Kolmogorov (1941) identified the dissipative scales through dimensional analysis. Given kinematic viscosity ν and energy dissipation rate ε, the unique length, time, and velocity scales are:
η = (ν³/ε)^(1/4) (Kolmogorov length) | τ_η = (ν/ε)^(1/2) (Kolmogorov time) | u_η = (νε)^(1/4) (Kolmogorov velocity)
The ratio of integral scale L to Kolmogorov scale η scales as L/η ~ Re^(3/4). At Re = 10⁶ (ocean), this is ~10^4.5 — a staggering range requiring ~10^13.5 mesh points for direct simulation!
| Scale | Formula | Value | Physical meaning |
|---|
| η (length) | (ν³/ε)^(1/4) | — | Smallest eddy size |
| τ_η (time) | (ν/ε)^(1/2) | — | Smallest eddy lifetime |
| u_η (velocity) | (νε)^(1/4) | — | Smallest eddy velocity |
| L/η (range) | Re^(3/4) | — | Cascade range width |
| DNS cost | ~ Re^(9/4) | — | Grid points needed |
Scale hierarchy: L → η (log scale), with inertial subrange
Kolmogorov scales vs Re (log-log) — highlighted: current Re