Interpretation of
two-dimensional turbulence spectrum
in terms of singularity in the vortex cores
Marie FARGE
LMD-CNRS, Ecole Normale Supérieure,
24, rue Lhomond. 75231 Paris Cedex 5, France
Matthias HOLSCHNEIDER
Centre de Physique Théorique, CNRS
Luminy, case 907. 13288 Marseille Cedex, France
PACS. 47.25.Cg - Fluid dynamics, isotropic turbulence.
Abstract
We propose a new geometrical interpretation of two-dimensional turbulence spectrum,
supposing the presence of at least one cusp-like axisymmetric coherent structure in the flow. We
compute a relation between the spectral slope and the exponent of the singularity. We predict a
sensibility of the two-dimensional flow dynamics to the regularity of the initial vorticity field: if
it is initially regular, the coherent structures formed will have a flat core, and if it is not regular
they will have a cusp-like shape with only the vortex core regularized by dissipation.
Introduction
A generic behaviour of two-dimensional turbulent flows is the emergence of coherent
structures which are observed in both laboratory /8/, /19/ and numerical experiments /12/, /3/,
/18/, /10/. The statistical theory of two-dimensional turbulence /16/, /4/ predicts a k-3 energy
spectrum, but most numerical experiments indeed observe a steeper slope, around k-4 /3/, /18/,
/10/. To explain this discrepancy some adjustments have been made to the statistical theory of
two-dimensional turbulence which suppose an intermittency of the enstrophy transfer in the
inertial range /17/, /3/, /2/. We propose here a geometrical theory of two-dimensional turbulence
based on a local scaling of some coherent structures, which does not refer to any statistical
mechanics arguments. This scaling seems to be confirmed by both numerical /6/ and laboratory
experiments /19/.
1.
Local scaling properties in axisymmetric vortices
Consider a rotational invariant solution {=(r), =(r)} of Euler equation describing the
time evolution of a perfect incompressible fluid:
(1a) ∂/∂t + J() = 0,
(1b) = 

where is defined as the vorticity, the streamfunction, and J the Jacobian

We assume that in the reference frame where this vortex solution is at rest, the
axisymmetric approximation will always be valid for small values of r. All scaling laws presented
herein hold only asymptotically, whereas at infinity it is assumed that all functions are rapidly
decreasing.
If a power-law behaviour is assumed for 
(2)
 ~ r,
equation (1b) implies that
(3)
 ~ r+2 + 
where 0 is an arbitrary constant.
By definition of the streamfunction,
(4)
v ~ r+1 + v0,
where v0 is an arbitrary constant.
Combining (2) and (3) leads to the function (), defined to be the coherent-structure function:
(5)
() ~ (-0)/(+2),
where 0 is the value of the streamfunction at the center of the vortex.
In order to have a finite circulation,
r
(6)
(r) = 2∫ (r')r'dr' ~ r
0
 should be restricted to values larger than -2.
In order to have a finite total enstrophy,
r
(7)
(r) = 2∫ (r')r'dr' ~ r
0
 should be restricted to values larger than -1.
In order to have a finite total energy
r
(8)
(r) = 2∫
v(r')r'dr' ~ r
0
 should be restricted to values larger than -2.
These constraints and (3) imply in particular that  is regular, having at worst a cusp-like
singularity, which in turn shows that 0, in the limit r  0, is finite.
The Fourier transform of the velocity (4) leads to the equation,
∞
(9)
v(k) = k-2 ∫ v(r'/k)J0(r') r'dr' ~ k--3,
0
where J0 is the zeroth-order Bessel function.
Taking the energy density as
(10)
v2(k) ~ k--6,
and integrating over shells of constant |k| leads to
(11)
E(k) ~ k-2-5.
2.
Experimental identification of the exponent 
It has been found that coherent structures can be characterized by a point-wise correlation
between vorticity  and streamfunction , that we have defined as the coherent-structure
function (). Results from both laboratory experiments /19/ and numerical simulations /6/
suggest that  ~ with <0, which implies a cusp-like shape for the coherent-structure
function. Benzi et al. /6/ have shown that for two-dimensional decaying turbulence the coherent
structures can be described by a universal axisymmetric function  ~ f(r/r0) with r0 a rescaling
constant. Benzi et al.'s /6/ figure 6 shows that /(2) scales as r-2, which corresponds to the
scaling obtained with our model by combining (6) and (7), scaling indeed independent of . In
Benzi et al.'s /6/ figure 7, the asymptotic behaviour (in the limit r  0) of 2 approximately
follows a r-1 law. Therefore ~ -1/2, which satisfies the finite energy (8), circulation (6) and
enstrophy (7) constraints and leads us to the coherent-structure function () scaling as ~ -1/3,
result confirmed by looking at Benzi et al.'s /6/ figure 4. Another reason supporting the singular
behaviour of coherent structures can be found in Nguyen Duc and Sommeria's /19/ figure 9,
which shows the same type of nonlinear coherent-structure function obtained from experimental
data on a two-dimensional decaying turbulent flow in a layer of mercury submitted to a transverse
magnetic field. In addition to Nguyen Duc and Sommeria's /19/ comment concerning the fact that
the first derivative of the coherent-structure function ∂/∂ reaches its maximum at the vortex
core, it is to be noted that the finite energy constraint (>-2) implies -∞<≤1. In
consequence, as tends to 0 i.e. in the vortex cores, ∂/∂ is either asymptotically constant
(=∞), corresponding to regular coherent structures with a flat core, or diverges, corresponding to
singular coherent structures. This is of course true only in the asymptotic region where the
axisymetric vortex hypothesis is valid.
