数理解析研究所講究録
第 1842 巻 2013 年 67-68

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Role of helicity in turbulenceTakeshi MATSUMOTO1

Fluid physics lab., department of physics, Kyoto university.
The energy is a conservative quantity of the inviscid (dissipationless) fluid, which plays the mostimportant role in the theory of turbulence although turbulence is a dissipative system. The helicity is
another inviscid conservative quantity, which is defined as$H= \int_{D}u\cdot\omega dx$.

(1)

Here $D$ is the domain where the fluid exists and we assume that there is no helicity input or output on theboundary (such condition holds for the periodic boundary and ). The andare the velocity
and the vorticity. The conservation of the helicity was found rather recently in 1960 and its roles has notbeen explored perhaps due to its lack of $sign$ definiteness. We discuss here possible roles of the helicity inthe theory of turbulence [1].
We start to explain what we mean by the important role of the energy. The central outcome of aturbulence theory concerns the exponent of the turbulent velocity $increment^{2}$ :$\mathbb{R}^{3}$$\omega=\nabla\cross u$

$u$

$s$

$\alpha$$u(x+r, t)-u(x, t)\propto r^{\alpha}$,(2)where we ignore the vectorial property of the velocity , the spatial coordinate
and for simplicity.
Furthermore we assume the turbulence reaches statistically steady state. In 1941 Kolmogorov predicts
that $\alpha=1/3$ , namely$u(x+r, t)-u(x, t)\propto r^{1/3}$ ,(3)
$u$

$x$

$r$

based on the fact that the nonlinearity of the Navier-Stokes equations conserves the energy [2]. There isof course the viscous term, which decrease the energy, in the incompressible Navier-Stokes equations$\partial_{t}u+(u\cdot\nabla)u=-\frac{1}{\rho_{0}}\nabla p+\nu\nabla^{2}u, \nabla\cdot u=0$.

(4)

However in the well-developed turbulence, the viscous term is smaller by several orders of magnitude than
the nonlinear term in a suitable range of spatial scale , where the law Eq.(2) holds. So we focus on thenonlinearity only. In any case, the prediction Eq.(3) has been confirmed to be true by various experiments
and computer simulations of turbulence.
How one can get Eq.(3)? The nonlinearity of the Navier-Stokes equations actually distributes theenergy among all the modes3, The first assumption is that the distribution occurs in a simple manner: theenergy flux 4 associated with the wavevector $k=|k|$ becomes constant and independent of . The second
assumption is that the statistics of the turbulence velocity is determined by the constant energy flux
The velocity has the dimension of [Length $/Time$]. The energy flux has the dimension [Energy $/Time$ ]$=[Length^{2}/Time^{3}]$ . We now make the dimension of the velocity (the left hand side of Eq.(3)) by using
and . The only dimensionally consistent combination is$r$$k$$\epsilon.$$\epsilon$$\epsilon$$r$(5)$u(x+r, t)-u(x, t)=C\epsilon^{1/3}r^{1/3}$ ,where is a non-dimensional constant. This is a way to get the Kolmogorov law Eq.(3).
The above argument can be easily applied to other conservative quantities to obtain a possible exponentin Eq.(2). So why not the helicity. The nonlinearity indeed does not decrease or increase the helicity but
distribute it among all the modes. By assuming the helicity flux , whose dimension is [Length $/Time^{3}$ ],becomes constant and independent of the length scale, we get$C$$\alpha$$\eta$$u(x+r, t)-u(x, t)=C’\eta^{1/3}r^{2/3}$ .
1 takeshi@kyoryu.scphys.kyoto-u.acjp2The velocity increment in th left hand side is understood as a suitable (spatial [average in$e$

nsemble) average(6)$x$], temporal [average in ] or3Let us imagine we have the Fourier transforms of the turbulent velocity and we call each Fourier transform$t$$\hat{u}(k, t)$of theas “modes”.
4This energy flux is a sort of the area integral of the energy transfer vector dotted the unit normal vector $k/|k|$ on thesphere of $k=|k|$ in the wavenumber space. For the precise definition, see Sec.6.2.2 and 6.2.3 of Ref.[2].
wavevector

$k$

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The question is whether or not the law Eq.(6) can be observed in experiments or simulations. The answer
is no so far. Nevertheless the assumption, the helicity flux becoming constant, is validated by numericalsimulations.
reasonable explanation is that there is no pure helicity-dominated turbulence leading toEq.(6) since the energy and the helicity are simultaneously transfered (so-called joint cascade) by thenonlinearity 5. Hence in reality we can expect that$\eta$$A$$u(x+r, t)-u(x, t)=C”\epsilon^{\beta}\eta^{\gamma}r^{\delta}$.

