Quantum Turbulence across Dimensions
Quantum turbulence (QT) is a disordered state of quantized vortices in superfluids and atomic Bose–Einstein condensates (BECs), offering a minimal and well-defined model of turbulence [1]. Its properties strongly depend on dimensionality.
1. Turbulence: 2D vs 3D
In three-dimensional (3D) classical and quantum turbulence, vortex stretching plays a central role, enabling a direct energy cascade from large to small scales (Richardson cascade), where energy is dissipated at short wavelengths. In contrast, two-dimensional (2D) turbulence lacks vortex stretching. Instead, both energy and enstrophy are conserved in the inviscid limit, leading to a dual-cascade scenario: an inverse energy cascade toward large scales and a direct enstrophy cascade toward small scales, as described by Kraichnan and Batchelor. The inverse cascade promotes the formation of large-scale coherent vortical structures, closely related to Onsager’s statistical theory of point vortices and negative-temperature states.
2. 2D Quantum Turbulence
In atomic BECs described by the Gross–Pitaevskii (GP) model, realizing ideal 2D turbulence is challenging. Vortex–antivortex pair annihilation breaks strict enstrophy conservation, compressibility introduces sound emission, and finite system size limits the inertial range. As a result, numerical and experimental studies often observe dominant direct cascades rather than clear inverse cascades. Nevertheless, under suitable conditions, vortex clustering and signatures of Onsager-like states have been reported, highlighting the interplay between vortex dynamics, dissipation, and finite-size effects in 2D QT.
3. Crossover across Dimensions
To clarify how QT changes across dimensions, we control the confinement height R2 in a cylindrical box potential within the GP framework. For small R2, the system shows 2D-like behavior: vortex–antivortex annihilation dominates and clustering consistent with inverse transfer develops. For large R2, Kelvin waves can be excited along vortex lines, inducing reconnections, phonon emission, and a direct energy cascade characteristic of 3D Vinen turbulence. The results demonstrate that the excitation of Kelvin waves, determined by the system height, governs the transition from 2D to 3D QT.
References:
[1] M. Tsubota and K. Kasamatsu, Quantum Hydrodynamics and Turbulence, Oxford University Press (2025).
[2] L. Onsager, Nuovo Cimento Suppl. 6, 279 (1949).
[3] R. H. Kraichnan, Phys. Fluids 10, 1417 (1967).
[4] G. K. Batchelor, Phys. Fluids Suppl. 12, II-233 (1969).
[5] W.-C. Yang, X. Wang, and M. Tsubota, arXiv:2502.06133 (2025).