3.
Interpretation of the power-law spectrum for two-dimensional
The scaling exponent  ~ -1/2 identified from Benzi et al.'s /6/ numerical results leads us
to the prediction that the coherent structures they observe have a cusp-like shape with a singular
core. This result seems confirmed by the wavelet analysis of coherent structures which showed
that the smallest scales of a two-dimensional flow are confined inside some vortex cores /9/,
/10/, /11/. This is also illustrated by visualizing the vorticity field, which shows the cusp-like
shape of some of the vortices (Figure 1). Such cusp-like vortices are likely to be stable, because
they present a monotonic distribution of vorticity /1/.
In a bounded domain, assuming the regularity (at least C1+) of the initial conditions, the
two-dimensional Euler flow preserves regularity (C ∞) and boundness (L ∞) for arbitrary finite
times /21/, /15/, /7/ and therefore if the initial vorticity field is regular the flow will not develop
singular coherent structures. But, if we consider functions L4-for the initial conditions, which
turbu
then present singularities in the vorticity field although circulation (6), enstrophy (7) and energy
(8) are bounded, due to the Lagrangian conservation of vorticity by Euler equation (1) in
dimension two it follows that, if N singular points are present in the initial vorticity field, they
will remain singular for all times, being only advected by the flow dynamics. If there is some
dissipation, we conjecture that those singular points will be the germ for the condensation of the
vorticity field into a set of N coherent structures, a number which will then be reduced due to the
merging process permitted by viscosity. The vortices thus formed will present a cusp-like shape
but their centers will no longer be singular due to the smoothing effect of dissipation.
Using (11) we predict that such a singular vortex alone will have an energy power-law
spectrum E(k) ~ k-4. This k-4 spectral law is steeper than the k-3 spectrum predicted by the
statistical theory of two-dimensional turbulence /16/, /4/, but corresponds to the energy spectrum
found in most of the numerical simulations /3/, /18/, /10/. Here the result is obtained without any
statistical argument but via a geometrical approach supposing the presence in the flow of at least
one cusp-like vortex scaling as r-1/2 (among a set of more regular structures, vortices and
vorticity filaments), because the spectral slope is conditioned by the scaling of the strongest
singularity supposing all singularities are isolated. In the case of accumulation of singularities
/13/, /14/, the spectral slope may be altered due to the fractal properties of the support of the
singularities. Such a geometrical interpretation in terms of singularities had previously been
devised for two-dimensional turbulence by Saffman /20/, giving also a k-4 spectrum, but the
singularities he proposes are not axisymmetric as the coherent structures observed in laboratory
and numerical experiments, and correspond instead to a distribution of the vorticity along some
fronts which are only observed numerically during the very early time evolution before the
emergence of coherent structures.
To conclude our analysis of Nguyen Duc and Sommeria's /19/ experiment: we predict that
the vortices having a linear coherence law correspond to smooth initial distributions of the
vorticity field, while the vortices presenting a nonlinear coherence law have a cusp-like shape and
have been created by a condensation of the vorticity field around some singularities that were
present in the initial conditions. This conclusion also applies to the interpretation of numerical
experiments of two-dimensional turbulent flows: if the initial enstrophy is confined in a limitedband spectrum /5/, then the initial vorticity field is smooth and the structures which may appear
in this field during the time evolution would be very regular with flat vortex cores, while if the
enstrophy is initially distributed among all the available scales, namely until the cut-off scale as it
is the case for most of the numerical experiments /18/, /6/, /10/ in two-dimensional turbulence,
then the initial distribution of vorticity will contain some singularities, which will then act as
germs for the condensation of the vorticity field into the cusp-like coherent structures we observe
after a long time evolution of the flow (Figure 1). This sensitivity of the Euler dynamics to the
initial non-regularity of the flow, may impair the quest for a universal behaviour of fullydeveloped turbulence in dimension two.
Conclusion
In conclusion, we propose a new 'picture' of two-dimensional turbulent flow, in which the
spectrum found in numerical experiments is interpreted in geometrical terms without any
statistical argument, supposing the presence in the flow of at least one singular coherent structure
scaling as r-1/2. We conjecture that these singular points are already present in the initial
conditions and that the flow dynamics selects and condenses the vorticity field around them,
giving rise to isolated coherent vortices with singular cores in the inviscid limit, or cusp-like
vortices with a core smoothed by dissipation otherwise.
k-4
Saint-Polgues, 17 April 1990
Acknowledgments
We thank Claude Bardos, Uriel Frisch, Robert Kiehn, Robert Kraichnan, Yves Pomeau,
Antonello Provenzale, Raoul Robert, Robert Sadourny and Jérôme Sirven for useful comments.
The visualization has been done at LACTAMME, Ecole Polytechnique, in collaboration with
Jean-François Colonna.
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