = 0.01, a + 2c = 2, and different h0.

Right:coordinated rescaled with hmax;same curve for collapses with a + 2c = 2,different for a + 2c = 0.2

Turbulence

Collapses

diagnostics

growth and decay

rescaled h(t)

maximum height

growth rate

ODE model

ODE terms

...side note

ODE test

Maximum height of collapses: (a + 2c) similarity

ε = 0.01

3 3.5 4 4.5

a+2c=2

a+2c=3

a = 0c = 0other

Turbulence

Collapses

diagnostics

growth and decay

rescaled h(t)

maximum height

growth rate

ODE model

ODE terms

...side note

ODE test

Growth rate of collapses: (a + 2c) similarity

3.63 4

1 102 104 106

(tc - t) h2max

ε(a+2c) = 1e-4ε(a+2c) = 2e-4ε(a+2c) = 5e-4ε(a+2c) = 1e-3ε(a+2c) = 2e-3

0.002 0.005 0.01 0.02 0.04

ε(a + 2c)

slope = 0.5

h ∝ (tc − t)−( 12 +α), α = α(a + 2c)

Left: h(t), compensated by (tc − t)−12 .

Right: coefficient α determined from dotted lines on the left panel.

Turbulence

Collapses

diagnostics

growth and decay

rescaled h(t)

maximum height

growth rate

ODE model

ODE terms

...side note

ODE test

ODE model for collapse growth(Fibich and Levy, 1998)

L2βt = −k1 − k2β + k3(LLt)2 − ν(β)

L3Ltt = −β

ν(β) = 45.1e−π/√

k1 = 2ε(a + 2c)N

k2 = εaP − N

k3 = εaP − N − 2M

N = 2π

ZρR2dρ ≈ 11.69

P = 2π

Zρ3R2

ρdρ ≈ 18.65

Zρ3R2dρ ≈ 3.46

Turbulence

Collapses

diagnostics

growth and decay

rescaled h(t)

maximum height

growth rate

ODE model

ODE terms

...side note

ODE test

Test of ODE model: term-by-term comparison

-0.008

-0.006

-0.004

-0.002

0.01 1 100 10000

L2 βt +

(tc - t) h2max

solid: k1 = ε(a+2c) 2N/ Mdashed: k1 = 0

-0.0015

-0.001

-0.0005

0.0005

0.0015

-0.06 -0.04 -0.02 0 0.02 0.04 0.06L2 β

k2 = -37ε(a+2c)

εa=1e-4εa=8e-5εa=6e-5εa=4e-5εa=2e-5ν(β)

L2βt = −k1 − k2β + k3(LLt)2 − ν(β)

L3Ltt = −β

Data suggest: k1 term describes first order effects well;k2 ≈ −37ε(a + 2c) instead of original k2 ≈ 2εa.

Turbulence

Collapses

diagnostics

growth and decay

rescaled h(t)

maximum height

growth rate

ODE model

ODE terms

...side note

ODE test

Side note on ODE model

-0.06 -0.04 -0.02 0 0.02 0.04 0.06

εa=1e-4εa=8e-5εa=6e-5εa=4e-5εa=2e-5

-2 -1 0 1 2

β k1-2/3

h/hmax = 1/cosh(x+1)

In slow variables keep first two terms of ODE model:

βτ = −k1 − k2β

Lττ − 2L2τ/L = −βL

Results in hhmax

= LmaxL

≈ cosh−1h−

“β − βmax

”|βmax|

where β = βk− 2

31 and βmax = −0.92

Turbulence

Collapses

diagnostics

growth and decay

rescaled h(t)

maximum height

growth rate

ODE model

ODE terms

...side note

ODE test

Test of ODE model: collapse evolution

-0.002 -0.001 0 0.001

t - tc

ε(a+2c) = 2e-4

NLS ODE, hic = 30FL, hic = 30FL, hic = 100

-0.002 -0.001 0 0.001β

t - tc

Turbulence

Collapses

parameter space

pdf of psi

spatial corr fun

time corr fun

spectra

critial collapse

beta and gamma

Parameter space: a, c , and b (ε = 0.01).

