We note that turbulence can arise in a wide range of systems governed by nonlinear field equations, not limited to hydrodynamic flows, with cascade phenomena serving as a hallmark of such turbulence. See, e.g., Ref. [16] for studies of turbulence in non-Abelian gauge theories. The turbulence driven by the nonlinear topological interaction investigated in this work falls within this broader class.

It should be noted that, at the quantum level, $T_{\varphi }(\mathcal{V})$ and $T_a(\mathcal{S})$ are not invariant under large gauge transformations. They can be made invariant under those transformations by rewriting them in terms of the topological quantum field theories (TQFTs), and the resulting symmetries become noninvertible [26, 27, 28, 33, 34]. However, as we do not consider the backgrounds with ’t Hooft lines or axionic vortices, this reformulation does not matter in the following discussion, and we write these topological quantities without introducing TQFTs.

We also have higher-form symmetries associated with the Bianchi identities ${\partial }_{\mu }{\tilde{f}}^{\mu \nu }=0$ and ${\epsilon }^{\mu \nu \rho \sigma }{\partial }_{\rho }{\partial }_{\sigma }\varphi =0$. Since these conservation laws do not involve the interaction terms, they do not contribute to the inverse cascade. Meanwhile, the presence of these symmetries implies the absence of dynamical magnetic monopoles and axionic strings.

While a full treatment within the Schwinger-Keldysh effective field theory framework [39, 40] would clarify the associated Kubo-Martin-Schwinger constraints and confirm the absence of other symmetry-allowed terms at the same order, such an analysis is beyond the scope of the present Letter. We therefore treat this term as a phenomenological assumption consistent with symmetry.

A similar approximation is employed in the analysis of the CPI in the presence of magnetic fields [42].

While this dimensional reduction simplifies the numerical analysis, we expect that the essential features of the inverse cascade should persist in the full ($3+1$)-dimensional case. This expectation is based on the general argument given earlier, where the interplay between dissipation and the conservation of the 1-form charge drives the transfer of the conserved quantity to larger spatial scales. This mechanism does not rely on the dimensional reduction and should apply in the full theory as well.