Conservation laws

The main difference with the three-dimensional case is the conservation of vorticity along fluid trajectories when viscosity, friction and external forcing are ignored.

The origin of this phenomenon is due to the vanishing in two dimensions of the so called ``vortex stretching term'' $({\mbox{\boldmath$\omega$}} \cdot \nabla) {\mbox{\boldmath$v$}}$ that appears as a forcing term in the evolution equation for vorticity in the three-dimensional case where it is responsible for the unbounded growth of enstrophy in the limit $Re \to \infty$.

In the inviscid limit $\nu =0$ and in absence of external forcing and friction, the vorticity equations simply states that the derivative of the vorticity along the fluid trajectories vanishes

$${D \omega \over D t } = {\partial \omega \over \partial t} + {\mbox{\boldmath$u$}} \cdot \nabla \omega = 0\;$$ (1.61)

which means that since the vortex stretching is absent the vorticity of a fluid parcel is conserved. Thus all the integrals of the form $\int f(\omega) d {\mbox{\boldmath$r$}}$ are inviscid invariants of the flow. In particular this properties yields to the conservation of the circulation $\Gamma$ defined as

$$\Gamma = \int_D \omega d^2 r = \oint_{\partial D} {\mbox{\b... ...ath$u$}}({\mbox{\boldmath$r$},t}) \cdot d{\mbox{\boldmath$s$}}$$ (1.62)

where $d{\mbox{\boldmath$s$}}$ denotes the length of an infinitesimal element of the boundary $\partial D$, and the total enstrophy

$$Z = \int d^2 r {1\over 2} \rho \vert\omega\vert^2$$ (1.63)

In two dimension the enstrophy is bounded by the energy balance equation, which is obtained from Eq. (1.60) in absence of external forcing $f = 0$ and friction $\alpha = 0$, and assuming periodic boundary conditions:

$${dZ \over dt} = - \nu \int d^2 r \rho \vert{\mbox{\boldmath$\nabla$}}\omega\vert^2$$ (1.64)

Therefore, at variance with the three-dimensional case, in two-dimensional turbulence the viscous dissipation of energy vanishes in the limit $\nu \to 0$

$$\lim_{\nu \to 0} {dE \over dt} = \lim_{\nu \to 0} - 2 \nu Z = 0$$ (1.65)

whereas there is dissipative anomaly for enstrophy when friction is not considered. One of the effects of friction that will be discussed in next chapter is the regularization of the vorticity field: in presence of friction also the enstrophy dissipation vanishes in the limit of vanishingly small viscosity [19].

Since the viscous energy dissipation vanishes it the limit $Re \to \infty$, in fully developed two-dimensional turbulence it is impossible to have a cascade of energy with constant flux toward small scales.

Moreover, the presence of two quadratic inviscid invariant, the energy and the enstrophy, modifies the picture of the turbulent cascade. Dividing the wavenumber space into shells of modulus $k_n = k_0 2^n$ the triad interactions between wavenumbers which produce the energy cascade in three dimensions can be thought of as pair interactions between the $n$-th shell and the $(n+1)$-th one. This is inadmissible in two dimensions because pair interaction between two neighbor shells cannot transfer both energy and enstrophy conservatively between equal wave-numbers. In order for both energy and vorticity to be conserved the net transfer by each triad interaction must be out of the middle wavenumber into both smallest and largest wave-numbers.

Figure 1.3: Schematic double cascading spectrum of forced two-dimensional turbulence

Starting from the hint that the interactions should act toward producing equilibrium, a state which is never reached because of the viscous dissipation, Kraichnan showed that in two-dimensional turbulence the enstrophy is mainly transferred to high wave-numbers where it is dissipated by viscosity, giving rise to the direct enstrophy cascade. On the contrary, the energy is transported to lower wave-numbers in the inverse energy cascade.