0 0.5 1 1.5 2h a

vg( N / area )1/2

0.85 xc=0, a=1 c=1, a=1 c=1, b=20a=1, b=20c=0, b=30

0 10 20 30 40

N/area = 0.09 b

I Notations: h ≡ |ψ|, havg = 〈|ψ|〉, N =∫|ψ|2d2r .

I No obvious scalings with a, c , or b......except N ∝ b and havg ∝

I We will show that havg well describes background turbulence.

I What determines havg?

Turbulence

Collapses

parameter space

pdf of psi

spatial corr fun

time corr fun

spectra

critial collapse

beta and gamma

Distribution of |ψ| in the field scales with havg

0 1 2 3

h / havg

b = 20, ε=0.01

c=0, a= 1 c=1, a=1/4c=1, a= 1 c=1, a= 2 c=2, a= 1 2x exp(-x2)

1 10 100P

h / havg

c=0, a= 1 c=1, a=1/4c=1, a= 1 c=1, a= 2 c=2, a= 1

I PDF of |ψ| scales with havg .

I Universal shape for h ∼ havg , power-law (?) tails for h�havg .

I The tails do not depend on b.

I The tails do depend on a and c .

Turbulence

Collapses

parameter space

pdf of psi

spatial corr fun

time corr fun

spectra

critial collapse

beta and gamma

1 10 100

h / havg

c = 0, a = 1

b = 30b = 20b = 10

1 10 100P

h / havg

c = 1, a = 1

b = 30b = 20b = 10

Turbulence

Collapses

parameter space

pdf of psi

spatial corr fun

time corr fun

spectra

critial collapse

beta and gamma

1 10 100

h / havg

c = 1, b = 20

a = 1/4a = 1/2a = 1 a = 2

1 10 100P

h / havg

a = 1, b = 20

c = 0 c = 1/4c = 1/2c = 1 c = 2

Turbulence

Collapses

parameter space

pdf of psi

spatial corr fun

time corr fun

spectra

critial collapse

beta and gamma

Spatial correlation function scales with havg

r havg

c = 0, a = 1

b = 30b = 20b = 10Gauss

0 1 2S

r havg

c = 1, a = 1

b = 30b = 20b = 10Gauss

I Spatial correlation function scales with havg .

I Correlation length does not depend on b.

I Correlation length depend on a and c .

I The shape, when rescaled to correlation length, is universal.

Turbulence

Collapses

parameter space

pdf of psi

spatial corr fun

time corr fun

spectra

critial collapse

beta and gamma

Spatial correlation function scales with havg

r havg

c = 1, b = 20

a = 1/4a = 1/2a = 1 a = 2

0 1 2S

r havg

a = 1, b = 20

c = 0 c = 1/4c = 1/2c = 1 c = 2

I Spatial correlation function scales with havg .

I Correlation length does not depend on b.

I Correlation length depend on a and c .

I The shape, when rescaled to correlation length, is universal.

Turbulence

Collapses

parameter space

pdf of psi

spatial corr fun

time corr fun

spectra

critial collapse

beta and gamma

Temporal correlation function scales with h2avg

0 0.2 0.4 0.6 0.8 1t h2

c = 0, a = 1

b = 30b = 20b = 10

0 0.2 0.4 0.6 0.8 1t h2

c = 1, a = 1

b = 30b = 20b = 10

I Temporal correlation function scales with h2avg .

I Correlation time does not depend on b.

I Correlation time depend on a and c .

Turbulence

Collapses

parameter space

pdf of psi

spatial corr fun

time corr fun

spectra

critial collapse

beta and gamma

Temporal correlation function scales with h2avg

0 0.2 0.4 0.6 0.8 1t h2

c = 1, b = 20

a = 1/4a = 1/2a = 1 a = 2

0 0.2 0.4 0.6 0.8 1t h2

a = 1, b = 20

c = 1/4c = 1/2c = 1 c = 2

I Temporal correlation function scales with h2avg .

