Realistic hydraulic geometry: Parker's tapered stable-bank cross-section (beyond rectangular)

[awickert]

Summary

Replace the rectangular channel cross-section used for the hydraulics with a
realistic tapered (Parker, 1978) stable-bank section. This is the physically
consistent endpoint of the narrow-channel thread: it dissolves the b/h = 2
rectangular artifact into a smooth taper, carries the bed/bank shear partition
natively, and makes the ε(b/h) taper of #21 fall out of the
geometry rather than being imposed.

The inconsistency to fix

The 2019 model borrows Parker's width closure (derived for a self-formed
channel with mobile, at-threshold banks) but uses a rectangular section for the
hydraulics (Q = u·b·h, R_h = b·h/(b+2h)). That pairing is internally
inconsistent: it is harmless when wide (banks a thin fraction of the perimeter) but
glaring as b/h → 2, where a rectangular half-pipe has near-uniform boundary shear
and therefore cannot hold same-gravel banks at threshold beneath a mobile bed.
A real self-formed channel is not rectangular.

The physical geometry

Parker's (1978) self-formed section: the banks curve/taper so the local
boundary shear equals τ*_c everywhere along them (marginally stable), while the
central bed sits at (1+ε)·τ*_c and transports; depth tapers to zero at the
margins. The mobile-bed width, R_h, and transport all follow from this shape.
The bank-stability constraint sacrifices hydraulic efficiency — it forbids the
efficient half-pipe precisely because uniform shear cannot keep same-gravel banks
stable under a mobile bed.

Approach

• Adopt Parker's stable-bank profile, or a tractable proxy (trapezoid, parabola,
  or Parker's cosine-type shape), as the cross-section.
• Compute R_h, mobile-bed width, and transport from the shape; the section
  becomes a shape problem, not just a width b.
• Recover the wide-channel and rectangular results as limits; check continuity with
  the Hydraulic-radius correction to the equilibrium-width transport law (+ competence-limited self-arrest) #20 cofactor in the wide regime.

Payoff

• Physically consistent; removes the rectangular b/h = 2 artifact — the mobile
  bed tapers smoothly to zero as forcing weakens (the "geometric taper toward
  0 driven by the Parker criterion").
• Carries the bed/bank shear partition natively (resolves the Hydraulic-radius correction to the equilibrium-width transport law (+ competence-limited self-arrest) #20 "Known
  limitation").
• The stress-multiplier taper ε(b/h) (Aspect-ratio-dependent excess shear: ε(b/h) tapering the multiplier 1.2 → 1 (bank-stability closure) #21) emerges from the geometry rather
  than being fit.

Scope / cost

Longer-term research. Turning the section into a shape reaches into cross-section
geometry beyond the width B, and interacts with the dynamic-B work (#19). Likely
warrants a tractable proxy first (trapezoid/parabola) before Parker's full profile.

Related

• Hydraulic-radius correction to the equilibrium-width transport law (+ competence-limited self-arrest) #20 — hydraulic-radius correction; its rectangular b/h=2 floor and bed/bank
  "Known limitation" are what this dissolves.
• Aspect-ratio-dependent excess shear: ε(b/h) tapering the multiplier 1.2 → 1 (bank-stability closure) #21 — ε(b/h) stress-multiplier taper (the rectangular-framework near-term
  version; this issue is the geometry it approximates).
• Valley realism: transient valley widening/narrowing + deposit (overbank) tracking #19 — dynamic valley width B(x,t).
• Sand-bed conductance singularity at S→0 — may be capped by the hydraulic-radius correction + competence self-arrest (cf. GRLP #20) FluvTree#1 — sand-bed application.

🤖 Generated with Claude Code