vortex self-similarity in unforced inviscid two...

Vortex self-similarity in unforcedinviscid two-dimensional turbulence

David Dritschel†, Chuong Tran†, Richard Scott†

Charlie Macaskill‡ and Georg Gottwald‡

Schools of Mathematics and Statistics

Universities of St Andrews† and Sydney‡

Dritschel, Tran, Scott, Macaskill & Gottwald

The Plan

⋄ A brief history ... cascade theories of Kraichnan, Batchelor & Charney

⋄ Small-scale vortex self-similarity and k−5 ... oops! — no cascade

⋄ Large-scale equipartition and Onsager’s vortex clustering

⋄ Equilibrium spectra and turbulent relaxation, not decay!!

⋄ Inviscid 3D quasi-geostrophic turbulence.


A brief history

Two-dimensional turbulence has long been considered an ideal testing ground fortheories, due to computational simplicity as well as the mathematical regularity of theNavier–Stokes and Euler equations (Davidson, 2004).

Most such theories until recent times have focused on the cascade of enstrophy, or

mean-square vorticity, to small scales.

Kraichnan (1967,1971) and Batchelor (1969) developed much of the theory for bothforced and freely-decaying flows at large but finite Reynolds numbers.

In the freely-decaying case, Batchelor predicted a k−3 small-scale energy spectrumover scales not directly affected by viscosity.

Tran & Dritschel, JFM (2006) showed that Batchelor’s assumption of finite enstrophydissipation in the inviscid limit is inconsistent.


Our starting assumptions

We focus here on idealised turbulence governed by the 2D Euler equations and theanalogous 3D quasi-geostrophic (QG) equations.

We don’t take the inviscid limit — we simply drop viscosity.

We consider an infinite, unbounded space.

Time and space scales are then irrelevant (we can take them to be unity).


Similarities between 2D Euler and 3D QG

The 3D QG equations govern the motion of a rapidly rotating, stably-stratifiedgeophysical fluid.

Both systems, 2D and 3D QG, have a materially conserved quantity, vorticity orpotential vorticity, respectively.

Both systems evolve purely by two-dimensional advection, which is layerwise in 3D

QG flow, with no vertical motion.

Both systems exhibit an inverse energy cascade to large scales and a direct enstrophycascade to small scales.

Both systems are predicted to have a k−3 small-scale energy spectrum (Batchelor,1969; Charney, 1971).


The 2D Euler equations in vorticity-streamfunction form

With a streamfunction ψ, an incompressible flow u = (u, v) can be represented by

u = −ψy & v = ψx

and ψ is found by inverting Laplace’s operator on the vorticity ω:

∇2ψ = ω

The dynamical evolution of the flow is controlled by material conservation of vorticity:

Dω

Dt≡ ωt + u · ∇ω = 0

The same equations govern a 3D QG fluid, only then ∇2 is the 3D Laplace operator.


Vortex self-similarity in 2D

We first consider moderate to small scales, where vortex interactions dominate.

We derive the form of the energy spectrum E(k) for a dilute gas of vortices.

The form of E(k) is uniquely determined by the vortex distribution —

only one power-law spectrum is both self-similar and physically realisable.


In order to work with finite forms of energy and enstrophy, the infinite plane is tiledby squares of area As. In each tile, we compute the kinetic energy |u|2/2, then averageit over all tiles to obtain the energy spectrum E(k):

∫ ∞

0

E(k) dk =1

2〈|u|2〉

Similarly, we may obtain the enstrophy spectrum Ω(k):

∫ ∞

0

Ω(k) dk =1

2〈ω2〉 .

Note that Ω(k) = k2E(k).


As

Now consider a dilute gas of vortices which have equal vorticity magnitude ωv andwhose sizes are distributed according to a number density n(A).


We expect that the vortices have scales smaller than the energy-enstrophy scaleL =

√

E/Q, where E =∫

Edk and Q =∫

Ωdk are the conserved energy and enstrophyper unit area.

Then, the average number of vortices in a subdomain of area A0, counting only

vortices with areas exceeding λ2A0, is

Nv =

∫ A0

λ2A0

n(A)dA . (1)

Likewise, the (average) fraction of the area covered by vortices in this subdomain is

fv =1

(1 − λ2)A0

∫ A0

λ2A0

An(A)dA . (2)


The average enstrophy in an area As is proportional to the area occupied by vorticesdivided by As, i.e.

