  Abstract Self-similar, fractal nature of turbulence is discussed in the
context of two dimensional turbulence, by considering the fractal structure of
the wave-number domain using spirals. In loose analogy with phyllotaxis in
plants, each step of the cascade can be represented by a rotation and a scaling
of the interacting triad. Using a constant divergence angle and a constant
scaling factor, one obtains a family of such fractals depending on the distance
of interactions. Scaling factors in such sequences are given by the square
roots of known ratios such as the plastic ratio, the supergolden ratio or some
small Pisot numbers. While spiral chains can represent mono-fractal models of
self-similar cascade, which can span a very large range in wave-number domain
with good angular coverage, it is also possible that spiral chains or chains of
consecutive triads play an important role in the cascade. As numerical models,
the spiral chain models based on decimated Fourier coefficients have problems
such as the dual cascade being overwhelmed by statistical chain equipartition
due to almost stochastic evolution of the complex phases. A generic spiral
chain model based on evolution of energy is proposed, which is shown to recover
the dual cascade behavior in two-dimensional turbulence.
