Influence of helicity on scaling regimes in the extended Kraichnan model
O G Chkhetiani$^1$
M Hnatich$^{2,3}$
E Jur\v{c}i\v{s}inov\'a$^{2,4}$
M Jur\v{c}i\v{s}in$^{2,5}$
A Mazzino$^{6}$ and M Repa\v{s}an$^{2}$
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\title[Influence of helicity on scaling regimes in ]{Influence of helicity on scaling regimes in the extended Kraichnan model}

\author{O G Chkhetiani$^1$, M Hnatich$^{2,3}$, E Jur\v{c}i\v{s}inov\'a$^{2,4}$,
        M Jur\v{c}i\v{s}in$^{2,5}$, A Mazzino$^{6}$ and M Repa\v{s}an$^{2}$}
\address{$^{1}$ Space Research Institute, Profsoyuznaya 84/32, 117997 Moscow, Russian Federation}
\address{$^{2}$ Institute of Experimental Physics, Slovak Academy of Sciences,
                Watsonova 47, 040 01, Ko\v{s}ice, Slovakia}
\address{$^{3}$  Department of Mathematics Faculty of Civil Engineering, Technical University
Vysoko\v{s}kolsk\'a 4,
040 01 Ko\v{s}ice, Slovakia}
\address{$^{4}$ Laboratory of Information Technologies, Joint Institute for Nuclear Research,
                141980 Dubna, Moscow Region, Russian Federation}
\address{$^{5}$ N.N. Bogoliubov laboratory of Theoretical Physics, Joint Institute for
       Nuclear Research, 141980 Dubna, Moscow Region, Russian Federation}
\address{$^{6}$ Department of Physics, University of Genova, National Institute of Nuclear Physics, Genova Section,
via Dodecaneso 33, I-16146 Genova, Italy}

\ead{hnatic@saske.sk}
%\begin{abstract}
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%\verb"iopart.cls" (the IOP \LaTeXe\ preprint class file).
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%journals published by Institute of Physics Publishing.
%Authors submitting to any IOP journal, i.e.\
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%\end{abstract}

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%\pacs{00.00, 20.00, 42.10}
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%\noindent{\it Keywords}: Article preparation, IOP journals
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\begin{abstract}
{We have investigated  the advection of  a passive scalar quantity
by incompressible helical turbulent flow in the frame of extended Kraichnan model.  Turbulent fluctuations of velocity field
are assumed to  have the Gaussian statistics with zero mean and   defined noise with finite time-correlation.
Actual calculations have been done up to two-loop approximation in the frame of field-theoretic renormalization group approach.
It turned out that space parity violation (helicity) of turbulent environment does not affect anomalous scaling
which is peculiar attribute of corresponding model without helicity. However, stability of asymptotic regimes, where
anomalous scaling takes place, strongly depends on the amount of helicity. Moreover, helicity gives rise to the turbulent
diffusivity, which has been calculated in one-loop approximation.}
%\keywords helicity, diffusivity, anomalous scaling,  renormalization group.
%\pacs{PACS numbers: 47.10.+g, 47.27.$-$i, 05.10.Cc}
%\pacs Put PACS numbers here.
\end{abstract}
\pacs{47.27.-i, 47.10.+g, 05.10.Cc}
%\maketitle
\submitto{\JPA}

\section{Introduction}

During the last decade much attention has been paid to the
inertial range of fully developed turbulence, which contains wave
numbers larger then those that pump the energy into the system and
smaller enough then those that are related to the dissipation
processes \cite{Monin}. Grounding of the inertial
range turbulence have been created in the well known
 Kolmogorov--Obukhov (KO) phenomenological theory
(see, e.g., \cite{Monin,McComb,Legacy}). One of the main problems in
the modern theory of fully developed turbulence is to verify the
validity of the basic principles of  KO theory and their
consequences
 within the framework of a
microscopic model. Recent experimental and theoretical studies
indicate  possible deviations from the celebrated Kolmogorov
scaling exponents. The scaling behavior  of the velocity
fluctuations with exponents, which values are different from
Kolmogorov ones, is called as anomalous and usually  is associated
with intermittency phenomenon. Roughly speaking, intermittency
means that statistical properties (for example, correlation or
structure functions of the turbulent velocity field) are dominated
by rare spatiotemporal configurations, in which the regions with
strong turbulent activity have exotic (fractal) geometry and are
embedded into the vast regions with regular (laminar) flow. In the
turbulence such phenomenon is believed to be related to the strong
fluctuations of the energy flux which, therefore, leads to
deviations from the predictions of the aforementioned KO theory.
The deviations, referred to as ``anomalous'' or
``non-dimensional'' scaling, manifest themselves in singular
(arguably power-like) dependence of correlation or structure
functions on the distances and the integral (external) turbulence
scale $L$. The corresponding exponents are certain nontrivial and
nonlinear functions of the order of the correlation function, the
phenomenon referred to as ``multiscaling''.
%(we have to note that the
%term ``anomalous scaling'' reminds of the critical scaling in models
%of equilibrium phase transitions).
%Even thought the great progress in the understanding of
%intermittency and anomalous scaling in turbulence was achieved as
%a result of intensive studies, their investigation in fully
%developed turbulence still remains one of the major theoretical
%problem.

Although the theoretical description of the fluid turbulence on
the basis of the "first principles", i.e., on the stochastic
Navier-Stokes (NS) equation \cite{Monin} remains essentially
an open problem, considerable progress has been achieved in
understanding simplified model systems that share some important
properties with the real problem: shell models \cite{Dyn},
stochastic Burgers equation \cite{Burgulence} and passive
advection by random ``synthetic'' velocity fields \cite{FGV}.

The crucial role in these studies are played by the models of
advected passive scalar field \cite{Obu49}.
%,{Cor51}.
A simple model of a passive scalar quantity advected by a random
Gaussian velocity field, white in time and self-similar in space
(the latter property mimics some features of a real turbulent
velocity ensemble), the so-called Kraichnan's rapid-change model
\cite{Kra68}, is an example. The interest to these models is based
on two important facts: first, as were shown by both natural and
numerical experimental investigations, the deviations from the
predictions of the classical Kolmogorov-Obukhov phenomenological
theory \cite{Monin,McComb,Legacy} is even more
strongly displayed for a passively advected scalar field than for
the velocity field itself (see, e.g.,
\cite{AnHoGaAn84,Sre91,HolSig94} and
references cited therein), and second, the problem of passive
advection is much more easier to be consider from theoretical
point of view. There, for the first time, the anomalous scaling
was established on the basis of a microscopic model
\cite{Kraich1}, and corresponding anomalous exponents was
calculated within controlled approximations
\cite{GK,Pumir} (see also review \cite{FGV} and
references therein).

The greatest stimulation to study the simple models of passive
advection not only of scalar fields but also of vector fields
(e.g., weak magnetic field) is related to the fact that even
simplified models with given Gaussian statistics of so-called
"synthetic" velocity field describes a lot of features of
anomalous behavior of genuine turbulent transport of some
quantities (as heat or mass) observed in experiments, see, e.g.,
\cite{HolSig94}--\cite{AveMaj90}
%,HolSig94,Pumir94,TonWar94,ElKlRo90te,Falk1,GK,Shraiman,Pumir,AveMaj90,
%Kraich1}
and references cited therein.



The term ``anomalous scaling'' reminds of the critical scaling in
models of equilibrium phase transitions. In those, the field
theoretic methods were successfully employed to establish the
existence of self-similar (scaling) regimes and to construct
regular perturbative calculational schemes (the famous $\epsilon$
expansion and its relatives) for the corresponding exponents,
scaling functions, ratios of amplitudes etc; see e.g.
\cite{Zinn,book}. Here and below, by ``field theoretic methods''
we mean diagrammatic and functional techniques, renormalization
theory and renormalization group, composite operators,
operator-product expansion and so on \cite{book}.



The feature specific to the theory of turbulence and simplified
models associated with it is the existence in the corresponding
field theoretical models of the composite operators with {\it
negative} scaling (critical) dimensions. Such operators, termed
``dangerous'' in \cite{RG,turbo,RG1,Novikov,RG3}, give rise to the
anomalous scaling, i.e., the singular dependence on the infrared
(IR) scale  $L$ with certain nonlinear anomalous exponents.

Important advantages of the RG approach are its universality and
calculational efficiency: a regular systematic perturbation
expansion for the anomalous exponents was constructed, similar to
the well-known $\epsilon$-expansion in the theory of phase
transitions, and the exponents were calculated in the first order
of expansion for passively advected vector fields
\cite{Lanotte2,amodel}  and in the second \cite{RG} and third
\cite{RG1} orders of that expansion for scalar fields.
Furthermore, the RG approach is not related only to the
rapid-change model and can also be applied to the case with finite
correlation time, anisotropy, the space parity violation, and,
moreover,  non-Gaussian advecting field \cite{RG3}.

The solution proceeds in two main stages. In the first stage, the
multiplicative renormalizability of the corresponding field
theoretic model is demonstrated and the differential RG equations
for its correlation functions are obtained. The asymptotic
behavior of the latter on their UV argument $(r/l)$ ($l$ is
internal  length) for $r\gg\ell$ and any fixed $(r/L)$ ($L$  is an
outer length) is given by IR stable fixed points of those
equations. It involves some ``scaling functions'' of the IR
argument $(r/L)$, whose form is not determined by the RG
equations. In the second stage, their behavior at $r\ll L$ is
found from the operator product expansion within the framework
of the general solution of
the RG equations. There, the crucial role is played by the
critical dimensions of various composite operators, which give
rise to an infinite family of independent scaling exponents (and
hence to multiscaling). Of course, these both stages (and thus the
phenomenon of multiscaling) have long been known in the RG theory
of critical behavior. The distinguishing feature, specific to
the models of turbulence  is the existence of composite operators with
aforementioned {\it negative} critical dimensions. Their
contributions to the operator product expansion  diverge at
$(r/L)\to 0$. In the models of critical phenomena, nontrivial
composite operators always have  positive dimensions and
  determine only corrections (vanishing for $(r/L)\to0$)
to the leading terms (finite for $(r/L)\to0$) in the scaling
functions.

Existence of  regular perturbation schemes and accurate numerical
simulations allows one to discuss, for the example of the
rapid-change model and its descendants, the issues that are
interesting within the general context of fully developed
turbulence: universality and saturation of anomalous exponents,
effects of compressibility, anisotropy and pressure, persistence
of the large-scale anisotropy and hierarchy of anisotropic
contributions and so on. Moreover, it is interesting and important
to study  the helicity (violation of space parity) effects because
many turbulence phenomena are directly influenced by them.


In \cite{RG3} the problem of a passive scalar advected by the
Gaussian self-similar velocity field with finite correlation time
\cite{all2} was studied by the field theoretic RG method. There, the
systematic study of the possible scaling regimes and anomalous
behavior was present at one-loop level. The two-loop corrections to
the anomalous exponents were obtained in \cite{Juha2}. It was shown
that the anomalous exponents are nonuniversal as a result of their
dependence on a dimensionless parameter, the ratio of the velocity
correlation time, and  turnover time of a scalar field.

In what follows, we shall continue with the investigation of this
model from the point of view of the influence of helicity on the
scaling regimes and the anomalous exponents within two-loop
approximation.

Helicity is defined as the scalar product of velocity and
vorticity and its non zero value expresses mirror symmetry
breaking of turbulent flow. It plays significant role in the
processes of magnetic field generation in electrically conductive
fluid \cite{dynamo1,Moffatt} and represents one of the most
important characteristics of large-scale motions as well
\cite{Etling85,Ponomar2003}.
%The presence of helicity is
%observed in various natural (like large air vortices in
%atmosphere) and technical flows
%\cite{Moffat92,Kholmyansky2001,Koprov2005}.
Despite of this fact
the role of the helicity in hydrodynamical turbulence is not
completely clarified up to now.

The Navier-Stokes equations conserve kinetic energy and helicity
in inviscid limit. Presence of two quadratic invariants leads to
the possibility of appearance of double cascade. It means that
cascades of energy and helicity take place in different ranges of
wave numbers analogously to the two-dimensional turbulence and/or
the helicity cascade  appears concurrently to the energy one in
the direction of small scales \cite{Brissaud73,Moiseev1996,koprovall}.
Particularly,  helicity cascade is closely connected with the
existence of exact relation between triple and double correlations
of velocity known as ``2/15'' law analogously to the ``4/5''
Kolmogorov law \cite{Chkhet96,kurien}. Corresponding to \cite{Brissaud73}
aforementioned scenarios of turbulent cascades  differ each other
by spectral scaling. Theoretical arguments given by Kraichnan
\cite{Kraich73}  and results of numerical calculations  of
Navier-Stokes equations \cite{Chen2003} support
the scenario of concurrent cascades. The appearance of helicity in
turbulent system leads to constraint of non-linear cascade to the
small scales. This phenomenon was firstly demonstrated by
Kraichnan \cite{Kraich73} within the modelling problem of
statistically equilibrium spectra and later in numerical
experiments.

Turbulent viscosity and diffusivity, which characterize influence of
small-scale motions on heat and momentum transport, are basic
quantities investigated in  the theoretic and applied models. The
constraint of direct energy cascade in helical turbulence  has to be
accompanied by decrease of turbulent viscosity. However, no
influence of helicity on turbulent viscosity was found in some works
\cite{Pouquet78,Zhou91}. Similar situation is observed for turbulent
diffusivity in helical turbulence. Although the modelling
calculations demonstrate intensification of turbulent transfer in
the presence of helicity \cite{Drum84} direct calculation of
diffusivity  does not confirm this effect
\cite{Knobl77,Lipscombe91}. Helicity is the pseudoscalar quantity
hence it can be easily understood, that its influence appears only
in quadratic and higher terms of perturbation theory or in the
combination with another pseudoscalar quantities (e.g., large-scale
helicity). Really, simultaneous consideration of memory effects and
second order approximation indicate effective influence of helicity
on turbulent viscosity \cite{Belian1998} and turbulent diffusivity
\cite{Drum84,Dolg87,Drummond2001} already in the limit of small or
infinite correlation time.

Helicity, as we shall see below, does not affect known results in
one-loop approximation and,  therefore, it is necessary to turn to
the second order (two-loop) approximation to be able to analyze
possible consequences. It is also important to say that in the
framework of classical Kraichnan model, i.e., the model of passive
advection by the Gaussian velocity field with $\delta$-like
correlations in time, it is not possible to study the influence of
the helicity because all potentially "helical" diagrams are
identically equal to zero at all orders in the perturbation
theory. In this sense, the investigation of the helicity in the
present model can be consider as the first step to analyze the
helicity in genuine turbulence.

