solver — two dimensions, live

Leray 1934 · Orr 1907 · Craik–Criminale 1986 · Beale–Kato–Majda 1984 · and the paper's §1–2

Why two dimensions cannot blow up

In two dimensions vorticity is a scalar, and taking the curl of the momentum equation gives ∂_tω + u·∇ω = νΔω: vorticity is carried by the flow and diffused, nothing else. A quantity that is only transported and diffused obeys a maximum principle — its largest value can never increase — so ‖ω(t)‖_∞ ≤ ‖ω(0)‖_∞ forever. Bounded vorticity is enough to keep the flow smooth (that is the Beale–Kato–Majda criterion, and in two dimensions the bound is free). This is why the classical theory settles the two-dimensional problem completely and the three-dimensional one not at all. Run the decaying-turbulence experiment and watch max|ω|: it does not rise. Energy and enstrophy fall monotonically, because in two dimensions dE/dt = −2νZ and dZ/dt = −2νP with P the palinstrophy ½⟨|∇ω|²⟩ — the solver's selftest checks both monotonicities and that the inviscid nonlinear term conserves E and Z to 10⁻¹⁰.

In three dimensions the vorticity equation is ∂_tω + u·∇ω = (ω·∇)u + νΔω. The new term is vortex stretching: a vortex line aligned with a stretching direction of the strain gets longer and, by conservation of angular momentum, spins faster. No maximum principle survives it. The paper's core is stretching made permanent: fluid converges on the axis and leaves along it, so axial vortex lines are stretched and the swirl grows like τ^−½−h (see core). Nothing in the two-dimensional equations can produce that.

The shear-plus-wave experiment

The base flow is the Kolmogorov shear U = sin y, with vorticity −cos y, plus a small wave ψ′ = ε cos(k_xx + k_yy). Where the shear dU/dy = cos y is positive (near y = 0), a wave with k_y/k_x > 0 leans against it and is stretched: its energy grows, peaks when its crests have been turned vertical, and then decays — Orr's mechanism, which the pulse page solves exactly for a uniform shear. Where the shear is negative the same wave is compressed immediately. The lower-right chart shows the horizontal mean flow U(y,t) and the Reynolds stress ⟨u′v′⟩(y) the wave carries; the dotted line is what U would have done under viscosity alone. The difference between the dashed and dotted lines is the wave's push on the mean flow: the divergence of a stress built from oscillations whose mean is zero. That is the whole of §2.2 of the paper, in a flow you can run.

Two honest caveats. The Kolmogorov shear is itself unstable at low viscosity and eventually breaks into vortices on its own — that is not the paper's effect, it is Kolmogorov flow being Kolmogorov flow; keep ν above about 10⁻³ for a clean run. And the amplitude here is small, so the mean-flow change is small too; the paper's pulses are large — A_wave ≍ q^−½−h/2 — because their leading self-interaction is a gradient and cancels, and what they must cancel is the whole leading residual of the annulus.

The method

• Spectral space. ω̂ on integer wavenumbers |k_x|, |k_y| < n/3 (the 2/3 rule: products of two retained modes cannot alias back onto a retained mode). ψ̂ = ω̂/|k|², û = ik_yψ̂, v̂ = −ik_xψ̂.
• The nonlinear term u·∇ω is formed pointwise in real space after four inverse transforms and transformed back once.
• Time stepping. Integrating-factor RK4: the viscous factor e^{−ν|k|²Δt} is applied exactly and only the nonlinear term is stepped, so the viscous term imposes no stability limit. The step is set from the advective CFL condition, Δt = 0.4·Δx/max|u|.
• The FFT is an iterative radix-2 Cooley–Tukey with precomputed twiddles and bit-reversal, applied to rows then columns. The selftest checks it against a naive O(n²) DFT to 10⁻¹⁴.
• Known answers. Taylor–Green vorticity 2cos x cos y is an exact solution decaying as e^−2νt — its self-advection is identically zero — and the solver reproduces it to 10⁻¹⁵. The selftest is ns/solver/ns2d.selftest.mjs and imports the exact file this page runs.