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High-order Simulation of Nearly Hydrostatic Atmospheric Flows
Shoyon Panday, Stéphane Gaudreault, Carlos Pereira, Philipp Birken
TL;DR
Low-Mach atmospheric motions and small perturbations around hydrostatic balance are difficult to simulate when numerical dissipation overwhelms the motions or discretization errors disturb equilibrium. This study combines DFR, exponential time integration, a modified AUSM+-up flux, and a deviation formulation; across the reported tests, the flux better preserves low-Mach structures, while the deviation method improves hydrostatic balance and robustness.
Problem
Low-Mach atmospheric motions can be excessively damped by conventional upwind fluxes, while discretization errors can generate spurious motion around hydrostatic balance.
Source: §1 · pp. 1–2Method
The study combines high-order DFR and exponential time integration with AUSM+-upH for low-Mach accuracy and a deviation formulation to preserve hydrostatic balance.
Source: §1 · p. 2, §6 · p. 17Results
AUSM+-upH preserves low-Mach structures more effectively than Rusanov, but is more sensitive to residual hydrostatic imbalance; the deviation formulation reduces these errors and improves robustness across the tested cases.
Source: §6 · pp. 17–18Takeaways & Limitations
Low-Mach correction and well balancing are complementary components for simulating compressible atmospheric flows with weak motions near hydrostatic balance.
Source: §6 · p. 18
Abstract
from arXiv · showhide
Accurate simulation of atmospheric flows is challenging. Low-Mach-number dynamics often need to be resolved in the presence of a dominant, nearly hydrostatic background state. Conventional upwind numerical fluxes can introduce excessive acoustic-speed-based dissipation in the low-Mach regime. Discretization errors in the balance law can also generate spurious motion. This work investigates these two issues within a high-order Direct Flux Reconstruction (DFR) framework coupled with a third-order exponential time integrator for the 3D compressible Euler equations with gravity. Low-Mach-number accuracy is addressed using a modified AUSM$^+$-up flux, denoted AUSM$^+$-upH. Preservation of the background equilibrium is improved through a deviation-based formulation. The resulting approach is assessed using a hierarchy of idealized atmospheric test cases. The results show that AUSM$^+$-upH preserves low-Mach flow structures more effectively than the more dissipative Rusanov flux. However, reduced numerical dissipation also makes AUSM$^+$-upH more sensitive to residual errors in the discrete hydrostatic balance. The deviation formulation substantially suppresses these errors by several orders of magnitude, with the strongest improvement observed for flows over orography. Overall, AUSM$^+$-upH exhibits greater robustness when combined with the deviation formulation across the atmospheric test cases considered.
1 Introduction
Atmospheric simulation must resolve weak, low-Mach motions while maintaining a nearly hydrostatic background, despite numerical dissipation and discretization errors that can distort them. The study combines low-Mach and well-balancing techniques in a high-order DFR framework for the 3D compressible Euler equations with gravity.
Motivation and context
- High-order methods are attracting NWP interest for accuracy, dispersion and dissipation properties, and efficient use of modern computing architectures.The study situates its approach among prior low-Mach and well-balancing work for Euler equations with gravity.Source: §1 · pp. 1–2
Low-Mach challenge
- Low-Mach compressible schemes face stiffness from fast acoustic waves and excessive upwind dissipation that can prevent recovery of the incompressible limit.As Mach number decreases, artificial diffusion can dominate even when physical flow scales remain unchanged.Source: §1 · p. 2
Hydrostatic balance
- Small perturbations around hydrostatic equilibrium can be comparable to or smaller than discretization error, allowing numerical errors to generate spurious departures from balance.Maintaining this stratified state over long periods and large domains is essential in atmospheric simulation.Source: §1 · pp. 1–2
Study approach
- A modified AUSM+-up flux, AUSM+-upH, is combined with exponential time integration and a deviation formulation in a DFR framework to address low-Mach accuracy, acoustic stiffness, and hydrostatic balance.The study assesses the resulting framework using idealized atmospheric test cases.Source: §1 · p. 2
2 Governing Equations
The governing system is the compressible Euler equations with gravity, expressed in conservative form and closed by a thermodynamic equation of state. The formulation uses potential temperature as a prognostic variable and assumes a dry, inviscid atmosphere without diabatic heating.
Governing system
- The three-dimensional compressible Euler system includes a gravity source term, with density, potential temperature, and the three velocity components among its conserved variables.Pressure and gravity are specified, with gravity acting in the negative z-direction.Source: §2 · pp. 2–3
Thermodynamic formulation
- The flux vectors contain mass, momentum, and potential-temperature fluxes, and the system is closed using an equation of state and a definition of potential temperature.Potential temperature, rather than total energy, is the prognostic thermodynamic variable.Source: §2 · p. 3
Assumptions
- The model assumes a dry, inviscid atmosphere with no diabatic heating.Source: §2 · p. 3
3 Numerical Method
The numerical method combines Direct Flux Reconstruction (DFR), third-order exponential time integration, and numerical interface fluxes. Its AUSM+-upH variant modifies AUSM+-up for nearly hydrostatic atmospheric flows while retaining low-Mach scaling of velocity-dependent dissipation.
