setupv.f

Prepares, once per call of the rotational-mixing step, the structural factors that the stability criteria and circulation velocities need at each interface between shells. Its comment explains the scheme: stability is computed for the interface $i/i-1$ and stored in element $i$; each criterion is "a combination of variables stored at each shell multiplied by a gradient between the shells," which must stay below a critical number. SETUPV computes the parts that "do not change during a diffusion timestep"; VCIRC multiplies them by the current $\Omega$ gradients.

• Who calls it: SECULR, once per time step, before the iterations.
• Inputs: the model's $\log\rho$, $g$ (HG), $L$ (solar units), $\log P$, $\log r$, enclosed mass (g), $\log T$, and the range IMIN–IMAX; from MDPHY_MOD the shell values of $\mu$ (AMUM), $c_P$ (CPM), $\nabla$, $\nabla_\mathrm{ad}$, $\nabla_\mathrm{rad}$ (DELM, DELAM, DELRM), $\epsilon$ (ESUMM), $\kappa$ (OM), $\partial\ln\rho/\partial\ln T$ (QDTM); from ROTDER_MOD the derivatives of $\kappa$ and $\epsilon$ (SQOT, SQET) and the nuclear fraction FRACNUC, set in COEFFT; the namelist switch LDIFAD from NMLPAR_MOD.
• Outputs: HRU (radii), QWRMAX (the largest stable $d\Omega/d\ln r$ for the dynamical shear), the interpolation weights FACI (an argument owned by SECULR, which passes it on to VCIRC), and, through modules, read by VCIRC, CODIFF, GETFC and DADCOEFT: the interface values in INTVAR_MOD (RM, PM, TM, DM, DELMI, DELAMI, QDTMI, HS3, EPSILM, ALM, HGM), the velocity factors FACT1…FACT5, FACT7, FACT8 (VFACT_MOD), FACT6, FES1–FES3, FGSFJ, FGSFK (TAUKH_MOD), DDELM, FESTIME, THM (OLDROT2_MOD), and with LDIFAD the higher-order terms in DIFAD3_MOD and EGRIDDER_MOD.

1. Interface values. At the midpoint $r_{m,i} = \tfrac12(r_i + r_{i-1})$, each quantity is found by 4-point Lagrange interpolation in $r$ over shells $i-2 \dots i+1$ (the first and last interfaces use the first or last four shells); $P$, $T$, $\rho$ are interpolated in the log. The actual gradient is $\nabla = \min(\nabla_\mathrm{rad}, \nabla_\mathrm{ad})$.

2. Scales. With $G$, $m$ the mass inside the interface, $H_p = P r^2/(\rho G m)$, $H_T = H_p/\nabla$, $\eta = \tfrac43\,r^3/(G m)$, and $\Delta\nabla = \max(\nabla_\mathrm{ad} - \nabla,\ \nabla_\mathrm{min})$ with a floor of $10^{-3}$ (larger at the ends of the range, including a $\mu$-gradient term).

3. Eddington–Sweet circulation (Endal and Sofia, paper II, as cited): $v_\mathrm{ES} = $ FACT1·FACT2·$\Omega^2$, with $$\texttt{FACT1} = \frac{1}{(\partial\ln\rho/\partial\ln T)\,\max(\nabla_\mathrm{ad}-\nabla,\,10^{-3})},\qquad \texttt{FACT2} = \frac{\nabla_\mathrm{ad}}{g}\,\eta\,\Bigl(\epsilon - \frac{L}{m}\Bigr).$$

4. GSF and $\mu$ currents. FACT3 $= |H_T r|$ for the Goldreich–Schubert–Fricke velocity; FACT4 $= \frac{r^2}{G m\rho}\,(P - aT^4/3)/\sqrt g$ for the "fictitious $\mu$ currents" (Kippenhahn, IAU 66).
5. Dynamical shear. Stable when the Richardson number $\frac{\rho}{P}(\nabla_\mathrm{ad}-\nabla)\,g^2/(d\Omega/d\ln r)^2 > \tfrac14$, that is when $$\left|\frac{d\Omega}{d\ln r}\right| < \texttt{QWRMAX} = 2g\sqrt{\frac{\rho}{P}\max(\nabla_\mathrm{ad}-\nabla,\,10^{-3})}.$$
6. Secular shear with $\mu$ gradients: FACT5 $= \frac{\rho g^2}{P\,\Delta\ln P}\cdot\frac{p_r - 1}{1 + 3p_r}$, where $p_r = aT^4/(3P)$ is the radiation-pressure fraction ($1 - p_r$ is the comment's "BETA") and the last factor is the comment's $-d\ln T/d\ln\mu$.
7. Zahn-type terms (FES1, FES2, FES3, FV*, and, with LDIFAD, FACT7, FACD2, FACD3, following Zahn and Maeder 1998 as the comments say), built from $F = P/(g\,\Delta\nabla\,c_P\,\rho\,T)$, $L/m$ and $\epsilon$; and THM $= r_i^2 r_{i-1}^2/(G\sqrt{m_i m_{i-1}}) \approx r^2/g$ for the $d\theta/dt$ term (lines 304–309↗). With LDIFAD, the block starts at shell IMIN-1 (lines 408–410↗); IMIN is at least 2 (midmod.f, lines 375–383↗), so no index 0 is read (settled from the code, Nov 9, 2026). In that block V0, QC1R, QQC1RR, QQCHITR, HTP2 and QHTR are computed but used only in the commented-out forms of FACT7 and FACT8, and FACT8 is always 0 (lines 418–450↗): unused?; so are QQQCOD (stored at line 410↗ and line 435↗, read by no routine) and FV1A, FV1B, FV2A, FV2B, FGSFJ and FACT3 (lines 350–355↗, line 374↗: read by no other routine). The current FACT7 is the one at lines 443–444↗.