\section{Field theoretic description of the model}

The advection of a passive scalar field
$\theta(x)\equiv\theta(t,\mathbf{x})$ in helical turbulent
environment is described by the stochastic equation
\begin{equation}
\partial_t \theta + v_i \partial_i \theta=\nu_0 \Delta
\theta + f,\label{eq:theta}
\end{equation}
\noindent where $\partial_{t}\equiv\partial/\partial t$,
$\partial_{i}\equiv\partial/\partial \mathrm{x}_{i}$, $\nu_{0}$ is the
molecular diffusivity coefficient (hereafter all parameters with a
subscript $0$ denote bare parameters of unrenormalized theory; see
below), $\triangle\equiv\partial^{2}$ is the Laplace operator, $v_i
\equiv v_i(x)$ is the $i$-th component of the divergence-free (owing
to the incompressibility) velocity field ${\bf v}(x)$, and $f\equiv
f(x)$ is an artificial the Gaussian random noise with zero mean and
correlation function
\begin{equation}
\langle f(x)f(x')\rangle  =  \delta(t-t')C(\mathbf{r}/L),\,\,\,
\mathbf{r} = \mathbf{x}-\mathbf{x}',\label{eq:corelf}
\end{equation}
where $L$ denotes an integral (outer) scale. It maintains the steady-state
of the system but the detailed form of the function
$C(\mathbf{r}/L)$
is unessential in our
consideration.
In
spite of the fact that in real problems the velocity field ${\bf
v}(x)$ satisfies the Navier-Stokes equation, in what follows, we suppose
that the statistics of velocity field is given in the form of Gaussian
distribution with zero mean and correlator
\begin{equation}
\langle v_i(x) v_j(x^{\prime}) \rangle  =  \int \frac{\mathrm{d}\omega d^d
k}{(2\pi)^{d+1}} P^{\rho}_{ij}({\bf k}) D_v(\omega,k)
\exp[-i\omega(t-t^{\prime})+i{\bf k}({\bf x}-{\bf x^{\prime}})],
\label{eq:corelv}
\end{equation}
with
\begin{equation}
D_v(\omega, k) = \frac{D_0
k^{4-d-2\varepsilon-\eta}}{(i\omega+u_0 \nu_0
{k}^{2-\eta})(-i\omega+u_0 \nu_0 k^{2-\eta})},\label{corrvelo}
\end{equation}
where $k=|\mathbf{k}|$,  $D_0=\mathrm{g}_0\nu_0^3$ is a positive amplitude factor,  $\mathrm{g}_0$  plays the role of
the coupling constant of the model, an  analog of the coupling
constant $\lambda_0$ in the $\lambda_0 \varphi^4$ model of critical
behavior \cite{Zinn,book}. In addition, $\mathrm{g}_0$ is a formal
small parameter of the ordinary perturbation theory. The positive
exponents $\varepsilon$ and $\eta$ ($\varepsilon=O(\eta)$) are small
RG expansion parameters, the analogs of the parameter
$\varepsilon=4-d$ in the $\lambda_0 \varphi^4$ theory. Thus we have
a kind of double expansion model in the $\varepsilon -\eta$ plane
around the origin $\varepsilon=\eta=0$. The correlator
(\ref{corrvelo}) is directly related to the energy spectrum via the
frequency integral \cite{RG3}
\begin{equation}
E(k)\simeq k^{d-1} \int \mathrm{d}\omega D^v(\omega, k) \simeq
\frac{\mathrm{g}_0 \nu_0^2}{u_0} k^{1-2\varepsilon}.
\end{equation}
Therefore, the coupling constant $\mathrm{g}_0$ and the exponent
$\varepsilon$ describe the equal-time velocity correlator or,
equivalently, energy spectrum. On the other hand, the constant $u_0$
and the second exponent $\eta$ are related to the frequency $\omega
\simeq u_0 \nu_0 k^{2-\eta}$  which characterizes the mode $\mathbf{k}$
\cite{Eyink96}. Thus, in our notation, the
value $\varepsilon=4/3$ corresponds to the well-known Kolmogorov
"five-thirds law" for the spatial statistics of velocity field, and
$\eta=4/3$ corresponds to the Kolmogorov frequency.
For completeness, we remain $d$-dependence in expressions
(\ref{eq:corelv}) and (\ref{corrvelo}) ($d$ is the dimensionality of
the ${\bf x}$ space), although, of course, when one investigates
system with helicity the dimension of the ${\bf x}$ space must be
strictly equal to three.
To include helicity the transverse projector $P^{\rho}_{ij}({\bf k})$ is
taken in the form
\begin{equation}
P^{\rho}_{ij}({\bf k})=P_{ij}({\bf k}) + H_{ij}({\bf k})=
\delta_{ij}-k_{i}k_{j}/k^{2}+i\rho\epsilon_{ijl}\frac{k_{l}}{k}.\label{eq:Pij}
\end{equation}
Here $P_{ij}({\bf k})=\delta_{ij}-k_{i}k_{j}/k^{2}$ represents
non-helical part of the total transverse projector
$P^{\rho}_{ij}({\bf k})$. On the other hand, $H_{ij}({\bf
k})=i\rho\epsilon_{ijl}k_{l}/k$ mimics the presence of helicity in
the flow. Thus, formally, the transition to the helical fluid
corresponds to the breaking  of spatial parity, and, technically,
this is expressed by the fact that the correlation function is
specified in the form of mixture of a true tensor and a
pseudotensor. The tensor $\varepsilon_{ijl}$ is Levi-Civita's
completely antisymmetric tensor of rank 3 and the real parameter
$\rho$, characterizes the amount of helicity. Due to the requirement
of positive definiteness of the correlation function the absolute
value of $\rho$ must be in the interval $|\rho| \in \langle
0,1\rangle$. Non-zero helical part proportional to $\rho$ physically
expresses existence of non-zero correlations $\langle{\bf v}\cdot
\mathrm{rot}\:{\bf v}\rangle$.

The general model (\ref{eq:corelv}), (\ref{corrvelo}) contains two
important special cases: rapid-change model limit when
$u_0\rightarrow \infty$ and $\mathrm{g}_0^{\prime}\equiv
\mathrm{g}_0/u_0^2=$ const, $D_v(\omega, k)\rightarrow
\mathrm{g}_0^{\prime} \nu_0 k^{-d-2\varepsilon + \eta},$ and
quenched (time-independent or frozen) velocity field limit which is
defined by $u_0\rightarrow 0$ and $\mathrm{g}_0^{\prime\prime}\equiv
\mathrm{g}_0/u_0=$ const, $D_v(\omega, k)\rightarrow
\mathrm{g}_0^{\prime\prime} \nu_0^2\pi \delta(\omega)
k^{-d+2-2\varepsilon},$ which is similar to the well-known models of
the random walks in random environment with long range correlations;
see, e.g., \cite{Bouchaud,Honkonen}.

Using Martin-Siggia-Rose mechanism \cite{Martin} the stochastic
problem (\ref{eq:theta})-(\ref{corrvelo}) can be treated as a
field theory with action functional
\begin{equation}
S(\theta,\theta',\mathbf{v})=\theta'D_{\theta}\theta'/2+
\theta'[-\partial_{t}+\nu_{0}\triangle-(v_i\partial_i)]\theta-
\mathbf{v}D_{v}^{-1}\mathbf{v}/2,\label{eq:Sucinok}
\end{equation}
where $\theta'$ is an auxiliary scalar field, and $D_{\theta}$ and
$D_{v}$ are correlators (\ref{eq:corelf}) and (\ref{eq:corelv}),
respectively. In the action (\ref{eq:Sucinok}) all the required
integrations over $x=(t,\mathbf{x})$ and summations over the
vector indices are understood. The first four terms in
Eq.\,(\ref{eq:Sucinok}) represent the Dominicis-Jansen type
action for the stochastic problem (\ref{eq:theta}),
(\ref{eq:corelf}) at fixed $\mathbf{v}$, and the last term
represents the Gaussian averaging over $\mathbf{v}$.

The model (\ref{eq:Sucinok}) corresponds to a standard Feynman
diagrammatic technique with the  bare propagators $\langle
\theta\theta^{\prime} \rangle_0$ and $\langle v_i v_j\rangle_0$
%(in the time-momentum representation)
%\begin{eqnarray}
%\langle \theta(t, {\mathbf{k}})\theta^{\prime}(t^{\prime}, {\mathbf{k}})
%\rangle_0 &=& \theta(t-t^{\prime})e^{-\nu_0 k^2 (t-t^{\prime})},
%\label{proptheta1} \\
%\langle v_i(t, {\mathbf{k}}) v_j(t^{\prime}, {\mathbf{k}})\rangle_0 &=&
%\frac{D_{0}}{2u_{0} k^{1+2\epsilon}} e^{-
%u_{0}\nu_{0}k^{2-\eta}(t-t')} P^{\rho}_{ij}(\mathbf{k})\, ,
%\end{eqnarray}
%where $\theta(t-t^{\prime})$ is the step function, or
(in the momentum-frequency representation)
\begin{equation}
\langle \theta(\omega, {\mathbf{k}})\theta^{\prime}(\omega, {\mathbf{k}})
\rangle_0 = \frac{1}{-i\omega+\nu_0
k^2}, \,
\langle v_i(\omega, {\mathbf{k}}) v_j(\omega, {\mathbf{k}})\rangle_0 =
P^{\rho}_{ij}(\mathbf{k}) D_v(\omega, k), \label{proptheta}
\end{equation}
where $D_v(\omega, k)$ is given directly by (\ref{corrvelo}). In the
Feynman diagrams, these propagators are represented by the lines
which are shown in figure \ref{fig1} (the end with a slash in the
propagator $\langle \theta\theta^{\prime} \rangle_0$ corresponds to
the field $\theta^{\prime}$, and the end without a slash corresponds
to the field $\theta$). The triple vertex (or interaction vertex)
$-\theta^{\prime} v_j\partial_j \theta = \theta^{\prime} v_j V_j
\theta $, where $V_j=i k_j$ (in the momentum-frequency
representation), is presented in figure \ref{fig1}, where momentum
$\mathbf{k}$ is flowing into the vertex via the auxiliary field
$\theta^{\prime}$.

\input epsf
%\bfg[t] %
 \begin{figure}[t]
     \vspace{0.0cm}
%\bc %
    \begin{center}
%\epsffile{fig1.eps}{\rule{0pt}{100mm}}
       \leavevmode
      \epsfxsize=10cm
     \epsffile{fig1.eps}
%\ec %
\end{center}
\vspace{0.0cm} \caption{(Left) Graphical representations of the
needed propagators of the model. (Right) The triple (interaction)
vertex of the model. Momentum ${\mathbf k}$ is entering into the vertex
via field $\theta^{\prime}$. \label{fig1}}
%\efg %
\end{figure}


\section{Renormalization group analysis}

The model (\ref{eq:Sucinok}) is logarithmic for $\epsilon=\eta=0$
(the coupling constant $\mathrm{g}_{0}$ is dimensionless) and, in this case,
possible ultraviolet (UV) divergences have the form of poles in
various linear combinations of $\varepsilon$ and $\eta$ in the
correlation functions. Using the standard analysis of quantum field
theory one finds that  all
divergences  can be removed by the only counterterm of the form
$\theta' \triangle \theta$ \cite{RG3}. Thus, the model is
multiplicatively renormalizable, which is expressed explicitly in
the multiplicative renormalization of the parameters $\mathrm{g}_0, u_0$, and
$\nu_0$ in the form
\begin{equation}
\nu_0=\nu Z_{\nu},\,\,\, \mathrm{g}_0=\mathrm{g} \mu^{2\varepsilon+\eta}
Z_{\mathrm{g}},\,\,\,u_0=u\mu^{\eta} Z_u. \label{zetka}
\end{equation}

Here the dimensionless parameters $\mathrm{g},u$ and $\nu$ are the
renormalized counterparts of the corresponding bare ones, $\mu$ is
the renormalization mass (a scale setting parameter),  an artefact
of dimensional regularization. Newly introduced quantities
$Z_i=Z_i(\mathrm{g},u;d,\rho;\varepsilon,\eta)=Z_i(\mathrm{g},u;d,\rho;\varepsilon)$ are
renormalization constants (note that if $\rho$ is non-zero then
$d=3$) and, in general, contain poles of linear combinations of
$\varepsilon$ and $\eta.$ However, as detailed analysis shows,  to
obtain all important quantities as the $\gamma$-functions,
$\beta$-functions, coordinates of fixed points, and the critical
dimensions, the knowledge of the renormalization constants  for
the special choice $\eta=0$ is sufficient up to two-loop
approximation (see details in \cite{RG3}).



The renormalized action functional has the following form
\begin{equation}
S_{R}(\theta,\theta',\mathbf{v})=\theta'D_{\theta}\theta'/2+\theta'[-\partial_{t}+\nu
Z_1\triangle-(v\partial)]\theta-
\mathbf{v}D_{v}^{-1}\mathbf{v}/2 ,\label{eq:Srenorm}
\end{equation}
where the correlator $D_v$ is written in renormalized parameters.
By comparison of the renormalized action (\ref{eq:Srenorm}) with
definitions of the renormalization constants $Z_i$, $i=\mathrm{g},u,\nu$
(\ref{zetka}) one comes to the relations among  them:
\begin{equation}
Z_{\nu}=Z_1,\,\,\, Z_{\mathrm{g}}=Z_{\nu}^{-3},\,\,\,
Z_u=Z_{\nu}^{-1}.\label{zetka1}
\end{equation}
The second and third relations are consequences of the absence of
the renormalization of the term with $D_v$ in renormalized action
(\ref{eq:Srenorm}). The parameter $\rho$ a the fields $\theta, \theta', \mathbf{v}$ are not renormalized therefore $Z_{\rho}=Z_{\theta}=Z_{\theta'}=Z_{\mathbf{v}}=1$.

The issue of interest is, in particular, the behavior of the
equal-time structure functions
\begin{equation}
S_{n}(r)\equiv\langle[\theta(t,{\bf x})-\theta(t,{\bf x'})]^{n}\rangle,
\qquad \, r\equiv |\mathbf{r}|=|\mathbf{x}-\mathbf{x}'|
\label{struc}
\end{equation}
in the inertial range, specified by the inequalities
$l <<r<<L$ ($l$ is internal length).
Here parentheses $< >$ mean the functional average over the fields $\theta, \theta', {\bf v}$
with the weight $exp(S_R).$ In the isotropic case, the
odd functions $S_{2n+1}$ vanish, while for $S_{2n}$ simple
dimensionality considerations give
\begin{equation}
S_{2n}(r)=  \nu_0^{-n}\, r^{2n}\, R_{2n} ( r/l, r/L, \mathrm{g}_0, u_0, \rho),
\label{strucdim}
\end{equation}
where  $R_{2n}$ are some functions of dimensionless variables. In
principle, they can be calculated within the ordinary perturbation
theory (i.e., as series in $\mathrm{g}_{0}$), but this is not useful
for studying inertial-range behavior: the coefficients are singular
in the limits $\ r/l \to\infty$  and/or $r/L\to 0$, which compensate
the smallness of $\mathrm{g}_{0}$, and in order to find correct
infrared behavior we have to sum the entire series. The desired
summation can be accomplished using the field theoretic
renormalization group (RG) and operator product expansion (OPE); see
\cite{RG,RG1,RG3} for details.

