1. science
2. physics

What is Control of Turbulence in Crossed
Fields? - Don’t Even Think of Eliminating All
Vortexes!
Dimitri Volchenkov1
Abstract
MSC2000:
Keywords:
Councilor Hamann: Down here, sometimes I think about all those people still
plugged into the Matrix and when I look at these machines I can’t help thinking that in a way we are plugged into them.
Neo: But we control these machines; they don’t control us.
Councilor Hamann: Of course not. How could they? The idea is pure nonsense.
But ... it does make one wonder... just... what is control?
Neo: If we wanted, we could shut these machines down.
Councilor Hamann: Of course. That’s it. You hit it. That’s control, isn’t it? If
we wanted we could smash them to bits! ... Although, if we did, we’d have to
consider what would happen to our lights, our heat, our air...
Neo: So we need machines and they need us. Is that your point, Councilor?
Councilor Hamann: No. No point. Old men like me don’t bother with making
points. There’s no point.
Neo: Is that why there are no young men on the council?
Councilor Hamann: ... Good point.
”The Matrix Reloaded”, the second film in The Matrix franchise,
Written and directed by Andy & Larry Wachowski.
The Center of Excellence Cognitive Interaction Technology (CITEC), University of Bielefeld,
Postfach 100131, D-33501, Bielefeld, Germany,
e-mail: [email protected]
1
2
D.Volchenkov
1 Introduction
The international project on magnetic confinement fusion is designed to make the
transition from today’s studies of plasma physics to future electricity-producing fusion power plants. A successful fusion device has to contain the particles in a small
enough volume for a long enough time for much of the plasma to fuse. Once fusion
has begun, neutrons having a vast kinetic energy radiate from the reactive regions of
the plasma, crossing magnetic field lines easily due to charge neutrality, barraging,
together with charged particles, the wall blanket of the containment chamber, and
degrading its structure. The reliable confinement (or control) of very energetic particles is one of the crucial problems arisen in course of the fusion project. Despite
lasting efforts, the strategy of effective plasma flow control of a turbulent boundary
layer is still mostly unclear that threatens our hopes for the successful implementation of the project in the near future.
Here, we show that control of turbulence being understood in the framework
of traditional paradigm as elimination of all long-living turbulent fluctuations in
plasma flows is by no means compatible with symmetry of the crossed-field system
and inevitably breaks down its stability. While trying to gain control over turbulent
patterns in crossed fields, we are perhaps plugged into vortexes keeping some of
them as tools for maintaining the stability of still an illusory construct of plasma
fusion. In the forthcoming section (Sec. 2), we demonstrate that while in crossed
fields, an alternative long-time, large-scale sate would exist in which the eddies of
some particular size are destined to persist during essentially long time. In Sec. 3,
we investigate the stochastic problem of the long-range turbulent transport in the
Scrape-Off Layer of thermonuclear reactors and calculate (in the one-loop approximation) the magnitude of poloidal drift required to override convective instability
in the cross-field system. We conclude in the last section.
In our study of the stochastic counterparts of models in nonlinear dynamics, deterministic trajectories are replaced by random trial trajectories of some well defined stochastic processes. The proposed approach is closely related to the Nelson stochastic mechanics, the probabilistic interpretation of dynamical equations,
and the critical phenomena theory. We thoroughly use the renormalization group
(RG) method – one of the most important non-perturbative techniques developed
in the framework of the quantum-field theory. Asymptotic solutions for the models
in stochastic dynamics are obtained in the form of a perturbation theory which can
be studied by means of Feynman functional integrals. Diagram series of the perturbation theory can sometimes be studied by means of renormalization group techniques. In statistical mechanics, the RG (which is, in fact, a semi-group since the
transformations are not invertible) forms an ensemble of transformations that map
a Hamiltonian into another Hamiltonian by the elimination of degrees of freedom
with respect to which the partition function of the system remains invariant. The RG
allows calculating the critical exponents related to phase transitions in renormalizable models.
What is Control of Turbulence?
3
2 Stochastic theory of turbulence in crossed fields: vortexes of all
sizes die out, but one
2.1 The method of Renormalization Group
Ultraviolet renormalization has been developed in the framework of quantum field
theory in 1953. An article by E.C.G. Stueckelberg and A. Peterman in 1953 and
another one by M. Gell-Mann and F.E. Low in 1954 opened the field by a study of
the fact of invariance of the renormalized quantum field action under the variation of
bare parameters at the subtraction point. In the framework of quantum field theory,
the renormalization group (RG) was developed to its contemporary form in the wellknown book of Bogoliubov and Shirkov, in 1959.
The technique was developed further by R. Feynman, J. Schwinger and S.-I.
Tomonaga, who received the Nobel prize for their contributions to quantum electrodynamics. However, these techniques have not been implemented in critical phenomena theory until the works of Leo Kadanoff who had proposed a simple blocking procedure in 1966. In 1974-1975, Kenneth Wilson had used it in order to solve
the famous Kondo problem. In 1982, he was awarded by the Nobel Prize for this
work. It is important to mention, in concern with the Kondo effect, the work of P.
W. Anderson, D.R. Hamman, and A. Yuval (1970), in which the techniques similar
to that of RG had been used in critical phenomena theory, independently of Wilson’s approach. The old-style RG in particle physics was reformulated in 1970 in
more physical terms by C.G. Callan and K. Symanzik. Later (in 1974), M. Fowler
and A. Zawadowski developed the method of multiplicative renormalization in the
framework of quantum-field theory.
Fig. 1 The compactification procedure.
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D.Volchenkov
It is remarkable that the mathematical background beyond the RG is quite simple
and has been known long time before Peterman and Kadanoff; it is called the compactification procedure. Logarithmic divergences arise since the integration domain
is not compact. If we find a way how to project the model onto a compact manifold
(in d + 1 dimensional space, the new dimensional parameter µ is called a renormalization mass), we gain finite amplitudes for integrals (see Fig. 2.1.1). In general,
such a ”projection” is irrelevant since it breaks the natural physical scales, however,
it may have a sense if the model possesses a property of scaling invariance.
Let us suppose that we have found a way how to redefine the model on a compact
manifold, and therefore the diagram series converges. Physically relevant results
should not depend upon µ . The interaction charges have to be independent of µ , and
the sum should be invariant with respect to uniform dilatations of all its arguments.
The Green function has to be an eigenfunction of the dilatation operator belonging
to some eigenvalue γ . If now we make a simultaneous rescaling of momentum and
mass by λ , then the Green function G would rescale with a power factor. If G has
a constant infrared asymptotic (in turbulence, it is called Kolmogorov constant), we
obtain the infrared scaling for the long time large scale asymptotic behavior.
Iterating the RG transformations R for the particular values of the initial bare
parameters, it may be possible to attain a fixed point such that
H ∗ = R(H ∗ )
(1)
where H is a Hamiltonian. In critical phenomena theory, the RG transformations R
rely upon the rescaling of the system variables described by the Hamiltonian H at
the fixed point H ∗ that has the same appearance whatever the scale at which it is
considered. This means that the correlation function of φ (x) (e.g. φ (x) may represent the magnetization density in a magnetic system or spins in the Ising model)
must be of the form
⟨φ (r)φ (0)⟩ ∼ r−(d−2+η ) ,
(2)
i.e. that the system is at the critical point, in which a correlation length ξ = ∞. If
we make a change to parameters of the Hamiltonian H, in the vicinity of the fixed
point,
(3)
H = H ∗ + ∑ gi Oi
i
where Oi are called operators and gi are called fields, then we can study how the
Hamiltonian evolved under the action of the RG transformations. In order to clarify
the idea of the method, let us imagine that transformations we like to study forms a
continuous group. Then, the fields gi have to obey the equations of motion,
dgi
= βi (g1 , . . . , gn ) .
ds
(4)
If we are interested in the stability analysis of the dynamical system described by
(4), we linearize the function β in its r.h.s. It is clear that linearized equations have
the solutions
What is Control of Turbulence?
5
gi = gi (0) exp (yi ) ,
(5)
for some parameter yi > 0, so that the field gi increases due to the renormalization
transformation; it is said that gi is an essential field (or a ”relevant field”). Otherwise,
for yi < 0, the field gi decreases under the action of renormalization transformations,
and called an inessential field (or an ”irrelevant field”). Finally, if yi = 0, the field gi
does not vary in the linear order and is called a marginal field. In the later case, to
investigate the stability of the fixed point we need to go beyond the linear order.
In critical phenomena theory, temperature and the magnetic field are those fields
pertinent near the critical point. Solving the renormalization group equations, we
obtain that in the vicinity of critical point the free energy is of the form
)
(
H
(2−α )
F = (T − Tc )
(6)
f
(T − Tc )γ +β
and therefore satisfies the Widom’s hypothesis of homogeneity. More generally, the
RG allows to predict all critical exponents pertinent to the system at the fixed point
by studying how its Hamiltonian is transformed by the RG at the fixed point. There
are many ways to implement the RG techniques for real-world models.
In large scales (small moments), the asymptotic behavior predicted by the RG can
be modified. The corrections are calculated by means of the Short Distance Expansion method. They are related to scaling behavior of composite operators, the local
averages with respect to a point. Namely these quantities can be measured experimentally. If their scaling dimensions are negative, they can alternate the asymptotic
behavior. Scaling dimensions are inherent not to composite operators themselves,
but to their certain linear combinations which have a physical meaning.
If the coupling constant in quantum field theory is not small, we have deal with
the essentially non-perturbative regime, and such a theory is said to be asymptotically free for low energies. The non-perturbative regime is difficult to study, because
of in addition to the problem of divergences of Feynman diagrams in perturbative
series we have to deal with the essentially non-perturbative contributions coming
from the instantons which cannot be neglected. In quantum mechanics and quantum
field theory, an instanton is a classic solution of equations of motion, i.e. one of local
minimums of the action functional, but not the global one. Mathematical methods
developed in quantum fields theory are beyond any doubt applicable also in Euclidean space to classical problems involving random fields. Models of stochastic
nonlinear dynamics can be reformulated as models of quantum fields theory, and
then the powerful techniques developed in that can be used.
2.2 Phenomenology of Fully Developed Isotropic Turbulence
Despite more than a century of work and a number of important insights, a complete understanding of turbulence remains elusive, as witnessed by the lack of
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D.Volchenkov
fully satisfactory theories of such basic aspects as transition and the Kolmogorov
”5/3−spectrum”.
In phenomenological theory of turbulence formulated by A.N. Kolmogorov in
1941, it was conjectured that the correlation functions of velocity in some intermediate scales (called the inertial interval) depend upon the only dimensional parameter,
the power of energy pump W . It was supposed that energy comes from large scale
eddies which bifurcate due to nonlinear interactions until the small scale vortexes
are dissipated in fluid at the minimal scale.
The only physically relevant combination of energy pumping rate W and momentum k gives the Kolmogorov asymptotic for the fully developed turbulence.
It follows then that the velocity of fluid has a formal dimension −1/3, and the
famous 1-dimensional energy spectrum is −5/3. This result has been justified in
the framework of RG techniques by many authors. The recent theoretical, computational and experimental results dealing with homogeneous turbulence dynamics
have been summarized in (Sagaut and Cambon, 2009).
In the present section, we follow the seminal work (Adzhemyan et al., 1996).
To describe the spectral properties of incompressible fluids in the inertial range of
developed turbulence, one considers the stochastic Navier-Stokes equation with an
external random force (Monin and Yaglom, 1971, 1975)-(Wyld, 1961).
∇t φi = ν0 ∆ φi − ∂i p + Fi ,
∇t ≡ ∂t + (φ∂ ).
(7)
Here φi is the vector velocity field, which is transverse due to the incompressibility condition, p and Fi are the scalar pressure field and transverse external random
force per unit mass (all these quantities depend on x ≡ (t, x)), ν0 is the kinematical
coefficient of viscosity, and ∇t is the Galilean-invariant covariant derivative.
The equation (7) is studied on the entire t axis and is supplemented by the retardation condition and by the condition that φ vanish asymptotically for t → −∞. We
take F to be a Gaussian distribution with zero average and correlator
∫
⟨
⟩
δ (t − t ′ )
′
Fi (x)Fj (x ) =
dk Pi j (k) dF (k) exp ik(x − x′ ),
(2π )d
where
(8)
ki k j
(9)
k2
is the matrix of transverse projector in the momentum (Fourier) representation,
dF (k) is some function of the momentum k ≡ |k| and the model parameters, and
d is the dimension of the physical space.
The introduction of a random force phenomenologically models the stochastic
drive (which in a real situation must arise spontaneously as a consequence of the
instability of laminar flow) and, at the same time, the injection of energy into the
system owing to the interaction with large-scale eddies. The average power W of
the energy injection is related to the function dF in (8) by the equation
Pi j (k) = δi j −
What is Control of Turbulence?
7
W =
d −1
2(2π )d
∫
dk dF (k).
(10)
In the stochastic problem we can also do away with specific initial and boundary
conditions and directly study homogeneous, fully developed turbulence (Monin and
Yaglom, 1971, 1975)-(Wyld, 1961). The field φ in (7) describes only to the chaotic
component of the actual velocity field (fluctuations).
Equation (7) is solved by iterating in the nonlinearity followed by averaging ⟨. . .⟩
over the distribution of the random force. The quantities calculated are the various
correlation functions
⟨ φ (x1 ) . . . φ (xn ) ⟩
and also the response functions, the variational derivatives of the correlation functions with respect to a pseudo-random external force introduced additively into the
right-hand side of (7). Borrowing quantum field terminology, we shall refer to all
these objects as Green functions. Due to the transversal invariance of the problem,
all they depend only on the time variable and coordinate differences, so that the
equal time correlation functions ⟨φ (x1 ) . . . φ (xn )⟩ with t1 = . . . = tn ≡ t are independent of the time t common to all the fields. Such objects will be termed static, in
contrast to the dynamic correlators with arbitrary times of the fields.
The independent parameters of the model are W , ν0 , the dissipation length lmin ,
−1 (the inverse largest characteristic length scale from
and a mass parameter m ≡ lmax
which the system is fed with energy) The main premises of the phenomenological Kolmogorov-Obukhov theory (Kolmogorov, 1941; Obukhov and Vasil’ev, 1998;
Monin and Yaglom, 1971, 1975) were formulated as two hypothesis. We use the formulation of the first hypothesis given in (Kraichnan, 1965)-(Kraichnan, 1968).
1. In the region k ≫ m a finite limit exists for m/k → 0 for the single -time distribution function of the spatial Fourier components φ (t, k) of the random velocity
field φ (t, x).
−1
−2
2. In the region k ≪ lmin
, ω ≪ ωmax = ν0 lmin
this distribution is independent of the
viscosity coefficient ν0 .
The first hypothesis guarantees the existence of a finite limit f (0) of the function
f (m/k) for m/k → 0, and the value f (0) has a simple relation to the well known
Kolmogorov constant. While, from the second hypothesis it follows, in particular,
that in this region of applicability the pair correlation function of the Fourier components of the velocity of the d-dimensional problem can be written as
⟨
⟩
φ (ω , k)φ (ω ′ , k′ ) = (2π )d+1 δ (ω + ω ′ ) δ (k′ + k) Pi j (k) D(ω , k),
(11)
in which
D(ω , k) = W 1/3 k−d−4/3 f (W k2 /ω 3 , m/k),
(12)
where f is some as yet unknown function of two independent dimensionless arguments.
Representations like the above one can also be written down for more complicated correlation functions involving any number of fields φ . They follow only the
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D.Volchenkov
hypothesis 2, and altogether imply the existence of an infrared (because the con−1
−2
ditions k ≪ lmin
, ω ≪ ν0 lmin
of hypothesis 2 do not suggest a lower limit) scale
invariance (scaling) with fully defined Kolmogorov dimensions of all infrared - relevant quantities F = {φ ≡ φ (x), m,t ∼ ω −1 , r ∼ k−1 } for irrelevant W , ν0 :
∆φ = −1/3,
∆t = −∆ω = −2/3;
∆k = −∆r = ∆m = 1.