I Correlation time does not depend on b.

I Correlation time depend on a and c .

Turbulence

Collapses

parameter space

pdf of psi

spatial corr fun

time corr fun

spectra

critial collapse

beta and gamma

Energy spectra scale with havg

0.1 1 10 100

| ψk|

k / (2π havg)

c = 0, a = 1

-5b = 30b = 20b = 10

0.1 1 10 100| ψ

k / (2π havg)

c = 1, a = 1

-6b = 30b = 20b = 10

I Spectra of |ψ|2k scale with havg .

Turbulence

Collapses

parameter space

pdf of psi

spatial corr fun

time corr fun

spectra

critial collapse

beta and gamma

0 0.2 0.4 0.6 0.8 1

| ψk|

k / (2π havg)

c = 0, a = 1b = 30b = 20b = 10

0 0.2 0.4 0.6 0.8 1| ψ

k / (2π havg)

c = 1, a = 1b = 30b = 20b = 10

Turbulence

Collapses

parameter space

pdf of psi

spatial corr fun

time corr fun

spectra

critial collapse

beta and gamma

0.1 1 10 100

| ψk|

k / (2π havg)

c = 1, b = 20

-6a = 1/4a = 1/2a = 1 a = 2

0.1 1 10 100| ψ

k / (2π havg)

a = 1, b = 20

-5c = 1/4c = 1/2c = 1 c = 2

Turbulence

Collapses

parameter space

pdf of psi

spatial corr fun

time corr fun

spectra

critial collapse

beta and gamma

Test: critcal collapses

1e-05 1e-04 1e-03 1e-02 1e-01

tc - t

h0 = 2.76h0 = 2.80h0 = 2.85

1e-05 1e-04 1e-03 1e-02 1e-01L

tc - t

Turbulence

Collapses

parameter space

pdf of psi

spatial corr fun

time corr fun

spectra

critial collapse

beta and gamma

1e-04 1e-03 1e-02 1e-01

tc - t

-L3 Ltt

(tc - t)2 L4/4

1e-04 1e-03 1e-02 1e-01

tc - t

LLt h0 = 2.76h0 = 2.80h0 = 2.85

Turbulence

Collapses

parameter space

pdf of psi

spatial corr fun

time corr fun

spectra

critial collapse

beta and gamma

-0.0003

-0.0002

-0.0001

0.07 0.071 0.072 0.073

L2 βt

h0 = 2.80-0.65 ν(β)

-0.003

-0.002

-0.001

0.09 0.1 0.11 0.12L2 β

h0 = 2.85-0.50 ν(β)

Turbulence

Collapses

parameter space

pdf of psi

spatial corr fun

time corr fun

spectra

critial collapse

beta and gamma

Beta and gamma with stabilization

-0.02 -0.01 0 0.01

t - tc

εa = 1e-4εa = 8e-5εa = 6e-5εa = 0

1e-05 1e-04 1e-03 1e-02 1e-01h

|t - tc|

|t-tc|-1/2

|t-tc|-1

Evolution of collapses with h0 = 2.8, r0 = 1 with c = 0, b = 0.

Here, ε(a + 2c) = 0.01. Notice that (a+2c) similarity is moreprononeced in L and its derivatives rather than in β and γ.

Turbulence

Collapses

parameter space

pdf of psi

spatial corr fun

time corr fun

spectra

critial collapse

beta and gamma

1e-05 1e-04 1e-03 1e-02 1e-01

tc - t

- L3 Ltt

1e-05 1e-04 1e-03 1e-02 1e-01

tc - t

εa = 1e-4εa = 8e-5εa = 6e-5εa = 0

Turbulence

Collapses

parameter space

pdf of psi

spatial corr fun

time corr fun

spectra

critial collapse

beta and gamma

0.1 1 10 100 1000

(tc - t) h2max

β, a = 0-L3Ltt, a = 0β, c = 0

-L3Ltt, c = 0

0.1 1 10 100 1000

(tc - t) h2max

γ, a = 0LLt, a = 0γ, c = 0LLt, c = 0

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