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arxiv: 2606.28303 · v1 · pith:NLKRMVY2new · submitted 2026-06-26 · 🧮 math.NA · cs.NA

A perfectly matched layer for damping vertically propagating waves in the compressible Boussinesq equations

Timothy C. Andrews , Kenneth Duru , David Lee This is my paper

Pith reviewed 2026-06-29 02:33 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords perfectly matched layerBoussinesq equationswave dampingatmospheric modelingfinite element methodacoustic wavesgravity waves
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The pith

The perfectly matched layer damps vertically propagating waves in the compressible Boussinesq equations without reflections at the damping layer onset.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes a new perfectly matched layer (PML) for the compressible Boussinesq equations to reduce wave reflections at the top of geophysical fluid models. Traditional sponge layers require many vertical levels or a high model top to be effective, but the PML is designed to be reflection-free at the continuous level where the damping begins. This property allows the PML to work well even with a thin damping layer. The authors derive PMLs for both linear and nonlinear versions of the equations, with the nonlinear one specifically damping perturbations around a hydrostatically balanced state, and test them using compatible finite elements.

Core claim

The paper claims that at the continuous level the PML is free of wave reflection at the onset of the damping layer for the Boussinesq system. This enables effective damping with thin layers. For the nonlinear system a novel formulation damps only perturbations from the reference state. Numerical tests confirm better performance than sponge layers in absorbing acoustic waves and avoiding standing waves from gravity wave reflections.

What carries the argument

The perfectly matched layer (PML) formulation that damps perturbations from a hydrostatically balanced reference state in the nonlinear Boussinesq equations, shown to be reflection-free at the continuous level.

Load-bearing premise

The PML equations remain stable and reflection-free after discretization by the compatible finite-element method, particularly when the nonlinear version damps only perturbations from the hydrostatically balanced reference state.

What would settle it

A numerical simulation of the discretized nonlinear Boussinesq system with a thin PML layer that exhibits measurable wave reflection or instability at the layer onset would falsify the central claim.

Figures

Figures reproduced from arXiv: 2606.28303 by David Lee, Kenneth Duru, Timothy C. Andrews.

Figure 1
Figure 1. Figure 1: Next-to-lowest order finite element function spaces used in this work. The vector Raviart [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A reference solution at t = 100 s of the linear Boussinesq test with a higher model top of 50 km. is weaker and the amplitudes of w are reduced, but there is still a noticeable spurious impact on the dynamics. With the PML, both the upwardly and downwardly propagating acoustic crests are successfully damped once they enter the PML, leaving no visual evidence of reflection. Accordingly, the solution is visu… view at source ↗
Figure 3
Figure 3. Figure 3: Vertical velocity (left column) and pressure (right column) fields from the linear Boussinesq [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: L2 vertical velocity error in the undamped domain, z ∈ [0, 18] km, for the linear Boussinesq test in simulations with no model top damping, a sponge layer, and a PML (with and without grid stretching). The vertical dashed lines indicate time intervals of 28.6 s, which is the time taken for the acoustic wave to travel 10 km. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Time series of energy components in the linear Boussinesq test, with no model top [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Time series of the PML power term, PPML (24), in the linear Boussinesq test. 4.2 Nonlinear Boussinesq equations We now move to the nonlinear Boussinesq equations, where we test the damping of both acoustic and orographic gravity waves. 4.2.1 Test setup We apply an analytical initial condition of u(x, z, t = 0) = u = [10, 0]T , (30a) b(x, z, t = 0) = b(z) = N 2 z, (30b) p(x, z, t = 0) = p(z) = N2 z 2 2 − p0… view at source ↗
Figure 7
Figure 7. Figure 7: Vertical velocity fields in the nonlinear Boussinesq acoustic test after 100 s (a) and the [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: L2 vertical velocity error in the undamped domain, z ∈ [0, 18] km, for the nonlinear Boussinesq acoustic test. Comparisons are made of simulations with no model top damping, a sponge layer, an unstretched PML with γ0 = 0 s, and a stretched PML with γ0 = 0.25 s. linear Boussinesq test, a nonzero vertical grid stretching factor reduces the error, although there is minimal difference between γ0 = 0 s and γ0 =… view at source ↗
Figure 9
Figure 9. Figure 9: Vertical velocities in the nonlinear Boussinesq orographic gravity wave test after 10,000 [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Investigating the integrated vertical kinetic energy of [PITH_FULL_IMAGE:figures/full_fig_p025_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The energy change from the PML in the nonlinear Boussinesq orographic gravity wave [PITH_FULL_IMAGE:figures/full_fig_p025_11.png] view at source ↗
read the original abstract