The RG analysis consists of two main stages. On the
first stage, the multiplicative renormalizability of the model is
demonstrated and the differential RG equations for its correlation
(structure) functions  are obtained. The asymptotic behavior of the functions
like (\ref{struc}) for $ r/l>>1$ and any fixed $r/L$ is given
by IR stable fixed points $\mathrm{g}_*, u_*$ (see below) of the RG equations and has the form
\begin{equation}
S_{2n}(r)= \nu_0^{-n}\, r^{2n}\,
(r/l)^{-\gamma_{n}} R_{2n} (r/L, \rho),  \quad r/l>>1
\label{strucdim2}
\end{equation}
with certain, as yet unknown, scaling functions $R_{2n} (r/L, \rho)\equiv R_{2n} (1, r/L, \mathrm{g}_*, u_*, \rho)$.
In the theory of critical phenomena \cite{Zinn,book} the quantity
$\Delta[S_{2n}]\equiv-2n + \gamma_{n}$ is termed the critical
dimension,
and the exponent $\gamma_{n}$, the difference between the critical
dimension $\Delta[S_{2n}]$ and the canonical dimension $-2n$,
is called the anomalous dimension. 
%In the case at hand, the latter
%has an extremely simple form: $\gamma_{n}=n\varepsilon$.
%Whatever be the functions
%$ R_{n} (r/L, \rho)$, the representation (\ref{strucdim2})
%implies the existence of a scaling (scale invariance) in
%the IR region ($r/l>>1$, $r/L$ fixed)
%with definite critical dimensions of all ``IR relevant''  parameters,
%$\Delta[S_{2n}] =-2n+n \epsilon$, $\Delta_{r}=-1$, $\Delta_{L^{-1}}=1$ and
%fixed ``irrelevant'' parameters $\nu_0$ and $l$.

On the second stage, the small $r/L$ behavior of the functions
$ R_{2n} (r/L, \rho)$ is studied within the general representation
(\ref{strucdim2}) using the operator product expansion (OPE).
It shows that, in the limit $r/L\to 0$,
the functions $ R_{2n} (r/L,\rho)$ have the asymptotic forms
\begin{equation}
 R_{2n} (r/L) = \sum_{F} C_{F}(r/L)\, (r/L)^{\Delta_n},
\label{ope}
\end{equation}
where $C_{F}$ are coefficients regular in $r/L$. In general,
the summation is implied over certain  renormalized composite
operators $F$  with critical dimensions $\Delta_n$.
In case under consideration the leading operators $F$ have the form
$F_n= (\nabla_i \theta \nabla_i \theta)^n. $

We have performed the complete two-loop calculation of
the critical dimensions  of the composite
operators $F_n$ for arbitrary values of $n$, $d$, $u$ and $\rho$ and obtain them
in the following form:
\begin{equation}
\Delta_n=\Delta_{n}^{(1)}\epsilon + \Delta_{n}^{(2)}\epsilon^2\, , \qquad \Delta_{n}^{(1)} = \frac{-n(n-2)(d-1)} {2(d-1)(d+2)}
\label{a00}
\end{equation}
where
%\begin{eqnarray}
%\Delta_{n}^{(1)} = \frac{-n(n-2)(d-1)} {2(d-1)(d+2)}
%\label{L=00}
%\end{eqnarray}
$\Delta_{n}^{(1)}$ is critical dimension  obtained in one-loop approximation.
Interesting technical details of these two-loop calculations will be present elsewhere.

Two-loop contribution $\Delta_{n}^{(2)}$ is rather cumbersome and
can be found in \cite{Juha2}.  The main and interesting result
consists in the fact that although separated two-loop Feynman graphs
of operators $F_n$ strongly depend on helicity parameter $\rho$,
such dependence disappears in their sum, which gives rise to the
critical  dimensions $\Delta_n$. We can conclude that in two-loop
approximation  anomalous scaling with {\it negative} exponents
(\ref{a00}) is not affected by the existence of non-zero helical
correlations $\langle{\bf v} rot {\bf v}\rangle$ in turbulent
incompressible flow. It turns out, however, that region of stability
of possible asymptotic  regimes governed by fixed points of RG
equations,
 where anomalous scaling
takes place,   and  effective
diffusivity  strongly depend on $\rho$.

Let us analyze asymptotic regimes in detail. The structure functions
and the other statistical averages of random fields $\theta,
\theta'$ satisfy linear differential RG equations with linear
differential operator ${\cal{D}}_{RG}:$
%For example, RG equation for
%pair structure function $S_2$ has the form
%\begin{equation}
%{\cal{D}}_{RG} S_2(r)=0, \label{RGE}
%\end{equation}
%with
\begin{equation}
{\cal{D}}_{RG} = {\cal{D}}_{\mu} +
\beta_{\mathrm{g}}(\mathrm{g},u)\partial_{\mathrm{g}}+\beta_u(\mathrm{g},u)\partial_u-\gamma_{\nu}(\mathrm{g},u){\cal{D}}_{\nu}\,
. \label{RGoper}
\end{equation}
Here ${\cal{D}}_x\equiv x\partial_x$ stands for any variable $x$ and
the RG functions (the $\beta$ and $\gamma$ functions) are given by
well-known definitions and in our case, using the relations
(\ref{zetka1}) for the renormalization constants, they acquire the
following form
\begin{eqnarray}
\gamma_{\nu}&\equiv& \tilde{\cal{D}}_{\mu} \ln Z_{\nu}, \label{gammanu}\\
\beta_\mathrm{g}&\equiv&\tilde{\cal{D}}_{\mu} \mathrm{g} =\mathrm{g}
(-2\varepsilon-\eta+3\gamma_{\nu}), \label{betag}\\
\beta_u&\equiv&\tilde{\cal{D}}_{\mu} u =u
(-\eta+\gamma_{\nu}).\label{betau}
\end{eqnarray}
The renormalization constant $Z_{\nu}$ is determined by the
requirement that
%the one-irreducible Green function
response function $G\equiv\langle \theta \theta'\rangle$ must be UV
finite when is written in renormalized variables. In our case it
means that it  has no singularities in the limit $\varepsilon,
\eta\rightarrow 0$. The response function $G$ is related to the
self-energy operator $\Sigma$, which is
expressed via Feynman graphs,  by the Dyson equation. In frequency-momentum
representation it has the following form
\begin{equation}
G(\omega, \mathbf{p})=\frac{1}{-i\omega+\nu_0 p^2 -
\Sigma(\omega, \mathrm{p})}.\label{Dyson}
\end{equation}
Thus, $Z_{\nu}$ is found from the requirement that the UV
divergences are canceled in (\ref{Dyson}) after substitution
$\nu_0=\nu Z_{\nu}$. This determines $Z_{\nu}$ up to an UV finite
contribution, which is fixed by the choice of the renormalization
scheme. In the MS scheme all the renormalization constants have the
form: 1 + {\it poles in $\varepsilon,\eta$ and their linear
combinations}. In contrast to the rapid-change model, where only
one-loop diagram exists (it is related to the fact that all
higher-order loop diagrams contain at least one closed loop which is
built on by only retarded propagators, thus are automatically equal
to zero), in the case with finite correlations in time of the
velocity field, higher-order corrections are non-zero. In two-loop
approximation the self-energy operator $\Sigma$ is defined by
diagrams which are shown in figure \ref{fig2}.


\input epsf
% \bfg[t] %
\begin{figure}[t]
     \vspace{0.0cm}
%\bc       %
\begin{center}
       \leavevmode
       \epsfxsize=12cm
       \epsffile{fig2a.eps}
%\ec  %
\end{center}
\vspace{0.0cm} \caption{The one and two-loops contributions to the
self-energy operator $\Sigma$.
\label{fig2}}
%\efg %
\end{figure}


As was already mentioned, in our calculations we can put $\eta=0$.
This possibility essentially simplifies the evaluations of all
quantities \cite{RG3,Juha2}.

 Two-loop calculations of divergent parts of diagrams in figure \ref{fig2} give the renormalization
 constant $Z_{\nu}$ and anomalous dimension $\gamma_{\nu}$ (\ref{gammanu}) in the form:

\begin{equation}
Z_{\nu} = 1 + \frac{g}{\varepsilon} {\cal A} +\frac{g^2}{\varepsilon} {\cal B}+ \frac{g^2}{\varepsilon^2} {\cal C}
\,, \, \gamma_{\nu}=-2(g{\cal A} + 2 g^2 {\cal B}).
\label{nove}
\end{equation}
Here
%\begin{equation}
${\cal A}= (1-d)/(2 d(1+u))$
%\label{AA}
%\end{equation}
is the one-loop contribution to the constant $Z_{\nu}$ and anomalous dimension $\gamma_{\nu}$
and the two-loop ones are
\begin{eqnarray}
{\cal B} &=& \frac{(d-1)(d+u)}{4 d^2(d+2)(1+u)^5}
\,\cdot\,{_2F_1}\left(1,1;2+\frac{d}{2};\frac{1}{(1+u)^2}\right) \nonumber
\\ & &
- \frac{\pi \rho^2}{36 (1+u)^3}
\,\cdot\,{_2F_1}\left(\frac12,\frac12;\frac52;\frac{1}{(1+u)^2}\right),\label{BB}\\
{\cal C}& = & -\frac{(d-1)^2}{d^2}\frac{1}{8(1+u)^3}\,,\nonumber
\end{eqnarray}
where
${_2F_1}(a,b,c,z)=1+\frac{a\,
b}{c\cdot1}z+\frac{a(a+1)b(b+1)}{c(c+1)\cdot1\cdot2}z^{2}+\ldots$
represents the hypergeometric function.
We substitute $d=3$ in the helical part (proportional to the $\rho$),
but for completeness we remain the $d$-dependence in the non-helical one.
In addition   we have introduced new notation $g =\mathrm{g} S_d/(2u(2\pi)^d) $
($S_d=2 \pi^{d/2}/\Gamma(d/2)$ denotes the $d$-dimensional
sphere).

From the expressions (\ref{gammanu}) - (\ref{betau}) and
(\ref{nove}) we are able to find and clasify all fixed points $g_*,
u_*$ which satisfy equations:
\begin{equation}
\beta_g(g_*,u_*)=\beta_u(g_*,u_*)=0.
\end{equation}
To investigate the infrared   stability of
a fixed point it is enough to analyze the eigenvalues of the  $2\times2$ matrix
$\Omega$ of first derivatives: $\Omega_{ij}=\partial \beta_{g_i}/\partial g_j$ $(g_i\equiv g, u).$
The anomalous scaling is governed by the infrared stable fixed
points, i.e., those for which both eigenvalues are positive.

Classification and detailed analysis of all fixed points, determination of region of their stability and influence of helicity
will be present elsewhere. Here we confine ourselves to the most interesting  IR stable fixed point, where both parameters $g_*, u_*$ acquire
non-trivial values at $\eta=\varepsilon$:
\begin{eqnarray}
g_*&=& \left((g_*^{(1)}+\left(g_*^{(2)}+g_*^{(3)}\rho^2\right)\varepsilon\right)\varepsilon,\quad
g_*^{(1)}= \frac{3}{2} (1+u_*)\,,
\nonumber \\
 g_*^{(2)}&=& \frac{3 (3+u_*)}{20(1+u_*)^2}\,\cdot\,
{_2F_1}\left(1,1;\frac{7}{2};\frac{1}{(1+u_*)^2}\right)\,,
\label{fix} \\
 g_*^{(3)}&=& - \frac{3 \pi}{8}\,\cdot\,
{_2F_1}\left(\frac12,\frac12;\frac{5}{2};\frac{1}{(1+u_*)^2}\right).
\end{eqnarray}
 Actually, the equation
(\ref{fix}) represents a line of fixed points in $g-u$ plane.
The competition between helical and non-helical terms appears
which  yields a nontrivial restriction for the fixed point values of
variable $u$ to have positive fixed values for variable $g$.




\section{Effective diffusivity \label{sec:EffDiff}}


One of the interesting object from the theoretical as well as
experimental point of view is so-called effective diffusivity $\bar
\nu$. In this section let us briefly investigate the effective
diffusivity $\bar \nu$, which replaces initial molecular diffusivity
$\nu_0$ in (\ref{eq:theta}) due to the interaction of the scalar
field $\theta$ with random velocity field $\mathbf{v}$. Molecular
diffusivity $\nu_0$ governs exponential dumping in time all
fluctuations in the system in the lowest approximation, which is
given by the propagator (response function) (\ref{proptheta}).
%$G_0(t-t', \mathbf{k})=\langle \theta(t,
%{\mathbf{k}})\theta^{\prime}(t^{\prime}, {\mathbf{k}}) \rangle_0 =
%\theta(t-t^{\prime})\exp(-\nu_0 k^2 (t-t^{\prime}))$.
Analogously, the effective diffusivity $\bar \nu$ governs
exponential dumping of all fluctuations described by full response
function, which is defined by Dyson equation (\ref{Dyson}). Its
explicit expression can be obtained by the RG approach. In
accordance with general rules of the RG (see, e.g., \cite{book}) all
principal parameters of the model $\mathrm{g}_0, u_0$ and $\nu_0$
are replaced by their effective (running) counterparts, which
satisfy  RG flow equations
\begin{equation}
s \frac{d \bar g}{d s}=\beta_{\mathrm{g}}(\bar g,\bar u)\,,\qquad
s \frac{d \bar u}{d s}=\beta_{u}(\bar g, \bar u)\qquad
s \frac{d \bar \nu}{d s}= -\bar \nu \gamma_{\nu}(\bar g,\bar u)
\label{nu}
\end{equation}
%\begin{equation}
%s \frac{d \bar \nu}{d s}= -\bar \nu \gamma_{\nu}(\bar g,\bar u)\, ,
%\label{nu}
%\end{equation}
with initial conditions $\bar g|_{s=1}=g, \bar u|_{s=1}=u, \bar
\nu|_{s=1}=\nu$. Here $s=k/\mu$,  $\beta$ and $\gamma$ functions are
defined in (\ref{gammanu})--(\ref{betau}) and  the running
parameters $\bar{g}, \bar{u}$, and $\bar{\nu}$ clearly depend on
variable $s$.
% Straightforward
%integration (at least numerical) of  equations (\ref{gu}) gives way
%how to find their fixed points. Instead of very often one solves the
%set of equations $\beta_{g}( g_*, u_*)=\beta_{u}( g_*, u_*)=0$
% which defines all fixed points $g_*, u_*$. Just
%last approach we used above when we classified all fixed points.
Due to special form of $\beta$-functions (\ref{betag}), (\ref{betau}) we are
able to solve the last equation (\ref{nu}) analytically.
Using the first equation (\ref{nu}) and (\ref{betag}) one immediately rewrites the equation for effective diffusivity
in the form
\begin{equation}
\frac{d\bar  \nu}{\bar \nu}= \frac{\gamma_{\nu}}{2\varepsilon+\eta - 3 \gamma_{\nu}}\frac{d \bar g}{\bar g}
\label{nu1}
\end{equation}
which can be easily integrated. Using initial conditions the
solution acquires the form:
\begin{equation}
\bar \nu = (\frac{g\nu^3}{\bar g s^{2\epsilon + \eta}})^{1/3}=
(\frac{D_0}{\bar g k^{2\epsilon + \eta}})^{1/3}\, .
\label{nue}
\end{equation}
%where to obtain the last expression we used the equations
%$g\mu^{2\epsilon+\eta} \nu^3=g_0\nu_0^3 =D_0$ (see (\ref{D0})).
We
emphasize that above solution is exact, i.e., the exponent
$2\epsilon+\eta$ is exact too. However, in infrared region $k\ll
\mu\sim l^{-1}$,  $\bar g \rightarrow g_*,$ which can be calculated
only pertubatively. In the two-loop approximation $g_*=
g^{(1)}_*\varepsilon + \left(g^{(2)}_* + g^{(3)}_* \right )\varepsilon^2$ and after Taylor
expansion of $g_*^{1/3}$ in (\ref{nue}) we obtain:
\begin{equation}
\bar \nu \approx \nu_*
\left(\frac{D_0}{g_*^{(1)}\varepsilon}\right)^{1/3}
k^{-\frac{2\epsilon + \eta}{3}}\,,\qquad \nu_*\equiv
1-\frac{(g_*^{(2)}+ g_*^{(3)})\varepsilon}{3 g_*^{(1)}}\,. \label{nue1}
\end{equation}
%
%\begin{equation}
%\bar \nu \approx \left(1-\frac{g_*^{(2)}\varepsilon}{3 g_*^{(1)}}\right) \left(\frac{D_0}{g_*^{(1)}\varepsilon}\right)^{1/3}
%k^{-\frac{2\epsilon + \eta}{3}}\,.
%\label{nue1}
%\end{equation}
Remind that for Kolmogorov values $\varepsilon =\eta=4/3$ the
exponent in (\ref{nue1}) becomes equal to $-4/3.$ Let us estimate
the contribution of helicity to the effective diffusivity in
the fixed point (\ref{fix}). In this point
$\varepsilon=\eta$ $((2\varepsilon+\eta)/3=\varepsilon)$ and
\begin{eqnarray}
\nu_* &=& 1- \varepsilon
\bigg[(\frac{(3+u_*)}{30(1+u_*)^3}\,\cdot\,{_2F_1}\left(1,1;\frac{7}{2};\frac{1}{(1+u_*)^2}\right)
\nonumber\\
 & - & \frac {\pi \rho^2}{12(1+u_*)}\,\cdot\,
{_2F_1}\left(\frac12,\frac12;\frac{5}{2};\frac{1}{(1+u_*)^2}\right)\bigg].\label{nukon}
\end{eqnarray}