3.1 Direct Flux Reconstruction
- DFR constructs a continuous flux polynomial within each element from interior flux values and numerical boundary fluxes, then differentiates it to form the semi-discrete system.The method uses Gauss-Legendre solution points and extends the one-dimensional basis by tensor products on regular multidimensional grids.Source: §3.1 · pp. 3–4
3.2 Time Integration
- The third-order EPI3 exponential integrator treats linear dynamics exactly, improving stability over explicit methods while avoiding the large nonlinear solves required by implicit methods.Exponential integration can propagate purely linear modes without artificial damping or phase errors.Source: §3.2 · pp. 4–5
3.3 Numerical Flux Functions
- The numerical flux defines a common interface flux from left and right extrapolated states; the study compares Rusanov with AUSM+-up.Source: §3.3 · p. 5
3.3.1 Rusanov Flux
- Rusanov dissipation scales with the largest characteristic speed, so as Mach number decreases its acoustic velocity scale can dominate the physical motion.The resulting pressure fluctuations can exceed the physical O(M^2) perturbation, contributing to low-Mach damping and loss of accuracy.Source: §3.3.1 · pp. 5–6
3.3.2 AUSM+-up Flux
- AUSM+-up separates advective and pressure contributions, with Mach-number scaling that reduces dissipation toward the incompressible limit while retaining pressure-velocity coupling.For AUSM+-upH, the p-term factor is set to unity to remove cutoff dependence, while Mach-number-dependent scaling is retained in the u-term; this modification follows hydrostatic flux tests.Source: §3.3.2 · pp. 6–7
4 Deviation Method
The deviation method evolves perturbations relative to a prescribed target state instead of evolving the full solution directly. This removes the large background balance from the residual and preserves a solution that matches the target exactly.
Motivation
- Discrete flux-gradient and gravity-source approximations may fail to cancel exactly, contaminating the small true imbalance in nearly hydrostatic flows and generating spurious motion.The method addresses this mismatch by evolving deviations from a prescribed target solution.Source: §4 · pp. 7–8
Formulation
- The deviation formulation subtracts the balance law evaluated at a prescribed reference solution from the original balance law and evolves the resulting difference.The target may be stationary or a known time-dependent solution.Source: §4 · pp. 7–8
Preservation property
- When the computed solution equals the target, the deviation and its flux and source differences vanish, so the reference state is preserved exactly.For small perturbations, removing the large background balance lets their dynamics primarily drive the numerical evolution.Source: §4 · p. 8
5 Numerical Experiments
The experiments test low-Mach accuracy and hydrostatic-balance preservation in five atmospheric cases, spanning 2D Cartesian and 3D cubed-sphere simulations. Across the cases, AUSM+-upH better preserves weak-flow structures, while the deviation formulation reduces errors from discrete imbalance, especially over orography.
5.1 Gresho Vortex
- AUSM+-upH maintains good Gresho-vortex accuracy across Mach numbers, whereas Rusanov becomes increasingly diffusive as Mach decreases.At Mmax = 10^-3, Rusanov develops a cross-shaped pattern; its resolution requirement is dictated by Mach number.Source: §5.1 · pp. 9–10, Figure 4 · p. 11
5.2 Isothermal Hydrostatic Equilibrium
- For first-order discretization, the deviation formulation reduces spurious hydrostatic-state velocities by several orders of magnitude for both fluxes; at third order, all methods remain near machine precision.The difference between using and omitting the deviation formulation is negligible at third order.Source: §5.2 · p. 11, Figure 5 · p. 12
5.3 Rising Thermal Bubble
- AUSM+-upH better preserves the rising thermal bubble’s weak buoyancy-driven structure, while Rusanov smooths the perturbation more strongly.No clear visual difference appears with versus without the deviation formulation in this case.Source: §5.3 · pp. 12, 14, Figure 6 · p. 13
5.4 Steady-State Atmosphere over Orography
- Without the deviation formulation, both fluxes generate growing spurious velocities and fail before six days; with it, both remain stable and Vmax is comparable at ∼10^-6 m s^-1.AUSM+-upH develops larger errors earlier without the formulation, consistent with its lower intrinsic dissipation.Source: §5.4 · p. 14, Figure 7 · p. 15
5.5 Non-Hydrostatic Gravity Waves
- In the gravity-wave test, deviation well-balancing removes AUSM+-upH’s spurious background noise and yields smooth contours with both fluxes; all configurations reproduce similar overall phase and amplitude.Without deviation well-balancing, AUSM+-upH has jagged outer wave-train contours and slight outer-lobe profile departures.Source: §5.5 · p. 16, Figure 9 · p. 16, Figure 10 · p. 17
6 Conclusion
Across the atmospheric tests, AUSM+-upH better preserves low-Mach flow structures, while the deviation formulation improves hydrostatic balance and robustness. The results show that low dissipation and well balancing are complementary, particularly for nearly hydrostatic flows.
6 Conclusion
- AUSM+-upH preserves low-Mach vortex structures more effectively than Rusanov, but residual hydrostatic imbalances can be amplified without the deviation method.The deviation method is needed to preserve hydrostatic balance accurately; Rusanov's numerical diffusion tends to damp the residual perturbations.Source: §6 · p. 17
- For the rising thermal bubble, deviation has limited influence on resolved dynamics, while AUSM+-upH retains a sharper structure than the more dissipative Rusanov flux.Source: §6 · p. 17
- Over orography, deviation reduces spurious-velocity errors by several orders of magnitude and lets both fluxes remain stable throughout the full integration period.Without deviation, both fluxes generate growing spurious velocities and fail before the intended integration is complete.Source: §6 · pp. 17–18
- In the gravity-wave test, configurations yield similar propagation, phase, and amplitude, but deviation lets AUSM+-upH capture the wave structure without relying on excessive diffusion.Without deviation, AUSM+-upH develops small-scale artefacts attributed to discretization errors against the larger hydrostatic background; Rusanov suppresses them through greater dissipation.Source: §6 · p. 18
- Low dissipation helps preserve weak atmospheric motions but exposes discretization errors, making well balancing a complementary component of low-Mach correction for nearly hydrostatic flows.Together with high-order DFR and exponential time integration, these techniques form a promising framework for flows with fast acoustic modes and slowly evolving, nearly balanced dynamics.Source: §6 · p. 18