Concentration of Stationary Measures onto Large Scales in the Fast-Advection Limit of the Stochastically Forced Two-Dimensional Navier-Stokes Equations
Abstract
We investigate the fast-advection limit (ε → 0) of the stochastically forced two-dimensional incompressible Navier-Stokes equations on the torus. Numerical simulations are performed at resolution N = 128 for four different forcing shells on both the square torus and a thin torus, with the advection parameter ε varied over approximately one decade. We find that the stationary distributions become increasingly concentrated on the lowest Fourier modes as ε → 0, with the energy-to-enstrophy ratio E/Ω and the fraction R of enstrophy in the lowest modes both increasing monotonically. At finite ε the degree of concentration depends on the forcing shell through the effective spectral value B1/B0, in agreement with the condensation bound of Sznitman and Widmayer. Linear extrapolation to ε = 0 suggests that the limiting values may be forcing-independent, with B1/B0 governing only the rate of convergence. The same qualitative behaviour persists on the thin torus.
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BibTeX
@misc{rohner2026concentration,
title = {{Concentration of Stationary Measures onto Large Scales in the Fast-Advection Limit of the Stochastically Forced Two-Dimensional Navier-Stokes Equations}},
author = {Tobias Rohner},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27512},
eprint = {2609.27512},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27512},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Concentration of Stationary Measures onto Large Scales in the Fast-Advection Limit of the Stochastically Forced Two-Dimensional Navier-Stokes Equations
AU - Tobias Rohner
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27512
UR - https://arxiv.org/abs/2609.27512
AB - We investigate the fast-advection limit (ε → 0) of the stochastically forced two-dimensional incompressible Navier-Stokes equations on the torus. Numerical simulations are performed at resolution N = 128 for four different forcing shells on both the square torus and a thin torus, with the advection parameter ε varied over approximately one decade. We find that the stationary distributions become increasingly concentrated on the lowest Fourier modes as ε → 0, with the energy-to-enstrophy ratio E/Ω and the fraction R of enstrophy in the lowest modes both increasing monotonically. At finite ε the degree of concentration depends on the forcing shell through the effective spectral value B_1/B_0, in agreement with the condensation bound of Sznitman and Widmayer. Linear extrapolation to ε = 0 suggests that the limiting values may be forcing-independent, with B_1/B_0 governing only the rate of convergence. The same qualitative behaviour persists on the thin torus.
ER -