This is a genuinely 2D-native solve of the same Ikeda–Parker–Sawyer excess-velocity mechanism used by the 3D version, but with a different numerical method. The 3D model tracks a closed chain of Lagrangian markers (an arc-length-parameterized centerline that can loop back on itself in space). This version instead solves the linearized theory directly: for a nearly-straight channel written as a single-valued graph z(x) of lateral offset vs. downstream distance, the curvature is simply the second derivative,
κ(x) ≈ ∂²z/∂x² (graph curvature, finite-difference)
κ̃(x) solves λ·∂κ̃/∂x = κ(x) − κ̃(x) (upstream-lag filter, Euler-integrated)
∂z/∂t = −E₀·U·κ̃(x)·taper(x) − c·z·|z|·taper(x) − D₄·∂⁴z/∂x⁴ (growth + saturation + stabilizer)
The first two lines are the exact same lag-filtered-curvature relation as the 3D theory box, just discretized on a fixed downstream grid (Eulerian) instead of a resampled marker chain. Because migration is proportional to curvature, any wiggle amplifies itself — the same positive-feedback instability that turns straight channels into meanders. A single-valued graph can never loop back on itself, so a literal neck cutoff is impossible in this representation; instead, a lobe cuts off whenever its local radius of curvature 1/|κ| shrinks below a small multiple of the channel width and its amplitude is large enough — the same "the neck has narrowed to channel width" criterion the 3D model checks geometrically, applied here through curvature instead of marker distance.
- Migration rate E₀ — the curvature→migration coupling constant; higher values grow bends faster.
- Curvature lag — the filter length λ in the ODE above; larger lag shifts the migration lobe further downstream of the bend apex, exactly like the 3D convolution kernel.
- Sediment deposition rate — how fast the point-bar band on the inner (convex) bank thickens.
The bottom panel is a spacetime (Hovmöller) diagram: each new simulated interval adds one row, plotting curvature κ(x) across the whole valley as a color at that instant, with new rows scrolling in at the top. Diagonal streaks reading down-and-across reveal bends migrating downstream over centuries — a plan-view river map can't show that directly, but a curvature field on a distance-vs-time grid does.