(13)
The scaling is generalized uniformly under a combined consistent dilatation F →
λ ∆F F with arbitrary parameter λ > 0 of all relevant quantities F for fixed irrelevant
quantities (zero dimensions can be formally assigned to the latter). The infrared (IR)
asymptote corresponds to λ → 0, and the statement about the existence of scaling,
strictly speaking, refers not to the exact Green functions, but only to the leading
term of their infrared asymptote λ → 0, because in representations like (12) the
2 ω /ν → 0 have already been discarded.
corrections vanishing for klmin → 0 and lmin
0
The aim of the theory is to justify scaling for a given microscopic model and
to calculate the corresponding critical dimensions of all IR-relevant quantities. In
the theory of critical behavior the analogous problem has been solved successfully
using the renormalization group method.
2.3 Quantum Field Theory Formulation of Stochastic
Navier-Stokes Turbulence
It is well known (Janssen, 1976)-(Phythian, 1977) that any model of stochastic dynamics with the t−local interaction (i.e., in which the interaction term does not
contain time derivatives) driven by the Gaussian random force is equivalent to the
quantum field theory model of the doubled set of fields, Φ = φ , φ ′ . This means that
statistical averages ⟨. . .⟩ of random quantities can be identified with functional averages with weight exp S(Φ ), so that the generating functionals of the full [G(A)] and
connected [W (A)] Green functions of the stochastic dynamical problem are represented by the functional integral
∫
G(A) = expW (A) =
DΦ exp [S(Φ ) + AΦ ]
(14)
with arbitrary sources A ≡ Aφ , Aφ ′ in linear form:
AΦ ≡
∫
]
[
dx Aφ (x)φ (x) + Aφ ′ (x)φ ′ (x) .
(15)
The generating functional of 1-irreducible Green functions is defined by the Legendre transform of W (A) with respect to A:
Γ (Φ ) = W (A) − AΦ ,
Φ (x) =
δ W (A)
.
δ A(x)
(16)
What is Control of Turbulence?
9
Here Φ is taken as the independent argument , and A = A(Φ ) is defined implicitly by
the second relation in (16). The source Aφ ′ is interpreted as a non-random external
′
force, so that, in particular, the
Green function ⟨φφ ⟩ coincides with the simplest
response function δ ⟨φ ⟩/δ Aφ ′ A=0 .
In particular, for the stochastic Navier-Stokes equation (7), one obtains the theory
of the transverse vector fields Φ = φ , φ ′ with action functional
1
S(Φ ) =
2
∫
+
∫ ∫
dx dx′ φ ′ (x)DF (x, x′ )φ ′ (x′ )
dx φ ′ (x) [−∂t φ (x) + ν0 ∆ φ (x) − (φ∂ )φ (x)] ,
(17)
in which DF is the random-force correlator (8). The auxiliary field φ ′ is transversal
that makes it possible to drop the purely longitudinal contribution ∂i p from (7) in
(17). The physically realistic injection function DF must be infrared: it must contain
the mass parameter m and the dominant contribution to the integral (10) must come
from scales k ∼ m. On the other hand, for the use of the standard quantum-field RG
technique it is important that the function DF have a power-law asymptote at large
k. This condition is satisfied by the function
DF (k) = D0 k4−2ε −d h(m/k),
h(0) = 1,
(18)
with arbitrary, sufficiently well behaved function h(x) ensuring convergence of the
integral (10) at small k and normalized to unity for k ≫ m. The parameter ε > 0
describes the deviation from logarithmic behavior. In this model it is independent
and completely unrelated to the space dimension d (in contrast to the theory of
critical behavior, where usually ε = 4 − d). The logarithmic theory corresponds to
the value ε = 0, and the injection (18) becomes infrared only for ε > 2. In the region
0 < ε < 2 the injection (18) is ultraviolet; the integral (10) for it diverges at large k.
−1
Then there is understood to be a cutoff at k ≤ lmin
, and the integral is dominated by
−1
contributions at scales k ∼ lmin .
Most studies on the RG theory of turbulence use a simpler, purely power-law
injection:
DF (k) = D0 k4−d−2ε ,
(19)
corresponding to m = 0 in (18). This is possible if one is interested only in establishing IR scaling and the critical dimensions (which must be independent of m for
any injection), and simple objects like the scaling functions are calculated using the
graphs of perturbation theory only in the form of ε -expansions. Then the passage to
the theory with m = 0 is consistent, because the coefficient of the ε −expansions of
the graphs always have finite limits for m → 0. For ε = 2 the parameter D0 in (19)
acquires the dimension of W .In addition, idealized injection by infinitely large eddies corresponds to DF (k) ∼ δ (k), and for suitable choice of the amplitude function
it can be assumed to be a power-law model of the d-dimensional δ −function.
The integral (14) is a standard construction of quantum field theory, and so all
the Green functions have the standard Feynman diagram representations (Vasil’ev,
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D.Volchenkov
1998). The lines in the graphs correspond to the elements of 2 × 2-matrix of bare
propagators ⟨ΦΦ ⟩0 , related as ⟨ΦΦ ⟩0 = K −1 to the matrix K in the free (quadratic
in Φ ) part of the action
1
S0 = − Φ K Φ ,
(20)
2
in which we imply the summation over indices and integration over continuous variables. In momentum -frequency representation, the bare propagators of the model
(17) have the form
⟨ ′⟩
⟨
⟩
φφ 0 = φ ′ φ 0 =
1
,
−iω + ν0 k2
⟨φφ ⟩0 =
⟨ ′ ′⟩
φ φ 0 = 0,
dF (k)
,
2
ω + ν02 k4
(21)
with the function dF (k) from (8). All lines of the diagram technique are proportional
in the vector indexess to the transverse projector Pi j , which was omitted in writing
(21) but is always understood to be present. The interaction in (17) is the three-leg
vertex
−φ ′ (φ∂ )φ = 1/2 · φi′Vi js φ j φs
with vertex factor
Vi js = i(k j δis + ks δi j ),
(22)
φ ′.
where k is the momentum flowing into the vertex via the field
As an illustration,
in Fig. 2.3.1 we give the graphs of the exact Green functions ⟨φφ ⟩ and ⟨φφ ′ ⟩ in the
one-loop approximation. The lines in the graphs correspond to the bare propagators
(21), the vertices correspond to the factors (22), the external legs with a slash correspond to the auxiliary field φ ′ , and those without a slash correspond to the field φ .
This diagram technique is known as the Wyld diagram technique (Wyld, 1961). The
Fig. 2 The Wyld diagram technique in the theory of turbulence.
expansion parameter in perturbation theory (the coupling constant or the charge in
the terminology of quantum field theory) is g0 ≡ D0 /ν03 with D0
What is Control of Turbulence?
11
The functional formulation (17) significantly simplifies the derivation of exact
functional relations like the Schwinger equations and, of particular importance, allows the standard quantum-field RG technique to be applied to the stochastic NavierStokes equation. Recently, the field theoretic renormalization group has been applied to the stochastic Navier Stokes equation in connection with the problem of
construction of the 1/d expansion for the fully developed fluid turbulence beyond
the scope of the standard ε −expansion in (Adzhemyan et al., 2008).
2.4 Analytical Properties of Feynman Diagrams
All the graphs of the pair velocity correlator converge at large momenta and frequencies for finite ε > 0, so that they can be calculated without a ultraviolet (UV) cutoff.
In the calculations with dimensional regularization, the UV divergences arising for
ε → 0 are manifested as poles in ε , and the perturbation series for the correlator has
the form
[
]
∞
⟨φφ ⟩ = ⟨φφ ⟩0 1 + ∑ (g0 k−2ε )n An (ω /ν0 k2 , m/k, ε ) ,
(23)
n=1
where g0 = D0 /ν03 with D0 from (18). The poles in ε are contained in the coefficients
An . We see from (23) that to define the k → 0 asymptote for fixed charge g0 and
coefficients An it is necessary to sum the entire series. This is the infrared problem
to be solved by the RG method. It is nontrivial for any ε > 0, including the region
0 < ε < 2, when the injection (19) is ultraviolet. Even for that the perturbation
series contains IR singularities, which will also be summed by the RG method. The
solution of this problem justifies the second Kolmogorov hypothesis.
However, the coefficients An in (23) can also contain singularities for m/k → 0.
These singularities present in the theory turbulence for finite ε > 0. This problem
cannot be solved just by the re-summation of the perturbation series using the ordinary RG technique. Analogous problems also arise in models of critical behavior
and are solves by the theory of renormalization of composite operators using the
Wilson operator expansion (Collins, 1992)-(Zinn-Justin, 1990).
2.5 Ultraviolet Renormalization and RG-Equations
Details on the RG procedure in the quantum field theory can be found in (Collins,
1992)-(Zinn-Justin, 1990) and in the comprehensive book of N.N. Bogolubov and
D.V. Shirkov (Bogolubov and Shirkov, 1980). The RG analysis of stochastic hydrodynamics had been discussed in (De Dominicis amd Martin, 1979),(Adzhemyan et
al., 1983)- (Adzhemyan et al., 1998), it contains anomalously few UV divergences
compared to the usual models of the critical phenomena theory like the famous φ 4 model.
12
D.Volchenkov
The standard analysis of UV divergences with regards to the canonical dimensions of fields shows that for d > 2 superficial divergences exist only in the 1irreducible functions ⟨φφ ′ ⟩ and ⟨φφ ′ φ ⟩, the corresponding counterterms must contain the symbol ∂ . Therefore, the first function generates only the counterterm φ ′ ∆ φ
without the addition φ ′ ∂t φ of the same dimension, and the second generates only the
counterterm φ ′ (φ∂ )φ when the the fact that all fields are transversal is taken into account. Although this counterterm is allowed by the conventional dimensional analysis, it is actually forbidden by Galilean invariance, which requires that the differential operators ∂t and φ∂ enter into the counterterm only as the covariant derivative
∇t = ∂t + φ∂ , (De Dominicis amd Martin, 1979). The absence of the counterterm
φ ′ ∂t φ implies the absence of φ ′ (φ∂ )φ . In the special case d = 2 a new superficial
divergence appears in the function ⟨φ ′ φ ′ ⟩ generating the local counterterm φ ′ ∆ φ ′ .
In this section, we suppose that d > 2.
All UV divergences can be removed from the perturbation theory of the renormalized action
1
SR (Φ ) = gµ 2ε ν 3 ϕ ′ k4−d−2ε φ ′ + φ ′ [−∂t φ + Zν ν∆ φ − (φ∂ )φ ],
2
(24)
in which Zν is the renormalization constant (the necessary summation over indexes
and integrations with respect to time and space are implied). It is completely dimensionless and can be therefore depend only on the completely dimensionless renormalized parameter g.
In renormalization using the minimal subtraction scheme, only the poles in ε are
subtracted from the divergent expressions without changing the finite contributions,
and the renormalization constants Z always have the following form:
∞
∞
n
k=1
n=1
k=1
Z = 1 + ∑ ak (g)ε −k ε −k = 1 + ∑ gn ∑ ank ε −k .
(25)
The coefficients ank in our models can depend only on the space dimension d, and
the absence of ε in the residues ak (g) is a feature specific to the minimal subtraction
scheme. One-loop graphs generate contributions of order g, two-loops ones generate
contributions of order g2 , and so on. The order of the pole in ε never exceeds the
number of loops in the graph. The renormalized action (25) is obtained from its
non-renormalized analog by the following dilatations of the parameters;
ν0 = ν Zν ,
g0 = gµ 2ε Zg ,
Zg = Zν−3 .
(26)
Fields do not require renormalization, ZΦ = 1. If the power-low injection is replaced
by the realistic model, the bare mass parameter m also does not require renormalization, m0 = Zm m, and Zm = 1.
The only independent renormalization constant Zν is calculated directly from
the graphs of the perturbation theory. It determines the corresponding RG functions
γF (g), the anomalous dimensions of a quantity F, and any charge g is used to introduce the corresponding β −function:
What is Control of Turbulence?
13
β = µ∂µ e g,
0
γF = β ∂g ln ZF ,
(27)
in which e0 = {ν0 , g0 } are the bare parameters of the theory. In particular, one has
γg = −3γν ,
β = g(−2ε − γg ) = g(−2ε + 3γν ),
(28)
and the RG operator takes the form
DRG = µ∂µ + βg ∂g − γν ν∂ν .
(29)
Calculation of the constant Zν for the model of stochastic hydrodynamics in the
one-loop approximation gives, (Adzhemyan et al., 1983),
Zν = 1 −
2π d/2 (d − 1)g
+ O(g2 ).
4(d + 2)Γ (d/2)(2π )d
(30)
The solutions of RG equation
DRGWnR = 0
(31)
are stable with respect to the infrared asymptotes if there exists a fixed point g∗ such
that
β (g∗ ) = 0, β ′ (g∗ ) > 0
in the physical region g > 0 for small ε > 0. When the IR-stable fixed point is
present, the leading terms of the IR asymptote of the Green functions WnR of any
single charge model satisfy the RG equation with the replacement g → g∗ . In particular, we obtain
[µ∂µ − γν∗ ν∂ν ]WnR = 0,
γν∗ ≡ γν (g∗ ) = 2ε /3.
(32)
Canonical scale invariance is expressed by the equations
[
]
[
]
∑ dFk DF dWk n
F
WnR = 0,
∑ dFω DF dWωn
WnR = 0,
(33)
F
in which F = {t, x, µ , ν , g, m} is the set of all arguments of WnR , n = {nφ , nφ′ } and
dFk,ω are the canonical dimensions of F in the action functional (17): dφk = −1,
k
k
ω
k
dφk ′ = d + 1, dm,
µ = 1, dν = −2, dg = 2ε (0 in the logarithmic theory), dφ = 1,
ω
ω
ω
ω
R
dφ ′ = −1,dm,µ = 0,dν = 1,dg = 0. The canonical dimensions of Wn are the sums
of canonical dimensions of their arguments.
We are interested in the scaling with dilatations of t, x, and m for fixed µ , ν , and
g. Substituting the canonical dimensions into (32) and (33), after eliminating µ∂µ
and ν∂ν we obtain the equation of critical scaling
[−x∂x + ∆t · t ∂t + ∆m · m∂m − ∆n ]WnR = 0
with the coefficients
(34)
14
D.Volchenkov
δt = −∆ω = −2 + γν∗ ,
k
ω
∆n = dW
,
+ ∆ω dW
n
n
∆m dmk ,
(35)
which are the corresponding critical dimensions. Substituting the known values γν∗ =
k = −n + (d + 1)n ′ , and d ω = n − n ′ (the dimension of the connected
2ε /3, dW
φ
φ
φ
φ
Wn
n
function WnR is equal to the sum of the dimensions of its fields) into (35), we obtain
the following expressions for the critical dimensions:
∆φ = 1 − 2ε /3,
∆φ ′ = d − ∆φ ,
∆m = 1,
(36)
∆t = −∆ω = −2 + 2ε /3.
They do not have terms of order ε 2 , ε 3 , and so on, and coincide with the Kolmogorov
dimensions (13) for the real value ε = 2. This main result has been reproduced in
(De Dominicis amd Martin, 1979) for the first time.
Influence of weak uniaxial small-scale anisotropy on the stability of inertialrange scaling regimes in a model of a passive transverse vector field advected by
an incompressible turbulent flow was investigated in (Jurcisinova et al., 2006) by
means of the field theoretic renormalization group. Weak anisotropy means that parameters which describe anisotropy are chosen to be close to zero, therefore in all
expressions it is enough to leave only linear terms in anisotropy parameters. Turbulent fluctuations of the velocity field are taken to have the Gaussian statistics
with zero mean and defined noise with finite correlations in time. In (Jurcisinova
et al., 2006), it was shown that stability of the inertial-range scaling regimes in the
three-dimensional case is not destroyed by anisotropy but the corresponding stability of the two-dimensional system can be destroyed even by the presence of weak
anisotropy.
Critical behavior of a fluid, subjected to strongly anisotropic turbulent mixing,
is studied by means of the field theoretic renormalization group in (Antonov and
Ignatieva, 2006) in a simplified model where relaxation stochastic dynamics of a
non-conserved scalar order parameter was coupled to a random velocity field with
prescribed statistics. Existence of a new, non-equilibrium and strongly anisotropic,
type of critical behavior (universality class) was established, and the corresponding
critical dimensions were calculated. The scaling behavior appears anisotropic in the
sense that the critical dimensions related to the directions parallel and perpendicular
to the flow are essentially different.
2.6 What do the RG Representations Sum?
In the renormalized theory all quantities are calculated as series in the charge g of
the type
∞
R(g, . . .) =
∑ gn Rn (. . .).
n=1
(37)
What is Control of Turbulence?
