Aspect-ratio-dependent excess shear: ε(b/h) tapering the multiplier 1.2 → 1 (bank-stability closure)

[awickert]

Summary

Replace Parker's constant excess-shear ratio ε = 0.2 with an aspect-ratio
function ε(b/h) that tapers the stress multiplier (1+ε) from 1.2 (wide) to
1 (narrow). This stays within the rectangular-hydraulics framework (cheap), gives
the model fixed-width flexibility, and replaces the hard b/h = 2 floor +
separate competence self-arrest of #20 with a single smooth taper.

Why ε should depend on aspect ratio

ε is the excess of the bed shear over the bank threshold: bed at (1+ε)·τ*_c,
banks (same gravel) at τ*_c (marginally stable). Maintaining bed above bank
requires a non-uniform boundary-shear distribution. As b/h → 2 the section
approaches a half-pipe, the shear flattens toward uniform, and the bed can no
longer sit above the same-gravel banks — so ε → 0. In the wide limit the shear
is strongly non-uniform and ε → 0.2 (Parker, 1978, self-formed value).

Because transport Q_s ∝ (τ*_b − τ*_c)^{3/2} = (ε·τ*_c)^{3/2} ∝ ε^{3/2}, a
tapering ε drives transport smoothly to zero as the channel narrows — the
competence-limited self-arrest of #20, but as a continuous closure rather than
a floor + branch.

Approach

• Take banks pinned at τ*_c; write ε(b/h) = τ_bed/τ_bank − 1 from the bed/bank
  boundary-shear ratio as a function of b/h.
• Source the shear ratio from known distributions — Lane (1955) tractive-force
  diagrams, Knight et al. boundary-shear partition, or a lateral-momentum /
  potential model — anchored to ε → 0.2 (wide) and ε → 0 (b/h → 2).
• Placeholder pending the real function (endpoints only, not derived):
  ε(b/h) ≈ 0.2·(1 − 2/(b/h))₊.
• Feed ε(b/h) into the variable-ε transport form; the fixed-width case (paper
  Eq. 28) then falls out with ε from the imposed aspect ratio.

Payoff

• One smooth closure spanning equilibrium-width (ε = 0.2), fixed-width
  (ε imposed), and self-arrest (ε → 0) — see the unification in Hydraulic-radius correction to the equilibrium-width transport law (+ competence-limited self-arrest) #20.
• Removes the artificial b/h = 2 cutoff; transport tapers to zero physically.
• Fixed-width flexibility with essentially no added cost (still rectangular).

Caveats

• Anchoring Parker's 0.2 (a tapered-channel result) onto a rectangular
  aspect ratio is approximate — the honest geometry is Realistic hydraulic geometry: Parker's tapered stable-bank cross-section (beyond rectangular) #22.
• The τ_bed/τ_bank(b/h) relation needs a defensible source/derivation, not the
  placeholder above.
• Still uses the reach-average R_h; the bed/bank-partition subtlety (Hydraulic-radius correction to the equilibrium-width transport law (+ competence-limited self-arrest) #20, "Known
  limitation") is only partially addressed until Realistic hydraulic geometry: Parker's tapered stable-bank cross-section (beyond rectangular) #22.

Related

• Hydraulic-radius correction to the equilibrium-width transport law (+ competence-limited self-arrest) #20 — hydraulic-radius correction + competence self-arrest (this taper replaces
  its b/h=2 floor with a continuous ε).
• Valley realism: transient valley widening/narrowing + deposit (overbank) tracking #19 — dynamic valley width B(x,t) (fixed/imposed-width partner).
• Realistic hydraulic geometry: Parker's tapered stable-bank cross-section (beyond rectangular) #22 — realistic tapered y–z cross-section (the ε(b/h) taper falls out of
  it natively).
• 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.

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