Orthogonal Four-Basis Decomposition for PV Tendency Analysis
We developed a unified framework to decompose potential vorticity (PV) tendencies into four physically interpretable and mutually orthogonal modes: intensification, zonal propagation, meridional propagation, and wave-breaking deformation.
The motivation comes from a long-standing problem in PV tendency diagnostics: traditional least-squares decompositions suffer from basis cross-correlations, making the resulting coefficients difficult to interpret physically. Our approach resolves this by constructing an explicitly orthogonal basis, allowing each dynamical contribution to be isolated cleanly.
The four modes
The decomposition is built from four template fields evaluated at the previous time step:
- PV anomaly — captures amplitude growth or decay (intensification)
- Zonal PV gradient — rigid east–west propagation
- Meridional PV gradient — rigid north–south propagation
- Cross-derivative of PV — diagonal strain associated with Rossby wave breaking
Each mode corresponds directly to an intuitive physical process acting on a PV anomaly.
From a 2×2 toy model to reanalysis data
We start with an idealized 2×2 matrix representation, where the four modes form an exact orthogonal basis under the Frobenius inner product. This toy problem builds intuition for how intensification, translation, and deformation appear as distinct spatial patterns.
The framework is then generalized to continuous fields using:
- latitude-weighted inner products,
- restriction to the negative-PV anomaly core,
- Gaussian pre-smoothing to suppress small-scale noise,
- and careful pre-normalization to ensure numerical stability.
Orthogonality is enforced via Gram–Schmidt, yielding a basis where each coefficient depends only on its corresponding mode.
Why orthogonality matters
With an orthogonal basis:
- projection coefficients are computed by direct inner products (no matrix inversion),
- each coefficient has a clean physical interpretation,
- propagation, intensification, and deformation are fully decoupled,
- and the residual captures only higher-order shape changes.
In the special case of PV tendency, the intensification coefficient directly links to PV enstrophy growth, providing a compact diagnostic for anomaly amplification.
What this enables
This framework provides a rigorous way to diagnose Rossby wave breaking and blocking dynamics by separating:
- how fast a PV anomaly is strengthening,
- how it is translating,
- and how much of its evolution comes from irreversible deformation.
It is designed to work seamlessly with reanalysis composites, making it suitable for both process studies and systematic climatological analysis.
This work was completed in Feb. 2026, with implementations tested on ERA5-based composites. Code for the orthogonal basis construction, projections, and visualization is available upon request.