15
For any initial value of charge g0 , the renormalized value g ∈ [0, g∗ ∼ ε ], and so
it can be assumed that g∗ ≃ ε . Owing to the smallness of g ∼ ε and the absence of
poles in ε in the coefficients of the series (which are eliminated by the UV renormalization), it may seem that in the ε scheme there is absolutely no need for any infinite
summation of contributions of the series (37). This is true, but not for the critical
region: direct calculations show that the coefficients Rn involve factors of the type
(s−2ε − 1)/ε , which are UV-finite (in the limit ε → 0, s = const) and of order unity
for s = k/µ ≃ 1, but become of order 1/ε and higher for |ε ln ε | ≥ 1. The maximum
number of such ”large” factors of order 1/ε in the terms of the perturbation series
never exceeds the number of ”small” factors g ∼ ε . These two numbers have to be
approximately equal that means at small s we have the new parameter
g
ξ ≡ (s−2ε − 1),
ε
(38)
all powers of which must be summed for each order in ε . This is the statement of the
first infrared problem in the language of the renormalized theory with small ε > 0.
2.7 Stochastic Magnetic Hydrodynamics
In this section, we consider the field-theoretic analysis of several problems in magneto hydrodynamic statistics. These problems include the inertial-range scaling laws
in incompressible fully developed turbulence of conductive fluid correspond to various regimes of physical scaling behavior in a model of magneto hydrodynamic
(MHD) equations supplemented with stochastic force terms and the scaling exponents of some composite operators.
The first attempt to study the MHD model has been performed in (Fournier et al.,
1982), but it was incomplete (see the details below). Then, the correct model had
been proposed in (Adzhemyan et al., 1985), but the renormalization had been made
erroneous. Here, we present the correct version of the renormalized magnetohydrodynamics.
The stochastic MHD equations for two transverse vector fields (v is the velocity
field and θ is the magnetic field) are written as
∇t vi = ν∆ vi − ∂i p + (θ ∂ )θi + fiv ,
∇t = ∂t + v∂ ,
(39)
∇t θi = λ ν∆ θi + (θ ∂ )vi + fiθ ,
(40)
where λ =
is the inverse magnetic Prandtl-type constant, and
and f θ
are the Gaussian random force and curl of the random current with mean zero values
and covariance
c2 /4πσ ν
fv
β
αβ
⟨ fiα (r,t) f j (r′ ,t ′ )⟩ = δ (t − t ′ )Di j (r − r′ ),
α , β = v, θ .
(41)
16
D.Volchenkov
√
where θ = B/ 4πρ , B is the magnetic induction, σ is the conductivity, and ρ is the
density of medium, p is the scalar pressure field, c is the speed of light. The transversal condition for the velocity field v follows from the incompressibility constraint:
∂ · v = 0.
The purely longitudinal contribution of the pressure in (39) can be eliminated by
inserting a transverse projection operator Pis = δis − ki ks /k2 onto solenoidal vector
fields in front of the longitudinal contributing factors: (v∂ )v to ∇t v and (θ ∂ )θ . The
correction to (39) is given by the Lorenz force which is proportional to
[curl B × B] = (B∂ )B − ∂ (B2 /2),
where the second term is included into the pressure p, and by (40) which follows
from Ohm’s law for a moving medium in the simplest form,
j = σ (E + [v × B]/c),
and from Maxwell’s equations without allowance for the displacement current. The
αβ
αβ
Fourier transform of Di j or force spectrum,Di j (k), is necessarily non-negative.
The canonical dimensions of these fields are the same, and the inverse magnetic
Prandtl constant is dimensionless.
In the massless models, the covariances for the random forces (41) are chosen in
the power-law forms:
3
Dvv
is = g1 ν Pis dvv ,
dvv = k4−d−2ε ,
Dθisθ = g2 ν 3 Pis dθ θ ,
dθ θ = k4−d−2aε ,
Dvisθ = g3 ν 3 εism km dvθ ,
(42)
dvθ = k3−d−(1+a)ε .
The amplitude factor gi , (i = 1, 2, 3) in the correlation functions play the role of
coupling constants (ν 3 is separated from the amplitude factors for the future convenience). The coupling constant g3 , in the mixed correlation function can be defined
√
as g3 = ξ g1 g2 where ξ is not in essence a charge but an arbitrary parameter of the
theory. This situation is analogous to the gauge parameter found in quantum electrodynamics. ξ is subject to the inequality |ξ | ≤ 1, which follows from the requirement
that the matrix of correlation functions be positive. The value ε = 2 corresponds in
the momentum representation to δ (k), this expressing the idea of “pumping of energy from the large-scale motion”. The theory is renormalizable and logarithmic for
ε = 0. The positive constant a in the exponent of the magnetic correlation function
is an arbitrary parameter of the theory and models the difference between the spectra
of magnetic and hydrodynamic energy pumping. εism is the completely antisymmetric pseudo-tensor. The index structure of the correlators (42) is determined by the
requirements that the fields be transverse and spatial parity be conserved: the field vi
is a vector, while θi is a pseudovector, and so the mixed correlator is a pseudo-tensor.
It is automatically transverse in the indexes i and s.
In calculations performed in the spirit of dimensional regularization, the symbols
δik and εism can formally be used for arbitrary dimension d, but in the final expressions the symbol εism , in contrast to δis , is meaningful only for the real dimension
What is Control of Turbulence?
17
d = 3. In other dimensions, say, in d = 2, there is no pseudo-tensor which is transverse in both indexes, and the mixed correlator must be taken to be zero. The mixed
correlator was not introduced in (Fournier et al., 1982) and had been introduced in
(Adzhemyan et al., 1985).
2.8 Renormalization Group in Magnetic Hydrodynamics
Stochastic dynamics allows for the path-integral representation of the probability
generating functionals, using the so-called MSR action (Martin et al., 1973). The
generating functional of renormalized correlation functions takes the form
∫
G(AΦ ) =
Φ = {v, v′ , θ , θ ′ }
D Φ det M exp(SR (Φ ) + Φ AΦ ),
(43)
with twice the number of fields and with the action functional given by
1
1
SR = g1 ν 3 µ 2ε v′ Dvv v′ + g2 ν 3 µ 2aε θ ′ Dθ θ θ ′ + g3 ν 3 µ (1+a)ε v′ Dvθ θ ′ +
(44)
2
2
[
]
]
[
+v′ −∂t v + Z1 ν∆ v − (v∂ )v + Z3 (θ ∂ )θ + θ ′ −∂t θ + Z2 ν ′ ∆ θ − (v∂ )θ + (θ ∂ )v
(the necessary integrations over x and summations over the indexes are understood).
The operator det M ∝ exp α H(0), where H(x) is the Heaviside function, it turns
into a constant if the convention is taken that H(0) = 0. The renormalization mass
parameter µ fixes the renormalization procedure.
Notice that action (44) possess the Galilean invariance, i.e., it is invariant under
the following field transformations:
vb (x,t) = v(x + u(t),t) − b(t), v′b (x,t) = v′ (x + u(t),t),
θb (x,t) = θ (x + u(t),t), θb′ (x,t) = θ ′ (x + u(t),t),
(45)
∫
t
dt ′ b(t ′ ).
where b(t) is a parameter of transformation, and u(t) = −∞
The structure of RG-transformation in the model of stochastic hydrodynamics is
given by the following relations between Z1,2,3 and the renormalization constants of
fields and charges:
Zv = Zv′ = 1,
Zθ = Zθ−1
′ = Z3 ,
Zg1 = Z1−3 ,
1/2
Zg2 = Z3 Z1−1 ,
Zν = Z1 ,
Zu = Z2 Z1−1 ,
Zg3 = Z3 Z1−3 .
1/2
(46)
The renormalization constants Z1,2,3 can be calculated from the corresponding 1irreducible functions (Adzhemyan et al., 1996) (all graphs needed for the one-loop
calculation are shown in Fig 2.8.1). In the diagrams shown on Fig. 2.8.1, the lines
(propagators) in the (k,t)-representation are
⟨vi (k,t)v′j (−k, 0)⟩0 = Pi j H(t) exp(−ν k2t),
18
D.Volchenkov
Fig. 3 The Feynman diagrams needed for the one-loop calculation of renormalization constants
Z1,2,3
⟨vi (k,t)v j (−k, 0)⟩0 = Pi j
g1 ν 2 µ 2ε 2−d−2ε
H(t) exp(−ν k2 |t|),
k
2
⟨θi (k,t)θ j′ (−k, 0)⟩0 = Pi j H(t) exp(−λ ν k2t),
⟨θi (k,t)θ j (−k, 0)⟩0 = Pi j
(47)
g2 ν 2 k1−d−(1+a)ε µ 2aε
H(t) exp(−νλ k2 |t|),
2λ
g3 ν 2 k1−d−(1+a)ε µ (1+a)ε [
H(t) exp(−λ ν k2t)
1+λ
]
+H(−t) exp(λ ν k2t) .
⟨θi (k,t)v j (−k, 0)⟩0 = εis j ks
All vertexes contain the the coordinate derivatives: v′ v∂ v and v′ θ ∂ θ are symmetric
with respect to the basic fields v and θ , the third vertex, θ ′ ∂ θ θ - antisymmetric.
What is Control of Turbulence?
19
From Fig. 2.8.1 and (47), one obtains (for d > 2):
Z1 = 1 −
Z2 = 1 −
g1 d(d − 1) g2 (d 2 + d − 4)
−
,
4Bε
4Bαελ 2
g1 (d + 2)(d − 1) g2 (d + 2)(d − 3)
−
,
2Bελ (λ + 1)
2Bαελ 2 (λ + 1)
g1
g2
Z3 = 1 +
−
,
Bελ Baελ 2
(48)
where B ≡ 2d(d + 2)(2π )d /Sd = d(d + 2)(4π )d/2Γ (d/2).
it is convenient to eliminate the ratios g1 /λ and g2 /λ 2 by changing from g1 and
g2 to new charges
g1
g2
.
g≡
, g′ ≡
(49)
Bλ
Bλ 2
In terms of new charges, the RG β -functions are the following:
βλ = λ (γ1 − γ2 ),
βg = g(−2ε + 2γ1 + γ2 ),
bg′ = g′ (−2aε + γ1 + 2γ2 − γ3 ).
(50)
Calculating the γi = µ∂µ ln Zi in the new variables (49), for d = 3, we find
γ1 = 2gλ + 4γ2 ,
γ2 =
10g
,
1+λ
γ3 = 2(g′ − g),
(51)
from which using (50) we find the β -functions in the one-loop approximation:
[
]
10g
βg = g −2ε + 6gλ + 8g′ +
,
λ +1
[
]
20g
βg′ = g′ −2aε + 3gλ + 2g + 2g′ +
,
(52)
1+λ
[
]
10g
′
.
βλ = λ 3gλ + 4g −
1+λ
In contrast to a single charge theories considered in the previous chapters, in a
multi-charge one there may be several fixed points gi∗ ∼ ε . The type of each point
is determined by the eigenvalues ωα of the corresponding stability matrix
∂ βi (g j ) ωi j =
(53)
.
∂ g j g j =g j ∗
A point g∗ = {gi∗ } will be IR-attractive (stable with respect to long time large scale
asymptotes) only if the real parts of all the eigenvalues ωα are strictly positive.
For such a point, any phase trajectory representing a solution of the RG differential
20
D.Volchenkov
equation passing through the neighborhood of g∗ will be subsequently attracted to
it.
The system of algebraic equations (52) has six non-trivial solutions βi (gi∗ ) = 0.
1.
2.
3.
4.
5.
6.
The line g∗ = g′∗ = 0 and λ∗ arbitrary;
g′∗ = λ∗ = 0, g∗ = ε /5;
g∗ = λ∗ = 0, g′∗ = aε ;
√
g′∗ = 0, g∗ = ε (1 + λ∗ /15), λ∗ ( 43/3 − 1)/2 ≃ 1.393;
λ∗ = 0, g∗ = ε (4a − 1)/39, g′∗ ε (11 − 5a)/39;
g∗ = ε (1 + λ∗ )/15, g′∗ = ε (10 − 3λ ∗ (λ∗ + 1)) /60, where λ∗ is the positive
root of the equation 3λ 2 + 7λ + 54 = 60a, which exists for g ≥ 0.9.
It is easy to check that the points 3 and 4 are stable. These points have been
found in (Fournier et al., 1982): the ”kinetic” point is stable for any a ≥ 0.25 and
the ”magnetic” fixed point is stable for any a ≤ 1.16. in the intermediate region
0.25 ≤ a ≤ 1.16 both points are stable, and the critical regime depends on which
of their basins of attraction the initial data of the RG-equations foe the invariant
charges are located in. In this sense the critical behavior in the stochastic magnetic
hydrodynamics is not universal. The basins of attraction of these points for the real
case a = 1 are studied in (Camargo and Tasso, 1992). However, the results quoted
there cannot be considered reliable, as the authors (Camargo and Tasso, 1992) did
not take into account the renormalization of the vertex v′ θ θ , erroneously assuming
that it is an effect of higher order in the charges.
Fully developed magneto-hydrodynamic turbulence near d = 2 up to d = 3 has
been investigated in (Jurcisin and Stehlik, 2006) by means of renormalization group
approach and double expansion regularization. A modification of standard minimal
subtraction scheme has been used to analyze the stability of the Kolmogorov scaling
regime which is governed by the renormalization group fixed point. In particular, it
has been demonstrated that the magnetic stable fixed point has been calculated and
its stability region has been also examined. This point losses stability: (1) below critical value of dimension dc = 2.36 (independently of the a-parameter of the magnetic
forcing) and, (2) below the value of ac = 0.146 (independently of the dimension).
2.9 Critical Dimensions in Magnetic Hydrodynamics
The problem of critical scaling in the magnetic hydrodynamic has not been discussed in (Fournier et al., 1982), but studied later in (Adzhemyan et al., 1996).
The RG equation for the model in question for any renormalizable Green function FR has the form:
]
[
3
µ∂ν + ∑ βi ∂gi − γν · ν∂ν + nθ γθ + nθ ′ γθ ′ FR = 0,
i=1
(54)
What is Control of Turbulence?
21
in which nθ and nθ ′ are the numbers of arguments θ and θ ′ in FR . Eq. (54) is satisfied
by any function of the following type:
{∫ s ′
}
ds ( ′ )
dF dFω
2
¯ exp
γ g(s
(55)
FR = k ν R(1, ω /ν k , g)
¯ , g) ,
′ F
1 s
where R is a scaling function of dimensionless arguments with a finite IR asymptote,
and g(s,
¯ g) is the solution of the Cauchy problem
s∂s g¯i = β (g),
¯
g¯i |s=1 = gi ,
∀i,
(56)
with s = k/µ . As the IR asymptote s → 0, in (55) we have g¯ → g∗ and ν → ν∗ , and
since R is assumed to be finite, it acquires a definite critical dimension:
∗
RγF → const · (k/µ )γF .
(57)
However, this is not the case for the magnetic hydrodynamics.
In the kinetic point, g′∗ = 0 and therefore the scaling function R⟨θ θ ⟩ vanishes as
being proportional to g′ . Introducing the correction exponent ωg′ characterizing the
falloff rate as g′ → 0,
ωg′ = ∂g′ βg′ g∗ = 2ε − 2aε − γ3∗ ,
one obtains the asymptote for the scaling function R⟨θ θ ⟩ as s → 0:
˜ ∗ , 0, λ∗ ).
R ∼ sωg′ R(g
(58)
Therefore, the scaling function R⟨θ θ ⟩ has a nonzero critical dimension ∆R⟨θ θ ⟩ = ωg′
that corresponds to the addition of ωg′ /2 to the critical dimension of the magnetic
field θ ,
2ε
∆θ = 1 −
(59)
+ ε (1 − a).
3
The critical dimensions for the velocity field v and time t coincide (in d = 3, ε = 2)
with their values in the Kolmogorov theory of turbulence, ∆v = −1/3 and ∆t =
−2/3. The value (59) is exact (has no corrections of the order O(ε 2 ), etc.) The
result is true for all the Green functions in the kinetic regime. In (Adzhemyan et
al., 1996), the correct critical dimension of magnetic field has been derived using
the usual
fields have been subjected to a suitable dilatation:
√ rules, but the magnetic
√
θ = θ / g′ and θ ′ = θ ′ g′ .
