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Energy Cascade Sandbox

Learning goal

Internalise the K41 picture — injection scale, inertial range with \(-5/3\) slope, dissipation cutoff at the Kolmogorov scale — and watch the range widen as \(Re^{3/4}\) while the cost of resolving it explodes as \(Re^{9/4}\).

Things to try

  1. Start at the laboratory jet and slide Re upward. The shaded inertial range stretches. Its width in decades is \(\tfrac34\log_{10}Re\) — so getting one extra decade of \(-5/3\) costs a factor of \(10^{4/3} \approx 22\) in Reynolds number. This is why clean inertial ranges are hard to produce in a laboratory, and why geophysical flows are the natural place to look for them.

  2. Check the scale-separation readout. \(L/\eta\) tracks \(Re^{3/4}\) exactly — it is not fitted, it follows from \(\eta = (\nu^3/\varepsilon)^{1/4}\) and \(\varepsilon \sim U^3/L\). Two lines of dimensional analysis, and it is the single most useful number in turbulence.

  3. Watch the grid count. Resolving every scale needs \(N \sim (L/\eta)^3 = Re^{9/4}\) points. At \(Re = 10^4\) that is \(10^9\) — a large but ordinary DNS. At \(Re = 10^8\), an airliner wing, it is \(10^{18}\) points and about \(2\times10^7\) terabytes for a single snapshot of the velocity field. Not "expensive": impossible, by many orders of magnitude, for the foreseeable future. This is the entire justification for turbulence modelling.

  4. Note where the Kolmogorov scale actually lands. For the atmospheric boundary layer, \(\eta\) comes out around a millimetre while the largest eddies are kilometres across. The atmosphere is genuinely doing physics across seven decades of scale, simultaneously, all the time.

  5. Check \(Re_\eta = u_\eta\eta/\nu\). It is exactly 1, always, by construction. That is the definition of the dissipation scale: where the local Reynolds number falls to unity and viscosity finally competes with inertia. Everything above it is inviscid cascade; below it, nothing survives.

  6. Compare the eddy cartoon with the decade count. Richardson's rhyme — big whorls have little whorls — is a statement about self-similarity, and the number of levels drawn is the number of decades the flow actually spans.

The dimensional argument, in three lines

In the inertial range the only available quantities are the flux \(\varepsilon\) (W/kg) and the wavenumber \(k\) (1/m). Demanding that \(E(k)\) have units m³/s²:

\[[E] = \mathrm{m^3/s^2},\quad [\varepsilon] = \mathrm{m^2/s^3},\quad [k] = \mathrm{m^{-1}} \;\Longrightarrow\; E(k) = C_K\,\varepsilon^{2/3}k^{-5/3}\]

The \(-5/3\) is forced. No dynamics, no model, no adjustable exponent — just the assumption that the inertial range knows nothing except how much energy is passing through it. That this crude argument matches measurements across decades in the ocean, the atmosphere and the laboratory is one of the more remarkable facts in classical physics.

What the model spectrum is and is not

The curve is the standard model form: a von Kármán roll-off at low \(k\), the exact \(-5/3\) power law in between, and an exponential dissipation cutoff. It is a fit to how real spectra look, not a solution of Navier–Stokes. Take the scaling from it — where the ranges sit and how they move with \(Re\) — not the detailed shape near the two crossovers, where real spectra differ between flows and where the (small but real) intermittency corrections to \(-5/3\) live.

Energy cascade · Turbulence · Reynolds number · Reynolds averaging · Dimensional analysis · Vorticity equation — stretching is the cascade's engine · Numerical diagnostics (PHY653B) — measuring spectra honestly