PENTRC: clar method inherits PENT formulas under a changed pitch convention — omega_b inflated by sqrt(bo), omega_D by bo

[logan-nc]

Follow-up to #281, which fixed the GAR precession normalization. The git archaeology done for that review shows the clar RLAR bounce-average branch has a defect with the same root cause, and makes the correction factors precise.

Origin

In PENT (pent/profile.f, 2013-2015), the pitch variable was dimensional: vpar = 1 - lmda*b, so lmda carried units of 1/Tesla. The CLAR formulas were consistent with that convention:

wbbar = pi*SQRT(2*epsr*lmda*bo)/(4*q*ro*ellipk)   ! lmda*bo dimensionless
wdbar = (2*q*lmda*(E/K - 0.5)/(ro**2*epsr))*wdfac ! lmda ~ 1/bo scale
bhat  = SQRT(2*T/mass)
dhat  = T/chrg

Both wbbar*bhat and wdbar*dhat are true frequencies under lmda ~ 1/T.

In the PENT->PENTRC rewrite (da0e6067, June 2015), the pitch became dimensionless: vpar = 1 - (lmda/bo)*b, i.e. Lambda = lmda_pent*bo. The clar formulas were carried over textually verbatim (pentrc/torque.F90:447-451). Substituting lmda_pent = Lambda/bo:

• kappa maps exactly (PENT's kk = (1/bo - lmda(1-eps))/(2*eps*lmda) becomes PENTRC's kk = (1 - lmda(1-eps))/(2*eps*lmda)), so the trapped-space geometry is correct, but
• wbbar retains a spurious sqrt(bo): SQRT(2*epsr*lmda*bo) should be SQRT(2*epsr*lmda)
• wdbar retains a spurious bo: the denominator should be bo*ro**2*epsr

So clar currently reports omega_b inflated by sqrt(bo) (~1.4-2.3x) and omega_D inflated by bo (~2-5.3x). The fs(:,3) action channel also scales with the inflated wbbar.

What this does NOT affect

• The rlar method's flux-surface estimators (wbhat/wdhat, torque.F90:335-336) are built from wtran/wgyro and are dimensionally sound — PENTRC: omega_D in the GAR branch carries a spurious major radius #281 used wdhat as its sanity check.
• The GAR methods, fixed by PENTRC: omega_D in the GAR branch carries a spurious major radius #281.

Verification status

This is an analytic + historical claim; it has not yet been checked against an independent reference operator the way #281 was. Prediction for anyone re-running #281's slope-vs-reference measurement on the clar branch: slope magnitude sqrt(bo) for omega_b and bo for omega_D before the fix, ~1 after. Noted in #281 as "the reduced-large-aspect-ratio branch has its own dimensional oddities"; this issue pins down the factors and their origin.

Fix PR incoming.