In the magnetic point, λ∗ = g∗ = 0, and g′∗ = O(aε ), and the variables λ¯ and g
fall off for s → 0 according to a power law:
λ¯ ∼ sωλ → 0,
g¯ ∼ sωg → 0
(60)
where ωλ = 4aε + . . . and ωg = −2ε + 8aε + . . . . In contrast to the charges g and
g′ which appear in the nominators of scaling functions, the charge λ would appear
in the denominators giving rise to the singularities in the scaling asymptote in the
22
D.Volchenkov
magnetic critical regime, while λ√→ 0. In (Adzhemyan
et al., 1996), the appropriate
√
dilatations, v = λ v, v′ = v′ , θ = λ θ , θ ′ = θ ′ / λ have been used. In the magnetic
critical regime it leads to the following critical dimensions of fields (in d = 3):
∆t = −2 + O(ε 2 ),
∆v = 1 + O(ε 2 ),
∆θ = 1 − aε + O(ε 2 ).
(61)
The critical dimensions (61) determine the IR asymptotes of the dynamical Green
functions and also of the corresponding static objects if they exist as g = 0 and λ = 0.
This result remains true for all Green functions excepting for the static functions
⟨vv⟩ and ⟨vvv⟩. When calculating the scaling functions R⟨vv⟩ , one can see that the
contribution of the bare graph ∫vanishes, while, in the internal 1-irreducible block,
the integral over the frequency, ω −1/2 d ω ∝ ω 1/2 , diverges for ω → ∞. This means
that it is impossible to go directly from dynamical objects to static one, so that one
has to analyze the asymptotes before taking this limit.
The divergences of the frequency integral gives rise to a crossover phenomenon
for the static correlation functions of the velocity field in the magnetic regime which
had been discovered first in (Fournier et al., 1982). The divergence factor ∝ λ −1/2 ∼
s−2aε arising from the 1-irreducible block should be compared with the contribution
of the bare propagator ∝ g/λ ∼ s−2ε +4aε . For a > 1/3 with s = k/µ → 0, the loop
contribution is more important, and for α < 1/3 the bare one is. This leads to a final
result (Fournier et al., 1982), (Adzhemyan et al., 1996):
{ −d+2−2aε
k
for a > 1/3
⟨vv⟩|stat ∼
(62)
k−d+2−2ε +4aε for a < 1/3.
2.10 Critical Dimensions of Composite Operators in Magnetic
Hydrodynamics
In phenomenological theory of turbulence, it is supposed that energy is pumped
into the inertial range from a large scale of Λ . We take this scale into account in the
infrared mass parameter m ≡ Λ −1 to consider corrections to fully developed turbulent spectra due to design of energy pump with the use of short distance expansion
method. According to this, one can establish that
ϕ (x1 ,t)ϕ (x2 ,t) ≃ ∑ ci (r)Fi (x,t),
ϕ ≡ {v, θ },
(63)
i
where x ≡ (x1 + x2 )/2, r ≡ x1 − x2 , and Fi are the various composite operators of the
theory. Averaging (63) over fluctuations, one obtains asymptotes, mr → 0, for the
1-partical correlation function
< ϕ (x1 ,t)ϕ (x2 ,t) >≃ ∑ ci (r)m∆Fi ,
i
ϕ ≡ {v, θ },
(64)
What is Control of Turbulence?
23
In this expression ∆Fi are the scaling exponents of the composite operators. Hence,
if there are any operators of negative scaling exponents, the contribution of the mass
parameter m to the asymptotes (m → 0) becomes essential.
In this section, we study the renormalization and compute the critical dimensions
of the simplest local composite operators of the theory: ϕi ϕk (x), ϕi′ ϕk (x), as well as
vector operators (ϕ ϕ )ϕi (x), and (∂ ϕ ϕ )i (x). We note that these operators are multiplicatively renormalizable, i.e., there exists a matrix of renormalization constants
Zik such that for the renormalized operator, Fi = Zik FkR (ΦR ), all the Greens functions
with one F R and any number of the fields Φ are finite (don’t have divergences) in
the logarithmic theory. If one adds the term ∑i ai FiR with arbitrary finite coefficients
ai to the action functional of the renormalized theory (44), then the corresponding
functional (44) is finite up to the terms linear in a inclusively. We now consider the
renormalization of composite operators with mixing. In accordance with the general rule, only the composite operators of the same total canonical dimension in the
logarithmic theory are mixed on renormalization.
The critical dimensions of Fi are found from the usual power counting (ZinnJustin, 1990),(Collins, 1992): ∆ikF = γik + dFk δik + ∆ω dFω δik , where dFk and dFω are the
momentum canonical dimensions of operator considered, γik = µ∂µ ln Zik , and ∆ω
is a critical exponent of frequency, ∆ω = 2 − 2ε /3 in the models with the power-law
correlators.
The entity ∆F is the matrix of critical dimensions of the system of the mixed
operators FiR , and the critical exponents are the eigenvalues of this matrix. Definite
critical dimensions belong to the linear combinations of operators make the matrix
∆F diagonal.
2.11 Operators of the canonical dimension d = 2.
The tensor operator ϕi ϕk is a sum of scalar and zero trace operator
1 2
1
ϕ δik + (ϕi ϕk − ϕ 2 δik ),
d
d
ϕ ≡ {v, θ },
they do not get mixed on renormalization. The operator G1 = 12 vi vk , which is not
Galilean invariant, is finite (Z11 = 1) because of the Galilean invariance property of
the theory (Lifschitz, 1989), and it does not get admixed to the operator G2 = 21 θi θk ,
hence Z21 = 0. Notice, that the operator G3 = vi θk is a pseudo-tensor, so that it
does not be admixed to G1 and G2 on renormalization: Z23 = Z32 = Z13 = Z31 = 0.
Furthermore, the 1-irreducible diagrams for G3 have no divergences: Z33 = Zθ−1 ,
where the renormalization constant of magnetic field can be expressed in term of
1/2
the known constant Z3 : Zθ = Z3 , (Adzhemyan et al., 1985)-(Adzhemyan et al.,
1996).
For fixed space dimension, d > 2, in the lowest order of perturbation theory, the
calculation yields
24
D.Volchenkov
(θi θk )(x) = (1 + λ C)(θi θk )R (x) −C(vi vk )R (x),
where
C=
(65)
g2 )
1 + δik (d + 2)(d − 1) (
g1 −
.
2(λ + 1)Bε
aλ
The operators g1 G1 and g2 G2 , in both ”kinetic” and ”magnetic” critical regimes,
have definite critical dimensions themselves (and not a linear combination of them)
The results about operator scaling exponents are shown in Tab. 1. Notice, that the
exponent for vv is exact.
Table 1 The critical dimensions of the composite operators of the ϕi ϕ j -type.
Composite Operator Kinetic regime Magnetic regime
vi vk , v2
2 − 4/3ε
2
θi θk
2 − 2(a − 3/10)ε
2 + 3aε
vi θ k , vθ
2 − (a + 1/3)ε
2 + aε
θ2
2 − 2aε
2 + 12aε
2.12 Vector operators of the canonical dimension d = 3.
The operators ϕ ′ ϕ do not get admixed to any other one and are not renormalized,
as soon as all the appropriate 1-irreducible functions are in reality equal to zero due
to the presence of closed loops of the advanced functions. Formally, these diagrams
are logarithmic, but in reality, the structure of interactions of action (44) provides
the removing of one derivative from the loop to the external ϕ ′ -line and reduces
the diagram index of divergence. Thus, the scaling dimension of these operators are
equal to the space dimension, ∆ (ϕ ′ ϕ ) = 3.
The operators, with given derivative, (∂ ϕ ϕ )i , can be reduced to a differential,
∂ (ϕ ϕ ), due to the fact that the fields ϕ ≡ {v, θ } are transversal. The scaling dimensions of such operators are
∆ (∂ (ϕ ϕ )) = ∆ (ϕ ϕ ) + 1,
ϕ ≡ {v, θ }.
The rest of vector operators, F1 = vi v2 and F2 = vi θ 2 , are the true tensors, yet
F3 = θi θ 2 and F4 = θ v2 are pseudotensors, so that these pairs do not be admixed
on renormalization. Due to Galilean invariance property of the action, one can prove
that F1 is finite and it does not be admixed to F2 , as well as F3 does not be admixed
to F4 , (Z11 = 1, Z21 = 0, Z34 = 0). At the “kinetic” fixed point these operators have
different powers of the coupling constant g2 . Yet at the ”magnetic” point all the
non-diagonal elements of a renormalization matrix are equal to zero as being proportional to λ∗ = 0. Thus, the scaling exponents are determined simply from the
diagonal elements of renormalization matrix just as in case of a simple multiplica-
What is Control of Turbulence?
25
tive renormalization, and we shall say that these dimensions are “associated” with
the corresponding composite operators. A deviation of the scaling dimensions from
the canonical dimensions are the following
γ11 = 0,
γ22 = 2(C1 − 1)γ ,
γ33 = −6C1 γ ,
γ44 = −2γ ,
where γ = g1 /Bλ −g2 /Bλ 2 , C1 = (d +2)(d −1)(λ +1)−1 . The scaling exponents
are listed in Table 2.
Table 2 The critical dimensions of the composite operators (ϕ ϕ )ϕi .
Composite Operator Kinetic regime Magnetic regime
vi v2
3 − 2ε
3
vi θ 2
3 − 2(a + 0.507)ε
3 + 18aε
θi θ 2
3 − 3(a + 1)ε
3 + 60aε
3 − (a + 0.68)ε
3 − 2aε
θi v2
2.13 Instability in Magnetic Hydrodynamics
In accordance with the SDE method, the inertial-range asymptote of a 1-partical
correlation function can be expressed as follows
(
)
( m )∆F
i
−d−∆ϕ1 −∆ϕ2
, {ϕ1 , ϕ2 } ≡ {v, θ },
< ϕ1 (k,t)ϕ2 (−k,t) >= Ak
1 + ∑ bi
k
i
(66)
where A is a Kolmogorov-type constant, and ∆ϕ are the scaling dimensions of the
fields. One can see that RG-predicted spectrum is secure as long as ∆Fi > 0. If ∆Fi is
negative, the appropriate contribution changes the scaling asymptote of correlation
function in the inertial range.
If we are interested in asymptote of static correlation functions,i.e., they do not
depend on time, we do not consider the contributions of those operators that are not
Galilean invariant. It is quite clear that such a contribution would depend on the parameter of Galilean transformation b(t), but these operators contribute to asymptote
of dynamic correlation functions.
For the real value of ε = 2 some Galilean invariant operators have negative scaling exponents at the ”kinetic” and ”magnetic” fixed points for some values of the
pump parameter a:
θ2
1
(a > ),
2
θi θk
4
(a > ),
5
θi θ 2 .
26
D.Volchenkov
Notice, that the last one does not contribute to the asymptotes of 1-partical static
correlation functions at the ”kinetic” point. This operator is a pseudotensor, it cannot contribute to real tensor correlation functions, yet its contribution to the mixed
1-partical correlation function, < vθ >, is O(g2 ∗ ) and can be neglected. For the
same reason, the operators θ 2 and θi θk do not contribute to asymptotic of 1-partical
hydrodynamic correlation function, < vv >. Hence, in the static case, for < vv >
and < vθ > the RG-predicted scaling asymptotes is secure for the both ”kinetic”
and ”magnetic” points. If a > 12 , the exponent for < θ θ > should be corrected at
the ”kinetic” point as follows
∆ (< θ θ >) = −d − 2∆θ − ∆ (θ 2 ).
This result has a simple physical meaning. At the ”kinetic” point the magnetic
field is passively advected to hydrodynamics. When the value of the parameter a is
comparable to 1, the spectrum of magnetic pump is infrared localized. It means that
the inertial-range motion is exposed to ambient magnetic field, which depends on
the hydrodynamics. In this case, some instabilities arise in MHD system, (Lifschitz,
1989), which are driven by the magnetic pressure gradient ∂ (B2 /2). Yet the velocity field correlation function is virtually unaffected by these instabilities, but they
contribute to the magnetic field correlation function.
Likewise the usual hydrodynamics, at the ”kinetic” critical regime the inertialrange asymptotes of dynamic correlation functions have a lot of essential contributions from Galilean non-invariant operators. In this sense, one can say, (Eyink,
1994), that there are infinitely many fixed points in the fully developed turbulence.
However, at the ”magnetic” critical regime, the scaling asymptotes for the
< vv >- and < θ θ >-functions still have the same value as predicted by RG. The
mixed function exponent has a correction associated with the operator θi v2 , which
becomes essential, while a > 43 . It is important to notice that in this case the contributions from each of the θi vn -type operators are also possible. We do not know their
scaling dimensions, so that this scaling exponent would be corrected as follows
∆ (< vθ >) = −d − ∆θ − ∆v − max[∆ (θ vn )].
2.14 Long life to eddies of a preferable size
In the present section, we compute the scaling asymptote of the spectral density tensor of energy dissipated in a unit time per unit mass by the magnetic hydrodynamical
system being in the ”kinetic” critical regime, (Landau and Lifshitz, 1995)-(Landau
et al., 1995):
ε = ∇σˆ v + λ ν [∇ × θ ]2 ,
(67)
averaged with respect to the statistic of Gaussian distributed random force f . Here
σˆ is a tension tensor of the incompressible fluid. Doing some basic calculations, we
arrive at
What is Control of Turbulence?
⟨
ε
ν0
⟨
⟩
=
27
⟩
1
2
2
∆ (v + λ θ ) + ∑ ∇i ∇k (vi vk + λ θi θk ) − ⟨v · v + λ ·
2
ik
⟩,
(68)
where the angular brackets denote the average with respect to configurations f (x,t).
The result of RG-transformation acting on a renormalized composite operator (the
local average of fields and their derivatives with respect to one point) is always a
linear combination of the renormalized composite operators having the same symmetry, structure, and canonical dimension. This fact is known as a miming of composite operators (Collins, 1992). Denoting the renormalized composite operators of
a mixing set as FiR , we write the RG equation for them in the form:
DRG FiR = ∆i j FjR
(69)
where ∆i j is the matrix of critical exponents. The linear combinations L {FiR }, for
which ∆i j has the diagonal form, have the definite physical meaning and correspond
to the certain physical processes.
The unique property of the energy dissipation composite operator ε is that two
different eigenvectors of the RG-operator have the same eigenvalue corresponding
to the zero anomalous dimension γ = 0. Consequently, the relevant critical dimension matrix ∆F can be transformed merely to the Jordan form
{
DRG L {FiR } = ∆L L {FiR }
(70)
DRG L ′ {FiR } = ∆L L ′ {FiR } + L {FiR },
in which L {FiR } is the eigenvector and L ′ {FiR } is the adjacent vector. ∆L is the
shared critical exponent of L and L ′ . Such a phenomenon has not been discussed
neither in the quantum fields theory literature nor in the statistical physics before.
In the asymptotic region k/µ → 0, one can solve (70) to obtain
L ∝ C1 (g, λ )
( )∆L
k
,
µ
L ′ ∝ C2 (g, λ )
( )∆L
( )
k
k
log
µ
µ
(71)
where C1 and C2 are the normalized amplitude factors.
Now we calculate the critical exponents ∆L and the relevant linear combinations
L of renormalized composite operators explicitly. Note that the energy dissipation
operator ε is a sum of local composite operators of the canonical dimension dF = 4.
The result of the cation of the DRG -operator on ε can be written in the form
DRG ε = K1 {FiR } + K2 {∇i ∇k GRik }
(72)
where K1 and K2 are some linear combinations of the renormalized composite operators ∇i ∇k GRik , in which dG = 2, and FiR with dF = 4, and there fore K1 and K2 are
renormalized separately with no mixing.
The set of composite operators Gik with dG = 2 reduces to the tensor operator
ϕi ϕk which is a sum of scalar and zero -trace operators,
28
D.Volchenkov
Gik =
(
)
ϕ2
ϕ2
δik + ϕi ϕk − δik ,
d
d
ϕ ≡ {v, θ }.
We have studied the renormalization of this family in the previous subsection. All
of them have their own critical dimensions independently of others and contribute
to the following part of the energy dissipation function:
⟨
⟩
1
ε2 ≡
∆ (v2 + λ θ 2 ) + ∑ ∇i ∇k (vi vk + λ θi θk ).
