Convergence of Zeitlin truncation for multilayer critical latitude predictions of quasi-geostrophic flow on an Earth-like planet

Arnout Franken; Sagy Ephrati; Erwin Luesink; Bernard Geurts

doi:10.3934/jcd.2026005

Article Contents

2026, Volume 14: 36-52. Doi: 10.3934/jcd.2026005

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Convergence of Zeitlin truncation for multilayer critical latitude predictions of quasi-geostrophic flow on an Earth-like planet

1.
Mathematics of Multiscale Modeling and Simulation, Faculty EEMCS, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
2.
Department of Mathematical Sciences, Chalmers University of Technology & University of Gothenburg, Göteborg, SE-412 96, Sweden
3.
Korteweg-de Vries Institute of Mathematics, Universiteit van Amsterdam, Amsterdam, The Netherlands
4.
Multiscale Physics, Faculty Technical Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
5.
Scientific Computing Group, Centrum voor Wiskunde en Informatica (CWI), Science Park 123, 1098 XG Amsterdam, The Netherlands

^*Corresponding author: Bernard Geurts
^*Corresponding author: Bernard Geurts

Received: August 25, 2025

Revised: January 16, 2026

Published online: March 5, 2026

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Abstract

Long-term predictions of large-scale flow features on an Earth-like planet are crucial for developing global atmospheric model systems. Such predictions require computational models, i.e., governing PDE systems combined with appropriate numerical methods, to preserve essential structures of the underlying physical model over extended simulation periods. We present a Lie–Poisson formulation of the multi-layer quasi-geostrophhic (QG) equations on the full globe, mimicking the dynamics in the troposphere extended over the first 10 km of the atmosphere. The chosen computational modeling ensures consistency with the underlying structure and enables long-term simulations without the need for additional regularization, forcing, or numerical dissipation. Recent advancements in Lie–Poisson discretization that preserve energy, enstrophy, and higher-order moments of potential vorticity are extended to stratified QG multilayer systems on the sphere. We adopt Zeitlin discretization, which yields a finite-dimensional dynamical system conserving all numerically resolved Casimirs with machine-precision. Particular attention is given to the convergence of critical latitude ($ \phi_{cl} $) predictions upon increasing the spatial resolution per layer ($ N $) and the number of layers in the model ($ M $). A systematic parameter study quantifies (ⅰ) that the dependency of $ \phi_{cl} $ on the resolution per layer $ N $ scales quadratically, showing near grid-independency for $ N \geq 96 $, and (ⅱ) the critical latitude decreases with the number of layers $ M $ for modest radial resolutions of $ M \leq 32 $.

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Mathematics Subject Classification: Primary: 76B70, 76E20, 76M60, Secondary: 86A10, 65F45, 65P10.

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Figure 1. Instantaneous velocity fields at $ N = 256 $ for the reference four-layer QG-model of the troposphere from the bottom to the top layer (a-d) after 1000 simulated days. Note that the amplitude of the flow structures decreases with increasing height of the layer; this is expressed by selecting different color-bars

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Figure 2. (a): Developing kinetic energy $ E_{kin} $ across the four layers of the reference model from bottom to top represented by layer 1 to layer 4, at a resolution per layer of $ N = 256 $ modes. The initial random state rapidly forms a statistically steady time-dependent solution in about 250 days; (b) Scaling of computational time $ T $ (in seconds, using a modern laptop) with the number of degrees of freedom $ N $ per horizontal layer. The measured simulation times are compared with linear (dash-dot) and quadratic (dash) scaling with $ N $, illustrating $ T\sim N^2 $

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Figure 3. Kinetic energy spectrum in the fully developed state after 1000 days, comparing results at $ N = 48 $ (a) with $ N = 256 $ (b). The bottom layer (solid) and the top layer (dash-dot) roughly provide an envelope for the spectrum of the two inner layers (dash)

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Figure 4. Median of the critical latitude $ \phi_{cl} $ of the reference four-layer system computed by averaging over 96 velocity fields, at different spatial resolutions 48-64-96-128-192-256. Averaged over all resolutions (dashed line), a value of $ \approx 44.9 $ degrees is found. The uncertainty in $ \phi_{cl} $ is expressed in terms of the interquartile range (IQR) shown as error bar. Overall, the accuracy with which the different randomly initiated simulations can be averaged over corresponds to an IQR of about 3 % for the three highest resolutions

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Figure 5. Velocity field in the first layer for models resolving the troposphere in $ M $ layers where $ M = 2^k $ for $ k = 0, 1, 2, ..., 5 $ at $ N = 96 $. The results at different $ k $ are shown in figures a, b, c, ..., f, respectively

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Figure 6. Critical latitude $ \phi_{cl} $ as a function of the number $ M $ of layers for $ M = 1, 2, 4, ..., 32 $, predicted at a layerwise resolution of $ 96 $

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