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Got it — let’s streamline your notes into a clean, structured, and practical reference. I’ll keep all the technical depth but organize it so it reads like a polished “cookbook” for Richardson number diagnostics and corrections.

🌬 Gradient & Bulk Richardson Numbers: Clean Reference

1. Definitions

  • Gradient Richardson number [ Ri_g = \frac{(g/\theta),\partial\theta/\partial z}{(\partial U/\partial z)^2} ]

  • Bulk Richardson number (layer (z_0 \to z_1)) [ Ri_b = \frac{g}{\theta}\frac{\Delta\theta,\Delta z}{(\Delta U)^2} ]

2. MOST Relation

[ Ri_g(\zeta) = \zeta \frac{\phi_h(\zeta)}{\phi_m(\zeta)^2}, \quad \zeta = \frac{z}{L} ]

3. Near-Neutral Series Expansion

[ \phi_{m,h} = 1 + a_{m,h}\zeta + b_{m,h}\zeta^2 + O(\zeta^3) ]

[ Ri_g = \zeta + \Delta \zeta^2 + \tfrac12(\Delta^2 + c_1)\zeta^3 + O(\zeta^4) ]

  • (\Delta = a_h - 2a_m)
  • (c_1 = b_h - 2b_m)

4. Inversion: ζ(Ri)

[ \zeta = Ri_g - \Delta Ri_g^2 + \Big(\tfrac32\Delta^2 - \tfrac12 c_1\Big)Ri_g^3 + O(Ri_g^4) ]

Use as a seed for Newton refinement when evaluating φ at given Ri.

5. Curvature Diagnostics

  • Log derivatives: [ V_{\log} = \frac{\phi_h'}{\phi_h} - 2\frac{\phi_m'}{\phi_m}, \quad W_{\log} = V_{\log}' ]

  • Curvature of (Ri_g(\zeta)): [ \frac{d^2Ri_g}{d\zeta^2} = F\big[2V_{\log} + \zeta(V_{\log}^2 - W_{\log})\big], \quad F = \frac{\phi_h}{\phi_m^2} ]

  • Neutral limit: [ \left.\frac{d^2Ri_g}{d\zeta^2}\right|_0 = 2\Delta ]

6. Bulk vs Point Bias

  • Concave-down ((\Delta < 0)): [ Ri_b < Ri_g(z_g), \quad z_g = \sqrt{z_0 z_1} ]
  • Bias factor: [ B = \frac{Ri_g(z_g)}{Ri_b} > 1 ]

7. Correction Principle

Introduce grid damping (G(\zeta,\Delta z)) with:

  • (G(0,\Delta z) = 1)
  • (\partial_\zeta G|_0 = 0)

This preserves neutral curvature (2Δ) while reducing bias at coarse Δz.

8. Critical Richardson Number

  • Fixed (Ri_c) vs dynamic (Ri_c^*).
  • Dynamic informed by curvature growth: (\zeta(V_{\log}^2 - W_{\log})).

9. Key Identities

  • Turbulent Prandtl number: [ Pr_t = \frac{\phi_h}{\phi_m} ]

  • Closure functions: [ f_m(Ri_g) = \frac{1}{\phi_m(\zeta(Ri_g))^2}, \quad f_h(Ri_g) = \frac{1}{\phi_m(\zeta(Ri_g))\phi_h(\zeta(Ri_g))} ]

10. Generic Scalar Closure

For scalar (q): [ f_q(Ri_g) = \frac{1}{\phi_m(\zeta(Ri_g)),\phi_q(\zeta(Ri_g))}, \quad K_q = l_m^2 S f_q ]

  • Schmidt number: [ Sc_t^{(q)} = \frac{\phi_q}{\phi_m} ]

  • Near-neutral series: [ f_q \approx 1 + a_q Ri_g + (b_q - a_q\Delta + 2a_m a_q)Ri_g^2 ]

11. Numerical Estimation

  • Point gradient (centered): [ \partial U/\partial z|{z_k} \approx \frac{U{k+1} - U_{k-1}}{z_{k+1} - z_{k-1}} ]

  • Bulk Ri_b (layer [z0,z1]):

    • Trapezoid: (\tfrac12[Ri_g(z_0) + Ri_g(z_1)])
    • Simpson: (\tfrac16[Ri_g(z_0) + 4Ri_g(z_g) + Ri_g(z_1)])

  • Representative heights:

    • (z_g = \sqrt{z_0 z_1}) (geometric mean)
    • (z_L = (z_1 - z_0)/\ln(z_1/z_0)) (log mean)
    • (z_a = (z_0 + z_1)/2) (arithmetic mean)

12. Practical Workflow

  1. Compute (Ri_g(z_g)).
  2. Compute (Ri_b).
  3. Bias factor (B = Ri_g(z_g)/Ri_b).
  4. Apply correction if (B > 1.05).
    • Mild damping if (1.05 < B \leq 1.3).
    • Strong damping + mixing-length reduction if (B > 1.3).

13. Mixed Concavity Handling

If (\tfrac{d^2Ri_g}{d\zeta^2}) changes sign:

  • Split layer at inflection (\zeta_{\text{inf}}).
  • Apply bias logic separately to concave-down and concave-up segments.
  • Recombine weighted averages.

14. φ‑Agnostic Surrogate

When φ-functions are unknown:

  • Use exponential Ri closures: [ f_m(Ri) = e^{-\gamma_m Ri / Ri_c^}, \quad f_h(Ri) = e^{-\gamma_h Ri / Ri_c^} ]
  • Suggested: (\gamma_m \approx 1.8, \gamma_h \approx 1.5).
  • (Ri_c^*) dynamic or default 0.25.

