Ertel PV Computation on Sigma Levels
Climate Dynamics, MIT EAPS, 2026
How to correctly compute Ertel Potential Vorticity (PV) for CMIP-class models that do not provide PV as a native output (unlike ERA5 or MPAS-Atmosphere).
Source repository: github.com/yanxingjianken/PV_Ertel_Computation
Motivation
ERA5 and MPAS output Ertel PV directly on model/pressure levels, but most CMIP models (CESM2, IPSL, MIROC, etc.) do not. Computing PV from the standard 3-D output $(u, v, T, p_s)$ seems straightforward, but the naive pressure-level finite-difference approach gives severely degraded PV near the surface and near steep topography.
The fix: compute PV on terrain-following $\sigma = p/p_s$ coordinates, where the lower boundary is a material surface and vertical derivatives are naturally aligned with the flow.
Formula
\[\text{PV}_\sigma = -\frac{g}{p_s}\left[ \frac{\partial v}{\partial\sigma}\frac{\partial\theta}{\partial x} - \frac{\partial u}{\partial\sigma}\frac{\partial\theta}{\partial y} + (f + \zeta)\frac{\partial\theta}{\partial\sigma} \right] \times 10^6 \ \text{PVU}\]where:
- $\sigma = p/p_s(x,y)$ — terrain-following vertical coordinate
- $\theta = T\,(p_0/p)^{R_d/c_p}$ — potential temperature ($p_0 = 1000$ hPa)
- $\zeta$ — relative vorticity via spherical-harmonic operators
- $f = 2\Omega\sin\phi$ — Coriolis parameter
- Horizontal derivatives: spectral (spherical harmonics) — no polar singularity
- Vertical derivatives: MPAS-style centred interior, one-sided boundaries
ERA5 Validation
We validate against ERA5’s native isobaric PV output. The sigma method dramatically improves near-surface PV:
| $\sigma$ level | Nominal hPa | RMSE (PVU) | Correlation |
|---|---|---|---|
| 0.850 | 861 | 0.30 | 0.92 |
| 0.500 | 507 | 0.37 | 0.94 |
| 0.250 | 253 | 1.12 | 0.97 |
Key result: 850 hPa correlation improves from 0.28 (isobaric finite-difference) to 0.92 (sigma coordinate).
Figure: Sigma-coordinate PV (left) vs ERA5 native PV (center) and their difference (right) at three representative levels. The sigma method captures the near-surface PV structure that isobaric finite-differences miss.
Usage
Quick start (ERA5)
cd era_sanity_check
micromamba run -n blocking python validate_era5.py
Quick start (CESM2-LENS2)
cd cesm2_compute
micromamba run -n blocking python compute_cesm2_pv.py
Python API
from src.ertel_pv import ertel_pv_sigma, DEFAULT_SIGMA_LEVELS
# sigma PV on 11 levels (primary)
pv_sigma, p_3d = ertel_pv_sigma(u, v, t, plev_Pa, ps, lat, lon)
# pv_sigma: (11, nlat, nlon) in PVU
Required Input Variables
| Variable | ERA5 name | CESM2 name | Units |
|---|---|---|---|
| Eastward wind | u | U | m s⁻¹ |
| Northward wind | v | V | m s⁻¹ |
| Temperature | t | T | K |
| Surface pressure | sp | PS | Pa |
| Pressure levels | (coordinate) | lev | Pa or hPa |
| Latitude | latitude | lat | °N |
| Longitude | longitude | lon | °E |
References
- Source code & documentation: github.com/yanxingjianken/PV_Ertel_Computation
- Uses
pvtend.sh_opsfor spherical-harmonic gradients - Linear vertical interpolation: NCL
int2p_n→ scipy/numpy port