2
ik
The family of composite operators dF = 4 consists of 7 items:
F1 = ⟨v4 ⟩,
F2 = ⟨v2 θ 2 ⟩,
F5 = ⟨θ · θ ⟩,
F3 = ⟨v · v⟩,
F6 = ⟨θ · (θ · ∇v)⟩,
F4 = ⟨v · (θ · ∇θ )⟩,
F7 = ⟨θ 4 ⟩.
Linear combinations of their renormalized analogs contribute to another part of the
energy dissipation function:
ε1 = ⟨v · v + λ θ · θ ⟩.
The renormalized operators FkR are related to the not renormalized ones, Fk , by the
linear equations
Fi = Zik FkR ,
in which Zik are the renormalization constants found from the requirements that all
correlation functions with one FiR and any number of fields v, θ , v′ , and θ ′ are finite
as the UV-cutoff Λ → ∞.
The problem of computation of the entries Zik can be substantially simplified by
the symmetry arguments. For instance, since the model of magnetic hydrodynamics is invariant under the Galilean transformation of fields, the composite operators
which break this symmetry is not renormalized being finite. Moreover, they do not
mix to any other Galilean invariant operator. Therefore,
Z11 = 1,
Zi1 = 0,
Z j2 = 0,
∀i > 1,
∀ j > 2,
(73)
Zk3 = Zk4 = 0, ∀k > 4.
Then, one can use the Schwinger functional equations and the Ward identities expressing the Galilean invariance of the MHD model:
⟨⟨ϕi (x)
δ SR (ϕ ϕ ′ )
δ WR (A)
⟩⟩ = −Aϕi ′ (x) i
′
δ ϕ (x)
δ Aϕ (x)
(74)
where ϕ ≡ {v, θ } and ϕ ′ ≡ {v′ , θ ′ } , Aϕ and Aϕ ′ are the relevant source functions.
The r.h.s. of (74) is UV-finite and has the definite critical exponents independently.
What is Control of Turbulence?
29
Therefore, the operators in the l.h.s. should also be UV-finite having the definite
critical exponents:
{
⟨ν Z1 v · v + Z3 v · (θ · ∇θ )⟩ < ∞
(75)
⟨λ ν Z2 θ · θ + Z3 θ · (θ · ∇v)⟩ < ∞
where Z1,2,3 are the renormalization constants of MHD-action. UV-finiteness means
that the divergent parts of the renormalization constants subtract each other in the
combinations (75), therefore,
Zi3 + α Zi4 = 0,
(i ̸= 5, 6),
Z55 = Z2−1 ,
Z j5 = α −1 Z j6 = −Z j3 ,
Z66 = Z3−1 ,
( j ̸= 5, 6)
where α = g(d − 1)/2d ε (1 + λ )(4π )d/2Γ (d/2). Other nontrivial entries of Z − ikmatrix require evaluation of the diverging parts of the relevant 1-irreducible diagrams and remain unknown. Nevertheless, we show that (73) and (75) provide us
enough information to define the critical exponents of the operator ε1 . The matrix
Zik appears to be triangular and its diagonal elements give us the complete set of
anomalous exponents, γik = −2ε · d ∂g log Zik + O(ε 2 ),
]
(
[
√ )
ε
43
2ε
γik = 0, 0, γ77 , −
(76)
1+
, − , γ22 , −α1 γ43 + γ33 .
15
3
3
The zero eigenvalue is twice degenerated, γ11 = γ22 = 0, the entries γ77 , γ22 , γ43 and
γ33 so not equal to zero and still unknown, however, they do not contribute to ε1 ,
ε1 = ∑ Wi j Fj ,
Wi j = diag[0, 0, 1, 0, λ , 0, 0].
i
Denote the matrix which transforms Wi j into the Jordan form as U, then, the linear
combinations of renormalized composite operators which correspond to ε1 in the
process of UV-renormalization are
−1 R
LiR = Wi j Z jkUkm
Fm .
There are two such combinations:
L1R = F3R + a1 F4R + λ F5R + a1 Z3−1 F6R + a2 F7R ,
L2R = F5R + Z2−1 F7R ,
in which a1,2 are some analytical coefficients expressed via the unknown entries
Zik−1 (which are obviously finite even as ε → 0).
We have to emphasize that L1R is a vector defined in the two-dimensional eigensubspace of the RG-operator having the single eigenvalue γ = 0 where
L1′′ R
L1′ R = ⟨v · v + Z3 Z1−1 v · (θ · ∇θ )⟩,
= ⟨λ b f θ · ∆ θ + Z2−1 θ · (θ · ∇v) + Z1−1 θ 4 ⟩
(77)
30
D.Volchenkov
are the linearly independent vectors spanning this eigensubspace.
The critical exponent relevant to L1R is ∆L1 = 4 − 4ε /3 = 4/3 (for ε = 2). The
anomalous exponent corresponding to the second combination, L2R , is −γ2 that gives
∆L2 = 4 − 2ε = 0 (ε = 2).
The critical dimensions of the composite operators Gik are ∆G = 2 + ∆ϕi ϕk where
∆ϕi ϕk are the critical dimensions of composite operators with dF = 2 studied in the
previous subsection. Certainly, we have:
4
∆∇i ∇k vi vk = ∆∇2 v2 = ,
3
6
∆∇i ∇k θi θk = ,
5
∆∇2 θ 2 = 0.
(78)
All values are computed for ε = 2.
We conclude this section collecting the results on the critical exponents of the
energy dissipation function ε . In Fig. 2.14.1, we have presented the different asymptotic contributions into the energy dissipation function E ≡ ε (k)/ε0 via the dimensionless scaling parameter s ≡ k/µ where ε0 is a constant energy dissipation rate.
The sum of all contributions is drawn with the black bold line. The uniform rate ε0
Fig. 4 The critical exponents of composite operators of the canonical dimension 4 in the model of
magnetic hydrodynamics in the kinetic critical regime.
is given by the thin gray horizontal line, two power-law asymptotic contributions are
represented by the dotted lines. At a decided disadvantage for the small scale eddied
(s ≫ 1) (in the far-dissipation range) the total dissipation rate increases considerably,
therefore, they dissipate very fast.
Fig. 2.14.1 displays that there are two opportunities for the long-time , largescale asymptotic behavior of ε . In the inertial range, indeed, the constant dissipation
rate ε0 dominates the dissipation process. Nevertheless, in the MHD model, the
alternative asymptotic ”steady state” exist, and it would come into play when the
What is Control of Turbulence?
31
regime characterized by the constant dissipation rate looses the stability. One can see
on Fig. 2.14.1 that both dissipating regimes meet precisely at the dissipation wave
number kd = µ ≃ Λ and at the pumping scale k µ −1 ≃ 0. The alternative dissipating
regime has a minimum somewhere in between these two points. The existence of
dissipation minimum in the sub-leading dissipation regime predicts essentially long
lifetime for the eddies of some preferable size ℓ.
The linear combination L1R is responsible for the long -time breaking of the size
equivalence of eddies in the inertial range. From (77) one can see that it describes
a kind of feedback controlling loop, accentuating the eddies of particular size and
suppressing the turbulence in other scales by shadowing one of the infinitely many
unstable periodic orbits embedded in the chaotic turbulent attractor. When the trajectory converges to the optimal orbit, the feedback term vanishes identically,
Z3 Z1−1
∫
dx v · (θ · ∇θ ) + Z2−1
∫
dx θ · (θ · ∇v) = 0.
The latter equation defines the configurations of fields θ and v relevant to the alternative dissipation regime.
3 In search of lost stability
3.1 Phenomenology of Long-Range Turbulent Transport in the
Scrape-Off Layer (SOL) of Thermonuclear Reactors
Turbulence stabilization in plasma close to the wall blanket of the ITER divertor
is the important technical problem determining the performance of the next step
device. Long range transport in the scrape-off layer (SOL) provokes the plasmawall interactions in areas that are not designed for this purpose. Evidence of the
strong outward bursts of particle density propagating ballistically with rather high
velocities far beyond the e-folding length in the SOL has been observed recently
in several experiments (see (Rudakov et al., 2002)-(Antar et al., 2001)) and in the
numerical simulations (Ghendrih et al., 2003). These events do not appear to fit into
the standard view of diffusive transport: the probability distribution function (pdf)
of the particle flux departs from the Gaussian distribution forming a long tail which
dominates at high positive flux of particles (Ghendrih et al., 2003).
Theoretical investigations of the reported phenomena remain an important task.
In the forthcoming sections, we consider a variety of two dimensional fluid models based on the interchange instability in plasma studied in (Nedospasov, 1989)(Garbet, 1991) and discussed recently in (Ghendrih et al., 2003) exerted to the
Gaussian distributed external random forces to get an insight into the properties of
turbulent transport in the cross-field system. The E × B drift motion of charged test
particle dynamics in the SOL was analyzed to investigate a transport control strategy
based on Hamiltonian dynamics in (Ciraolo et al., 2007). A method of control which
32
D.Volchenkov
is able to create barriers to magnetic field line diffusion by a small modification of
the magnetic perturbation has been proposed in (Chandre et al., 2006). This method
of control is based on a localized control of chaos in Hamiltonian systems.
Neglecting for the dissipation processes in plasma under the constant temperatures Te ≫ Ti , this problem is reduced to the interactions between the normalized
particle density field n(x, y,t) and the normalized vorticity field w(x, y,t) related to
the electric potential field ϕ (x, y,t),
∂t n = [n, ϕ ],
∂t w = [w, ϕ ] − g∂y log n,
w = ∆⊥ ϕ ,
g ≥ 0,
(79)
defined in the 2D plane transversal to ez , the direction of axial magnetic field B0 .
In (79), x and y are the normalized radial and poloidal coordinates respectively. The
Poisson’s brackets are defined by
[p, q] = ∂x p∂y q − ∂x q∂y p.
When g = 0, the equations (79) describe the 2D-rotations of the density and
vorticity gradients around the cross-field drift
v = −c/B0 ∇ϕ × ez ,
in which ∇ ≡ (∂x , ∂y ). Their laminar solutions (with w = 0) are given by any spatially homogeneous electric potential ϕ = ψ1 (t) and any stationary particle density
distribution n = ψ2 (x, y). Other configurations satisfying (79) at g = 0 are characterized by the radially symmetric stationary vorticity fields
w = ∂x2 ϕ (x)
with the electric potentials invariant with respect to the Galilean transformation
ϕ (x) → ϕ (x) + xφ1 (t) + φ2 (t)
where the parameters of transformations φ1,2 (t) are the arbitrary integrable functions of time decaying at t → −∞. The relevant density configurations
(
)
∫ t
′
′
n = φ3 x, y −
vy (x,t ) dt
−∞
have the form of profile-preserving waves convected in the poloidal direction by the
poloidal cross field drift vy (x,t). The poloidal component of cross field drift itself
remains invariant with respect to the Galilean transformation vy → vy + φ1 (t), while
its radial component vx = 0.
Configurations that satisfy (79) for g > 0 have the Boltzmann density distribution of particles in the poloidal direction. In particular, those solutions compatible
with the Galilean symmetry discussed above (with vy ̸= 0) are the solitons (solitary
waves) of density convected by the poloidal electric drift,
What is Control of Turbulence?
33
n ∝ exp −
1
g T (x, y)
t
′
′
vy (x, y,t ) dt ,
−∞
∫
y −
where T (x, y) is an arbitrary function twice integrable over its domain. In addition
to them, for g > 0, Eq. (79) allows for the radially homogeneous configurations
∂x n = 0, w˙ = U(y) mod 2π with vy = 0 which do not fit into the Galilean symmetry,
these are the steady waves,
n ∝ exp −
1
g
∫ y
U(y′ ) dy′ .
0
The latter solution does not possess a reference angle and can be considered as an
infinitely degenerated state of the system since the relevant configurations {n, w}
can be made equal at any number
of points by∫ the appropriate choice
of U:
∫
∫
U(y1 ) = U(y2 ) . . . = U(yn ), and 0y1 U(y′ ) dy′ = 0y2 U(y′ ) dy′ . . . = 0yn U(y′ ) dy′ .
For instance, it can be represented by the periodic lattice potential controlled by
the spokes of high particle density radiating from the center. With two concurrent
symmetries there can occur either the frustration of one of them or the vanishing of
both with the consequent appearance of a complicated dynamic picture that is most
likely stochastic. The latter case corresponds to a maximally symmetric motion resulting from the destruction of unperturbed symmetries (Sagdeev and Zaslavsky,
1986). In particular, instability in the system (79) occurs with respect to any small
perturbation either of density or vorticity.
Accounting for the dissipation processes in plasma smears the picture, so that the
small scale fluctuations would acquire stability. We demonstrate that the small scales
fluctuations can be stable provided there exist the reciprocal correlations between
the stochastic sources of density and vorticity in the dynamical equations. The large
scale stability of a fluctuation can be characterized by the order parameter
/(
)
ξ = ky kx2 + ky2
in the momentum space where kx and ky are the radial and poloidal components of
momenta respectively. For the uncorrelated random forces (under the white noise
assumption), a fluctuation with ξ > 0 is unstable with respect to the large scale
asymptote in the stochastic problem.
The accounting for the convection of particles by the random vortexes introduces
a finite reciprocal correlation time τc ( |r − r′ | ) between the density and vorticity
random forces. Then there exists the critical scale ξc , in the stochastic model, such
that a fluctuation with ξ < ξc vanishes with time, but its amplitude grows up unboundedly with time, for ξ > ξc .
Biasing of wall components can locally modify turbulent transport and is considered to be beneficial if one aims to insulate the Tokamak main chambers from the
bursts of density (Ghendrih et al., 2003). Indeed, the generation of a uniform electric
drift in the poloidal direction, vy → vy − V, would frustrate one of the symmetries
in (79) reestablishing the Galilean invariance in the system. For instance, those configurations characterized by the trivial poloidal component of electric drift vy = 0
34
D.Volchenkov
would be eradicated. We investigate the problem of turbulence stabilization close to
the divertor wall in the first order of perturbation theory and shown that there exists a critical value |Vc | < ∞ of the poloidal electric drift which would suppress the
large scale instability in the stochastic system with the correlated statistics of random forces, ξc > 0. However, for the uncorrelated random sources in the stochastic
problem, ξc = 0 and |Vc | → ∞ as k → 0.
Correlations between the unstable fluctuations of density at different points are
described by the advanced Green’s functions which are trivial for t > 0. In particular, these functions determine the concentration profile of the unstable fluctuations
of density which increases steeply toward the wall. The size of such fluctuations
grows linearly with time. In this case, the statistics of the transport events responsible for the long tail of the flux pdf is featured by the distribution of the characteristic
wandering times of growing blobs convected by the highly irregular turbulent flow
in the close proximity of the divertor wall. In our model, we have replaced this complicated dynamics with the one dimensional (the radial symmetry is implied in the
problem) discrete time random walks. Such a discrete time model would have another interpretation: the advanced Green’s function is a kernel of an integral equation which relates the amplitudes of the growing fluctuations apart from the wall
with those on the wall, in the stochastic dynamical problem. Indeed, this equation
is rather complicated and hardly allows for a rigorous solution. Therefore, being interested in the qualitative understanding of statistics of the turbulent transport in the
SOL, we develop a Monte Carlo discrete time simulation procedure which would
help us to evaluate the asymptotic solutions of the given integral equation.
General approach to the probability distributions of arrival times in such a discrete time model has been developed recently in (Floriani et al., 2003). In general,
its statistics can exhibit the multi-variant asymptotic behavior. Referring the reader
to (Floriani et al., 2003) for the details, we have shown that the statistics of arrival
times for the unstable fluctuations is either exponential or bounded by the exponentials (in particular, the latter would be true in the case of the randomly roaming
wall) that is in a qualitative agreement with the data of numerical simulations and
experiments (Ghendrih et al., 2003).
3.2 Stochastic Models of Turbulent Transport in Cross-Field
Systems.
The stochastic models of cross field turbulent transport used in the forthcoming
sections refers to the effectively two-dimensional fluid model of plasma based on
the interchange instability in the SOL (Nedospasov, 1989; Garbet, 1991) recently
discussed in (Ghendrih et al., 2003). In this model, one assumes the temperatures of
ions and electrons to be constant, Ti ≪ Te . Then the problem is reduced to that of
two coupled fields, the fluctuations of normalized particle density n(x, y,t) and that
of vorticity field w(x, y,t), governed by the following equations
What is Control of Turbulence?