\input epsf
%\bfg[t] %
\begin{figure}[t]
 %   \vspace{-4.5cm}
    \begin{flushleft} \leavevmode
    \epsfxsize=6.5cm
    \epsffile{fig3.eps}
  \end{flushleft}
\vspace{-10.1cm}
 \begin{flushright}
    \leavevmode
    \epsfxsize=6.5cm
    \epsffile{fig4.eps}
\end{flushright}
\vspace{-1.5cm}\caption{(Left) The dependence of $\nu_*$ on the
helicity parameter $\rho$ for definite IR fixed point values $u_*$
of the parameter $u$. (Right) The dependence of $\nu_*$ on the IR
fixed point $u_*$ for the concrete values of the helicity parameter
$\rho$. The value $\rho_c=4/\sqrt{3}$. It is a special value related
to the analysis of the stability of the scaling regime which is not
discussed here. \label{fig5}}
%\efg %
\end{figure}

In figure \ref{fig5}, the dependence of the $\nu_*$ on the helicity
parameter $\rho$ and the IR fixed point $u_*$ for Kolmogorov value
of parameter $\varepsilon$ is shown . As one can see from these
figures when $u_*\rightarrow\infty$ (the rapid change model limit)
the two-loop corrections to $\nu_*=1$ are vanishing. Such behavior
is related to the fact that within the rapid change model there are
no two and higher loop corrections at all. On the other hand, the
largest two-loop corrections to the $\nu_*$ are given in the frozen
velocity field limit ($\nu_*\rightarrow0$).
% It is interesting that for
%all finite values of the parameter $u_*$ there exists a value of the
%helicity parameter $\rho$ for which the two-loop contribution to
%$\nu_*$ are canceled. For example, for the frozen velocity field
%limit ($u_*=0$) such situation arises when the helicity parameter
%$\rho$ is equal to its ``critical'' value $\rho_c=4/(\sqrt{3}\pi)$
%(this situation can be seen in the right figure of the
%Fig.\,\ref{fig5}). It is again the result of the competition between
%the non-helical and helical parts of the the two-loop corrections as
%is shown in Eq.\,(\ref{nukon}). Further important feature of the
%expression (\ref{nukon}) is that it is linear in the parameter
%$\varepsilon$. Thus, when one varies the value of $\varepsilon$ the
%picture is the same as in Fig.\,\ref{fig5} and only the scale of
%corrections is changed. In Fig.\,\ref{fig5} we have shown the
%situation for the most interesting case when $\varepsilon$ is equal
%to its ``Kolmogorov'' value, namely, $\varepsilon=4/3$.



Finally, let us analyze  time-behaviour of retarded response function
$G\equiv<\theta\theta'>$  in the limit $t \to \infty$.
%which characterizes response of the physical system on external regular forcing.

In  frequency-wave vector representation $G(\omega, {\mathbf p})$ satisfies  Dyson equation (\ref{Dyson}).
%\begin{equation}
%G(\omega,\mathbf{k})=(-i\omega+\nu_{0}k^{2}-\Sigma_{\theta'\theta}(\omega,\mathbf{k}))^{-1},
%\label{eq:Dyson}
%\end{equation}
 Self-energy operator $\Sigma$  is expressed via multi-loop Feynman graps and
 can be calculated perturbatively.
 We have found its divergent part up to two-loop approximation and calculated
 its finite part  with  the one-loop precision.




Using Dyson equation we find response function in time-wave vector representation:
\begin{equation}
G(t,\mathbf{p})=\int \frac{\mathrm{d}\omega}{2\pi} e^{-i\omega t}G(\omega,\mathbf{p}) =
\int \frac{\mathrm{d}\omega}{2\pi}\frac{e^{-i\omega t}}{-i\omega+\nu_{0}p^{2}-\Sigma(\omega,\mathbf{p})}\, .
\label{eq:Dyson1}
\end{equation}

In the lowest approximation  $\Sigma(\omega,\mathbf{p}) = 0$,  thus
the integral can be easily calculated:
$G_{0}(t,\mathbf{p})=\theta(t) \exp^{-i\omega_r t}.$ Here
$\theta(t)$ denotes usual step function and $\omega_r$ is residuum
in complex plain $ \omega$ in point $-i\nu_0 p^2 $. According to
\cite{Loran}, where analogical problem have been analyzed for turbulent viscosity,  
we suppose that this situation remains the same for
the full response function $G$, i.e., the leading contribution to
its  asymptotic behavior for $t \to \infty$ is determined by the
residuum $\omega =\omega_r$, which corresponds to the smallest root
of dispersion relation
\begin{equation}
G^{-1}(\omega,\mathbf{p})=-i\omega_r+\nu_{0}p^{2}-\Sigma(\omega_r,\mathbf{p})=0.
\label{eq:Dyson2}
\end{equation}
It is advantageous  to rewrite  the last relation in dimensionless
form:
\begin{equation}
1-z - I(1,z)= 0\,,\quad  z\equiv \frac{i\omega_r}{\nu_0 p^2} \,, \quad  I(1,z,g)\equiv \frac{ \Sigma(\omega,\mathbf{p})}{\nu_0 p^2}\,,
\end{equation}
which after renormalization can be rewritten in the fixed point $g_*$ (\ref{fix}) as follows
\begin{equation}
1-z_* - I_{*}= 0\, , \qquad \quad  z_*\equiv \frac{i\omega_r}{\bar{\nu} p^2} \,,
\label{ura}
\end{equation}
where $\bar{\nu}$ is effective diffusivity (\ref{nue1}) and
$I_*\equiv I_{*}(1,z_*, g_*)$ is renormalized (finite) part of
dimensionless self-energy operator $I$ at the fixed point $g_*$.


Hence decay law $G_{0}(t,\mathbf{p})\sim \exp^{-\nu_0 p^2 t}$  is changed into
\begin{equation}
G(t,\mathbf{p}) \sim  \exp^{-i\omega_r t}= \exp^{-z_*\bar{\nu} p^2 t}\, \qquad  t \to \infty \,.
\label{inf}
\end{equation}
To find the residuum $\omega_r$ (or, equivalently, $z_*$) it is necessary to calculalate quantity $I_*$. In
one-loop (linear in $\varepsilon$) approximation it can be written in the form:



%where  turbulent diffusivity $\nu_{turb}(k)\equiv i \omega_r/k^2$ has been introduced.
%We have found this  turbulent diffusivity in the scaling regime governed by the fixed point $g_*$ (\ref{fix}):
%\begin{equation}
%\nu_{turb}(k)= \chi D_0^{1/3} g_*^{-1/3} k^{-2\epsilon/3} ,
%\end{equation}
%where $\chi=1 + (\chi_1\pm\chi_2 i)\epsilon)$ are two complex roots of dispersion relation (\ref{eq:Dyson2}).
%In one-loop approximation we obtain
%\begin{equation}
%\nu_{turb}(k)= \left( 1 + \left(\chi_1-\frac{g_*^{(2)} + g_*^{(3)}\rho^2}{3g_*^{(1)}}\right)\epsilon
%\pm \chi_2\, i\, \epsilon \right) \left(\frac{D_0}{g_*^{(1)}}\right)^{1/3} k^{-2\epsilon/3},
%\label{dif3}
%\end{equation}
%where $g_*^{(i)}$ are defined in (\ref{fix}) and $\chi_1=4/3-3 \ln{[(1+u_*)/2}],$ $\chi_2 = \pi/2$
%for arbitrary $u_* < \infty.$


%It is convenient to subtract both $\Sigma_{1}$, $\Sigma_{2}$ and
%perform angular integration
\begin{equation}
I_*=-g_*\int_{-1}^{1}(1-x^{2})^{\frac{d-1}{2}}\mathrm{d}x\,{\cal I}\,
\label{eq:B.I}\end{equation}
with
\begin{equation}
{\cal I}=\int_{0}^{\infty}\mathrm{d}k\left[\frac{k}{ 1- z_*+(1+u_*)k^{2}-2kx}-\frac{\theta(k-1)}{(1+u_*)k}\right]\,.
\label{integral}
\end{equation}
Generally, the root $z_*$ can be complex and in one-loop  approximation it has form
\begin{equation}
z^{*}  =  z_{1}^{*}+\mathrm{i}z_{2},\quad  z_{1}^{*}  =  1+x_{1}\varepsilon,\quad
z_{2}^{*}  =  x_{2}\varepsilon\,. \label{eq:B.zp}\end{equation}

With our guaranteed precission %$g_*=g_*^{(1)}\varepsilon$ (see (\ref{fix}))
$I_*$ is linear in $\varepsilon$, therefore on the first sight  it
seems that in the last integral it is enought to take $z_*=1$ ($g_*
\sim \varepsilon$), but, actually, for its correct calculation  we
need to remain imaginary part $ x_2\varepsilon.$ Then the integral
(\ref{integral}) can be easily calculated by means of Sokhotsky's
formula:
\begin{equation}
\lim_{\varepsilon\rightarrow 0^{+}}\frac{1}{y\pm\mathrm{i}\epsilon}=\mp\mathrm{i}\pi\delta(y)+P(\frac{1}{y}),
\end{equation}
where
$P(\frac{1}{y})$ denotes the principal value of the integral.
Integration over angle $x$
%We obtain the following expression
%\begin{equation}
%{\cal I}  =  \overbrace{\pm\frac{\mathrm{i}\pi}{1+u_*}}^{x>0}+\frac{1}{1+u_*}\ln|\frac{1+u_*}{2x}|\label{eq:B.Int}
%\end{equation}
%Substitution (\ref{eq:B.Int}) to (\ref{eq:B.I}) and integration
%over angle $x$
gives final result for dimensionless self-energy operator $I_*$:
\begin{eqnarray}
I_*  =  -\frac{g_*}{(1+u_*)}\left(\frac{\sqrt{\pi}\Gamma(d+1)
\left(\gamma+\psi(1+\frac{d}{2})+2\ln|1+u_*|\right)}{d\,2^{d}\Gamma\left(\frac{d-1}{2}\right)\Gamma\left
(\frac{d}{2}+1\right)}\pm\mathrm{i}\pi\frac{d-1}{2d}\right),\nonumber \\
\label{eq:B.vysledok}\end{eqnarray}
where $\gamma$ is Euler's constant
and $\psi(z)$ is digamma function defined as $\psi(z)=\Gamma^{\prime}(z)/\Gamma(z)$.

Successful calculation of integral $I$ allows one to determine  the residue
$z_*$ (\ref{eq:B.zp}).
% by substituting (\ref{eq:B.zp}) and (\ref{eq:B.vysledok})
%to (\ref{eq:B.rovnicaRes}), that leads to equation
%\begin{equation}
%x_{1}+\mathrm{i}x_{2}=-I.
%\label{eq:B.x1x2}
%\end{equation}
Comparison of real and complex part of both sides of (\ref{ura})
gives the following terms in real space $d=3$ and in the fixed point  $g_*=3(1+u_*)/2$
(see (\ref{fix}))
%\begin{eqnarray}
%x_{1} & = & %\frac{g_*}{1+u_*}\left(\frac{\sqrt{\pi}\Gamma(d+1)\left(\gamma+\psi(1+\frac{d}{2})+2\ln|1+u_*|\right)}{d\,2^{d}\Gamma\left(\frac{d-1}{2}\right)%\Gamma\left(\frac{d}{2}+1\right)}\right),\qquad\,\,\,\\
%x_{2} & = & \pm\frac{g_*}{1+u_*}\frac{d-1}{2d}\pi,\end{eqnarray}
% In special case, when spatial dimension $d$ is equal to three, previous
%formulae simplifies to
\begin{equation}
x_{1}  = \frac{8}{3}+2\ln\frac{1+u_*}{2},\qquad
x_{2} =  \pm \frac{\pi}{2}.
\end{equation}

Due to the existence of two complex conjugate values $z_*$  the response function $G(t,p^{2})$ can be written in the asymptotic limit $t\rightarrow\infty$ in the following final form
\begin{equation}
G(t,p^{2})\cong e^{-\nu_{eff} p^{2-\varepsilon}t}\sin(\nu_f p^{2-\varepsilon}t),
\end{equation}
where
\begin{eqnarray}
 \nu_{eff}&\equiv& \bigg[1- \varepsilon \bigg[ \frac{8}{3}+2\ln\frac{1+u_*}{2}+
\frac{(3+u_*)}{30(1+u_*)^3}\,\cdot\,{_2F_1}\left(1,1;\frac{7}{2};\frac{1}{(1+u_*)^2}\right)
\nonumber\\
 & - & \frac {\pi \rho^2}{12(1+u_*)}\,\cdot\,
{_2F_1}\left(\frac12,\frac12;\frac{5}{2};\frac{1}{(1+u_*)^2}\right)\bigg]\bigg]
 \left(\frac{2 D_0}{3(1+u_*)\varepsilon}\right)^{1/3}\\
 \nu_f &\equiv& \frac{\pi\varepsilon}{2} \left(\frac{2 D_0}{3(1+u_*)\varepsilon}\right)^{1/3}
\nonumber
\end{eqnarray}
It's clear that the exponential
damping is accompanied by the oscilations.











