Running Held-suarez on a reduced-radius earth

DCMIP2016 steady-state initialization to (hopefully) speed up spinup

  REAL(8), PARAMETER ::               &   
       T0E        = 310.d0     ,      & ! temperature at equatorial surface (K)
       T0P        = 240.d0     ,      & ! temperature at polar surface (K)
       B          = 2.d0       ,      & ! jet half-width parameter
       K          = 3.d0       ,      & ! jet width parameter
       lapse      = 0.001d0             ! lapse rate parameter

We will use a reduction of $r=a/10$ (Consistent with smallest earth used in Yessad and Wedi (2009)). Reduction of lapse rate from 0.005 to 0.001 K/m is to address static instability observed in Skamarock DA MPAS paper.

Output plan:

• u, v, w
• T, p, geo

Output frequency: every 6 hour.

idea: adapt hybrid to small earth DA

Assume reference column with known $T(z)$ (I'll assume $T(z) = 273 K - 0.005 \textrm{ K m}^{-1} \cdot z$). Assume that the atmosphere is quasi-hydrostatic, $$\pder{p}{z} = -\rho g \implies \log(p/p_s) = - \int \frac{g}{R_d T} \intd{z}$$ which we calculate by quadrature and invert via rootfinding

Given reference interface values for $\eta_i, A_i, B_i$, we use the fact that $A_i + B_i = \eta_i$, then we set $\eta'_{K+0.5} = \eta'_{K+0.5}$, then set $$\frac{(\eta'_{K-0.5} - \eta'_{K+0.5})}{\hat{r}^2(\eta'_K)} = \eta_{K-0.5} - \eta_{K+0.5}$$