(7)

But now the exponents are not dimensionally determined uniquely.
Or it may be that the naive velocity difference in the left hand sides should be modified to observe thehelicity-related scaling law Eq.(6). We need to come back to vectorial properties of the velocity field now,as
In fact the left hand side of the Kolmogorov law (5) holds for the velocity component$u_{L}$$u_{L}(x+r, t)-u_{L}(x, t)=C\epsilon^{1/3}|r|^{1/3}$

(8)

,

where
$u_{L}(x, t)=u(x, t) \cdot\frac{r}{|r|}, u_{T}(x, t)=u(x, t)-u(x, t)\cdot\frac{r}{|r|}$.

(9)

Probably a helicity-flux related scaling law such as Eq.(6) or (7) holds among suitable components ofThis may be seen by analyzing the isotropic form of the third order correlation functions of the velocity
field. Certainly a simple dimensional analysis \‘a la Kolmogorov is not able to capture this kind of subtle
geometrical structure. If the helicity or its related quantity is able to capture a geometrical structureotherwise impossible, it can be a useful quantity to rectify geometrical properties of the turbulent velocity
or vorticity field.
It has been known for a long time that turbulence spontaneously generates conspicuous geometricalstructures like vortex sheets and tubes, whose dynamical origin and impact on statistics are not wellunderstood [2]. Several properties of helicity suggest its usefulness. Firstly, the helicity density
measures how an instantaneous streamline (an integral curve of the velocity) is close to a right-hand screw(favorable in positive helicity) or a left-hand one (favorable in negative helicity). Secondly, in the caseof idealized vortex lines, the helicity measures knottedness of the vortex lines [1]. Lastly, change in thehelicity may be associated to a certain event which is called vorticity reconnection. With the presence ofthe viscosity, the vortex lines can touch and re-connect (without the viscosity, they cannot). When this
topological change of vortex lines occurs, the viscous term (the last term) in the helicity budget equation$u_{T}.$$u\cdot\omega$$\frac{d}{dt}[\int u\cdot\omega dx]+\int\nabla\cdot[u(u\cdot\omega)]dx=-\frac{1}{\rho 0}\int\nabla\cdot(p\omega)dx+\int\nabla\cdot[\omega\frac{|u|^{2}}{2}]dx$$- \nu[\int\nabla^{2}(u\cdot\omega)dx+\int.\sum_{j=1}^{3}(\frac{\partial u_{*}}{\partial x_{j}})(\frac{\partial\omega}{\partial x_{j}})dx]$.

(10)

can generate (not dissipate) the helicity [3] 6. We believe that the helicity is a good starting point tounderstand the relation between turbulence statistical law like Eq.(2) and dynamical events happening inturbulence.

References[1] For authoritative reviews on helicity, see e.g., H.K. Moffatt and A. Tsinober, Helicity in laminar andturbulent flow, Annual Reviews of Fluid Mechanics, 24, 281-312 (1992).

[2] See e.g., U. Frisch, Turbulence: the legacy of A.N. Kolmogorov, (Cambridge university press 1996).
[3] H. Aref and I. Zawadzki, Linking of vortex rings, Nature, 354, 50-53 (1991).

$\overline{5}$On the contrary there can beapure energy-dominated turbulence, where the helicity is zero and the helicity flux isnegligibly small.
6This means that the absolute value of the helicity can increase (as opposed to the energy can decrease) with dissipation
but we do not call it ae エシ イ”, ajoke due to Prof. Katayama$\overline{\tau}$