Two-dimensional turbulent condensates without bottom drag
Abstract
The extent to which equilibrium statistical theory is applicable to driven dissipative dynamics remains an important open question in many systems. We use extensive direct numerical simulations (DNS) of the incompressible two-dimensional (2D) Navier–Stokes equation to examine the steady state of large-scale condensates in 2D turbulence at finite Reynolds number Re in the absence of bottom drag. Large-scale condensates appear above a critical Reynolds number Rec≈4.19. Close to this onset, we find a power-law scaling of the energy with Re-Rec, with the energy spectrum at large scales following the absolute equilibrium form proposed by Kraichnan. At larger Re, the energy spectrum deviates from this form, displaying a steep power-law range at low wavenumbers with exponent -5, with most of the energy dissipation occurring within the condensate at large scales. We show that this spectral exponent is consistent with the logarithmic radial vorticity profile of the condensate vortices predicted by quasi-linear theory for a viscously saturated condensate. Our findings shed new light on the classical problem of large-scale turbulent condensation in forced dissipative 2D flows in finite domains, showing that the large scales are close to equilibrium dynamics in weakly turbulent flows but not in the strong condensate regime with Re≫1. This article is part of the theme issue ‘Frontiers of turbulence and statistical physics’.
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BibTeX
@article{kan2026twodimensional,
title = {{Two-dimensional turbulent condensates without bottom drag}},
author = {Adrian van Kan and Alexandros Ens Alexakis and Edgar Knobloch},
journal = {Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences},
year = {2026},
month = sep,
doi = {10.1098/rsta.2025.0019},
eprint = {2504.02978},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05726555v1},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Two-dimensional turbulent condensates without bottom drag
AU - Adrian van Kan
AU - Alexandros Ens Alexakis
AU - Edgar Knobloch
PY - 2026
DA - 2026/09/04
JO - Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
DO - 10.1098/rsta.2025.0019
UR - https://hal.science/hal-05726555v1
AB - The extent to which equilibrium statistical theory is applicable to driven dissipative dynamics remains an important open question in many systems. We use extensive direct numerical simulations (DNS) of the incompressible two-dimensional (2D) Navier–Stokes equation to examine the steady state of large-scale condensates in 2D turbulence at finite Reynolds number Re in the absence of bottom drag. Large-scale condensates appear above a critical Reynolds number Rec≈4.19. Close to this onset, we find a power-law scaling of the energy with Re-Rec, with the energy spectrum at large scales following the absolute equilibrium form proposed by Kraichnan. At larger Re, the energy spectrum deviates from this form, displaying a steep power-law range at low wavenumbers with exponent -5, with most of the energy dissipation occurring within the condensate at large scales. We show that this spectral exponent is consistent with the logarithmic radial vorticity profile of the condensate vortices predicted by quasi-linear theory for a viscously saturated condensate. Our findings shed new light on the classical problem of large-scale turbulent condensation in forced dissipative 2D flows in finite domains, showing that the large scales are close to equilibrium dynamics in weakly turbulent flows but not in the strong condensate regime with Re≫1. This article is part of the theme issue ‘Frontiers of turbulence and statistical physics’.
SN - 1364-503X
SN - 1471-2962
ER -