This paper introduces a new application of the perfectly matched layer (PML) for mitigating model top wave reflections in geophysical fluid models. Typically, a strong Laplacian or Rayleigh damping sponge layer is used near the upper boundary, but these often need many vertical levels or a high model top to be sufficiently effective. An advantage of the PML is that, at the continuous level, it is free of wave reflection at the onset of the damping layer. This enables the PML to be effective even with a thin damping layer. We derive PMLs for the linear and nonlinear versions of the Boussinesq equations, which are a simplified model for vertical dynamics in the atmosphere. In the nonlinear system, we define a novel PML that damps perturbations from a hydrostatically balanced reference state. We approximate the PML equations using the compatible finite element method for numerical experiments. First, tests with the linear Boussinesq system show that the PML is more effective than a typical sponge layer in absorbing acoustic waves near the model top. Next, tests in the nonlinear system show that i) the PML can damp acoustic waves even when they are under-resolved by the time discretisation, and ii) the PML can avoid the standing wave pattern caused by model top reflection of orographic gravity waves. We propose that the PML is worth further development and investigation as a sponge layer alternative in dynamical cores for atmospheric modelling.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper derives perfectly matched layer (PML) formulations for the linear and nonlinear compressible Boussinesq equations to absorb vertically propagating acoustic and gravity waves at the model top. At the continuous level the PML is constructed to be reflection-free at the interface with the physical domain, enabling effective damping with thin layers. For the nonlinear system a novel variant damps only perturbations from a hydrostatically balanced reference state. The equations are discretized with a compatible finite-element method and tested on linear acoustic-wave absorption and nonlinear orographic gravity-wave cases, where the PML outperforms a standard sponge layer.

Significance. If the continuous-level reflection-free property is preserved under the chosen discretization, the approach supplies a mathematically systematic alternative to ad-hoc Laplacian or Rayleigh sponges. This could reduce the vertical extent required for damping regions in atmospheric dynamical cores, lowering computational cost while maintaining hydrostatic balance in the reference state. The perturbation-damping construction for the nonlinear system is a concrete, reusable extension of standard PML techniques.

major comments (2)
  1. [Derivation sections (linear and nonlinear PML)] The central continuous-level claim (reflection-free onset) is standard for PML constructions and appears to follow from the usual complex-coordinate stretching applied to the Boussinesq system; however, the manuscript supplies no explicit verification (e.g., plane-wave analysis or energy-flux calculation) that the chosen stretching functions indeed cancel the reflected component at the interface for the compressible case.
  2. [Numerical experiments] Numerical experiments report qualitative improvement over sponges but provide no quantitative absorption metrics (reflection coefficient, transmitted energy, or L2-norm error against a reference solution with a higher model top). Without these, the claim that the PML remains effective for thin layers after discretization cannot be assessed.
minor comments (2)
  1. [Nonlinear PML formulation] The definition of the reference hydrostatic state and the precise form of the perturbation variables in the nonlinear PML should be stated explicitly (e.g., as an equation) rather than described only in prose.
  2. [Figures] Figure captions and axis labels for the wave-propagation snapshots should include the nondimensional time and the precise location of the PML interface so that the thin-layer claim can be visually verified.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive assessment and constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [Derivation sections (linear and nonlinear PML)] The central continuous-level claim (reflection-free onset) is standard for PML constructions and appears to follow from the usual complex-coordinate stretching applied to the Boussinesq system; however, the manuscript supplies no explicit verification (e.g., plane-wave analysis or energy-flux calculation) that the chosen stretching functions indeed cancel the reflected component at the interface for the compressible case.

    Authors: We agree that an explicit verification would strengthen the presentation. Although the PML is constructed via the standard complex stretching (which is known to yield a reflection-free interface for hyperbolic systems), we will add a brief plane-wave analysis confirming cancellation of the reflected component for the compressible Boussinesq equations in the revised manuscript. revision: yes

  2. Referee: [Numerical experiments] Numerical experiments report qualitative improvement over sponges but provide no quantitative absorption metrics (reflection coefficient, transmitted energy, or L2-norm error against a reference solution with a higher model top). Without these, the claim that the PML remains effective for thin layers after discretization cannot be assessed.

    Authors: We agree that quantitative metrics would allow a more rigorous assessment. In the revision we will add reflection-coefficient estimates and L2-norm errors relative to reference solutions on domains with extended vertical extent. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper constructs PML equations for linear and nonlinear Boussinesq systems by direct extension of standard PML techniques (complex coordinate stretching or auxiliary variables) to the target PDEs. The reflection-free property at the continuous level follows mathematically from the PML ansatz itself and is verified by the derivation rather than assumed or fitted. The novel nonlinear variant damps perturbations around a hydrostatic reference by explicit construction, not by redefining inputs. No self-citations are invoked to justify uniqueness or load-bearing steps; numerical tests with compatible FEM are independent validation. The central claim reduces to the standard PML property applied to this model, with no reduction to fitted parameters or self-referential definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Central claim rests on applicability of standard PML theory to Boussinesq equations without additional fitted constants beyond the damping profile; no free parameters or invented entities are identifiable from the abstract.

axioms (1)
  • domain assumption Boussinesq equations model vertical atmospheric dynamics
    Stated as the target system for PML application in the abstract.

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Reference graph

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