Turbulent Diffusion of Lines and Circulations
7 Turbulent Diffusion of Lines and Circulations
Gregory L. Eyink
Department of Applied Mathematics & Statistics
The Johns Hopkins University
3400 N. Charles Street
Baltimore, MD 21218
Tel: 410-516-7201, Fax: 410-516-7459
e-mail: eyink@ams.jhu.edu
Abstract
We study material lines and passive vectors in a model of turbulent flow at infinite-
Reynolds number, the Kraichnan-Kazantsev ensemble of velocities that are white-noise in
time and rough (Hölder continuous) in space. It is argued that the phenomenon of “spon-
taneous stochasticity” generalizes to material lines and that conservation of circulations
generalizes to a “martingale property” of the stochastic process of lines.
PACS: 47.27.Jv, 52.65.Kj, 02.50.Fz, 05.45.Df
keywords: turbulence, material lines, circulations, Kraichnan model, dynamo, fractals
http://arxiv.org/abs/0704.0263v1
The evolution of material lines and surfaces passively carried by turbulent flow has long
been a subject of interest [1]. This is motivated in part by questions surrounding dynamically
relevant objects, such as vortex lines [2, 3, 4] and magnetic field-lines [5, 6], which have been
argued to behave similar to material lines at high Reynolds numbers. However, in that limit, the
turbulent velocity field is no longer differentiable in space but only Hölder continuous [7, 8, 9].
Observations from experiments and simulations suggest that material objects advected by such a
rough velocity become fractal, with a Hausdorff dimension strictly greater than their topological
dimension [10, 11, 12, 13, 14, 15]. This poses a difficulty to the view that vortex lines behave
as material lines—a consequence of the Kelvin-Helmholtz theorem [16, 17]—since circulations
are a priori not defined for non-rectifiable loops. It has recently been argued that the Kelvin
theorem in fact breaks down in turbulent flows, in the sense that the circulation is not strictly
conserved for every loop [18, 19]. Similar breakdown of Alfvén’s theorem on magnetic-flux
conservation [20] is expected in plasma turbulence at high magnetic Reynolds numbers [21].
These questions have been sharpened by recent work on the Kraichnan model of advection
by a Gaussian random velocity field that is delta-correlated in time [22]. A novel phenomenon
has been discovered there called spontaneous stochasticity: Lagrangian particle trajectories for
a non-Lipschitz advecting velocity are non-unique and split to form a random process in path-
space for a fixed velocity realization [23, 24, 25, 26, 27, 28, 29]. This phenomenon raises many
fundamental questions, including whether material objects such as lines and surfaces can even
exist in the limit of infinite Reynolds number. It is the purpose of this Letter to outline a
new approach to the evolution of such geometric objects in the Kraichnan model. We focus on
material lines and passive vectors, which are dual objects in the same sense as material particles
and passive scalar fields [30]. In particular, we shall sketch the proof of a “martingale property”
that has previously been proposed [18] as a generalization of the conservation of circulations
for a rough velocity field.
We consider stochastic flows [31] on a d-dimensional manifold M driven by Brownian vector
fields that are not Lipschitz regular in space. To simplify the presentation, we use Euclidean
space M = Rd or the torus M = Td to illustrate the main ideas. More precisely, u(x, t) is a
Gaussian random vector field, with mean u(x, t) and fluctuating part ũ(x, t) with covariance
〈ũi(x, t)ũj(x
′, t′)〉 = Dij(x,x
′; t)δ(t − t′). (1)
for x,x′ ∈ M. We are mainly interested in the case that u(x, t) ≡ 0 and u(x, t) ≡ ũ(x, t) is a
homogeneous random field, with D(x,x′; t) = D(x− x′, t). The quantity
∆(x,x′; t) = tr[D(x,x; t) +D(x′,x′; t)− 2D(x,x′; t)] (2)
is 〈‖u(x) − u(x′)‖2〉, the mean of the Euclidean norm squared, for a random field u(x) with
covariance D(x,x′; t). The case of greatest interest to us has ∆(x,x′; t) ∝ ‖x − x′‖2α with
0 < α < 1. In that case, u(x) is Hölder continuous with exponent α at every point in space.