15. Minimal Pseudocode

z_g = sqrt(z0*z1)
Ri_g_zg = compute_point_Ri(z_g)
Ri_b = compute_bulk_Ri(z0,z1)
B = Ri_g_zg / Ri_b

if B > 1.1:
    G = exp(-D*(dz/dz_ref)**p * (zeta/zeta_ref)**q)
    K_star = K * G
else:
    K_star = K

# φ-agnostic fallback
f_m = exp(-gamma_m * Ri / Ric_star)

Curvature in ζ versus z

Curvature in ζ is natural in MOST because the similarity functions are defined in terms of the non-dimensional height (\zeta=z/L). If (L) is treated as locally constant across a thin layer, then derivatives transform simply: (\frac{d}{dz}=\frac{1}{L}\frac{d}{d\zeta}) and (\frac{d^2}{dz^2}=\frac{1}{L^2}\frac{d^2}{d\zeta^2}). In that case, the sign and relative magnitude of curvature are preserved between (z) and (\zeta). When (L) varies with height, curvature in (z) picks up extra terms involving (dL/dz), so (\zeta)-based diagnostics cleanly separate profile shape from coordinate effects.

Near-neutral coefficients from Businger–Dyer

For the classic Businger–Dyer (BD) stable formulations near (\zeta\to 0^+): [ \phi_m = 1 + a_m \zeta,\quad \phi_h = 1 + a_h \zeta, ] with commonly used values (a_m\approx 4.7) and (a_h\approx 7.8). If you retain only the linear terms (i.e., (b_m=b_h=0)), then [ \Delta = a_h - 2a_m = 7.8 - 9.4 = -1.6,\qquad c_1 = b_h - 2b_m = 0. ] Implications:

  • Neutral curvature: (\left.\frac{d^2 Ri_g}{d\zeta^2}\right|_0 = 2\Delta = -3.2) → concave-down.
  • Bias direction: concave-down implies (Ri_b < Ri_g(z_g)) and a bulk-versus-point bias factor (B>1).
  • ζ inversion (cubic): [ \zeta \approx Ri_g - \Delta Ri_g^2 + \Big(\tfrac32\Delta^2 - \tfrac12 c_1\Big)Ri_g^3 = Ri_g + 1.6,Ri_g^2 + 3.84,Ri_g^3. ]

For unstable BD, (\phi) functions are non-linear (e.g., (\phi_m=(1-16\zeta)^{-1/4}), (\phi_h=(1-16\zeta)^{-1/2})). A Taylor expansion about (\zeta=0^-) yields finite linear coefficients as well: [ \phi_m \approx 1 + 4\zeta + 10\zeta^2 + \dots,\quad \phi_h \approx 1 + 8\zeta + 48\zeta^2 + \dots, ] so near-neutral on the unstable side you’d have (a_m\approx 4), (a_h\approx 8), giving (\Delta\approx 0) (specifically (\Delta = 8 - 2\cdot 4 = 0)), i.e., weak curvature in the immediate neutral limit and rapidly increasing nonlinearity at larger (|\zeta|). If you prefer other empirical constants (e.g., 5 and 5 for modified BD), update (a_m,a_h) and recompute (\Delta,c_1) accordingly.

Curvature mapping: ζ to z

  • If L is uniform in the layer:

    • Label: Derivative scaling
    • [ \frac{d^2 Ri_g}{dz^2} = \frac{1}{L^2}\frac{d^2 Ri_g}{d\zeta^2} ]
    • Result: Same concavity and bias logic; only magnitude rescales by (1/L^2).

  • If L varies with z:

    • Label: Extra terms
    • [ \frac{d^2 Ri_g}{dz^2} = \frac{1}{L^2}\frac{d^2 Ri_g}{d\zeta^2}
      • 2\frac{1}{L^3}\frac{dL}{dz}\frac{d Ri_g}{d\zeta}
      • \text{terms with } \frac{d^2 L}{dz^2} ]
    • Result: Curvature in (z) combines profile shape and stability variation; using (\zeta) isolates the MOST shape. Practically, use (\zeta)-curvature for diagnostics and treat (L(z)) variability via layer splitting or effective (L).

Practical guidance

  • Near-neutral diagnostics: Use BD-derived (a_m,a_h) to compute (\Delta) and neutral curvature (2\Delta). This sets the expected sign of the bulk vs point bias.
  • Bias correction: Apply damping (G(\zeta,\Delta z)) designed so (G(0)=1) and (G'(0)=0), preserving neutral curvature while reducing coarse-grid bias.
  • Representative height: Evaluate point (Ri_g) at (z_g=\sqrt{z_0z_1}) and use Simpson/trapezoid integration for (Ri_b) when curvature is non-negligible.
  • ζ inversion for closures: Use (\zeta(Ri_g)) from the series as a seed for Newton to evaluate (\phi_m,\phi_h) robustly when Ri is the control variable.

Quick check with BD-stable

  • Coefficients: (a_m=4.7,\ a_h=7.8 \Rightarrow \Delta=-1.6).
  • Neutral curvature: (2\Delta=-3.2) → concave-down; expect (B>1).
  • ζ seed: (\zeta \approx Ri_g + 1.6,Ri_g^2 + 3.84,Ri_g^3).
  • Action: Prefer Simpson for (Ri_b); if (B>1.05), apply mild damping; if (B>1.3) with strong inversion, reduce mixing length and consider raising (Ri_c^*).

If you’re using a different BD constant set, share them and I’ll plug them in to give you the updated (\Delta), curvature, and ζ-inversion coefficients.