\section{Conclusion}
We have studied  the advection of scalar field by turbulent flow in the frame of extended Kraichnan model and
investigated the influence of helicity  on anomalous scaling, stability of asymptotic regimes and effective
diffusivity. Such investigation is useful for understanding of efficiency of simplified  models
to study  the real turbulent motions by means of modern theoretical methods including renormalization group approach.
Actually, we performed two-loop calculations of divergent parts of Feynman graphs, which are necessary to achieve
multiplicative renormalization of equivalent field theoretic model. We have shown that anomalous scaling, which
is typical for the Kraichnan model and its numerous extensions \cite{Juha2,Pressure}, is not violated by inclusion of helicity to
the incompressible fluid.
%We expect that helicity gives rise to anomalous scaling exponents in case of  compressible turbulent flows.
On the other hand, stability of asymptotic regimes, values of fixed RG points and turbulent diffusivity strongly depend on amount of helicity.
It can be easily see  from (\ref{nukon}) that helicity enlarges turbulent diffusivity and high order contributions lead to the
appearance of oscillations in response function (\ref{eq:Dyson1}).
%We emphasize that all aforementioned effects can be observed only starting from two-loop approximation.

\section{Acknowledgement}
M.H. is thankful to N.V.Antonov and L.Ts. Adzhemyan for discussion.
The work was supported in part by VEGA grants 3211 and 2/6193/26 of
Slovak Academy of Sciences, by Science and Technology Assistance
Agency under contract No. APVT-51-027904, by  grant RFFI - RFBR
05-05-64735, by  grant RFFI - RFBR 05-02-17603, and by COFIN 2003
"Sistemi Complessi e Problemi a Molti Corpi.".


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they can cause difficulties when processing your files.
\end{itemize}
\subsubsection{\label{fname}Naming your figure files} In addition to the above points, please give each figure file a name which indicates the number of the figure it contains; for example, figure1.eps, figure2a.eps etc. If the figure file contains a figure with multiple parts, for example figure 2(a) to 2(e), give it a name such as figure2a\_2e.eps, and so forth.
\subsection{Submitting your article files to an IOP journal}
For most IOP journals you send your files directly to us by web, e-mail or FTP; however, there are a number of exceptions to this. For certain IOP journals you need to send your files to a different institution or to an external editor. How to find the submission address for a particular journal is discussed in section 1.4, below.
\subsection{Where to send your files \label{agide}}
Full details of how to submit files to a particular IOP journal are contained in IOP's {\it Guidelines for authors} which can be accessed online by going to authors.iop.org. From this document's table of contents, select the `How to submit' link to access full details of the submission address for a particular journal.

\subsection{Obtaining the class file and documentation}
The  \LaTeXe\ class file and guidelines can be downloaded from \verb"authors.iop.org".
Files are available in zipped format for PCs, tar compressed format files for
Unix or as a stuffit archive for Macintosh.

\section{Preparing your article}

Please follow these guidelines as closely as possible, particularly with regard to the preparation of the reference list.

At the start of the \LaTeX\ source please include
commented material to identify the journal, author and reference number if
known. The first non-commented line should be
\verb"\documentclass[12pt]{iopart}"  to load the preprint class
file. Omitting \verb"[12pt]" produces an article with the normal journal
page and type sizes. The start of the article text is signalled by
\verb"\begin{document}".
Authors of very long articles may find it convenient to separate
their article into a series of files each containing a section, each of which is called
in turn by the primary file.

Authors may add their own macros
at the start of an article provided they do not overwrite existing definitions and
that they send copies of their new macros with their text file. Macros for the individual paper not included
in a style file should be inserted in the preamble to the paper with comments to
describe any complex or non-obvious ones.
\verb"iopart" can be used with other package files such
as those loading the AMS extension fonts
\verb"msam" and \verb"msbm" (these fonts provide the
blackboard bold alphabet and various extra maths symbols as well as symbols useful in figure
captions); an extra style file \verb"iopams.sty" is provided to load these
packages and provide extra definitions for bold Greek letters.

\subsection{Double-column journals}
Authors writing for double-column journals should use the \verb"iopart" class file. Conversion from the single-column format to the double-column output required for printing will be done during the production process.
However, authors should bear in mind that all mathematical formulae will need to be fitted
into the width of a single column, so individual lines of equations should not occupy more than two thirds of the line width in this preprint form.

\section{The title and abstract page}
The code for setting the title page information is slightly different from
the normal default in \LaTeX.

\subsection{Titles and article types}
The title is set using the command
\verb"\title{#1}", where \verb"#1" is the title of the article. The
first letter
of the title should be capitalized with the rest in lower case.
Mathematical expressions within the title may be left in light-face type.

If the title is too long to use as a running head at the top of each page (apart from the
first) a short
form can be provided as an optional argument (in square brackets)
before the full title, i.e.\ \verb"\title[Short title]{Full title}".

For article types other than papers, \verb"iopart.cls"
has a generic heading \verb"\article[Short title]{TYPE}{Full title}"
and the specific definitions given in table~\ref{arttype}. In each case (apart from Letters
to the Editor) an
optional argument can be used immediately after the control sequence name
to specify the short title; where no short title is given the full title
will be used as the running head.
For Letters use \verb"\letter{Full title}", no short title is required as
the running head is automatically defined to be {\it Letter to the Editor}.
The generic heading could be used for
articles such as those presented at a conference or workshop, e.g.
\small\begin{verbatim}
\article[Short title]{WORKSHOP ON HIGH-ENERGY PHYSICS}{Title}
\end{verbatim}\normalsize
Footnotes to titles may be given, though acknowledgment of
funding should be included in the acknowledgments section rather than here. A footnote can be included by using \verb"\footnote{Text of footnote.}" immediately after the title.

\begin{table}
\caption{\label{arttype}Types of article defined in the {\tt iopart.cls}
class file.}
\footnotesize\rm
\begin{tabular*}{\textwidth}{@{}l*{15}{@{\extracolsep{0pt plus12pt}}l}}
\br
Command&Type&Heading on first page\\
\mr
\verb"\title{#1}"&Paper&---\\
\verb"\review{#1}"&Review&REVIEW\\
\verb"\topical{#1}"&Topical review&TOPICAL REVIEW\\
\verb"\comment{#1}"&Comment&COMMENT\\
\verb"\note{#1}"&Note&NOTE\\
\verb"\paper{#1}"&Paper&---\\
\verb"\prelim{#1}"&Preliminary communication&PRELIMINARY COMMUNICATION\\
\verb"\rapid{#1}"&Rapid communication&RAPID COMMUNICATION\\
\verb"\letter{#1}"&Letter&LETTER TO THE EDITOR\\
\verb"\article{#1}{#2}"&Other articles&Whatever is entered as {\tt
\#1}\\
\br
\end{tabular*}
\end{table}

\subsection{Authors' names and addresses}
The next information required is the list of authors' names and
their affiliations. For the authors' names type \verb"\author{#1}",
where \verb"#1" is the
list of all authors' names. The style for the names is initials then
family name, with a comma after all but the last
two names, which are separated by `and'. Initials should {\it not} be followed by full stops. First (given) names may be used if
desired and Chinese-style names included in the form they should be printed in. If the authors are at different addresses a superscripted number, e.g. $^1$, \verb"$^1$", should be used after each
name to reference the author to his/her address.
If an author has additional information to appear as a footnote, such as
a permanent address, a normal \LaTeX\ footnote command
should be given after the family name and address marker
with this extra information.

The addresses of the authors' affiliations follow the list of authors.
Each address is set by using
\verb"\address{#1}" with the address as the single parameter in braces.
If there is more
than one address then the appropriate superscripted number, followed by a space, should come at the start of
the address.

Please also add the e-mail addresses for at least one of the authors. This is done by inserting the
command \verb"\ead{#1}" after the postal address(es) where \verb"#1" is the e-mail address.
See section~\ref{startsample} for sample coding. For more than one e-mail address, please use the command
\verb"\eads{\mailto{#1}, \mailto{#2}}" with \verb"\mailto" surrounding each e-mail address.

\subsection{The abstract}
The abstract follows the addresses and
should give readers concise information about the content
of the article and indicate the main results obtained and conclusions
drawn. It should be self-contained with no reference to
figures, tables, equations, bibliographic references etc
included and should not normally exceed 200
words.
To indicate the start
of the abstract type \verb"\begin{abstract}" followed by the text of the
abstract (not in braces).  The abstract should normally be restricted
to a single paragraph and is terminated by the command
\verb"\end{abstract}"

\subsection{Subject classification numbers}
Any Physics and Astronomy Classification System (PACS)
codes or Mathematics Subject Classification (MSC) scheme
numbers should come immediately after the abstract.
Classification codes can greatly help in the choice of suitable
referees and allocation of articles to subject areas.
For {\it Inverse Problems} and {\it Nonlinearity} authors may
use either PACS or MSC codes.

PACS or MSC numbers are included after the abstract
using \verb"\pacs{#1}" and \verb"\ams{#1}" respectively.

After any classification numbers the command
\verb"\submitto{#1}" can be inserted, where \verb"#1" is the journal name written in full or the appropriate control sequence as
given in table~\ref{jlab1}. This command is not essential to the running of the file.

\subsubsection{Information on PACS and MSC}
For more information on PACS and MSC see
\begin{itemize}
\item MSC: http://www.ams.org/msc
\item PACS: http://www.aip.org/pacs
\end{itemize}

\subsection{Keywords}
Keywords should be provided for submissions to {\it Measurement Science and Technology, Physical Biology, Physiological Measurement}, and both parts of {\it Journal of Optics\/}. Add these as a new paragraph starting \verb"\noindent{\it Keywords\/}:" after the end of the abstract.

\subsection{Making a separate title page}
The command \verb"\maketitle" forces a page break after the point where it
is inserted and so to keep the header material on a separate page from the
body of the text insert \verb"\maketitle" or \verb"\newpage" before the start of the text.
If \verb"\maketitle" is not included the text of the
article will start immediately after the abstract.

\subsection{Sample coding for the start of an article}
\label{startsample}
The code for the start of a title page of a typical paper might read:
\small\begin{verbatim}
\documentclass[12pt]{iopart}
\begin{document}
\title[The anomalous magnetic moment of the
neutrino]{The anomalous magnetic moment of the
neutrino and its relation to the solar neutrino problem}

\author{P J Smith$^1$, T M Collins$^2$,
R J Jones$^3$\footnote{Present address:
Department of Physics, University of Bristol, Tyndalls Park Road,
Bristol BS8 1TS, UK.} and Janet Williams$^3$}

\address{$^1$ Mathematics Faculty, Open University,
Milton Keynes MK7~6AA, UK}
\address{$^2$ Department of Mathematics,
Imperial College, Prince Consort Road, London SW7~2BZ, UK}
\address{$^3$ Department of Computer Science,
University College London, Gower Street, London WC1E~6BT, UK}
\ead{williams@ucl.ac.uk}

\begin{abstract}
...
\end{abstract}
\pacs{1315, 9440T}
\submitto{\JPG}
\maketitle
\end{verbatim}
\normalsize

\section{The text}
\subsection{Sections, subsections and subsubsections}
The text of articles may be divided into sections, subsections and, where necessary,
subsubsections. To start a new section, end the previous paragraph and
then include \verb"\section" followed by the section heading within braces.
Numbering of sections is done {\it automatically} in the headings:
sections will be numbered 1, 2, 3, etc, subsections will be numbered
2.1, 2.2,  3.1, etc, and subsubsections will be numbered 2.3.1, 2.3.2,
etc.  Cross references to other sections in the text should, where
possible, be made using
labels (see section~\ref{xrefs}) but can also
be made manually. See section~\ref{eqnum} for information on the numbering of displayed equations. Subsections and subsubsections are
similar to sections but
the commands are \verb"\subsection" and \verb"\subsubsection" respectively.
Sections have a bold heading, subsections an italic heading and
subsubsections an italic heading with the text following on directly.
\small\begin{verbatim}
\section{This is the section title}
\subsection{This is the subsection title}
\end{verbatim}\normalsize

The first section is normally an introduction,  which should state clearly
the object of the work, its scope and the main advances reported, with
brief references to relevant results by other workers. In long papers it is
helpful to indicate the way in which the paper is arranged and the results
presented.

Footnotes should be avoided whenever possible and can often be included in the text as phrases or sentences in parentheses. If required, they should be used only for brief notes that do not fit conveniently into the text. The use of
displayed mathematics in footnotes should be avoided wherever possible and no equations within a footnote should be numbered.
The standard \LaTeX\ macro \verb"\footnote" should be used.

\subsection{Acknowledgments}
Authors wishing to acknowledge assistance or encouragement from
colleagues, special work by technical staff or financial support from
organizations should do so in an unnumbered `Acknowledgments' section
immediately following the last numbered section of the paper. The
command \verb"\ack" sets the acknowledgments heading as an unnumbered
section. For Letters
\verb"\ack" does not set a heading but leaves a line space and does not
indent the next paragraph.

\subsection{Appendices}
Technical detail that it is necessary to include, but that interrupts
the flow of the article, may be consigned to an appendix.
Any appendices should be included at the end of the main text of the paper, after the acknowledgments section (if any) but before the reference list.
If there are
two or more appendices they should be called Appendix A, Appendix B, etc.
Numbered equations will be in the form (A.1), (A.2), etc,
figures will appear as figure A1, figure B1, etc and tables as table A1,
table B1, etc.

The command \verb"\appendix" is used to signify the start of the
appendices. Thereafter \verb"\section", \verb"\subsection", etc, will
give headings appropriate for an appendix. To obtain a simple heading of
`Appendix' use the code \verb"\section*{Appendix}". If it contains
numbered equations, figures or tables the command \verb"\appendix" should
precede it and \verb"\setcounter{section}{1}" must follow it.

\subsection{Some matters of style}
It will help the readers if your article is written in a clear,
consistent and concise manner. The Production Department at Institute of Physics
Publishing will try to make sure that your work is presented to its
readers in the best possible way without sacrificing the individuality of
your writing. Some recommended
points to note, however, are the following.
\begin{enumerate}
\item Authors are often inconsistent in the use of `ize' and `ise' endings.
We recommend using `-ize' spellings (diagonalize,
renormalization, minimization, etc) but there are some common
exceptions to this, for example: devise,
promise and advise.

\item English spellings are mandatory (colour, flavour, behaviour, tunnelling, artefact, focused, focusing, fibre, etc) except in the following journals where either American or English spellings are acceptable: {\it Physical Biology}, {\it Smart Materials and Structures} and {\it Journal of Micromechanics and Microengineering}.

\item The words table and figure should be written
in full and {\bf not} abbreviaged to tab. and fig. Do not include `eq.'\, `equation' etc before an equation number or `ref.'\, `reference' etc before a reference number.
\end{enumerate}

Please check your article carefully for accuracy, consistency and clarity before
submission. Remember that your article will probably be read by many
people whose native language is not English and who may not therefore
be aware of many of the subtle meanings of words or idiomatic phases
present in the English language. It therefore helps if you try to keep
sentences as short and simple as possible.

