The broken link between space and time in elastic turbulence

Elastic turbulence (ET), observed in flows of sufficiently elastic polymer solution at small inertia, is characterized by chaotic motions and power-law scaling of energy spectrum ($E$) in both wavenumber ($k$) and frequency ($\omega$): $E(k) \sim k^{-\alpha}$ and $E(\omega) \sim \omega^{-\beta}$. Experiments of ET have obtained a vast range of values for the exponent $\beta$. In inertial turbulence, Taylor's frozen-flow hypothesis implies $\alpha = \beta$, i.e., spatial and temporal scales are linearly related to each other. In contrast, from high-resolution simulation in three different setups, a tri-periodic box, a channel, and a planar jet, we show that in ET $\alpha \approx 4$ while $\beta$ varies significantly. Our analysis shows that in general Taylor's hypothesis does not hold in ET as there is no universal relation, linear or otherwise, between space and time. We thus clear the confusion of the different scaling exponents found in ET, and focus the attention of future research on understanding $\alpha$. Our analysis also implies that waves-like dynamics with a linear dispersion relation (e.g., Alfv\'en waves) can not play a role in determining the scaling behavior of ET. The techniques introduced here can be useful for studying smooth chaotic flows in general, e.g., active turbulence.