1

2〈ω2〉 =

1

2ω2v

∫ As

0An(A) dA

As. (3)

Following Benzi et al (1992), we can relate this to Ω(k) if we identify k with A−1/2:

Ω(k)dk ∼ω2v

Asn(A)k−5 dk , (4)

using dA ∼ k−3dk and dropping O(1) factors. As Ω(k) = k2E(k), we find

E(k) ∼ ω2vA

−1s k−7n(A) . (5)


We now consider several powerlaw forms for n(A) in order to explore the possibilityof self-similar behaviour.

We assume that the mean vorticity of a vortex ωv is independent of A.

— ωv cannot increase with decreasing A since then the vorticity would be unboundedin the limit A→ 0.

— ωv cannot decrease with decreasing A since then very small vortices would alsobe very weak, and would be overwhelmed by the strain coming from large scales >

∼ L

(Kida 1981, Dritschel 1990).

Consider first n(A) = c/A2 where c is a constant. Then, using A ∼ k−2, we find from(5) that this number density corresponds to

E(k) ∼ c ω2vA

−1s k−3 , (6)

— the classical small-scale spectrum (Batchelor 1969, Kraichnan 1971).


However, the average number of vortices in a subdomain

Nv =c

λ2A0

−c

A0

(7)

increases with decreasing area A0. Moreover, the area fraction occupied by vortices

fv =c log(λ−2)

(1 − λ2)A0

(8)

also increases with decreasing area A0 — indeed it diverges as A0 → 0.This is physically impossible (fv ≤ 1).

The classical k−3 spectrum does not correspond to a self-similar vortex population.


Consider next n(A) = c/A. We find from (5) that this number density corresponds to

E(k) ∼ c ω2vA

−1s k−5 , (9)

— significantly steeper than the classical k−3 spectrum, but not at all inconsistentwith the results of McWilliams (1984), Santangelo, Benzi & Legras (1989), Braccoet al (2000), and many others. The average number of vortices in a subdomain is

Nv = c log(λ−2) (10)

and now is independent of the area A0.

Furthermore, the area fraction occupied by vortices is

fv = c (11)

and is also independent of the area of the subdomain.


This means that — on average — one would count the same number of vorticesin any subdomain, irrespective of its area, and one would find the same area fractioncovered by the vortices.

This is self similar. No other form of n(A) exhibits this property.

Any steeper form n(A) = c/Ap, p > 1 implies an unbounded area fraction fv as thesubdomain area A0 → 0. This is unphysical.

The implication is that any energy spectrum shallower than k−5 at small scales cannotbe due to vortices; instead it must be due to filamentary debris.


k1.

k−5.

k−3.

log10 E .

log10 k .

−1

−3

−4

−5

−6

−7

0 1 2 3

−2

Energy spectra at

t = 24 (thin solid line),

t = 26 (long dashed line),

t = 28 (bold solid line) and

t = 30 (short dashed line)

eddy turnaround times

in a high-resolution CASL simulation

of 2D turbulence in a 2π × 2π

doubly-periodic box starting from

E = Ck3 exp[−2k2/k20], with k0 = 32.


Large-scale dynamics

The enstrophy cascade has preoccupied research on 2D turbulence. The large-scaledynamics has seen relatively little attention, and yet is intimately connected.

Until recently, for example, no rigourous constraint on the inverse cascade was known(Tran & Dritschel, Phys. Fluids 2006). This paper proved that

E(k, t) ≤ Ck3t2

for some constant C proportional to the square of the total energy, starting fromE(k, 0) = 0 over this wavenumber range.

It is widely accepted that the inverse cascade implies that turbulence seeks everlarger scales

— but is this true for an inviscid fluid?


Consider scales greater than the largest vortex size Lm.

For a sufficiently dilute population of vortices, it is plausible that the largest vortexdoes not grow forever but reaches a finite limit, or fluctuates about a finite value dueto rare interactions— many of which do not result in a growth in vortex size,cf. Dritschel & Waugh (1992) and Dritschel & Zabusky (1996).

Yet, scales larger than Lm continue to be dynamically active, and may exhibitcoherent motions associated with e.g. clusters of like-signed vortices, as forseen byOnsager (1949).

Onsager used a thermodynamical analogy to predict clustering depending onproperties of the initial vortex distribution. He considered point vortices having finitecirculations but infinitesimal size.


We revisit this model to study the development of large-scale order from an initialsmall-scale reservoir containing disorganised or incoherent motions.

We used a standard particle-in-cell algorithm which interpolates the vorticity field to

a grid in order to compute the velocity (at grid points) efficiently (using FFTs).

The point vortices are advected by the (bi-linearly) interpolated velocity field.


Experimental Set-up

In each grid box, we uniformly placed 4 × 4 = 16 point vortices of alternating signs.