\section{Mathematics}
\subsection{Two-line constructions}
The great advantage of \LaTeX\
over other text processing systems is its
ability to handle mathematics to almost any degree of complexity. However,
in order to produce an article suitable for publication both within a print journal and online,
authors should exercise some restraint on the constructions used. Some equations using very small characters which are clear in a preprint style article may be difficult read in a smaller format or online at a relatively low resolution.
For simple fractions in the text the solidus \verb"/", as in
$\lambda/2\pi$, should be used instead of \verb"\frac" or \verb"\over",
care being taken to use parentheses where necessary to avoid ambiguity,
for example to distinguish between $1/(n-1)$ and $1/n-1$. Exceptions to
this are the proper fractions $\frac12$, $\frac13$, $\frac34$,
etc, which are better left in this form. In displayed equations
horizontal lines are preferable to solidi provided the equation is
kept within a height of two lines. A two-line solidus should be
avoided where possible; the construction $(\ldots)^{-1}$ should be
used instead; for example use:
\begin{equation*}
\frac{1}{M_{\rm a}}\left(\int^\infty_0{\rm d}
\omega\;\frac{|S_o|^2}{N}\right)^{-1}\qquad\mbox{instead of}\qquad
\frac{1}{M_{\rm a}}\biggl/\int^\infty_0{\rm d}
\omega\;\frac{|S_o|^2}{N}.
\end{equation*}

\subsection{Roman and italic in mathematics}
In mathematics mode \LaTeX\ automatically sets variables in an italic
font. In most cases authors should accept this italicization. However,
there are some cases where it is better to use a Roman font; for
instance, IOP journals use a Roman d for a differential d, a Roman e
for an exponential e and a Roman i for the square root of $-1$. To
accommodate this and to simplify the  typing of equations we have
provided some extra definitions. \verb"\rmd", \verb"\rme" and \verb"\rmi"
now give Roman d, e and i respectively for use in equations,
e.g.\ $\rmi x\rme^{2x}\rmd x/\rmd y$
is obtained by typing \verb"$\rmi x\rme^{2x}\rmd x/\rmd y$".


Certain other common mathematical functions, such as cos, sin, det and
ker, should appear in Roman type. \LaTeX\ provides macros for
most of these functions
(in the cases above, \verb"\cos", \verb"\sin", \verb"\det" and \verb"\ker"
respectively); we have also provided
additional definitions for $\Tr$, $\tr$ and
$\Or$ (\verb"\Tr", \verb"\tr" and \verb"\Or", respectively).

Subscripts and superscripts should be in Roman type if they are labels
rather than variables or characters that take values. For example in the
equation
\[
\epsilon_m=-g\mu_{\rm n}Bm
\]
$m$, the $z$ component of the nuclear spin, is italic because it can have
different values whereas n is Roman because it
is a label meaning nuclear ($\mu_{\rm n}$
is the nuclear magneton).


\subsection{Displayed equations in double-column journals}
Authors should bear in mind that all mathematical formulae in double-column journals will need to be fitted
into the width of a single column, so individual lines of equations should not occupy more than two thirds of the line width in this preprint form.

\subsection{Special characters for mathematics}
Bold italic characters can be used in our journals to signify vectors (rather
than using an upright bold or an over arrow). To obtain this effect
use \verb"\bi{#1}" within maths mode, e.g. $\bi{ABCdef}$. If upright
bold characters are required in maths use \verb"\mathbf{#1}" within maths
mode, e.g. $\mathbf{XYZabc}$. The calligraphic (script) uppercase alphabet
is obtained with \verb"\mathcal{AB}" or \verb"\cal{CD}"
($\mathcal{AB}\cal{CD}$).

The American Mathematical Society provides a series of extra symbol fonts
to use with \LaTeX\ and packages containing the character definitions to
use these fonts. Authors wishing to use Fraktur
\ifiopams$\mathfrak{ABC}$ \fi
or Blackboard Bold \ifiopams$\mathbb{XYZ}$ \fi can include the appropriate
AMS package (e.g. amsgen, amsfonts, amsbsy, amssymb) with a
\verb"\usepackage" command or add the command \verb"\usepackage{iopams}"
which loads the four AMS packages mentioned above and also provides
definitions for extra bold characters (all Greek letters and some
additional other symbols).

The package iopams uses the definition \verb"\boldsymbol" in amsbsy
which allows individual non-alphabetical symbols and Greek letters to be
made bold within equations.
The bold Greek lowercase letters \ifiopams$\balpha \ldots\bomega$,\fi
are obtained with the commands
\verb"\balpha" \dots\ \verb"\bomega" (but note that
bold eta\ifiopams, $\bfeta$,\fi\ is \verb"\bfeta" rather than \verb"\beta")
and the capitals\ifiopams, $\bGamma\ldots\bOmega$,\fi\ with commands
\verb"\bGamma" \dots\
\verb"\bOmega". Bold versions of the following symbols are
predefined in iopams:
bold partial\ifiopams, $\bpartial$,\fi\ \verb"\bpartial",
bold `ell'\ifiopams, $\bell$,\fi\  \verb"\bell",
bold imath\ifiopams, $\bimath$,\fi\  \verb"\bimath",
bold jmath\ifiopams, $\bjmath$,\fi\  \verb"\bjmath",
bold infinity\ifiopams, $\binfty$,\fi\ \verb"\binfty",
bold nabla\ifiopams, $\bnabla$,\fi\ \verb"\bnabla",
bold centred dot\ifiopams, $\bdot$,\fi\  \verb"\bdot", other
characters are made bold using
\verb"\boldsymbol{\symbolname}".

Please do not use the style file \verb"amsmath.sty" (part of the AMSTeX package) in conjunction with \verb"iopart.cls". This will result in several errors. To make use of the macros defined in \verb"amsmath.sty", we have provided the file \verb"setstack.sty" which reproduces the following useful macros from \verb"amsmath.sty":
\small\begin{verbatim}
\overset \underset \sideset \substack \boxed   \leftroot
\uproot  \dddot    \ddddot  \varrow   \harrow
\end{verbatim}\normalsize

Table~\ref{math-tab2} lists some other macros for use in
mathematics with a brief description of their purpose.

\begin{table}
\caption{\label{math-tab2}Other macros defined in IOP macros for use in maths.}
\begin{tabular*}{\textwidth}{@{}l*{15}{@{\extracolsep{0pt plus
12pt}}l}}
\br
Macro&Result&Description\\
\mr
\verb"\fl"&&Start line of equation full left\\
\verb"\case{#1}{#2}"&$\case{\#1}{\#2}$&Text style fraction in display\\
\verb"\Tr"&$\Tr$&Roman Tr (Trace)\\
\verb"\tr"&$\tr$&Roman tr (trace)\\
\verb"\Or"&$\Or$&Roman O (of order of)\\
\verb"\tdot{#1}"&$\tdot{x}$&Triple dot over character\\
\verb"\lshad"&$\lshad$&Text size left shadow bracket\\
\verb"\rshad"&$\rshad$&Text size right shadow bracket\\
\br
\end{tabular*}
\end{table}

\subsection{Alignment of displayed equations}
The normal style for aligning displayed equations is to align them left rather than centre them. The class file automatically does this and indents each line of a display. To make any line start at the left margin of the page, add \verb"\fl" at its start (to indicate full left).

Using the \verb"eqnarray" environment equations will naturally be aligned left without the use of any ampersands for alignment, see equations (\ref{eq1}) and (\ref{eq2})
\begin{eqnarray}
\alpha + \beta =\gamma^2 \label{eq1}\\
\alpha^2 + 2\gamma + \cos\theta = \delta \label{eq2}
\end{eqnarray}
This is the normal equation style for our journals.

Where some secondary alignment is needed, for instance a second part of an equation on a second line, a single ampersand is added at the point of alignment in each line  (see  (\ref{eq3}) and (\ref{eq4})).
\begin{eqnarray}
\alpha &=2\gamma^2 + \cos\theta + \frac{XY \sin\theta}{X+ Y\cos\theta} \label{eq3}\\
 & = \delta\theta PQ \cos\gamma. \label{eq4}
\end{eqnarray}

Two points of alignment are possible using two ampersands for alignment (see  (\ref{eq5}) and (\ref{eq6})).  Note in this case extra space \verb"\qquad" is added before the second ampersand in the longest line (the top one) to separate the condition from the equation.
\begin{eqnarray}
\alpha &=2\gamma^2 + \cos\theta + \frac{XY \sin\theta}{X+ Y\cos\theta}\qquad& \theta > 1 \label{eq5}\\
 & = \delta\theta PQ \cos\gamma &\theta \leq 1\label{eq6}
\end{eqnarray}

For a long equation which has to be split over more than one line the first line should start at the left margin, this is achieved by inserting \verb"\fl" (full left) at the start of the line, the use of the alignment parameter \verb"&" is not necessary unless some secondary alignment is needed.
\begin{eqnarray}
\fl \alpha + 2\gamma^2 = \cos\theta + \frac{XY \sin\theta}{X+ Y\cos\theta} +  \frac{XY \sin\theta}{X- Y\cos\theta} +
+ \left(\frac{XY \sin\theta}{X+ Y\cos\theta}\right)^2 \nonumber\\
+  \left(\frac{XY \sin\theta}{X- Y\cos\theta}\right)^2\label{eq7}
\end{eqnarray}

The Plain \TeX\ command \verb"\eqalign" can be used within an \verb"equation" environment to obtain a multiline equation with a single centred number, for example
\begin{equation}
\eqalign{\alpha + \beta =\gamma^2 \cr
\alpha^2 + 2\gamma + \cos\theta = \delta.}\label{eq2}
\end{equation}

\subsection{Miscellaneous points}
Exponential expressions, especially those containing subscripts or
superscripts, are clearer if the notation $\exp(\ldots)$ is used except for
simple examples. For instance $\exp[\rmi(kx-\omega t)]$ and $\exp(z^2)$ are
preferred to $\e^{\rmi(kx-\omega t)}$ and $\e^{z^2}$, but
$\e^x$
is acceptable. Similarly the square root sign $\sqrt{\phantom{b}}$ should
only be used with relatively
simple expressions, e.g.\ $\sqrt2$ and $\sqrt{a^2+b^2}$;
in other cases the
power $1/2$ should be used; for example, $[(x^2+y^2)/xy(x-y)]^{1/2}$.

It is important to distinguish between $\ln = \log_\e$ and $\lg
=\log_{10}$. Braces, brackets and parentheses should be used in the
following order: $\{[(\;)]\}$. The same ordering of brackets should be
used within each size. However, this ordering can be ignored if the
brackets have a
special meaning (e.g.\ if they denote an average or a function).  Decimal
fractions should
always be preceded by a zero: for example 0.123 {\bf not} .123. For long
numbers commas are not inserted but instead a thin space is added after
every third character away from the position of the decimal point unless
this leaves a single separated character: e.g.\ $60\,000$, $0.123\,456\,78$
but 4321 and 0.7325.

Equations should be followed by a full stop (periods) when at the end
of a sentence.

\subsection{Equation numbering}
\label{eqnum}
\LaTeX\ provides facilities for automatically numbering equations
and these should be used where possible. Sequential numbering (1), (2),
etc, is the default numbering system although, if the command
\verb"\eqnobysec" is included in the preamble, equation numbering
by section is obtained, e.g.\
(2.1), (2.2), etc. In articles with several appendices equation numbering
by section is useful in the appendices even when sequential numbering has
been used throughout the main body of the text and is switched on by the
\verb"\appendix" command. Equation numbering by section {\it must}
be used for {\it Reports on Progress in
Physics}. When referring to an equation in the text it is not normally necessary to include the word equation before the number,
which should be parentheses. Do not
use abbreviations such as eqn or eq.
When cross-referencing is used, \verb"(\ref{<label>})"
 will produce `(\verb"<eqnum>")',
\verb"\eref{<label>}" produces `(\verb"<eqnum>")' and
\verb"\Eref{<label>}" produces `Equation (\verb"<eqnum>")',
where \verb"<label>" is the label to produce equation number \verb"<eqnum>".

Sometimes it is useful to number equations as parts of the same
basic equation. This can be accomplished by inserting the
commands \verb"\numparts" before the equations concerned and
\verb"\endnumparts" when reverting to the normal sequential numbering.
The equations below show the previous equations numbered as separate parts
using \verb"\numparts ... \endnumparts" and the \verb"eqnarray"
environment
\numparts
\begin{eqnarray}
T_{11}&=(1+P_\e)I_{\uparrow\uparrow}-(1-P_\e)
I_{\uparrow\downarrow}\\
T_{-1-1}&=(1+P_\e)I_{\downarrow\downarrow}-(1-P_\e)I_{\uparrow\downarrow}\\
S_{11}&=(3+P_\e)I_{\downarrow\uparrow}-(3-P_e)I_{\uparrow\uparrow}\\
S_{-1-1}&=(3+P_\e)I_{\uparrow\downarrow}-(3-P_\e)
I_{\downarrow\downarrow}
\end{eqnarray}
\endnumparts


\subsection{Miscellaneous extra commands for displayed equations}
The \verb"\cases" command of Plain \TeX\ is available
for use with \LaTeX\ but has been amended slightly to
increase the space between the equation and the condition.
\Eref{cases}
demonstrates simply the output from the \verb"\cases" command
\begin{equation}
\label{cases}
X=\cases{1&for $x \ge 0$\\
-1&for $x<0$\\}
\end{equation}
%The code used was:
%\small\begin{verbatim}
%\begin{equation}
%\label{cases}
%X=\cases{1&for $x \ge 0$\\
%-1&for $x<0$\\}
%\end{equation}
%\end{verbatim}
%\normalsize

To obtain text style fractions within displayed maths the command
\verb"\case{#1}{#2}" can be used (see equations (2) and (5)) instead
of the usual \verb"\frac{#1}{#2}" command or \verb"{#1 \over #2}".

When two or more short equations are on the same line they should be
separated by a `qquad space' (\verb"\qquad"), rather than
\verb"\quad" or any combination of \verb"\,", \verb"\>", \verb"\;"
and \verb"\ ".

\section{Referencing\label{except}}
Two different styles of referencing are in common use:
the Harvard alphabetical system and the Vancouver numerical system.
All IOP journals allow the use of the Harvard or Vancouver system
(authors should use the style they are comfortable with),
apart from the following journals

\begin{itemize}
\item {\it Physics in Medicine and Biology}
\item {\it Physiological Measurement}
\end{itemize}

\noindent for which authors must use the Harvard referencing style.

\subsection{Alphabetical (Harvard) system}
In the Harvard system the name of the author appears in the text together
with the year of publication. As appropriate, either the date or the name
and date are included within parentheses. Where there are only two authors
both names should be given in the text; if there are more than two
authors only the first name should appear followed by `{\it et al}'
(which can be obtained by typing \verb"\etal"). When two or
more references to work by one author or group of authors occur for the
same year they should be identified by including a, b, etc after the date
(e.g.\ 1986a). If several references to different pages of the same article
occur the appropriate page number may be given in the text, e.g.\ Kitchen
(1982, p 39).

The reference list at the end of an article consists of an
unnumbered `References' section containing an
alphabetical listing by authors' names and in date order for each
author or group of identical authors. The reference list in the
preprint style is started by including the command \verb"\section*{References}" and then
\verb"\begin{harvard}".
Individual references start with \verb"\item[]" and the reference list is completed with \verb"\end{harvard}".
There is also a shortened form of the coding; \verb"\section*{References}"
and \verb"\begin{harvard}" can be replaced by the single command
\verb"\References" and \verb"\end{harvard}" can be shortened to
\verb"\endrefs".