35
∇t w = u0 ν ∆⊥ w −
(−1)k gk
∂y nk + fw ,
k
k≥1
∑
∇t n = ν ∆⊥ n + fn
(80)
/
written in the
/ polar frame of reference with the normalized radial x = (r − a) ρs
and y = aθ ρs poloidal coordinates. Time and space are normalized respectively
to Ωi−1 , the inverse ion cyclotron frequency, and to ρs , the hybrid Larmor radius.
The covariant derivative is
∇t ≡ ∂t + ez · v × ∇,
in which ∇ ≡ (∂x , ∂y ), and ∆⊥ is the Laplace operator defined( on the/ plane
) transversal to the axial magnetic field. The effective drive ∝ ∂y log 1 + n n¯ risen in the
cross-field system due to the curvature/ of magnetic lines is represented by the series
in ∂y nk with the coefficients gk ∼ ρs Rn¯ k where R is the major radius of torus and
n¯ is the mean normalized particle density. The curvature coefficients gk averaged
along the lines of magnetic field are considered to be constant and small parameters in the problem. The diffusion coefficients ν and u0 ν both are normalized to
the Bohm’s value Te /eB and govern the damping in the small scales, herewith u0 is
the dimensionless Prandtl number where the knot distinguishes its value in the free
theory from its effective value u in the renormalized theory.
The Gaussian distributed random forces fn and fw in (80) play the role of stochastic sources in the dynamical problem maintaining the system out of equilibrium and
simultaneously modeling the Bohm’s boundary conditions at the sheath which have
not been explicitly included in (80), in contrast to the original models (Ghendrih
et al., 2003)-(Garbet, 1991). Herewith, the physically important effect of particle
escape at the sheath is replaced with a quenched loss of particles in the SOL at the
points for which fn (x, y, t) < 0. Simultaneously, the particles are supposed to arrive
in the SOL in the areas where fn (x, y, t) > 0 modeling the injection of particles from
the divertor core along with the perturbations risen in the system due to the Langmuir probes (Gunn, 2001)-(Labombard, 2002). For a simplicity, we assume that the
processes of gain and loss of particles are balanced in average therefore ⟨ fn ⟩ = 0.
The stochastic source of particles is used instead of the continuously acting radial
Gaussian shaped source (localized at x = 0) studied in the numerical simulations
(Ghendrih et al., 2003). Similarly, we impose the random helicity source fw exerting onto the vorticity dynamics in (80).
Furthermore, the random sources fn and fw account for the internal noise risen
due to the microscopic degrees of freedom eliminated from the phenomenological
equations (80). From the technical point of view, the random forces help to construct
a forthright statistical approach to the turbulent transport in the SOL. In particular, it
allows for the quantum field theory formulation of the stochastic dynamical problem
(80) (based on the Martin-Siggia-Rose (MSR) formalism (Martin et al., 1973)) that
gives a key for the use of advanced analytical methods of modern critical phenomena
theory (Ma, 1976).
36
D.Volchenkov
The Gaussian statistics of random forces in (80) is determined by their covariances,
⟨
⟩
⟨
⟩
Dnn (r − r′ ,t − t ′ ) ≡ fn (r,t) fn (r′ ,t ′ ) , Dww (r − r′ ,t − t ′ ) ≡ fw (r,t) fw (r′ ,t ′ ) ,
where r ≡ (x, y). describing the detailed microscopic properties of the stochastic
dynamical⟨system. /
We discuss⟩the large
⟨ scale/asymptotic⟩behavior of the response
functions δ n(r,t) δ fn (0, 0) and δ n(r,t) δ fw (0, 0) quantifying the reaction
of system onto the external perturbation and corresponding to the r-distributions
of particle density fluctuations expected at time t > 0 in a response to the external disturbances of density and vorticity occurring at the origin at time t = 0. The
high order response functions are related to the analogous multipoint distribution
functions Fn (r1 ,t1 , . . . rn ,tn ; r′ 1 ,t1′ . . . r′ n ,tn ) as
⟨ n
⟩
(
)
δ [n(r1 ,t1 ) . . . n(rn ,tn )]
=
Fn r1 ,t1 . . . rn ,tn ; r′ 1 ,t1′ . . . r′ n ,tn′
∑
′
′
δ fn (r 1 ,t1 ) . . . fn (r n ,tn )
permutations
with summation over all n! permutations of their arguments r1 ,t1 . . . rn ,tn .
We consider a variety of microscopic models for the random forces fn and fw in
the stochastic problem (80). Under the statistically simplest ”white noise” assumption, these random forces are uncorrelated in space and time,
Dnn (r−r′ , t −t ′ ) = Γn δ (r′ −r) δ (t −t ′ ),
Dww (r−r′ , t −t ′ ) = Γw δ (r′ −r) δ (t −t ′ ),
(81)
in which Γn and Γw are the related Onsager coefficients.
Recent studies reported on the statistics of transport events in the cross-field systems (Ghendrih et al., 2003),(Carreras, 1996) pointed out the virtual importance
of correlations existing between density and vorticity fluctuations in the dynamical
problem. In particular, this effect is referred to the formation of large density blobs
of particles close to the divertor walls by attracting particles via the cross field flow,
the latter being the larger for strong blobs with strong potential gradients (Ghendrih et al., 2003). Indeed, in the physically realistic models of turbulent transport
in the SOL, it seems natural to assume that the random perturbations enter into the
system in a correlated way. To be specific, let us suppose that there exists a finite
reciprocal correlation time τc (| r′ − r |) > 0 between the random sources fw (r, τc )
and fn (r′ , 0) in the stochastic problem (80). For a simplicity, we suppose that the
relevant relaxation dynamics is given by the Langevin equation,
∂ fn
fn
fw
=
+√ ,
∂t
τc
β
(82)
⟨ ⟩
in which β ≃ fw2 > 0. In the momentum representation, the non-local covariance
operator τc−1 can be specified by the pseudo-differential operator with the kernel
τc−1 ( k ) = λ ν k2−2γ ,
0 < 2γ < 1,
(83)
What is Control of Turbulence?
37
which specifies the characteristic viscoelastic interactions between the ”fast” modes
of density and vorticity fluctuations. The coupling constant λ > 0 naturally establishes the time scale separation between ”fast” and ”slow” modes. In the case of
2γ ≪ 1, the Langevin equation (82) with the kernel (83) reproduces the asymptotic
dispersion relation typical for the Langmuir waves traveling in plasma, ω ∼ k2−η∗
as k → 0 with η∗ ≃ 0.0804 (in three dimensional space) (Pelletier, 1980). Alternatively, for the exponents 2γ → 1, it corresponds to the √
ion-acoustic waves traveling in the collisionless plasma with the velocity λ ν ∼ Te / M where M is the
ion mass. Intermediate values of γ correspond to the various types of interactions
between these two types of plasma waves described by the Zaharov’s equations (Zaharov, 1972).
The relaxation dynamics (82 - 83) establishes the relation between the covariances of random sources in (80),
(
)
∫
∫ ∞
J0 (ρ r′ ) exp −λ ν ρ 2−2γ t ′
1
′ ′
Dnn (r,t) =
dr dt
dρ
Dww (r′ − r,t ′ − t)
4πβ
λ ν ρ 1−2γ
0
(84)
where r ≡ | r |, and J0 is the Bessel function of the first kind. In the present section,
we choose the covariance of random vorticity source,
⟨
⟩ ∫ d ω ∫ dk
[
]
fw (r,t) fw (r′ ,t ′ ) =
Dww (ω , k) exp −iω (t − t ′ ) + ik (r − r′ ) ,
2
2π
(2π )
k ≡ | k |, in the form of white noise (81), in which the relevant Onsager coefficient
Γw is found from the following physical reasons. Namely, the instantaneous spectral
balance of particle flux,
W(k) =
1
2
∫
dω
⟨ fn (k, ω ) fn (−k, ω ) ⟩ ,
2π
(85)
derived from (84) should be independent from the reciprocal correlation time τc ( k )
at any k that is true provided Dww (ω , k) ∝ λ k−2γ . Furthermore, the Onsager coefficient Γw has to fit into the appropriate physical dimension which is assembled
from the relevant dimensional parameters, u0 ν and k. Collecting these factors, one
obtains the Ansatz Dww ∝ λ u30 ν 3 k6−d−2γ , in which d = 2 is the dimension of space.
The power law model for the covariance of random helicity force ∝ k6−d−2γ does
not meet the white noise assumption since δ (r − r′ ) ∼ k0 and therefore calls for
another control parameter 2ε > 0. Eventually, we use the model
Dww (k, ω ) = Γw k6−d−2ε −2γ ,
Γw ∝ λ u30 ν 3 ,
d = 2,
(86)
with the actual value of regularization parameter 2ε = 4, for d = 2.
Let us note that the Ansatz (86) is enough flexible to include the various particular
models of particle pump into the SOL. For instance, the alternative to the white noise
assumption spatially uniform particle pump for which the covariance
38
D.Volchenkov
Dww ≃ δ ( k ) = lim
ξ →0
(
∫
dx
x
ρs
)−ξ
eikx = k−d
Γ (d/2)
lim (ξ k ρs ) ,
2π d/2 ξ →0
in the large scales, can be represented by the Ansatz (86) with the actual value
2ε = 3.
In the rapid-change limit of the stochastic model, λ → ∞ (i.e., τc → 0), the
covariance (84) turns into
⟨ fn (k, ω ) fn (−k, ω ) ⟩ ≃
ν 2−d−2ε +2γ
k
,
λβ
(87)
and recovers the white noise statistics (81) along the line ε = γ , in d = 2. Alternatively, in the case of λ → 0 (that corresponds to τc → ∞), the time integration is
effectively withdrawn from (84), so that the resulting configuration relevant to (86)
appears to be static ∝ k4−d−2ε and uncorrelated in space (at d = 2) for 2ε = 2.
The power-law models for the covariances of random forces has been used in
the statistical theory of turbulence (Adzhemyan et al., 1998) (see also the references
therein). The models of random walks in random environment with long-range correlations based on the Langevin equation (82) have been discussed in concern with
the problem of anomalous scaling of a passive scalar advected by the synthetic compressible turbulent flow (Antonov, 1999), then in (Volchenkov et al., 2002), for the
purpose of establishing the time scale separation, in the models of self organized
criticality (Bak et al., 1987)-(Bak, 1996).
Recently, the renormalization-group methodology have been applied in order to
prove the breakdown of magnetic flux conservation for ideal plasmas, by nonlinear
effects (see (Eyink and Hussein, 2006)). The analysis of (Eyink and Hussein, 2006)
is based upon an effective equation for magneto-hydrodynamic (MHD) modes; it
is proven that flux-conservation can be violated for an arbitrarily small length-scale
that is similar to the decay of magnetic flux through a narrow superconductive ring,
by phase-slip of quantized flux lines. Being analogous to Onsagers result on energy
dissipation anomaly in hydrodynamic turbulence, this result gives analytical support
to and rigorous constraints on theories of fast turbulent reconnection.
3.3 Iterative Solutions in Crossed Fields
The linearized homogeneous problem, for the fluctuations of density n and vorticity
w vanishing at t → ∞,
←
−
[ ∂t − ν∆⊥ ] ∆ n = δ ( r ) δ (t ),
←
−
←
−
[ ∂t − u0 ν ∆⊥ ] ∆ w (r, t) + g1 ∂y ∆ n (r, t) = δ ( r ) δ (t ),
is satisfied by the retarded Green’s functions,
(88)
What is Control of Turbulence?
39
(
)
θ (t )
r2
←
−
∆ n (r,t) =
exp −
,
4π ν t
4ν t
(
)
θ (t )
r2
2g1 θ (t ) x
←
−
∆ w (r,t) =
,
exp −
+
4π uν t
4u ν t
ν (u + 1) r2
←
−
←
−
fitting into the retarding conditions, ∆ n (r,t) = ∆ w (r,t) = 0, for t < 0, that express
the casualty principle in the dynamical problem. Nonlinearities in (88) can then be
taken into account by the perturbation theory,
∫
n(r,t) =
[
]
←
−
dr′ dt ′ ∆ n (r − r′ , t − t ′ ) fn (r′ ,t ′ ) − v(r′ ,t ′ ) × ∇n(r′ ,t ′ ) ,
∫
w(r,t) =
dr
′
←
−
dt ′ ∆ w (r − r′ ,t − t ′ )
[
(−1)k gk
∂y nk (r′ ,t ′ )−
k
k≥1
fw (r′ ,t ′ ) + ∑
∫
∫
]
←
−
←
−
v(r′ ,t ′ ) × ∇w(r′ ,t ′ ) + g1 dr′ dt ′ ∆ w (r − r′ ,t − t ′ ) dr′′ dt ′′ ∆ n (r′ − r′′ ,t ′ − t ′′ )
[
]
×∂y fn (r′′ ,t ′′ ) − v(r′′ ,t ′′ ) × ∇n(r′′ ,t ′′ ) .
(89)
The solutions (89) allow for the diagram representation (see Fig. 5), where the external line (a tail) stands for the field n, the double external line denotes the field w,
and the bold line represents the magnetic flux v. The triangles stay for the random
←
−
force fn , and the filled triangles represent fw . The retarded Green functions ∆ n are
′
marked by the lines with an arrow which corresponds to the arguments (r ,t ′ ) and
(r′′ ,t ′′ ) (the direction of arrows marks the time ordering of arguments in the lines).
Similarly, the double lines with an arrow correspond to the retarded Green functions
←
−
∆ w . Slashes correspond to the differential operator ∇. Circles surrounding vertexes
representing the antisymmetric interaction v × ∇, squares present the vertexes proportional to the poloidal gradient ∂y . All correlation functions of fluctuating fields
and functions expressing the system response for the external perturbations could
be found by the multiplication of trees (89) displayed on Fig. 5 followed by the
averaging over all possible configurations of random forces fn ( r,t ) and fw ( r,t ).
In diagrams, this procedure corresponds to all possible contractions of lines ended
with the identical triangles. Thereat, the diagrams having an odd number of external
triangles (correspondent to the random forces) give zero contributions in average.
As a result of these contractions, the following new elements (lines) appear in the
diagrams of perturbation theory:
∆nn (r − r′ ,t ′ − t) =
∆ww (r − r′ ,t ′ − t) =
∫
∫
dr1 dt1
∫
→
−
dr2 dt2 ∆ n (r − r1 ,t1 − t)Dnn (r1 − r2 ,t1 − t2 )
←
−
× ∆ n (r2 − r′ ,t2 − t ′ ),
∫
dr1 dt1
→
−
dr2 dt2 ∆ w (r − r1 ,t1 − t)Dww (r1 − r2 ,t1 − t2 )
←
−
× ∆ w (r2 − r′ ,t2 − t ′ ) + g21 ∂y2 ∆nn (r − r′ ,t ′ − t),
40
D.Volchenkov
Fig. 5 The diagrammatic representation for Eq. (89).
∆wn (r − r′ ,t ′ − t) = g1
←
−
∆ wn (r − r′ ,t ′ − t) = g1
∫
∫
→
−
dr1 dt1 ∆ n (r − r1 ,t1 − t)∂y ∆ww (r1 − r′ ,t1 − t ′ ),
←
−
←
−
dr1 dt1 ∆ n (r − r1 ,t1 − t)∂y ∆ w (r1 − r′ ,t1 − t ′ ),
(90)
which are the free propagators of particle density and vorticity fluctuations, the
mixed correlator, and the retarded mixed Green’s function. In diagrams, we present
the free propagators (90) by the correspondent lines without an arrow, and the retarded mixed Green’s functions by the composite directed lines (see Fig. 6). The
cross-field drift function v( r,t ) is not involved into the linear homogeneous problem (88) and, therefore, it does not appear in the free propagators (90), however, it is
presented in the nonlinear part of dynamical equations and therefore appears in the
diagrams of perturbation theory. Due to the simple relation w = ∇ × v, the propa←
−
gators containing the field v are the same as those with w: ∆vv , ∆nv , and ∆ nv . The
bold lines representing v in Fig. 5 can be replaced in the diagrams of perturbation
theory with the double lines (which correspond to the field w) with the additional
factor (in the momentum representation) −i εzms km /k2 , where m, s ≡ x, y and εzxy is
the antisymmetric pseudo-tensor, for each v.