We consider oriented lines (1-cells) given parametrically as continuous, one-to-one maps
C : [0, 1] → M. A material line satisfies
(d/dt)C(σ, t) = u(C(σ, t), ◦ t). (3)
for σ ∈ [0, 1] and t ∈ R. The circle “◦” means that we interpret equation (3) in the Stratonovich
sense. The (forward) Ito equation (d/dt)C(σ, t) = u(C(σ, t), t) equivalent to (3) has the mean
changed to u∗i (x, t) = ui(x, t) + (1/2)(∂/∂x
k)Dik(x,x
′; t)|
′=x ([31], section 3.4.) If M = R
d and if u(x, t) is a homogeneous random field, then the Ito and Stratonovich interpretations of
equation (3) are equivalent. Now let P
[C, t] denote the conditional probability distribution of
lines for a fixed velocity realization u. This distribution satisfies a stochastic Liouville equation:
(d/dt)P
[C, t] = −
δCi(σ)
(ui(C(σ), ◦ t)Pu[C, t]) . (4)
Equation (4) is a direct consequence of equation (3) and must also be interpreted in the
Stratonovich sense. It is formally equivalent to the Ito equation:
(d/dt)P
[C, t] = −
δCi(σ)
([u∗i (C(σ), t) + ũi(C(σ), t)]Pu[C, t])
δCi(σ)δCj(σ′)
Dij(C(σ), C(σ
′); t)P
[C, t]
. (5)
Averaging equation (5) over the Gaussian ensemble of velocities ũ yields a functional Fokker-
Planck equation for distributions in the space of free lines C on the manifold M :
(d/dt)P [C, t] = −
δCi(σ)
([u∗i (C(σ), t)P [C, t])
δCi(σ)δCj(σ
Dij(C(σ), C(σ
′); t)P [C, t]
. (6)
The first term on the righthand side represents a drift with the mean velocity u∗ and the
second term represents a diffusion arising from the velocity covariance D. Similar diffusions
on the path- and loop-spaces of a manifold M have been much studied, motivated in part by
questions from quantum field theory [32, 33].
The above considerations are rigorously justifiable for the case of a Lipschitz velocity with
α = 1 but are only formal when α < 1. A more careful (and also more physically realistic)
approach in the latter case is to replace the advecting velocity u with a “coarse-grained” or
smoothed velocity uλ = ϕλ ∗ u, by convolution with a smooth filter kernel ϕλ(r) = λ
−dϕ(r/λ).
The length-scale λ can be interpreted as a mathematical representation of the viscous cutoff in
a true turbulent velocity field [25, 26]. The exact solution of the Liouville equation (4) for such
a smoothed velocity is
(dC, t|C0, t0) = δ(C − ξ
(C0))dC, (7)
with initial condition P
(dC, t0|C0, t0) = δ(C − C0)dC. Here ξ
: M → M is the stochastic
flow of diffeomorphisms generated by the smoothed velocity-field uλ ([31], section 4.6). Despite
(7), a nontrivial diffusion process in line-space can be obtained if the limit λ → 0 is taken
appropriately. Consider a “nice” distribution Gρ(dC) which is supported on lines entirely
contained in the ball B(0, ρ) of radius ρ at the origin 0 and take the weak limit
Gρ(dC
(dC, t|C0 + C
0, t0)Ψ(C) =
(dC, t|C0, t0)Ψ(C) (8)
for bounded, continuous functionals Ψ(C) and t > t0. In the “weakly compressible regime”
[24, 25, 28]— and, in particular, for a divergence-free velocity field satisfying ∇·u = 0— this
limit should yield a non-degenerate diffusion. 1 This is a generalization of the phenomenon
of spontaneous stochasticity to the turbulent advection of lines, with initial line C0 at time t0
splitting into a random ensemble of lines C at time t.