\subsection{Numerical (Vancouver) system}
In the numerical system references are numbered sequentially
throughout the text. The numbers occur within square brackets and one
number can be used to designate several references. A numerical
reference list in the \verb"iopart" style is started by including the
command \verb"\section*{References}" and then
\verb"\begin{thebibliography}{<num>}", where \verb"<num>" is the largest
number in the reference list (or any other number with the same number
of digits).  The
reference list gives the references in
numerical, not alphabetical, order and is completed by
\verb"\end{thebibliography}". Short forms of the commands are again
available: \verb"\Bibliography{<num>}" can be used at the start of the
references section and \verb"\endbib" at the end.

A variant of this system is to use labels instead of numbers within
square brackets, this method is allowed although not recommended.

\subsection{BiBTeX\label{bibtex}}
Note that for the journals lised at the start of section {\ref{except}} authors should use the
appropriate \verb".bst" file to produce references in the correct style---see section \ref{bibby} for
a list of reccommended \verb".bst" files.

\subsection{References, general}
A complete reference should provide the reader with enough information to
locate the article concerned. Titles of journal articles can also be included and are required for {\it Inverse problems, Journal of Neural Engineering, Measurement Science and Technology, Physical Biology, Physics in Medicine and Biology\/} and {\it Physiological Measurement}. Final page numbers of references are required for {\it Reports on Progress in Physics\/} and {\it Physiological Measurement}.

Up to ten authors may be given in a particular reference; where
there are more than ten only the first should be given followed by
`{\it et al}'. If an author is unsure of an abbreviation and the
journal is not given in appendix B, it is best to leave the title in full.

The terms {\it loc.\ cit.}\ and {\it ibid}.\ should not be used.
Unpublished conferences and reports should generally not be included
in the reference list if a published version of the work exists. Articles in the course of publication should
include the article title and the journal of publication, if known.
A reference to a thesis submitted for a higher degree may be included
if it has not been superseded by a published
paper but the institution where the work was done should be included.

The basic structure of a reference in the reference list is the same in both the alphabetical and numerical systems, the only difference the code at the start of the reference. Alphabetic references are preceded by \verb"\item[]", numeric by \verb"\bibitem{label}" or just \verb"\item" to generate a number or \verb"\nonum" where a reference is not the first in a group of references under the same number.

Cross referencing between the text and the
reference list is most useful in the numeric system but is not necessary for alphabetic referencing
in the Harvard system as adding or deleting a reference
does not normally change any of the other references.
Care should, however, be taken to make sure that the text citations match the reference list and vice versa.

Note that footnotes to the text should not be
included in reference list, but should appear at the bottom of the relevant page by using the \verb"\footnote" command.

\subsection{References to journal articles}
A normal reference to a journal article contains three changes of
font:
the authors and date appear in Roman type, the journal title in
italic, the volume number in bold and the page numbers in Roman again.
A typical journal entry would be:

\smallskip
\begin{harvard}
\item[] Cisneros A 1971 {\it Astrophys.\ Space Sci.} {\bf 10} 87
\end{harvard}
\smallskip

\noindent which would be obtained by typing, within the references
environment
\small\begin{verbatim}
\item[] Cisneros A 1971 {\it Astrophys. Space Sci.} {\bf 10} 87
\end{verbatim}\normalsize

Features to note are the following.

\begin{enumerate}
\item The authors should be in the form surname (with only the first
letter capitalized) {\bf followed} by the initials with {\bf no}
periods after the initials. Authors should be separated by a comma
except for the last two which should be separated by `and' with no
comma preceding it. For journals that accept titles of articles in the
reference list,  the title should be in Roman (upright)
lower case letters, except for an initial
capital, and should follow the date.

\item The year of publication follows the authors and is not in parentheses. An article title (in Roman) would follow the year.

\item The journal is in italic and is abbreviated. \ref{jlabs}
gives a list of macros that will give the correct abbreviation for
many of the common journals. If a journal has several parts denoted by
different letters the part letter should be inserted after the journal in Roman type, e.g.\
{\it Phys.\ Rev.\ \rm A}.

\item The volume number is bold; the page number is Roman.
 Both the initial and final page numbers should be given where possible. The final page number should be in
the shortest possible form and separated from the initial page number by an en rule (\verb"--"), e.g.\ 1203--14.

\item Where there are two or more references with identical authors,
the authors' names should be repeated for the second and subsequent references. Each individual publication should be presented as a separate reference, although in the numerical system one number can be used for several references. This facilitates linking in the electronic version of the journal.
\end{enumerate}
\subsection{Electronic journal references}
These often do not always follow the conventional {\it year-journal-volume-page numbers} pattern. Some examples are:

\numrefs{1}
\item Carlip S and Vera R 1998 {\it Phys. Rev.} D {\bf 58} 011345
\item Davies K and Brown G 1997 {\it J. High Energy Phys.} JHEP12(1997)002
\item Hannestad S 2005 {\it J. Cosmol. Astropart. Phys.} JCAP02(2005)011
\item Hilhorst H J 2005 {\it J. Stat. Mech.} L02003
\item Gundlach C 1999 {\it Liv. Rev. Rel.} 1994-4
\endnumrefs

\subsection{Preprint references}
Preprint may be referenced but if the article concerned has been published in a peer-reviewed journal, that reference should take precedence. If only a preprint reference can be given, it is helpful to include the article title. Examples are:
\vskip6pt
\numrefs{1}
\item Neilson D and Choptuik M 2000 {\it Class. Quantum Grav.} {\bf 17} 761 ({\it Preprint} gr-qc/9812053)
\item Harrison M 1999 Dipheomorphism-invariant manifolds {\it Preprint} hep-th/9909196
\endnumrefs

\subsection{References to books, conference proceedings and reports}

References to books, proceedings and reports are similar, but have only two
changes of font. The authors and date of publication are in Roman, the
title of the book is in italic, and the editors, publisher,
town of publication
and page number are in Roman. A typical reference to a book and a
conference paper might be

\smallskip
\begin{harvard}
\item[] Dorman L I 1975 {\it Variations of Galactic Cosmic Rays}
(Moscow: Moscow State University Press) p~103
\item[] Caplar R and Kulisic P 1973 {\it Proc.\
Int.\ Conf.\ on Nuclear Physics (Munich)} vol~1 (Amsterdam:
North-Holland/American Elsevier) p~517
\end{harvard}
\smallskip

\noindent which would be obtained by typing
\small\begin{verbatim}
\item[] Dorman L I 1975 {\it Variations of Galactic Cosmic Rays}
(Moscow: Moscow State University Press) p~103
\item[] Caplar R and Kulisic P 1973 {\it Proc. Int. Conf. on Nuclear
Physics (Munich)} vol~1 (Amsterdam: North-Holland/American
Elsevier) p~517
\end{verbatim}\normalsize
\noindent respectively.


Features to note are the following.
\begin{enumerate}
\item Book titles are in italic and should be spelt out in full with
initial capital letters for all except minor words. Words such as
Proceedings, Symposium, International, Conference, Second, etc should
be abbreviated to Proc., Symp., Int., Conf., 2nd,
respectively, but the rest of the title should be given in full,
followed by the date of the conference and the
town or city where the conference was held. For
Laboratory Reports the Laboratory should be spelt out wherever
possible, e.g.\ {\it Argonne National Laboratory Report}.

\item The volume number as, for example, vol~2, should be followed by
the editors, if any, in a form such as ed~A~J~Smith and P~R~Jones. Use
\etal if there are more than two editors. Next comes the town of
publication and publisher, within brackets and separated by a colon,
and finally the page numbers preceded by p if only one number is given
or pp if both the initial and final numbers are given.

\item If a book is part of a series (for examples, {\it Springer Tracts in Modern Physics\/}), the series title and volume number is given in parentheses after the book title. Whereas for an individual volume in a multivolume set, the set title is given first, then the volume title.
\end{enumerate}
\smallskip
\begin{harvard}
\item[]Morse M 1996 Supersonic beam sources {\it Atomic Molecular and Optical Physics\/} ({\it Experimental Methods in the Physical Sciences\/} vol 29) ed F B Dunning and R Hulet (San Diego: Academic)
\item[]Fulco C E, Liverman C T and Sox H C (eds) 2000 {\it Gulf War and Health\/} vol 1 {\it Depleted Uranium, Pyridostigmine Bromide, Sarin, and Vaccines\/} (Washington, DC: The National Academies Press)
\end{harvard}

\section{Cross referencing\label{xrefs}}
The facility to cross reference items in the text is very useful when
composing articles as the precise form of the article may be uncertain at the start
and  revisions and amendments may subsequently be made.
\LaTeX\ provides excellent facilities for doing cross-referencing
and these can be very useful in preparing articles.

\subsection{References}
\label{refs}
Cross referencing is useful for numeric reference lists because, if it
is used, adding
another reference to the list does not then involve renumbering all
subsequent references. It is not necessary for referencing
in the Harvard system where the final reference list is alphabetical
and normally no other changes are necessary when a reference is added or
deleted.
Two passes are necessary initially to get the cross references right
but once they are correct a single run is usually sufficient provided an
\verb".aux" file is available and the file
is run to the end each time.
\verb"\cite" and \verb"\bibitem" are used to link citations in the text
with their entry in the reference list;
if the
reference list contains an entry \verb"\bibitem{label}",
then \verb"\bibitem{label}"
will produce the correct number in the reference list and
\verb"\cite{label}" will produce the number within square brackets in the
text. \verb"label" may contain alphabetic letters,
or punctuation characters but must not contain spaces or commas. It is also
recommended that the underscore character \_{} is not used in cross
referencing.
Thus labels for the form
\verb"eq:partial", \verb"fig:run1", \verb"eq:dy'",
etc, may be used. When several
references occur together in the text \verb"\cite" may be used with
multiple labels with commas but no spaces separating them;
the output will be the
numbers within a single pair of square brackets with a comma and a
thin space separating the numbers. Thus \verb"\cite{label1,label2,label4}"
would give [1,\,2,\,4]. Note that no attempt is made by the style file to sort the
labels and no shortening of groups of consecutive numbers is done.
Authors should therefore try to use multiple labels in the correct
order.

The numbers for the cross referencing are generated in the order the
references appear in the reference list, so that if the entries in the
list are not in the order in which the references appear in the text
then the
numbering within the text will not be sequential. To correct this
change the ordering of the entries in the reference list and then
rerun {\it twice}.

\subsection{Equation numbers, sections, subsections, figures and
tables}
Labels for equation numbers, sections, subsections, figures and tables
are all defined with the \verb"\label{label}" command and cross references
to them are made with the \verb"\ref{label}" command.

Any section, subsection, subsubsection, appendix or subappendix
command defines a section type label, e.g. 1, 2.2, A2, A1.2 depending
on context. A typical article might have in the code of its introduction
`The results are discussed in section\verb"~\ref{disc}".' and
the heading for the discussion section would be:
\small\begin{verbatim}
\section{Results}\label{disc}
\end{verbatim}\normalsize
Labels to sections, etc, may occur anywhere within that section except
within another numbered environment.
Within a maths environment labels can be used to tag equations which are
referred to within the text.

In addition to the standard \verb"\ref{<label>}" the abbreviated
forms given in the \tref{abrefs}
are available for reference to standard parts of the text

\Table{\label{abrefs}Alternatives to the normal references $\backslash${\tt ref}
and the text generated by
them. Note it is not normally necessary to include the word equation
before an equation number except where the number starts a sentence. The
versions producing an initial capital should only be used at the start of
sentences.}
\br
Reference&Text produced\\
\mr
\verb"\eref{<label>}"&(\verb"<num>")\\
\verb"\Eref{<label>}"&Equation (\verb"<num>")\\
\verb"\fref{<label>}"&figure \verb"<num>"\\
\verb"\Fref{<label>}"&Figure \verb"<num>"\\
\verb"\sref{<label>}"&section \verb"<num>"\\
\verb"\Sref{<label>}"&Section \verb"<num>"\\
\verb"\tref{<label>}"&table \verb"<num>"\\
\verb"\Tref{<label>}"&Table \verb"<num>"\\
\br
\endTable

\section{Tables and table captions}
Tables are numbered serially and referred to in the text
by number (table 1, etc, {\bf not} tab. 1). Each table should have an
explanatory caption which should be as concise as possible. If a table
is divided into parts these should be labelled \pt(a), \pt(b),
\pt(c), etc but there should be only one caption for the whole
table, not separate ones for each part.

In the preprint style the tables may be included in the text
or listed separately after the reference list
starting on a new page.

\subsection{The basic table format}
The standard form for a table is:
\small\begin{verbatim}
\begin{table}
\caption{\label{label}Table caption.}
\begin{indented}
\item[]\begin{tabular}{@{}llll}
\br
Head 1&Head 2&Head 3&Head 4\\
\mr
1.1&1.2&1.3&1.4\\
2.1&2.2&2.3&2.4\\
\br
\end{tabular}
\end{indented}
\end{table}
\end{verbatim}\normalsize

Points to note are:
\begin{enumerate}
\item The caption comes before the table. It should have a period at
the end.

\item Tables are normally set in a smaller type than the text.
The normal style is for tables to be indented. This is accomplished
by using \verb"\begin{indented}" \dots\ \verb"\end{indented}"
and putting \verb"\item[]" before the start of the tabular environment.
Omit these
commands for any tables which will not fit on the page when indented.

\item The default is for columns to be aligned left and
adding \verb"@{}" omits the extra space before the first column.

\item Tables have only horizontal rules and no vertical ones. The rules at
the top and bottom are thicker than internal rules and are set with
\verb"\br" (bold rule).
The rule separating the headings from the entries is set with
\verb"\mr" (medium rule).

\item Numbers in columns should be aligned on the decimal point;
to help do this a control sequence \verb"\lineup" has been defined
which sets \verb"\0" equal to a space the size of a digit, \verb"\m"
to be a space the width of a minus sign, and \verb"\-" to be a left
overlapping minus sign. \verb"\-" is for use in text mode while the other
two commands may be used in maths or text.
(\verb"\lineup" should only be used within a table
environment after the caption so that \verb"\-" has its normal meaning
elsewhere.) See table~\ref{tabone} for an example of a table where
\verb"\lineup" has been used.
\end{enumerate}

\begin{table}
\caption{\label{tabone}A simple example produced using the standard table commands
and $\backslash${\tt lineup} to assist in aligning columns on the
decimal point. The width of the
table and rules is set automatically by the
preamble.}

\begin{indented}
\lineup
\item[]\begin{tabular}{@{}*{7}{l}}
\br
$\0\0A$&$B$&$C$&\m$D$&\m$E$&$F$&$\0G$\cr
\mr
\0\023.5&60  &0.53&$-20.2$&$-0.22$ &\01.7&\014.5\cr
\0\039.7&\-60&0.74&$-51.9$&$-0.208$&47.2 &146\cr
\0123.7 &\00 &0.75&$-57.2$&\m---   &---  &---\cr
3241.56 &60  &0.60&$-48.1$&$-0.29$ &41   &\015\cr
\br
\end{tabular}
\end{indented}
\end{table}

\subsection{Simplified coding and extra features for tables}
The basic coding format can be simplified using extra commands provided in
the \verb"iopart" class file. The commands up to and including
the start of the tabular environment
can be replaced by
\small\begin{verbatim}
\Table{\label{label}Table caption}
\end{verbatim}\normalsize
this also activates the definitions within \verb"\lineup".
The final three lines can also be reduced to \verb"\endTable" or
\verb"\endtab". Similarly for a table which does not fit in when indented
\verb"\fulltable{\label{label}caption}" \dots\ \verb"\endfulltable" or \verb"\endtab"
can be used. \LaTeX\ optional positional parameters can, if desired, be added after
\verb"\Table{\label{label}caption}" and \verb"\fulltable{\label{label}caption}".


