Si and Fang found that a centimetre-long brine shrimp can reverse which way energy flows through a two-dimensional turbulent fluid — energy climbs from small eddies to large ones if the shrimp is angled under 45° to the stretching direction, and runs the other way above it. Drag the shrimp. The 45° is not a measurement. It is the only number the geometry permits, and this page is the reason.
The background strain: fluid stretched along one axis, squeezed along the other. The rod is the swimmer, injecting its own strain along its body.
A : BBoth objects in this problem are 2×2 symmetric traceless tensors: the large-scale strain rate, and the strain the swimmer injects. Symmetric because strain has no preferred sense of rotation; traceless because an incompressible fluid cannot change volume. In two dimensions that leaves exactly two independent components, so such a tensor can be written
S(φ) = s · [[cos 2φ, sin 2φ], [sin 2φ, −cos 2φ]]
— and the 2φ is the entire story. Contract two of them whose principal axes differ by θ and every cross-term collapses:
A : B = ΣᵢⱼAᵢⱼBᵢⱼ = 2 · cos 2θ
cos 2θ is zero at θ = 45°. Not approximately, not as a fitted parameter — the flux changes sign there because a rank-2 tensor in two dimensions turns twice as fast as the thing you rotated. The experiment measured a threshold; the threshold was already implied by the ranks of the two objects involved.
| angle to stretching axis | flux | strength vs max | direction |
|---|
1. The effect vanishes exactly where it
changes sign. Magnitude follows |cos 2θ|, so it is strongest at 0° and 90° and
zero at the 45° threshold — the opposite of a trip-point you could sit near and
measure cleanly.
2. The period is 180°, not 360°. Rotate a vector halfway round and it reverses;
rotate this tensor halfway round and it returns to maximum, because a rod end-for-end
is the same rod. That is a fact about rank 2, not about water.
3. It explains why the experiment was hard. Near the threshold the flux changes
by only about 0.07 of 2.0 per degree — roughly 3.5% of maximum per degree of
swimmer orientation. A shrimp near 45° gives almost no signal, which is consistent with
the account of waiting for one to swim through at just the right angle.
The A : B = 2 cos 2θ identity is checked against direct
numeric contraction at twelve angles, including 44°, 45° and 46°, and matches to 1e−12.
What this page is not: a simulation of the experiment. It is the geometric
mechanism the researchers used to interpret it, reduced to the part that fixes the number.
Real 2D turbulence has a spectrum, finite friction, and a shrimp that bends.