Sand-bed conductance singularity at S→0 — may be capped by the hydraulic-radius correction + competence self-arrest (cf. GRLP #20)

[awickert]

Summary (hypothesis)

The sand-bed sediment conductance diverges as S → 0 (∝ S^(p−1) = S^(−1/6)).
The hydraulic-radius reframing + competence-limited self-arrest developed for
GRLP — full derivation in MNiMORPH/GRLP#20
(MNiMORPH/GRLP#20) — may cap this divergence,
converting it into a physical Q_s → 0 shutoff. But a quick estimate suggests
the self-arrest onset overlaps realistic lowland sand-bed slopes, so this is not
obviously a free fix — it needs derivation and testing.

The singularity

In fluvtree/alluvial_core/closures.py, SandClosure:

• conductance_exponent = -1/6 → the implicit sediment-flux conductance goes as
  S^(−1/6) → ∞ as S → 0.
• channel_depth(S) = (1+ε)·τ_crit_bank / (ρg·S) → ∝ 1/S → ∞ as S → 0.

The connection to GRLP #20

GRLP #20's finding: the length the shear/threshold closure returns and calls
"depth" is actually the hydraulic radius R_h; the true flow depth differs,
and the width b from continuity (∝ Q·S^(7/6)) is really the wetted
perimeter. The correction is a single linear cofactor f = R_h/h_true
applied to Q_s = q_s·b, plus a competence-limited self-arrest at the
narrow-channel limit.

Sand's channel_depth = (1+ε)·τ_crit_bank/(ρg·S) has the identical ∝1/S
bank-shear-closure form — so it too is R_h, not the true flow depth, and the
same reframing/cofactor/self-arrest apply.

Mechanism at S → 0 (sand): R_h ∝ 1/S → ∞ while width b ∝ S^(7/6) → 0,
so b/R_h ∝ Q·S^(13/6) → 0, dropping below the existence floor b ≥ 8·R_h. The
channel can no longer hold the bank-shear threshold → competence-limited
self-arrest → Q_s → 0, which caps the S^(−1/6) conductance divergence
before it reaches S = 0. That is the sense in which #20 may resolve the
singularity.

Caveat — why this may not be a clean fix

Rough estimate of the self-arrest onset slope S_min (existence floor
b = 8·R_h; sand constants n = 0.02, τ_crit_bank = 2–10 — test values):

Q (m³/s)	S_min (τ_c=2)	S_min (τ_c=10)
1	5.7×10⁻⁴	4.1×10⁻³
100	6.8×10⁻⁵	4.9×10⁻⁴
10000	8.1×10⁻⁶	5.9×10⁻⁵

Realistic lowland sand-bed slopes are ~10⁻⁴–10⁻⁵, so these overlap. The
correction would therefore change real low-gradient sand reaches, not merely
remove the numerical singularity at S = 0. Whether that is physically correct
(real low-gradient sand transport genuinely is competence/supply-limited) or an
artifact of porting gravel-derived geometry to sand is the open question.
(Numbers preliminary: test-value constants and the gravel-derived 8·R_h floor.)

To investigate / do

• Port the R_h naming + cofactor + self-arrest from GRLP #20 to SandClosure,
  re-deriving the sand-specific constants (Manning n, τ_crit_bank).
• Derive the sand existence floor / Q_min / onset S_min properly — the
  gravel quadratic geometry may not carry over unchanged (sand width closure
  is bank-cohesion, not grain-threshold).
• Test conductance behaviour near S = 0 with/without the correction;
  confirm it caps the S^(−1/6) divergence.
• Check the impact on realistic lowland sand-bed slopes — is the self-arrest
  onset physical, or an artifact to be pushed below the realistic range?

Reference

Full derivation, numerical verification, gravel implementation plan, and the
b = A/R_h = P identity: MNiMORPH/GRLP#20
(MNiMORPH/GRLP#20).

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