Estimates for the 2D Navier-Stokes equations: the effects of forcing

Mathematical estimates for the Navier-Stokes equations are traditionally expressed in terms of the Grashof number, which is a dimensionless measure of the magnitude of the forcing and hence a control parameter of the system. However, experimental measurements and statistical theories of turbulence are based on the Reynolds number. Thus, a meaningful comparison between mathematical and physical results requires a conversion of the mathematical estimates to a Reynolds-dependent form. In two dimensions, this was achieved under the assumption that the second derivative of the forcing is square integrable. Nonetheless, numerical simulations have shown that the phenomenology of turbulence is sensitive to the degree of regularity of the forcing. Therefore, we extend the available estimates for the energy and enstrophy dissipation rates as well as the attractor dimension to forcings in the Sobolev space of order $s$; i.e. forcings whose Fourier coefficients decay with the wavenumber $k$ faster than $k^{-s-1}$. We consider the range $-1\leqslant s\leqslant 2$, where $s=2$ corresponds to the known estimates, and $s=-1$ is the smallest value of $s$ for which weak solutions are known to exist. The main result is the existence of three distinct regimes as a function of the regularity of the forcing.

💡 Research Summary

The paper addresses a long‑standing gap between rigorous mathematical estimates for the two‑dimensional Navier–Stokes equations (NSE) and the quantities used by experimentalists and turbulence theorists. Traditionally, mathematical bounds are expressed in terms of the Grashof number (Gr), a nondimensional measure of the forcing magnitude. In contrast, physical turbulence studies are formulated with the Reynolds number (Re), which depends on the flow response (velocity or energy dissipation). Converting Gr‑based estimates to Re‑dependent forms is straightforward when the forcing is sufficiently smooth (its second derivative is square‑integrable, i.e., belongs to the homogeneous Sobolev space \dot H²). However, numerical simulations and renormalisation‑group (RG) analyses have shown that the regularity of the forcing dramatically influences turbulent statistics, especially when the forcing spectrum extends over a wide range of scales.

The authors extend existing Re‑dependent bounds for the time‑averaged energy dissipation rate ε, the enstrophy dissipation rate χ, and the fractal dimension of the global attractor d_f to forcings belonging to the Sobolev space \dot H^s with −1 ≤ s ≤ 2. The parameter s characterises how fast the Fourier coefficients of the forcing decay: coefficients must fall off faster than k^{−s−1}. The case s = 2 reproduces known results; s = −1 is the minimal regularity for which weak solutions of the 2‑D NSE are known to exist.

The main contribution is the identification of three distinct regimes, each associated with a different interval of s:

1. High regularity (s ≥ 2). This regime coincides with earlier work (Doering & Foias, Alexakis & Doering, Gibbon & Pavliotis). The bounds are

   • ε ℓ U³ ≤ Re^{−1/2}(c₁ + c₂ Re)^{1/2},
   • χ ℓ³ U³ ≤ c₁ + c₂ Re,
   • d_f ≤ c₀ Re (1 + ln Re)^{1/3}. These estimates are consistent with the Kraichnan–Leith–Batchelor (KLB) picture: enstrophy dissipates at a finite rate while energy dissipation vanishes as ν→0, and the attractor dimension scales like (ℓ/η_χ)².

2. Intermediate regularity (0 ≤ s < 2). Here the forcing is less smooth, and the Re‑dependence becomes more intricate. The authors derive

   • ε ∼ Re^{2−s}/(2 + s) · Re^{−s/(s+2)},
   • χ ∼ Re^{2−s}/(2 + s) · Re^{−s/(s+2)},
   • d_f ∼ Re^{(2−s)/(s+2)} (1 + ln Re)^{1/3}. As s decreases, ε and χ decay more slowly with Re, reflecting that a broader forcing spectrum injects energy over many scales, weakening the classic scale separation required for a clean inverse‑energy cascade and direct enstrophy cascade.

3. Low regularity (−1 ≤ s < 0). This is the most singular admissible forcing (only \dot H^{−1} regularity). The bounds become

   • ε ∼ Re^{−s} · (s + 2)^{−1} · Re^{−s/(s+2)},
   • χ ∼ Re^{−s} · (s + 2)^{−1} · Re^{−s/(s+2)},
   • d_f ∼ Re^{(2−s)/(s+2)} (1 + ln Re)^{1/3}. In this regime the forcing spectrum reaches into the viscous range, so the dissipation rates grow with Re rather than decay, and the attractor dimension increases sharply. This mirrors the three‑dimensional results of Cheskidov, Doering, and Petrov, where the loss of square‑integrability of the forcing leads to a “forcing‑dominated” turbulence with ε scaling as a power of Re.

The authors obtain these results by adapting the techniques used for three‑dimensional NSE with \dot H^s forcing (Cheskidov et al.) to the two‑dimensional vorticity equation. Key steps include:

• Using the vorticity formulation to exploit the two quadratic invariants (energy and enstrophy).
• Applying interpolation inequalities and Gagliardo–Nirenberg estimates to control the nonlinear advection term for arbitrary s.
• Deriving a Grashof‑to‑Re conversion that depends only on the forcing shape (the dimensionless field Φ) and not on other parameters.
• Translating the enstrophy bound into an upper bound for the attractor dimension via the Constantin–Foias–Temam framework.

Physical interpretation of the three regimes is emphasized. For s ≥ 2, the classic KLB picture holds: η_χ ∼ ℓ Re^{−1/2} and η_ε ∼ ℓ Re^{−3/4}. For 0 ≤ s < 2, the enstrophy dissipation scale shrinks more slowly, η_χ ∼ ℓ Re^{−(2−s)/(2(s+2))}, indicating a weaker direct cascade. For s < 0, η_χ becomes even smaller, reflecting that the forcing injects enstrophy directly at small scales, destroying the clean cascade picture.

The paper also discusses implications for numerical modelling and experimental design. Since the regularity of the forcing can be tuned (e.g., by using fractal grids, broadband or power‑law forcings), the results provide a rigorous guide for predicting how changes in forcing spectra will affect dissipation rates and the effective number of degrees of freedom. Moreover, the identification of a critical regularity threshold (s = 0) where the scaling of ε and χ switches from decaying to growing with Re offers a clear target for future investigations of “forcing‑dominated” turbulence in two dimensions.

In summary, the authors deliver a comprehensive, mathematically rigorous extension of Re‑dependent turbulence estimates to a broad class of forcings, revealing three distinct scaling regimes governed by the Sobolev regularity exponent s. This work bridges the gap between deterministic PDE analysis and the phenomenological/statistical descriptions used in turbulence research, and it highlights the pivotal role of forcing regularity in shaping two‑dimensional turbulent dynamics.