• • • •

• • • •

• • • •

• • • •

The vortices then cancel each other at the level of the grid, so that initially there isno enstrophy (or energy) above grid scale.

The vortex strengths are chosen so that the gridded vorticity would equal 4π forvortices all of the same sign.

To get things going, the vortices are randomly displaced in x and in y by a randomnumber between −0.001∆x and +0.001∆x, where ∆x is the grid length.

Grid resolutions of 2562, 5122 and 10242 were used.


Energy Spectra E(k, t) (log-log scales): 10242 resolution

t = 0 t = 100 to 400

log10 k .

log10 E .

k1.

0 1 2 3

−18

−17

−16

−15

−14

−13

−12k1

.

k3.

log10 E .

log10 k .

0 1 2 3

−10

−11

−12

−13

−9

−8

−7


a long time later ...

t = 500 to 2000 t = 2000

k1.

k3.

log10 E .

log10 k .

0 1 2 3

−10

−11

−12

−13

−9

−8

−7k1

.

k3.

log10 E .

log10 k .

0 1 2 3

−10

−11

−12

−13

−9

−8

−7


105E .

t.

Q.

1000 20000

1

2

3

4

0

Total energy E

and enstrophy Q

in wavenumbers k ≤ 512

versus time t.


Streamfunction Evolution

ψ(x, 200) ψ(x, 500) ψ(x, 2000)

Onsager’s clusters or a dipole gas?


The invasion of the k3 spectrum by a k1 spectrum is a consequence of equipartition,

in which a linear combination of energy |u|2 and enstrophy k2|u|2 spreads itself uniformlyamong the Fourier modes (Kraichnan, 1967).

This gives rise to the equipartition spectrum

Eeq(k) =a1k

k2 + c2(12)

where a1 and c are determined from

E =

∫ kmax

0

Eeqdk & Q =

∫ kmax

0

k2Eeqdk

and kmax is the maximum wavenumber in the truncated inviscid dynamical model(Fox & Orszag, 1973).


The point vortex simulation also has truncated dynamics.

This appears to be sufficient for the flow to approach equipartition.

However the equilibrium energy spectrum (12) does not explain the inverse energycascade at very large scales. We may model this by

E(k, t) ∼ak3

(k2 + b2)(k2 + c2)(13)

and find a(t), b(t) and c(t) from E, Q and the integral P (t) =∫ kmax

0k−2Edk = 〈ψ2〉.


log10 E .

log10 k .

−7

−8

−9

−11

−10

3210

300

800

2100 Actual Energy spectra E(k, t)

at t = 300, 800 and 2100

(thin lines)

compared to the

hybrid-equipartition spectra

ak3/[(k2 + b2)(k2 + c2)]

(bold lines)


0.0196 + 0.000204t.

t.

b−1.

0 20001000

0.4

0.3

0.2

0.1

0

t.

3 × 106a.

c.

0 20001000

220

210

200

180

190

a(t) and c(t) rapidly approach constantswhile 1/b(t) is bounded by a linear function of time

(Tran & Dritschel, Phys. Fluids 2006)


A model for all scales

The central assumption of our model is that the largest vortex size ∼ Lmasymptotes to a constant value as t→ ∞.

Vortex interactions are expected to become rare and inefficient as the populationbecomes dilute and intermingled filamentary debris cascades to ever finer scales.

In time, we anticipate that the small-scale spectrum steepens to k−5

... while the large-scale spectrum extends its k1 range.


k1.

log10 E .

log10 k .

k3.

k−5.

k−3.

−1

−3

−4

−5

−6

−7

0 1 2 3

−2 Energy spectra at

t = 2 (thin solid line),

t = 6 (long dashed line),

t = 14 (bold solid line) and

t = 30 (short dashed line)

eddy turnaround times

in a CASL simulation

of 2D turbulence.


Equilibrium spectrum

In the limit t→ ∞, we propose that the energy spectrum E(k, t) → E∞(k), where

E∞(k) =ak

(k2 + c2)(k2 +m2)2. (14)

Here m ∝ 1/Lm is the wavenumber associated with the maximum vortex size.

For k ≫ m, E∞(k) ∝ k−5, the self-similar vortex spectrum.

For fixed energy and enstrophy, say E = Ω = 1/2, (14) describes a one-parameterfamily of spectra, depending only on m, the ratio of the energy-enstrophy scale

(here 1) to the maximum vortex size.


The coefficients A few spectra

c. a.

m.

a.

c.

1

1.0

0.8

0.6

0.4

0.2

0

0

2

4

6

8

10

2 3

log10 E .

log10 k .