\verb"\centre{#1}{#2}" can be used to centre a heading
\verb"#2" over \verb"#1"
columns and \verb"\crule{#1}" puts a rule across
\verb"#1" columns. A negative
space \verb"\ns" is usually useful to reduce the space between a centred
heading and a centred rule. \verb"\ns" should occur immediately after the
\verb"\\" of the row containing the centred heading (see code for
\tref{tabl3}). A small space can be
inserted between rows of the table
with \verb"\ms" and a half line space with \verb"\bs"
(both must follow a \verb"\\" but should not have a
\verb"\\" following them).

\Table{\label{tabl3}A table with headings spanning two columns and containing notes.
To improve the
visual effect a negative skip ($\backslash${\tt ns})
has been put in between the lines of the
headings. Commands set-up by $\backslash${\tt lineup} are used to aid
alignment in columns. $\backslash${\tt lineup} is defined within
the $\backslash${\tt Table} definition.}
\br
&&&\centre{2}{Separation energies}\\
\ns
&Thickness&&\crule{2}\\
Nucleus&(mg\,cm$^{-2}$)&Composition&$\gamma$, n (MeV)&$\gamma$, 2n (MeV)\\
\mr
$^{181}$Ta&$19.3\0\pm 0.1^{\rm a}$&Natural&7.6&14.2\\
$^{208}$Pb&$\03.8\0\pm 0.8^{\rm b}$&99\%\ enriched&7.4&14.1\\
$^{209}$Bi&$\02.86\pm 0.01^{\rm b}$&Natural&7.5&14.4\\
\br
\end{tabular}
\item[] $^{\rm a}$ Self-supporting.
\item[] $^{\rm b}$ Deposited over Al backing.
\end{indented}
\end{table}

Units should not normally be given within the body of a table but
given in brackets following the column heading; however, they can be
included in the caption for long column headings or complicated units.
Where possible tables should not be broken over pages.
If a table has related notes these should appear directly below the table
rather than at the bottom of the page. Notes can be designated with
footnote symbols (preferable when there are only a few notes) or
superscripted small roman letters. The notes are set to the same width as
the table and in normal tables follow after \verb"\end{tabular}", each
note preceded by \verb"\item[]". For a full width table \verb"\noindent"
should precede the note rather than \verb"\item[]". To simplify the coding
\verb"\tabnotes" can, if desired, replace \verb"\end{tabular}" and
\verb"\endtabnotes" replaces
\verb"\end{indented}\end{table}".

If all the tables are grouped at the end of a document
the command \verb"\Tables" is used to start a new page and
set a heading `Tables and table captions'. If the tables follow an appendix then add the command \verb"\noappendix" to revert to normal style numbering.

\section{Figures and figure captions}

Figures (with their captions) can be incorporated into the text at the appropriate position or grouped together
at the end of the article. If the figures are at the end of the article and follow an appendix then add the command \verb"\noappendix" to revert to normal style numbering.

\subsection{Using copyright material}
If you wish to illustrate your paper using material for which you
do not own the copyright then you must seek permission from
the copyright holder, usually both the author and the publisher of the previous work. It is the author's responsibility to obtain
copyright permissions and this should be done
prior to submitting your paper.
If you have obtained permission, please provide full details of
the permission granted---for example, copies of the text of any e-mails
or a copy of any letters you may have received. Figure captions must include an acknowledgment of the original source of the material even when permission to use has been obtained.

\subsection{Inclusion of graphics files\label{figinc}}
Using the \verb"graphicx" package graphics files can
be included within figure and center environments at an
appropriate point within the text using code such as:
\small\begin{verbatim}
\includegraphics{file.eps}
\end{verbatim}\normalsize
The \verb"graphicx" package supports various optional arguments
to control the appearance of the figure. Other similar
packages can also be used (e.g. \verb"graphics", \verb"epsf").
For more detail about graphics inclusion see the documentation
of the \verb"graphicx" package, refer to one of the books on \LaTeX\ {\cite{book1}}
or download some of the excellent free documentation available via the Comprehensive
TeX Archive Network (CTAN) http://www.ctan.org---{in particular see \cite{eps}}.
IOP's graphics guidelines, \gguide, provide further information on preparing EPS files---a copy
should have accompanied this document but it may be download from authors.iop.org.

\subsection{Captions}
Each figure should have a brief caption describing it and, if
necessary, interpreting the various lines and symbols on the figure.
As much lettering as possible should be removed from the figure itself
and included in the caption. If a figure has parts, these should be
labelled ($a$), ($b$), ($c$), etc and all parts should be described
within a single caption. \Tref{blobs} gives the definitions for describing
symbols and lines often used within figure captions (more symbols are
available when using the optional packages loading the AMS extension fonts).

\subsection{Multimedia}
All of our journals encourage authors to submit multimedia attachments to
enhance the online versions of published research articles. Multimedia
enhancements typically consist of video clips, animations or supplementary
data such as data files, tables of extra information or extra figures. They can
add to the reader's understanding and present results in attractive ways that go
beyond what can be presented in the print version of the journal.
The printed journal remains the archival version, and multimedia items are
supplements which enhance a reader's understanding of the paper but are not
essential to that understanding. For electronic-only journals, multimedia attachments
may be used to convey essential information. Guidelines on multimedia file formats
are contained in the document {\it Guidelines for authors} which can be accessed
online by going to authors.iop.org.

\begin{table}[t]
\caption{\label{blobs}Control sequences to describe lines and symbols in figure
captions.}
\begin{indented}
\item[]\begin{tabular}{@{}lllll}
\br
Control sequence&Output&&Control sequence&Output\\
\mr
\verb"\dotted"&\dotted        &&\verb"\opencircle"&\opencircle\\
\verb"\dashed"&\dashed        &&\verb"\opentriangle"&\opentriangle\\
\verb"\broken"&\broken&&\verb"\opentriangledown"&\opentriangledown\\
\verb"\longbroken"&\longbroken&&\verb"\fullsquare"&\fullsquare\\
\verb"\chain"&\chain          &&\verb"\opensquare"&\opensquare\\
\verb"\dashddot"&\dashddot    &&\verb"\fullcircle"&\fullcircle\\
\verb"\full"&\full            &&\verb"\opendiamond"&\opendiamond\\
\br
\end{tabular}
\end{indented}
\end{table}

\clearpage

\appendix
\section{List of macros for formatting text, figures and tables}

\begin{table}[hb]
\caption{Macros available for use in text. Parameters in square brackets are optional.}
\footnotesize\rm
\begin{tabular}{@{}*{7}{l}}
\br
Macro name&Purpose\\
\mr
\verb"\title[#1]{#2}"&Title of article and short title (optional)\\
\verb"\paper[#1]{#2}"&Title of paper and short title (optional)\\
\verb"\letter{#1}"&Title of Letter to the Editor\\
\verb"\comment[#1]{#2}"&Title of Comment and short title (optional)\\
\verb"\topical[#1]{#2}"&Title of Topical Review and short title
(optional)\\
\verb"\review[#1]{#2}"&Title of review article and short title (optional)\\
\verb"\note[#1]{#2}"&Title of Note and short title (optional)\\
\verb"\prelim[#1]{#2}"&Title of Preliminary Communication \& short title\\
\verb"\author{#1}"&List of all authors\\
\verb"\article[#1]{#2}{#3}"&Type and title of other articles and
short title (optional)\\
\verb"\address{#1}"&Address of author\\
\verb"\pacs{#1}"&PACS classification codes\\
\verb"\pacno{#1}"&Single PACS classification code\\
\verb"\ams{#1}"&Mathematics Classification Scheme\\
\verb"\submitto{#1}"&`Submitted to' message\\
\verb"\maketitle"&Creates title page\\
\verb"\begin{abstract}"&Start of abstract\\
\verb"\end{abstract}"&End of abstract\\
\verb"\nosections"&Inserts space before text when no sections\\
\verb"\section{#1}"&Section heading\\
\verb"\subsection{#1}"&Subsection heading\\
\verb"\subsubsection{#1}"&Subsubsection heading\\
\verb"\appendix"&Start of appendixes\\
\verb"\ack"&Acknowledgments heading\\
\verb"\References"&Heading for reference list\\
\verb"\begin{harvard}"&Start of alphabetic reference list\\
\verb"\end{harvard}"&End of alphabetic reference list\\
\verb"\begin{thebibliography}{#1}"&Start of numeric reference list\\
\verb"\end{thebibliography}"&End of numeric reference list\\
\verb"\etal"&\etal for text and reference lists\\
\verb"\nonum"&Unnumbered entry in numerical reference list\\
\br
\end{tabular}
\end{table}


\begin{table}
\caption{Macros defined within {\tt iopart.cls}
for use with figures and tables.}
\begin{indented}
\item[]\begin{tabular}{@{}l*{15}{l}}
\br
Macro name&Purpose\\
\mr
\verb"\Figures"&Heading for list of figure captions\\
\verb"\Figure{#1}"&Figure caption\\
\verb"\Tables"&Heading for tables and table captions\\
\verb"\Table{#1}"&Table caption\\
\verb"\fulltable{#1}"&Table caption for full width table\\
\verb"\endTable"&End of table created with \verb"\Table"\\
\verb"\endfulltab"&End of table created with \verb"\fulltable"\\
\verb"\endtab"&End of table\\
\verb"\br"&Bold rule for tables\\
\verb"\mr"&Medium rule for tables\\
\verb"\ns"&Small negative space for use in table\\
\verb"\centre{#1}{#2}"&Centre heading over columns\\
\verb"\crule{#1}"&Centre rule over columns\\
\verb"\lineup"&Set macros for alignment in columns\\
\verb"\m"&Space equal to width of minus sign\\
\verb"\-"&Left overhanging minus sign\\
\verb"\0"&Space equal to width of a digit\\
\br
\end{tabular}
\end{indented}
\end{table}

\clearpage
\section{Control sequences for journal
abbreviations}
\label{jlabs}
\begin{table}[hb]
\caption{\label{jlab1}Abbreviations for the IOP journals.}
\begin{indented}
\item[]
\begin{tabular}{@{}lll}
\br
Macro name&{\rm Short form of journal title}&Years relevant\\
\mr
{\it Current journals}\\
\mr
\verb"\CQG"&Class. Quantum Grav.\\
\verb"\EJP"&Eur. J. Phys.\\
\verb"\IP"&Inverse Problems\\
\verb"\JHEP"&J. High Energy Phys.&1999 and onwards\\
\verb"\JMM"&J. of Michromech. and Microeng.\\
\verb"\JNE"&J. Neural Eng.\\
\verb"\JOA"&J. Opt. A: Pure and Applied Opt.&1998 and onwards\\
\verb"\JOB"&J. Opt. B: Quantum and Semiclass. Opt.&1999 and onwards\\
\verb"\JPA"&J. Phys. A: Math. Gen.\\
\verb"\jpb"&J. Phys. B: At. Mol. Opt. Phys.&1988 and onwards\\
\verb"\JPCM"&J. Phys: Condens. Matter&1989 and onwards\\
\verb"\JPD"&J. Phys. D: Appl. Phys.\\
\verb"\jpg"&J. Phys. G: Nucl. Part. Phys.&1989 and onwards\\
\verb"\MSMSE"&Modelling Simul. Mater. Sci. Eng.\\
\verb"\MST"&Meas. Sci. Technol.&1990 and onwards\\
\verb"\NJP"&New J. Phys.&1999 and onwards\\
\verb"\NL"&Nonlinearity\\
\verb"\NT"&Nanotechnology\\
\verb"\PB"&Phys. Biol.\\
\verb"\PM"&Physiol. Meas.\\
\verb"\PMB"&Phys. Med. Biol.\\
\verb"\PPCF"&Plasma Physics and Controlled Fusion\\
\verb"\PSST"&Plasma Sources Sci. Technol.\\
\verb"\RPP"&Rep. Prog. Phys.\\
\verb"\SST"&Semicond. Sci. Technol.\\
\verb"\SMS"&Smart Mater. Struct.\\
\verb"\SUST"&Supercond. Sci. Technol.\\
\mr
{\it No Longer Published}\\
\mr
\verb"\JPB"&J. Phys. B: At. Mol. Phys.&1968--1987\\
\verb"\JPC"&J. Phys. C: Solid State Phys.&1968--1988\\
\verb"\JPE"&J. Phys. E: Sci. Instrum.&1968--1989\\
\verb"\JPF"&J. Phys. F: Met. Phys.&1971--1988\\
\verb"\JPG"&J. Phys. G: Nucl. Phys.&1975--1988\\
\verb"\PAO"&Pure and Applied Opt.&1992--1998\\
\verb"\QO"&Quantum Opt.&1989--1994\\
\verb"\QSO"&Quantum and Semiclass. Opt.&1995--1998\\
\br
\end{tabular}
\end{indented}
\end{table}

\begin{table}

\caption{\label{jlab2}Abbreviations for some more
common non-IOP journals.}
\begin{indented}
\item[]\begin{tabular}{@{}ll}
\br
Macro name&{\rm Short form of journal}\\
\mr
\verb"\AC"&Acta Crystallogr.\\
\verb"\AM"&Acta Metall.\\
\verb"\AP"&Ann. Phys., Lpz\\
\verb"\APNY"&Ann. Phys., NY\\
\verb"\APP"&Ann. Phys., Paris\\
\verb"\CJP"&Can. J. Phys.\\
\verb"\GRG"&Gen. Rel. Grav.\\
\verb"\JAP"&J. Appl. Phys.\\
\verb"\JCP"&J. Chem. Phys.\\
\verb"\JJAP"&Japan. J. Appl. Phys.\\
\verb"\JMMM"&J. Magn. Magn. Mater.\\
\verb"\JMP"&J. Math. Phys.\\
\verb"\JOSA"&J. Opt. Soc. Am.\\
\verb"\JP"&J. Physique\\
\verb"\JPhCh"&J. Phys. Chem.\\
\verb"\JPSJ"&J. Phys. Soc. Japan\\
\verb"\JQSRT"&J. Quant. Spectrosc. Radiat. Transfer\\
\verb"\NC"&Nuovo Cimento\\
\verb"\NIM"&Nucl. Instrum. Methods\\
\verb"\NP"&Nucl. Phys.\\
\verb"\PF"&Phys. Fluids\\
\verb"\PL"&Phys. Lett.\\
\verb"\PR"&Phys. Rev.\\
\verb"\PRL"&Phys. Rev. Lett.\\
\verb"\PRS"&Proc. R. Soc.\\
\verb"\PS"&Phys. Scr.\\
\verb"\PSS"&Phys. Status Solidi\\
\verb"\PTRS"&Phil. Trans. R. Soc.\\
\verb"\RMP"&Rev. Mod. Phys.\\
\verb"\RSI"&Rev. Sci. Instrum.\\
\verb"\SSC"&Solid State Commun.\\
\verb"\SPJ"&Sov. Phys.--JETP\\
\verb"\ZP"&Z. Phys.\\
\br
\end{tabular}
\end{indented}
\end{table}

\section*{References}
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\end{document}

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\end{thebibliography}