Spectral Exactness and Quantitative Instability Approximation in the Euler-Zeitlin Model for Zonal Equilibria
Abstract
Building on finite-time convergence results for geometric matrix hydrodynamics, we establish global spectral exactness for Zeitlin's approximation of the two-dimensional Euler equation on the sphere linearized about polynomial zonal vorticities. By separating each azimuthal block into a skew-adjoint transport operator and a compact perturbation, we obtain a dimension-uniform estimate |KN−K|=O(N−1) and the sharper fixed-block estimate |KN,m−Km|=O(N−2). These bounds, together with strong-resolvent convergence of transport, yield exact approximation of the isolated spectrum off the imaginary axis: algebraic multiplicities are preserved, Riesz projections converge in norm, and spurious off-axis spectral limits are excluded. Every algebraically simple off-axis eigenvalue has an O(N−2) approximation error. For eigenvalues of higher multiplicity we establish second-order convergence of arithmetic means of eigenvalue clusters and O(N−2/α) individual eigenvalue bounds, where α is the maximal Jordan-block size. For any prescribed positive growth margin, the numbers of unstable and stable eigenvalues beyond that margin are eventually correct. The argument combines an exact matrix-harmonic coordinate recurrence, uniform bounds for high azimuthal modes, and resolvent factorizations involving compact operators. An explicit degree-three zonal equilibrium has a rigorously enclosed simple unstable eigenvalue, providing a concrete instance of the quantitative theory.
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BibTeX
@misc{vrabel2026spectral,
title = {{Spectral Exactness and Quantitative Instability Approximation in the Euler-Zeitlin Model for Zonal Equilibria}},
author = {Robert Vrabel},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01664},
eprint = {2611.01664},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01664},
}
RIS
TY - UNPB
TI - Spectral Exactness and Quantitative Instability Approximation in the Euler-Zeitlin Model for Zonal Equilibria
AU - Robert Vrabel
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01664
UR - https://arxiv.org/abs/2611.01664
AB - Building on finite-time convergence results for geometric matrix hydrodynamics, we establish global spectral exactness for Zeitlin's approximation of the two-dimensional Euler equation on the sphere linearized about polynomial zonal vorticities. By separating each azimuthal block into a skew-adjoint transport operator and a compact perturbation, we obtain a dimension-uniform estimate |K_N−K|=O(N^−1) and the sharper fixed-block estimate |K_N,m−K_m|=O(N^−2). These bounds, together with strong-resolvent convergence of transport, yield exact approximation of the isolated spectrum off the imaginary axis: algebraic multiplicities are preserved, Riesz projections converge in norm, and spurious off-axis spectral limits are excluded. Every algebraically simple off-axis eigenvalue has an O(N^−2) approximation error. For eigenvalues of higher multiplicity we establish second-order convergence of arithmetic means of eigenvalue clusters and O(N^−2/α) individual eigenvalue bounds, where α is the maximal Jordan-block size. For any prescribed positive growth margin, the numbers of unstable and stable eigenvalues beyond that margin are eventually correct. The argument combines an exact matrix-harmonic coordinate recurrence, uniform bounds for high azimuthal modes, and resolvent factorizations involving compact operators. An explicit degree-three zonal equilibrium has a rigorously enclosed simple unstable eigenvalue, providing a concrete instance of the quantitative theory.
ER -