In this framework, the exact correlation functions of fields and the response functions can be found from the Dyson equations,
⟨
δn
δ fn
⟩−1
←
−−1
= ∆ n − Σn ,
⟨
δw
δ fw
⟩−1
←
−−1
= ∆ w − Σw ,
(91)
What is Control of Turbulence?
41
Fig. 6 First diagrams for the simplest response functions ⟨δ n/δ fn ⟩ and ⟨δ w/δ fw ⟩.
where Σn and Σw are the infinite diagram series, in which the first diagrams are
shown in Fig. 6. The diagram technique introduced in the present section is suitable
for the system preserving the continuous symmetry of (79), apart from the sheath.
3.4 Functional Integral Formulation of Cross-Field Turbulent
Transport
In the present section, we study the properties of diagram series resulting from the
iterations of the stochastic dynamical equations with the consequent averaging with
respect to all possible configurations of random forces. The diagrams for some correlation functions diverge in small scales. The use of conventional arguments borrowed from the quantum field renormalization group (Zinn-Justin, 1990) helps to
prove the consequent subtraction of the logarithmic divergent terms in all orders of
perturbation theory out from the diagrams.
The set of diagrams arisen in the perturbation theory by the iterations of (80) is
equivalent to the standard Feynman diagrams of some quantum field theory with
the doubled set of stochastic fields: the fluctuations n and w, the flux function v,
the auxiliary fields n′ , w′ functionally conjugated to the Gaussian distributed random forces fn and fw in (80), and the Lagrange multiplier v′ for the binding relation
w = ∇ × v. The coincidence of diagrams is a particular consequence of the general equivalence between the t-local stochastic dynamical problems (in which the
interactions contain no time derivatives) and the relevant quantum field theories (De
42
D.Volchenkov
Dominicis amd Martin, 1979) with the action functional S found in accordance to
the MSR formalism (Martin et al., 1973). Statistical averages with respect to all admissible configurations of random forces in a stochastic dynamical problem can be
identified with the functional averages with the weight exp S . In particular, for the
stochastic problem (80), the generating functional of the Green functions, G (AΦ ),
with the arbitrary source fields AΦ (r,t) where Φ ≡ {n, n′ , w, w′ , v, v′ } can be represented by the functional integral
[
]
∫
∫
G (AΦ ) = D Φ exp S (Φ ) + dr dt AΦ (r,t)Φ (r,t) ,
(92)
in which
S (Φ ) =
∫
1
2
∫
[
dr dr′ dt dt ′ n′ (r,t)Dnn (r − r′ ,t − t ′ )n′ (r′ ,t ′ )
(93)
]
+w′ (r,t)Dww (r − r′ ,t − t ′ )w′ (r′ ,t ′ ) −
dr dt n′ (r,t) [ ∂t n(r,t) + v(r,t) × ∇n(r,t) − ν∆⊥ n(r,t) ]
[
∫
(−1)k gk
′
∂y nk (r,t)
− dr dt w (r,t) ∂t w(r,t) + v(r,t) × ∇w(r,t) + ∑
k
k≥1
−u0 ν∆⊥ w(r,t)] +
∫
dr dt v′ (r,t) [ w(r,t) − ∇ × v(r,t) ] .
The source functions An′ ( r,t ) and Aw′ ( r,t ) in (92) are interpreted as the not random
external forces, so that the Green functions
⟨
⟩ ⟨
⟩ ⟨
⟩
w(r,t) w′ (r′ ,t ′ ) ,
n(r,t) n′ (r′ ,t ′ ) ,
n(r,t) w′ (r′ ,t ′ ) ,
⟨
/
⟩ ⟨ /
⟩
′
for
⟨ t/ < t,⟩ coincide with the response functions δ w δ fw , δ n δ fn , and
δ n δ fw respectively. All possible boundary conditions and the damping asymptotic conditions for the fluctuation fields n and w at t → −∞ are included into the
functional integration domain in (92).
The functional integral formulation (92) of the stochastic dynamical problem
(80) allows for the use of various techniques developed in the quantum field theory
to study the long-time large-scale asymptotic behavior of the quantum and stochastic
systems. The integral (92) has the standard Feynman diagram representation which
is equivalent to the iterative solution of (80) with an exception of the self-contracted
lines
)
(
(
)
←
−
←
−
Tr ∆ ΦΦ ′ = Tr ∆ Φ ′ Φ = 0,
which are proportional to ∝ θ (t −t ′ ) in time-representation and therefore discontinuous at t = t ′ . Stipulating that θ (0) = 0, one can exclude all redundant graphs from
the perturbation theory of (92). Lines and vertexes in the graphs of perturbation
theory are defined by the conventional Feynman rules and correspond to the free
propagators (equivalent to (90)) readily calculated from the free (quadratic) part of
What is Control of Turbulence?
43
functional (93) and the nonlinear interactions of fields respectively. In the actual calculations, it is convenient to use the propagators (90) in their momentum-frequency
representation,
←
−
∆ nn′ (k, ω ) = (−iω + ν k2 )−1 ,
∆wv′ (k) = −
εzms ks
,
k2
←
−
∆ ww′ (k, ω ) = (−iω + u0 ν k2 )−1 ,
→∗
−
←
−
∆ nn′ (k, ω ) = ∆ nn′ (k, ω ),
→
−
←
−
→
−
←
−
∆nn (k, ω ) = ∆ nn′ Dnn (k, ω ) ∆ n′ n , ∆ww (k, ω ) = g21 ky2 ∆nn (k, ω )+ ∆ ww′ Dww (k, ω ) ∆ w′ w ,
←
−
←
−
←
−
←
−
∆nw (k, ω ) = ig1 ∆ nn′ (k, ω )ky ∆ww (k, ω ), ∆ nw′ (k, ω ) = ig1 ∆ nn′ (k, ω )ky ∆ ww′ (k, ω ),
←
−
→∗
−
←
−
→∗
−
∆ ww′ (k, ω ) = ∆ ww′ (k, ω ), ∆ nw′ (k, ω ) = ∆ w′ n (k, ω ).
Propagators including the field v coincide with those of the field w up to the
multiplicative factor −i εzms km /k2 for each field v. Propagators of auxiliary fields
∆Φ ′ Φ ′ (k, ω ) = 0.
The action functional (93) is invariant with respect to the generalized Galilean
transformations of fields in the poloidal direction,
vy ( r,t ) → vy ( r,t ) − a(t ),
n(x, y,t) → n ( x, y − b(t ),t )
(94)
with the parameter of ∫transformations a(t ) (the integrable function decaying at
t
t → −∞) and b(t ) = −∞
a(t ′ ) dt ′ . Furthermore, any quantity Q in (93) can be
characterized with respect to the independent scale transformations, in space and
time, by its momentum dimension dQk and the frequency dimension dQω . In the ”logarithmic” theory (which is free of interactions that is analogous to the linearized
problem (88)), these scale transformations are coupled due to the relation ∂t ∼ ∇2
in the dynamical equations.
This allows for the introduction of the relevant total canonical dimension
dQ = dQk + 2dQω
and the analysis of UV divergences arisen in the diagrams of perturbation theory based on the conventional dimension counting arguments (Zinn-Justin, 1990;
Collins, 1992). In dynamical models, dQ plays the same role as the ordinary (momentum) dimension in the critical static problems. Let us note that the poloidal
gradient term ∝ ∂y in (88) is responsible for the stationary contributions into the
Green’s function as t > 0, so that the above definition of dQ remains unambiguous.
Stipulating the natural normalization conventions,
dkk = −drk = 1,
dkω = drω = 0,
dωk = dtk = 0,
dωω = −dtω = 1,
one can find all relevant canonical dimensions from the simultaneous momentum
and frequency scaling invariance of all terms in (93) (see Tab. 3.4.1).
Integrals correspondent to the diagrams of perturbation theory representing the
1-irreducible Green’s functions G diverge at the large momenta (small scales) if the
44
D.Volchenkov
Table 3 Canonical dimensions of fields and parameters in the action functional (93)
ν u0 n
d k -2 0 0
dω 1 0 0
d 0 0 0
w
0
1
2
v
-1
1
1
v′
d
0
d
n ′ w′
d d
0 2
d d +4
ϕ
-2
1
0
gk
−1
2
3
correspondent UV-divergence index δG is a nonnegative integer in the logarithmic
theory,
δG = d + 2 − (NΦ dΦ + NΦ ′ dΦ ′ ) ≥ 0,
(95)
where d is the dimension of space, dΦ ,Φ ′ are the total canonical dimensions of fields
Φ and Φ ′ , and NΦ ,Φ ′ are the numbers of relevant functional arguments in G. As a
consequence of the casualty principle, in the dynamical models of MSR-type (Martin et al., 1973), all 1-irreducible Green’s functions without the auxiliary fields Φ ′
vanish being proportional to θ (0) and therefore do not require counterterms. Furthermore, the dimensional parameters and external momenta occurring as the overall
factors in graphs also reduce their degrees of divergence (95).
In spite of the following 1-irreducible Green functions could be superficially divergent at large momenta:
⟨n′ nv⟩,
⟨n′ v⟩,
⟨n′ vv⟩,
⟨n′ n . . . n⟩
(with an arbitrary number of fields n ), the only Galilean invariant Green’s function
admissible in the theory (93) which actually diverges at large momenta is ⟨n′ n⟩. The
inclusion of the relevant counterterm subtracting their superficially divergent contribution is reproduced by the multiplicative renormalization of the Prandtl number
u0 = uZu
where u0 and u are the bare and renormalized values of Prandtl’s number. In principle, the relevant renormalization constant Zu can be calculated implicitly from the
graphs of perturbation theory up to a finite part of the relevant counterterm.
However, the standard approach of the critical phenomena theory is useless for
the determining of the large scale asymptotes in the problem in question, since the
severe instability frustrates the critical behavior in the system preventing its approaching to the formal asymptote predicted by the conventional renormalization
group method. As a consequence, the critical dimensions of fields and parameters
which can be computed from the renormalization procedure would have just a formal meaning.
What is Control of Turbulence?
45
3.5 Large-scale Instability of Iterative Solutions
The iterative solutions for the stochastic problem (80) constructed in the previous
subsections would be asymptotically stable in the large scales provided all small
perturbations of both density and vorticity damp out with time. In particular, the
exact response functions found from the Dyson equations (91) should have poles
located in the lower half-plane of the frequency space.
The stability of free response function
/
⟩
⟨ δ n δ fn = (−iω + ν k2 )−1
which effectively corresponds to the linearized problem (88) is ensured in the large
scales by the correct sign of the dissipation term ν k2 > 0. In a ”proper” perturbation
theory, apart from a crossover region, the stability of exact response functions is also
secured by the dissipation term which dominates over the dispersion relation in the
large scales,
ω ( k ) = −i ν k2 + i Σnn ( k, ω )
(96)
where the self-energy operator Σnn ( k, ω ) is the infinite series of all relevant 1− irreducible diagrams of perturbation theory. However, in the perturbation theory, the
leading contribution into the self-energy operator is Σnn ≃ O( ky ) that could lift up
the pole of the response function into the upper half-plane of the complex ω −plane
rising the instability in the system as k → 0. Such an anomalously strong contribution comes from the diagrams which simultaneously include both the antisymmetric
interaction vertex ≃ i ez · εi jz vi k j together with the poloidal gradient ≃ i gk ky and the
free propagator of particle density ∆nn ( k, ω ). Such /diagrams
⟩ appear/in all⟩orders of
perturbation theory for the response functions ⟨ δ n δ fn and ⟨ δ n δ fw indicating that the instability could arise due to the random fluctuations of both density
and vorticity. The first order 1-irreducible diagrams for them are displayed in Fig. 7.
The small parameters gn with n > 2 appear in the forthcoming orders of perturbation theory. In particular, the last diagrams in both series shown in Fig. 7 generate
the ”anomalous” contributions in the large
To be certain, let us consider the
/ scales.
⟩
second diagram in the series for ⟨ δ n δ fn which corresponds to the following
analytical expression:
Σnn |1−loop ≃ g2 ky
∫
dp
(2π )2
∫
dω
2π
∫
d ω ′ kx px + ky py − p2x − p2y
2π
(k − p)2
(97)
←
−
×∆nn (p, ω ′ ) ∆ w (k − p, ω − ω ′ ).
Being interested in the O( ky )-contribution into (97), one can neglect the k−dependence
in the integrand. The analytical properties of this contribution depends very much
upon the certain assumption on the covariances of random forces since it changes
the free propagator ∆nn . For instance, under the white noise assumption (81) ∆nn ≃
( ω 2 + ν 2 p4 )−1 , the integral in (97) diverges at the small scales (large momenta)
for ε < 1 and diverges at the large scales (small momenta) for ε > 1. Introducing
46
D.Volchenkov
Fig. 7 The simplest 1-irreducible diagrams contributing into the self-energy corrections Σnn and
Σnw for the response functions ⟨δ n/δ fn ⟩ and ⟨δ n/δ fw ⟩. The field indexes denote the type of propagators and simultaneously the type of vertexes. The slashes mark the positions of derivatives ∇,
and the skewed triangles denotes the inverse operator curl−1 .
the relevant cut-off parameters, one obtains the anomalous contribution
(
)
ν g2 Λ 2−2 ε − m2−2 ε
Σnn |1−loop ≃ − ky
,
8π (−1 + ε ) (u + 1)
(98)
in which u is the renormalized value of Prandtl’s number. In the preceding section,
we have shown
/ that⟩the logarithmic divergencies risen in diagrams for the response
function ⟨ δ n δ fn in the small scales (large momenta) can be eliminated from
the perturbation theory by the appropriate renormalization. The singularity in (98)
arisen at the small momenta m → 0 for ε > 1 would compensate the smallness of
g2 , so that any density fluctuation with ky > 0 appears to be unstable.
Accounting for the finite reciprocal correlation time τc > 0 between the vorticity
and density random sources in (80) introduces the new dimensional parameter λ > 0
(
) )−1
(
. Then,
into the particle density propagator, ∆nn ≃ ω 2 + ν 2 p4 1 + λ p−2γ
the integral (97) can be computed by its analytic continuation for any momenta
excepting for the isolated points, −1 + ε = γ mod 1,
)
(
( u )(−1+ε )/γ g
π (−1 + ε )
2
.
Σnn = ν A ( ε , γ ) ky , A ( ε , γ )|1−loop ≃
csc
λ
8uγ
γ
(99)
What is Control of Turbulence?
47
The dispersion relation (96) determines the region of asymptotic stability in the
phase space of cross-field transport system. Namely, a density fluctuation arisen in
the SOL with some random momenta (kx , ky ) would be asymptotically stable with
respect to the large scales k → 0 if
ky
1
<
kx2 + ky2
A ( ε, γ )
(100)
and be unstable otherwise. In the first order of perturbation theory, the amplitude
factor A(ε , γ ) is given by (99). For different values A ( ε , γ ), the stability condition
(100) determines the set of circles (see Fig. 8) osculating at the origin which bound
the unstable segments of phase space. One can see that the density fluctuations with
Fig. 8 . The admissibility condition (100) defines the set of elliptic curves which bound the unstable segments in the phase space. Density fluctuations with ky → 0, (i.e. extended in the poloidal
direction) appear to be stable in the large scales for any | kx | > 0. Those fluctuations characterized
by | ky | > 0 would also be stable in the large scales provided sign ( ky ) = − sign ( A(ε , γ ) ) for the
given values ε and γ .
ky → 0 (i.e. extended in the poloidal direction) are asymptotically stable for any
| kx | > 0. Density fluctuations characterized by | ky | > 0 would be asymptotically
stable in a certain stochastic model provided sign ( ky ) = − sign (A( ε , γ )) , for the
given values of ε and γ . The signature of the 1-loop order contribution (99) into
A( ε , γ ) is displayed on the diagram in Fig. 9 (black is for +1, white is for −1)
at different values of γ and d = 6 − 2ε − 2γ , the space dimension related to the
actual value of regularization parameter ε under the statistical assumption (86). It
48
D.Volchenkov
Fig. 9 . The signature of 1-loop order contribution (99) into A(ε , γ ) (black is for +1, white is for
−1) at different values of γ and d = 6 − 2ε − 2γ , the space dimension related to the actual value of
regularization parameter ε under the statistical assumption (86).
is important to note that the stability condition
(100) can be formulated as an upper
/
bound for the order parameter ξ = ky k2 : ξ < ξc where ξc = A ( ε , γ )−1 . For
the uncorrelated statistics of random forces, in the stochastic dynamical problem,
A( ε , γ ) (98) diverges as k → 0 and therefore ξc → 0.