As for the case of smooth advection, an unconditional diffusion satisfying equation (6)
may be obtained by averaging over the velocity u. The instantaneous realizations C of this
diffusion process should be fractal objects when the advecting velocity is Hölder continuous
with exponent 0 < α < 1 and rigorous estimates of their Hausdorff dimensions would be of
much interest. These questions may also be addressed numerically using Lagrangian Monte
Carlo techniques [34, 35, 36]. In such a study, the material line C(t) would be represented by a
discrete approximation CN (t) constructed from N +1 Lagrangian particles xa(t), a = 0, ..., N :
CN (σ, t) = (1− θN (σ))xaN (σ)(t) + θN (σ)xaN (σ)+1(t). (11)
Here aN (σ) = [Nσ] with [x] the greatest integer less than or equal to x (modulo N for loops)
and θN (σ) = (Nσ) where (x) = x− [x] is the fractional part of x. Thus, (11) corresponds to a
piecewise-linear curve with linear segments connecting the successive Lagrangian particles. So
long as δN (t) = maxa |xa(t)−xa+1(t)| . λ, the discrete approximation CN (t) represents well the
material line C(t) and the approximation becomes better as N → ∞ and δN (t) ≪ λ. However,
the same is not true in the opposite limit, where λ ≪ δN (t). The phenomenon of “spontaneous
stochasticity” for a rough velocity field makes it very doubtful that material lines even exist if
the limit λ → 0 is taken before evolving in time and an initial line then presumably “explodes”
1We note that the same diffusion process should also be obtained by using Duhamel’s formula to solve equation
(5) as the Ito integral
Pu[C, t] = S
0 (t, t0)P [C, t0]−
(t, t
δCi(σ)
[eui(C(σ), t
)]Pu[C, t
, (9)
where S∗0 (t, t
′) = Texp
∗(t′)
(t) = −
δCi(σ)
i (C(σ), t)·) +
δCi(σ)δCj(σ′)
Dij(C(σ), C(σ
); t)·
is the Fokker-Planck operator of the (mean) diffusion in line-space. Equation (9) can be solved iteratively to
generate a representation Pu[C, t] = S
(t, t0)P [C, t0] as a Wiener chaos expansion in white-noise eu; cf. [28, 29].
into a disconnected cloud of particles at any time t > 0. Thus, the velocity smoothing in (8)
appears to be necessary to define appropriately a line-diffusion for a rough (Hölder) velocity.
Alternatively, a stochastic regularization might be employed that adds a white-noise κdW (t)
to the evolution equation of Lagrangian particles [30].
We now turn to the dual problem of a passive vector (1-form) A advected by the velocity
u = u+ ũ:
∂tA(x, t) + (u(x, ◦t)·∇)A(x, t) + (∇u(x, ◦t))A(x, t) = 0, (12)
This stochastic equation is interpreted again in the Stratonovich sense. Equation (12) for d = 3
is equivalent by vector calculus identities to ∂tA+∇(u·A)− u×(∇×A) = 0. The latter has
the form of Ohm’s law,
E+ u×B = ηJ, (13)
in the ideal limit of zero resistivity (η = 0) for an electric field E = −∂tA − ∇(u·A) and
magnetic field B = ∇×A given by a vector potential A, with the electric current J = ∇×B.
Taking the curl of (13) yields an induction equation ∂tB = ∇×(u×B)+η△B for the magnetic
field. With this interpretation, the passive vector equation was introduced by Kazantsev [37]
as a soluble model of the kinematic dynamo. (See also [38, 39, 40].) The “circulation” (or
“holonomy”) of A along C is defined in Lagrangian form as
ΦL(C, t) =
A(t)·dx, (14)
where C(t) is the material line advected by u(t) which started as line C at the initial time
t = t0. Conservation of “circulation”, (d/dt)ΦL(C, t) = 0, follows formally from (12) for any
space dimension d ≥ 1. It is rigorously true for the case of a smooth advecting velocity with
α = 1 ([31], section 4.9). If C is a closed loop (1-cycle), then the line-integral (14) represents
gauge-invariant magnetic flux and the conservation law corresponds to Alfvén’s theorem [20].
We now consider the generalization of this result for α < 1. For this purpose, it is useful to
reformulate the passive vector equation (12) as a passive scalar in line-space:
∂tΦE(C, t) +
dσ [ui(C(σ), t) + ũi(C(σ), ◦t)]
δCi(σ)
ΦE(C, t) = 0. (15)
In this equation, ΦE(C, t) =
A(t)·dx is the Eulerian circulation of A along a fixed (non-
advected) line C. An exactly analogous reformulation of the incompressible Euler equation
(as an active scalar in loop-space) was advanced some time ago by Migdal [41]. Note that
conservation of circulations is just the formal solution of (15) by the method of characteristics.
