Spiral chain models of two-dimensional turbulence

Reduced models, mirroring self-similar, fractal nature of two-dimensional turbulence, are proposed, using logarithmic spiral chains, which provide a natural generalization of shell models to two dimensions. In a turbulent cascade, where each step can be represented by a rotation and a scaling of the interacting triad, the use of a spiral chain whose nodes can be obtained by scaling and rotating an original wave vector provides an interesting perspective. A family of such spiral chain models depending on the distance of interactions can be obtained by imposing a logarithmic spiral grid with a constant divergence angle and a constant scaling factor and imposing the condition of exact triadic interactions. Scaling factors in such sequences are given by the square roots of known ratios such as the plastic ratio, the super-golden ratio, or some small Pisot numbers. While spiral chains can represent monofractal models of a self-similar cascade, which can span a large range of wave numbers and have 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 the usual problems of shell models of two-dimensional turbulence such as the dual cascade being overwhelmed by statistical chain equipartition due to an 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.

Figure 1

The triad ${▵}_1$ defined as $\ell =1$, $m=2$, $g=\sqrt{\phi }$. Scaling ${▵}_1$ by $g$ and rotating by ${\alpha }_{qp}=141.{83}^0$, we obtain ${▵}_2$. Scaling ${▵}_2$ by $g$ and rotating by $\pi /2$, we obtain ${▵}_3$. Note that the three triads share the common wave vector ${\mathbf{q}}_1={\mathbf{p}}_2={\mathbf{k}}_3$, which we can call ${\mathbf{k}}_n$. The energy inverse cascades via ${\mathbf{p}}_3\to {\mathbf{k}}_n\to {\mathbf{k}}_2$ (blue arrows, pointing from ${\mathbf{p}}_i$'s to ${\mathbf{k}}_i$'s), while enstrophy forward cascades via ${\mathbf{p}}_1\to {\mathbf{k}}_n\to {\mathbf{q}}_2$ (red arrows, pointing from ${\mathbf{p}}_i$'s to ${\mathbf{q}}_i$'s).

Figure 2

The spiral chain $\ell =2$, $m=3$ with $g=\sqrt{\rho }$. The counterclockwise primary spiral chain is shown by black dashed lines, while the clockwise secondary spirals are shown by blue dashed lines. Note that as the energy travels along the primary chain, it gets exchanged between the five secondary chains. Finally an interacting triad with $\mathbf{k}={\mathbf{k}}_n$ (black arrow, pointing right), $\mathbf{p}={\mathbf{k}}_{n-2}$ (red arrow, pointing up), and $\mathbf{q}={\mathbf{k}}_{n+1}$ (blue arrow, pointing left) is shown (i.e., $\mathbf{k}+\mathbf{q}-\mathbf{p}=0$).

Figure 3

The four-spiral chain grid shown explicitly. The original spiral chain is shown by black squares, while its reflection with respect to the origin is shown by red squares (if in color). The full system is symmetric with respect to reflection $\mathbf{k}\to -\mathbf{k}$, and therefore one can actually use only half of the $k$-plane (e.g., the upper half) and obtain the rest of the points by reflection.

Figure 4

Wave-number spectra for the two variants of the $\ell =1$, $m=3$ spiral chain model. The red line (if in color) is the model for the complex amplitude ${\mathrm{\Phi }}_n$, whereas the black line is the model for $E_n$. While the model for $E_n$ is driven with constant forcing $P_n=2.5\times {10}^{-4}$, the model for ${\mathrm{\Phi }}_n$ is driven with random forcing such that $〈P_n〉=2.5\times {10}^{-4}$. The spectrum for the ${\mathrm{\Phi }}_n$ model is averaged over a long stationary phase, where $E\left(k_n\right)=〈{\left|{\mathrm{\Phi }}_n\right|}^2〉k_n$, which is integrated up to $t=10000$ and the average is computed over $t=\left[5000,10000\right]$, whereas the spectrum for the $E_n$ model is averaged over $t=\left[190,200\right]$ (in fact the instantaneous solution is not that different from the averaged result).

Figure 5

Energy and enstrophy fluxes for the two variants of the $\ell =1$, $m=3$ spiral chain model. The red solid and dashed lines (if in color) are the energy and enstrophy fluxes for the complex amplitude model, whereas the black solid and dashed lines are the energy and enstrophy fluxes for the $E_n$ model, respectively, normalized to their maximum values. We can see that rapid oscillations of the phases observed in the complex model cause the suppression of the fluxes and result in statistical chain equipartition solutions instead of proper dual cascade solutions.

Figure 6

Two-dimensional “log-log,” i.e., $\left\{{log}_{10}\left(k\right)cos\left({\theta }_k\right),{log}_{10}\left(k\right)sin\left({\theta }_k\right),{log}_{10}\left[E\left(k\right)\right]\right\}$, plot of the wave-number spectrum for the four-spiral chain model discussed in Sec. 4b. The energy injection is located around $k_x=0$, $k_y=\pm 2\times {10}^3$, shown as black $\times$'s. The resulting spectrum consists of a clear inverse energy cascade range of $E\left(k\right)\propto k^{-5/3}$ (the red central region) and a forward enstrophy cascade range of $E\left(k\right)\propto k^{-3}$ (the blue peripheral region). A one-dimensional spectrum, which can be obtained by plotting $\mathrm{E}\left(k_n\right)=E_n/k_n$ as a function of $k_n=\left|{\mathbf{k}}_n\right|$, is also shown with guiding lines showing the theoretical predictions.

Figure 7

Time evolution of the one-dimensional $k$-spectrum, showing how it gets established in time in an asymmetric nonlinear diffusion where the small scales are rapidly filled while large scales take a while to populate. Here the colors show different levels of $E\left(k\right)$, where as before the red region between $log\left(k\right)=0$ and 3 corresponds to the inverse cascade and the blue region between 4 and 6 corresponds to the forward cascade region as in Fig. 6.

Figure 8

Energy and enstrophy fluxes ${\mathrm{\Pi }}_n^E$ and ${\mathrm{\Pi }}_n^W$, normalized to their maximum values, for the four-spiral chain model. This is averaged over 10 time steps, but even instantaneously, they are extremely flat and stationary.

Figure 9

Snapshot of the stream function $\mathrm{\Phi }\left(\mathbf{x},t\right)$ obtained from the spiral chain model with $\ell =1$, $m=3$ for the evolution of $E_n$. The fractal structure that we observe consist of discrete logarithmic spirals, where each point of the spiral is a spiral in itself centered around that point. The weights are such that in the end we obtain a $k^{-5/3}$ spectrum for energy.

Figure 10

Snapshot of the vorticity field $\omega \left(\mathbf{x},t\right)$ obtained from the spiral chain model with $\ell =1$, $m=3$ for the evolution of $E_n$. Low-pass filtered vorticity field is shown at the top left plot. The box in the center (which is 1/10 of the original box) is then expanded to show the band-passed filtered vorticity field on the top right. The box in the center of this plot is then expanded to show the band-pass filtered vorticity field on the bottom left and so on. The fractal structure, which consists of discrete logarithmic spirals, where each point of the spiral is a spiral centered around that point, is now visible mainly at the smallest spatial scales (lower right plot). This is actually the other end of the same fractal structure visible for the large scales in Fig. 9 for the stream function. The weights of the fractal form at those smaller scales are such that we have a $k^{-3}$ spectrum for energy.