3.6 Turbulence Stabilization by the Poloidal Electric Drift
To promote the stochastic cross-field turbulent transport system (80) from the instability to a stable regime, it seems natural to frustrate the symmetry which breaks
the Galilean invariance in (80). This can be achieved by generating a constant
uniform drift in the poloidal direction vy → vy − V (by biasing the limiter surface, ϕ (x) → ϕ (x) + xV ) that would eradicate those configurations with the trivial
poloidal component of electric drift vy = 0. In general, the relevant dispersion equation ω ( k,V ) = 0 could have many solutions Vk for k ≪ 1. Herewith, the turbulence
stabilization is achieved for the drifts V from the intervals Vk−1 < V < Vk for which
Im(ω ) < 0.
To be certain, let us consider
/ the
⟩ dispersion equation correspondent to the simplest response function ⟨ δ n δ fn , in the 1-loop order. The leading contribution
into the dispersion equation is given in the large scale region by the diagram (97).
What is Control of Turbulence?
49
Under the white noise assumption (81), the free propagator accounting for the uni(
)−1
form electric drift V is ∆nn ≃ ω 2 + ν 2 k4 + V 2 ky 2
. Then, for | ky | < 1, the
dispersion relation reads as following,
)
( (
)
√ V 2 (u + 2) Γ (1/2 + ε ) log m
ν m2−2 ε
g1 g2
2
ω ( k, V ) ≃ −i ν k + i ky
− π
.
u + 1 8π (−1 + ε )
8ν (1 + u) Γ (2 + ε )
(101)
The latter relation shows that for any finite m > 0 there exists finite Vc < ∞ such
that for any V > Vc one obtains Im(ω ) < 0 in (101), however, Vc → ∞ as m → 0.
In contrast to it, in the case of correlated statistics (82 - 83, 86), the dispersion
relation is not singular for k → 0 excepting some particular values of ε and γ , and
the correspondent stabilizing electric drift, in the 1-loop order, equals to
)
(
/
(uλ )1/2γ (1 + u)1 + (2ε +1)/2γ ( 1 + u )1+ ε /γ − 1 sin π ( 2ε + 1 ) 2γ
3
Vc = π ν u
.
sin π ε /γ
(1 + u)2 + (2ε +1)/2γ + u (2 ε + 1) − 2 γ (1 + 2 u)
(102)
In the range 0 < γ < 1/2, u > 0, this expression is singular at the points ε /γ ∈ Z.
3.7 Qualitative Discrete Time Model of Anomalous Transport in
the SOL
Large scale instability developed in the cross-field model (80) is related to the appearance and unbounded growth of fluctuations of particle density close to the wall.
In accordance to the fluctuation-dissipation theorem, the fluctuations arisen in the
stochastic dynamical system are related to its dissipative properties. In particular,
the matrix of the exact response functions R( k, ω ) expressing the perturbations of
fields n and w risen due to the random sources fn and fw determines the matrix of
exact dynamical Green’s functions G( k, ω ),
R ( k, ω ) − R† ( −k, −ω ) = i ω G ( k, ω )
(103)
where R† is the transposed R. In the large scale limit k → 0, we take into account
for the leading contributions into the self-energy operators in the elements of R,
(
)−1
Rnn ≃ −iω − ν A1 ky + ν k2
,
(
)−1
Rnw ≃ −iω − ν u A2 ky + ν u k2
, (104)
(
)−1
Rwn ≃ −iω + ν k2
,
(
)−1
Rww = −iω + ν u k2
,
in which A1,2 are the amplitudes of the anomalous contributions competing with the
dissipation ∝ O(k2 ) in the large scales. Fluctuations of particle density arisen in the
model (80) grow up unboundedly provided either A1 ky > k2 or A2 ky > k2 for the
given values of ε and γ . The correspondent advanced Green’s functions appear to
50
D.Volchenkov
be analytic in the lower half-plane of the frequency space,
ν · Gann ( k,t ) =
A1 ky θ (−t)
,
k4 − A21 ky2
ν u · Ganw ( k, t ) = −
θ (−t)
θ (−t)
−
, (105)
2
(A2 ky − k )
k2
being trivial for t > 0.
For instance, let us consider the advanced Green’s function Gann which relates
the density of particles δn( r, t ) in the fluctuations characterized with A1 ky > k2
and arisen at the point r ∈ Ω inside the divertor at time t with the particle density
δn( r′ , t ′ ) of those achieved the divertor wall at some subsequent moment of time
t ′ > t at the point r′ ∈ ∂ Ω :
∫
δn( r′ , t ′ )∂ Ω =
where
Gann
∫
t <t ′
dt
Ω
dr δn( r, t ) Gann ( r′ − r )
[
(
)
1
A1 (y′ − y)
A1 |r′ − r|
sin
J0
2ν A1
2
2
(
)]
A1 (y′ − y)
A1 |r′ − r|
+ cos
H0
,
2
2
(106)
( ′
)
r −r =
(107)
√
in which |r − r′ | ≡ (x − x′ )2 + (y − y′ )2 , J0 and H0 are the Bessel and Struve functions respectively. The integral in the r.h.s. of (106) is finite for any Ω provided
| A1 | > 1, but for the compact Ω as | A1 | < 1. To be specific, let us consider the
circle CR of radius R as the relevant domain boundary and suppose for a simplicity
that the density of particles incorporated into the growing fluctuations inside the domain is independent of time and maintained at the stationary rate δn0 ( r). Then the
r−integral in the r.h.s of (106) can be calculated at least numerically and gives the
growth rate B(R) for those density fluctuations,
δn( R, τ ) = τ · B(R) ,
(108)
where τ is the traveling time of the density blob to achieve the divertor wall that
can be effectively considered as a random quantity. It is the distribution of such
wandering times that determines the anomalous transport statistics described by the
flux pdf in our simplified model. The discrete time model we discuss below is similar
to the toy model of systems close to a threshold of instability studied in (Floriani et
al., 2003) recently. Despite its obvious simplicity (the convection of a high density
blob of particles by the turbulent flow of the cross field system is substituted by the
discrete time one-dimensional (in the radial direction) random walks characterized
with some given distribution function), its exhibits a surprising qualitative similarity
to the actual flux driven anomalous transport events reported in (Ghendrih et al.,
2003).
We specify the random radial coordinate of a growing fluctuation by the real
number x ∈ [0, 1]. Another real number R ∈ [0, 1] is for the coordinate of wall.
The fluctuation is supposed to be convected by the turbulent flow and grown as
What is Control of Turbulence?
51
long as x < R and is destroyed otherwise (x ≥ R). We consider x as a random
variable distributed with respect to some given probability distribution function
P{x < u} = F(u). It is natural to consider the coordinate of wall R as a fixed number,
nevertheless, we discuss here a more general case when R is also considered as a
random variable distributed over the unit interval with respect to another probability
distribution function (pdf) P {R < u} = Q(u). In general, F and Q are two arbitrary left-continuous increasing functions satisfying the normalization conditions
F( 0 ) = Q( 0 ) = 0, F( ∞ ) = Q( ∞ ) = 1.
Given a fixed real number η ∈ [0, 1], we define a discrete time random process
in the following way. At time t = 0, the variable x is chosen with respect to pdf F,
and R is chosen with respect to pdf Q. If x < R, the process continues and goes to
time t = 1. Otherwise, provided x ≥ R, the process is eliminated. At time t ≥ 1,
the following events happen:
i) with probability η , the random variable x is chosen with pdf F, but the threshold
R keeps the value it had at time t − 1. Otherwise,
ii) with probability 1 − η , the random variable x is chosen with pdf F, and R is
chosen with pdf Q.
If x ≥ R, the process ends; if x < R, the process continues and goes to time t + 1.
Eventually, at some time step τ , when the coordinate of the blob, x, drops ”beyond” R, the process stops, and the integer value τ resulted from such a random
process limits the duration of convectional phase. The new blob then arises within
the domain, and the simulation process starts again.
While studying the above model, we are interested in the distribution of durations
of convection phases Pη (τ ; F, Q) (denoted as P(τ ) in the what following) provided
the probability distributions F and Q are known, and the control parameter η is
fixed. The motionless wall corresponds to η = 0. Alternatively, the position of wall
is randomly changed at each time step as η = 1.
The proposed model resembles to the coherent-noise models (Newman and
Sneppen, 1996)-(Sneppen and Newman, 1997) discussed in connection with a standard sandpile model (Bak et al., 1987) in self-organized criticality, where the statistics of avalanche sizes and durations take power law forms.
We introduce the generating function of P( τ ) such that
∞
ˆ s) 1 d τ P(
,
ˆ
P(s)
= ∑ sτ P( τ ), P(τ ) =
(109)
τ ! dsτ s=0
τ =0
and define the following auxiliary functions
∫ ∞
K(n) =
0
F(u)n dQ(u),
δK(n) = K(n) − K(n + 1),
52
D.Volchenkov
p(l) = η l K(l + 1) ,
q(l) = (1 − η )l K(1)l−1 ,
r(l) = η l [η δK (l + 1) + (1 − η ) K(l + 1) δK(0)] ,
ρ = η δK(1) + (1 − η ) K(1) δK(0) .
for l ≥ 1 ,
for l ≥ 1 ,
for l ≥ 1,
p(0) = 0 ,
q(0) = 0 ,
r(0) = 0 ,
(110)
Then we find
s
[ˆr(s) + ρ p(s)
ˆ q(s)
ˆ + ρ K(1)q(s)
ˆ + K(1) q(s)ˆ
ˆ r(s)] ,
1 − p(s)
ˆ q(s)
ˆ
(111)
where p(s),
ˆ
q(s),
ˆ
rˆ(s) are the generating functions corresponding to p(l), q(l), r(l),
respectively.
In the marginal cases η = 0 and η = 1, the probability P(τ ) can be readily calculated,
Pη =0 (τ ) = K(1)τ δK(0), Pη =1 (τ ) = δK(τ ).
(112)
ˆ = δK(0)+ ρ s+
P(s)
The above equation shows that in the case of η = 0, for any choice of the pdf F
and Q, the probability P(τ ) decays exponentially. In the opposite case η = 1, many
different types of behavior are possible, depending upon the particular choice of F
and Q.
To estimate the upper and lower bounds for P(τ ) for any η , one can use the fact
that
K(1)n ≤ K(n) ≤ K(1) and 0 ≤ δK(n) ≤ K(1) , n ∈ N.
Then the upper bound for K(n) is trivial, since 0 ≤ F(u) ≤ 1 for any u ∈ [0, 1]. The
upper bound for K(n) exists if the interval of the random variable u is bounded and
therefore can be mapped onto [0, 1] (as a consequence of Jensen’s inequality, and of
the fact that the function u :→ un is convex on the interval ]0, 1[ for any integer n).
The calculation given in (Floriani et al., 2003) allows for the following estimation
for the upper bound,
Pη (τ ) ≤ η τ δK(τ ) + (1 − η )K(1) δK(0) [η + (1 − η ) K(1)]τ −1
{
}
+ η K(1) [η + (1 − η ) K(1)]τ −1 − η τ −1 ,
(113)
and, for the lower bound,
Pη (τ ) ≥ η τ δK(τ ) + (1 − η ) K(1)τ δK(0)
(114)
τ
= η Pη =1 (τ ) + (1 − η )Pη =0 (τ ) .
We thus see that, for any 0 ≤ η < 1, the decay of distribution P(τ ) is bounded by
exponentials. Furthermore, the bounds (113) and (114) turns into exact equalities,
in the marginal cases η = 0 and η = 1.
The simpler and explicit expressions can be given for P(τ ) provided the densities
are uniform dF(u) = dQ(u) = du for all u ∈ [0, 1]. Then the equations (112) give,
Pη =0 ( τ ) = 2− (τ + 1) ,
Pη =1 ( τ ) =
1
.
(τ + 1)(τ + 2)
(115)
What is Control of Turbulence?
For the intermediate values of η , the upper and lower bounds are
(
)
ητ
1 1+η τ
.
+ (1 − η ) 2− (τ + 1) ≤ P( τ ) ≤
(τ + 1)(τ + 2)
2
2
53
(116)
The above results are displayed in Fig. 10.
Fig. 10 . The distributions of wandering times near the wall in the discrete time model, in the
case of the uniform densities dF(u) = dG(u) = du for all u ∈ [0, ∞) at different values of control
parameter η .
The
accounting
/(
) for the dissipation processes introduces the order parameter
ξ = ky kx2 + ky2 and its critical value ξc such that the particle density fluctuation δn(ξ ) grows unboundedly with time as ξ > ξc and damps out otherwise. We
compute the value of ξc , in the first
/ order of perturbation theory developed with respect to the small parameter ρs Rn¯ where ρs is the Larmor radius, R is the major
radius of torus, and n¯ is the mean normalized density of particles.
Our results demonstrate convincingly that the possible correlations between density and vorticity fluctuations would drastically change the value ξc modifying the
stability of model. Characterizing the possible reciprocal correlations between the
density and vorticity fluctuations by the specific correlation time τc , we demonstrate that any fluctuation of particle density grows up with time in the large scale
limit (k → 0) as τc → ∞ (the density and vorticity fluctuations are uncorrelated) and
therefore ξc = 0. Alternatively, ξc > 0 provided τc < ∞.
54
D.Volchenkov
The reciprocal correlations between the fluctuations in the divertor is of vital
importance for a possibility to stabilize the turbulent cross field system, in the large
scales, by biasing the limiter surface discussed in the literature before (Ghendrih et
al., 2003). Namely, if ξc > 0, there would be a number of intervals [Vk−1 ,Vk ] for the
uniform electric poloidal drifts V such that all fluctuations arisen in the system are
damped out fast. In particular, in the first order of perturbation theory, there exists
one threshold value Vc such that the instability in the system is bent down as V > Vc .
However, Vc → ∞ as ξc → 0.
To get an insight into the statistics of growing fluctuations of particle density that
appear as high-density blobs of particles close to the reactor wall, we note that their
growth rates are determined by the advanced Green’s functions analytical in the
lower half plain of the frequency space. We replace the rather complicated dynamical process of creation and convection of growing density fluctuations by the turbulent flow with the problem of discrete time random walks concluding at a boundary.
Such a substitution can be naturally interpreted as a Monte Carlo simulation procedure for the particle flux. Herewith, the wandering time spectra which determine
the pdf of the particle flux in such a toy model are either exponential or bounded by
the exponential from above. This observation is in a qualitative agreement with the
numerical data reported in (Ghendrih et al., 2003).
4 Conclusion
Applications of methods developed in quantum field theory to the problems of statistical physics and critical phenomena have a long history. These powerful methods
became an important tool in studies of nonlinear dynamical systems. In this report,
we have developed a strategy of use the RG method in purpose of study the longtime large-scale asymptotic behaviors in stochastic magneto-hydrodynamics.
The main conclusion of the study in magneto-hydrodynamics is that the RG
transformations are characterized by two different fixed points stable with respect
to long-time large-scale asymptotic behavior that can be naturally interpreted as
”kinetic” and ”magnetic” critical regimes, in which fields and parameters of the
MHD theory acquires different critical dimensions. We have investigated long-time
large-scale asymptotic behavior of correlation functions and composite operators
(the local averages of fields and their derivatives, which can be observed in real
experiments) in both critical regimes. The immediate observation of our study is
that the MHD system is thoroughly unstable. Perhaps, the most fascinating result
of our approach to MHD is the prediction of ”optimal size” eddies that could survive in cross-fields much longer than others. In fact, we claim that if the cross-field
system losses stability, it becomes transparent for certain plasma vortexes.We have
considered two -dimensional models of the cross-field turbulent transport close to
the ”scrape-off layer” (SOL) in thermonuclear reactors. Stochastic perturbations of
electron density and vorticity are responsible for the aggregation of electrons into
bulbs which then propagate ballistically towards the wall blanket. The operation
What is Control of Turbulence?
55
stability of the ”next step” device crucially depends upon correlations between the
fluctuations of electron density and vorticity.
We have studied possible mechanisms which break the operation stability and
proposed a simple discrete-time ”toy model” resembling a Markov chain that correctly reproduces the statistics of the burst-like events observed experimentally in
the ITER facility, in Cadarache (France).
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