We shall take the equation (15) as our primitive formulation of the passive vector; one of
the immediate advantages is that we can avoid (for the moment) the question how to define
line-integrals over fractal lines. We then convert (15) to Ito formulation:
∂tΦE(C, t) = −
dσ [u∗i (C(σ), t) + ũi(C(σ), t)]
δCi(σ)
ΦE(C, t)
dσ′ Dij(C(σ), C(σ
′), t)
δCi(σ)δCj(σ′)
ΦE(C, t) (16)
This stochastic equation is solved by the method of LeJan and Raimond [28, 29] (cf. footnote
#1), writing it as a (backward) Ito integral and iterating to obtain ΦE(C, t) = Su(t, t
′)ΦE(C, t
where the Markov operator semi-group S
(t, t′) is defined by a Wiener chaos expansion. More
intuitively, this solution is expressed as
ΦE(C, t) =
(dC ′, t′|C, t)ΦE(C
′, t′), t′ < t, (17)
in terms of the turbulent diffusion of lines (backward in time). We see that the circulations
are not conserved, except on average. This is precisely the “martingale property” that was
conjectured (for solutions of incompressible Euler equations) in [18]. Note that this property
imposes an irreversible arrow of time, since Eulerian circulations are given as averages over
their past values, not future ones. This “generalized Alfvén theorem” should be related to
dynamo action in the Kazantsev model [37, 38, 39, 40]. In the physical context of the dynamo,
there is an additional resistive term η
dxj∂i
δσij (x)
ΦE(C, t) on the righthand side of (15),
where δ/δσij(x) is the “area derivative” in the loop calculus of Migdal [41]. This term breaks
time-reversal symmetry and should select the backward-martingale solution (17) in the ideal
limit η → 0. Of course, it cannot be ruled out a priori that the η → 0 limit of the resistive
regularization and the λ → 0 limit for regularized velocity, as in (8), shall yield distinct weak
solutions of the loop-equation (15), as occurs in the intermediate compressibility regime of the
passive scalar problem [24, 25, 26, 28].
These results can be generalized to turbulent diffusion processes of higher-dimensional ma-
terial objects, k-dimensional oriented submanifolds of M or k-cells Ck(t). The dual object is
the passive k-form ωk, which satisfies (in Stratonovich sense)
k + L
ωk = 0 (18)
with L
the Lie-derivative along the vector field u ([31], section 4.9). This equation is formally
equivalent to conservation of the integral invariants
I(Ck, t) =
Ck(t)
ωk(t) (19)
for any k-cell Ck(t) comoving with u [42]. Then k = 0 is the passive scalar, k = 1 the passive
vector and k = d the passive density [24]. A theory similar to that developed here for k = 1
applies for any integer k. A unified approach to all these results is to consider directly the
turbulent diffusion of the Lagrangian flow maps ξt,t
, which satisfy the stochastic equation
(d/dt)ξt,t
(a) = u(ξt,t
(a), ◦t), ξt
′,t′(a) = a. (20)
In this framework one can derive for the distribution P
[ξ, t] on maps exact analogues of the
Liouville equation, in Stratonovich form (4) or Ito form (5). It is natural to formulate the
problem as an infinite-dimensional diffusion in the Hilbert space H = L2(M,Rd) . It is known
for the cases M = Rd or Td that the semigroup S(M) of Borel volume-preserving maps is a
closed subset of this Hilbert space, and that the group G(M) of volume-preserving diffeomor-
phisms is dense in S(M) for the L2-topology [43]. This construction is a close analogue of the
“generalized Euler flows” of Brenier, but for the Cauchy initial-value problem.
To summarize: We have outlined an approach to the study of material lines in a model
of turbulent flow at infinite Reynolds number and to the dual problem of a passive vector in
the same flow. The main conclusions are (1) that a non-degenerate diffusion should exist for
material lines, generalizing the phenomenon of “spontaneous stochasticity” of material points,
and (2) that the Kelvin/Alfvén theorem on conservation of circulations should generalize to a
“martingale property”. Although the approach sketched here depends heavily on the white-
noise character of the velocity field in time, we expect that similar results hold for more realistic
velocity ensembles with the crucial property that realizations are rough (Hölder continuous) in
space. See [18, 21] for related rigorous results on the solutions of incompressible fluid equations.
The two properties discussed in the context of this model problem should be an essential feature
of real fluid turbulence in the high Reynolds number limit.
Acknowledgements. We thank S. Chen, M. Chertkov, L. Chevillard, R. Ecke, C. Meneveau,
K. R. Sreenivasan and E. T. Vishniac for useful conversations. This work was supported by the
NSF grant # ASE-0428325 at the Johns Hopkins University and by the Center for Nonlinear
Studies at Los Alamos National Laboratory, where the research was begun.
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http://arxiv.org/abs/hep-th/9303130
http://arxiv.org/abs/hep-th/9306152
http://arxiv.org/abs/hep-th/9310088