1

0

−1

−2

−3

−4

−5

m = 1

m = 3m = 2

−3 −2 −1 0 1

We conjecture that these spectra arise asymptotically, from all initial conditions, solong as the fluid is inviscid and unbounded, and has finite energy and enstrophy perunit area.


The approach to equilibrium

We argue that the classical k−3 spectrum is swept to ever smaller scales but carrieswith it a finite amount of enstrophy ∆Q.

We model this cascade by a transition in E(k, t) from k−5 to k−3 around a wavenumberf(t), and by truncating the spectrum at an ‘enstrophy front’ d(t).

A simple spectral form combining this with the large-scale evolution is

E(k, t) =ak3

(k2 + b2)(k2 + c2)(k2 +m2)2+

ek3

(k2 + f 2)3(15)

for k < d(t), and E(k, t) = 0 for k ≥ d(t).

For b, e→ 0 and f, d→ ∞, this spectrum converges to the equilibrium spectrum E∞(k).


b c log(k)

log E( )

m f


CASL palinstrophy spectra log10(k4E(k, t)) (thin) versus model spectra (bold)

t = 18 t = 24 t = 30

log10 k .

(a).

0 1 2 3

2

1

3

4

log10 k .

(b).

0 1 2 3

2

1

3

4

log10 k .

(c).

0 1 2 3

2

1

3

4


We now determine the time evolution of the coefficients.

The two terms in

E(k, t) =ak3

(k2 + b2)(k2 + c2)(k2 +m2)2+

ek3

(k2 + f 2)3

are comparable at k = f only if

e ≈a

f 2(16)

— referred to as the ‘transition condition’.

The second term carries away a finite amount of enstrophy, implying

∆Q =

∫ d

0

ek5dk

(k2 + f 2)3≈ e[log(d/f) + O(1)] (17)

assuming d/f ≫ 1.


Combining these two, we have

e ≈ ∆Q [log(d/f)]−1 (18)

f ≈ (a/∆Q)1/2 [log(d/f)]1/2 (19)

valid, as always, for sufficiently large t.

Hence, if ∆Q is truly a constant in the limit t→ ∞, and if d ∼ eγt, then

e ∼ t−1 (20)

f ∼ t1/2 (21)

(modulo log t factors). Here, the large-scale strain γ ∼ Q1/2.

These conclusions do not depend on our specific transition from the k−5 to the k−3

spectrum, but only on [1] the invariance of the k−5 part of the spectrum (constant a),[2] finite enstrophy dissipation ∆Q, and [3] the transition condition e ≈ a/f 2.


We can go further still and predict the evolution of the inverse scale of the largestvortex cluster, b(t).

To do this, we ensure that the total energy remains constant.

Balancing the energy decrease at small scales with the energy growth at large scalesgives

e

4f 2≈

ab2

c2m4log(c/b) . (22)

Inserting the forms of e and f from before and re-arranging implies

b ≈cm2∆Q

a[log(d/f)]−1 [log(c/b)]−1/2 . (23)

⇒ b ∼ t−1 for d ∼ eγt.


Notably, the same prediction b ∼ t−1 follows (without any log t factor) if one assumes

a piecewise-linear transition in log E expressed as a function of log k.

To summarise, [1] we have used the assumption of finite cumulative enstrophydissipation together with a transition from k−5 to k−3 in the energy spectrum around awavenumber k = f(t) to predict that f ∝ [log(d/f)]1/2. This implies f ∼ t1/2 if we furtherassume d(t) ∼ eγt.

[2] We then balanced the energy decay associated with the k−3 spectrum at smallscales with the growth at large scales to predict b ∝ [log(d/f)]−1, or b ∼ t−1 againassuming d(t) ∼ eγt.

Effectively, energy lost due to filament roll-up and thinning is gained by a wideningk1 range, corresponding to an increase in the size of the largest vortex cluster.

[3] Finally, we showed that this model is consistent with the mathematical boundon the growth of the energy spectrum E(k, t) at large scales.


In short, the proposed model is internally self-consistent.

It is also consistent with numerical results we have obtained at large and small scales,though a proper verification at much higher resolutions remains to be carried out.

Nevertheless, our model of the approach to equilibrium demonstrates that it ispossible to conserve energy and enstrophy and yet arrive, by relaxation and not decay,

to a non-trivial equilibrium spectrum E∞(k).

At any fixed k > m, we predict that E(k, t) steepens to k−5, the self-similar vortexspectrum, as the filaments roll up into vortices at progressively smaller scales.

Meanwhile, at large scales k < c, we predict that the k1 spectrum extends to everlarger scales, as vortex clusters grow in size.


vortex self-similarity in unforced inviscid two...

Documents