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arxiv: 2605.15451 · v1 · pith:VPDD4AGZnew · submitted 2026-05-14 · ⚛️ physics.flu-dyn

On the fundamental solution for viscous internal waves and Brinkman flows. Part 1. Two dimensions

Saikumar Bheemarasetty , Stefan G. Llewellyn Smith This is my paper

Pith reviewed 2026-05-19 14:29 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords internal wavesviscous flowsfundamental solutionsBrinkman flowsstratified fluidsasymptotic expansionsPrandtl numberdensity diffusion
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0 comments X

The pith

The fundamental solutions for viscous internal waves and Brinkman flows are single integrals with logarithmic singularities.

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

This paper derives the fundamental solutions for monochromatic internal waves that incorporate viscosity and density diffusion in a uniformly stratified fluid. It also obtains the corresponding solutions for anisotropic Brinkman flows in two dimensions. These representations appear as single integrals containing logarithmic singularities and support efficient numerical evaluation for boundary integral methods. For Prandtl numbers of order one or larger the work supplies a uniform asymptotic expansion that quantifies the attenuation and broadening produced by density diffusion.

Core claim

The viscous and diffusive fundamental solution for monochromatic internal waves in a uniformly stratified medium takes the form of a single integral with logarithmic singularities. A parallel integral form holds for anisotropic Brinkman flow. For Prandtl numbers greater than or equal to order one the wave field is a superposition of wave-like and Stokeslet-like terms, with density diffusion attenuating amplitude by the factor (1 + Pr^{-1})^{-2/3} and broadening beam width by the factor (1 + Pr^{-1})^{1/3}. Evanescent waves and anisotropic Brinkman flows display analogous integral structures, the latter remaining purely real and increasingly confined to the direction of least resistance as an

What carries the argument

The single-integral representation of the fundamental solution obtained by Fourier or integral-transform methods applied to the linearized viscous and diffusive governing equations.

If this is right

  • The integral forms permit efficient numerical quadrature and direct insertion into boundary integral formulations for solving more complex boundary-value problems.
  • Far-field asymptotics recover and extend earlier results such as those of Thomas and Stevenson for wave beams both inside and outside the characteristic angle.
  • Evanescent waves in stratified fluids and anisotropic Brinkman flows admit the same integral structure and exhibit single dominant circulation cells.
  • Increasing anisotropy in Brinkman flow progressively confines the circulation to the direction of least resistance.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The explicit scaling of beam width and amplitude with Prandtl number supplies a testable prediction for laboratory experiments that vary the diffusivity ratio.
  • These fundamental solutions can serve as kernels for modeling wave propagation through regions with spatially varying stratification or weak nonlinearity.
  • The two-dimensional integral construction offers a template for deriving analogous three-dimensional representations in a subsequent study.

Load-bearing premise

The medium is assumed to be uniformly stratified with constant buoyancy frequency and the governing equations are linearized for monochromatic small-amplitude waves.

What would settle it

Direct numerical evaluation of the single-integral expression for a chosen Prandtl number followed by comparison against a finite-difference solution of the linearized point-forced equations would confirm or refute the predicted amplitude and beam-width scalings.

Figures

Figures reproduced from arXiv: 2605.15451 by Saikumar Bheemarasetty, Stefan G. Llewellyn Smith.

Figure 1
Figure 1. Figure 1: Phase portraits of the roots of 𝛼 in the 𝑘-plane for 𝜔/𝑁 = 0.8. The colour bar corresponds to the phase angle. Viewed as an equation in 𝑚, this is a biquadratic with four roots in the complex plane: 𝑚1 = i √︄ 𝑘 2 − i + i √︁ 1 + 4i𝑘 2 (𝑁/𝜔) 2 2 , 𝑚2 = i √︄ 𝑘 2 − i − i √︁ 1 + 4i𝑘 2 (𝑁/𝜔) 2 2 , (2.12) along with −𝑚1 and −𝑚2. We pick the branch of square root in the complex plane with positive real part so as … view at source ↗
Figure 2
Figure 2. Figure 2: Singularities in the 𝜃-integral for two different observer locations. The circles corresponds to 𝛩𝑎 and the rectangles to 𝛩𝑑. The solid curves correspond to nonzero 𝐽, i.e. sgn 𝑑 arg 𝑎 ∈ (−𝜋/2, 0), and the dotted curves to 𝐽 = 0 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Phase portrait of G for 𝜔/𝑁 = 0.8 plotted against the dimensionless displacement from the origin x˜ = (x − x0) √︁ 𝜔/𝜈. for modest precision. The singularities can be removed using the results ∮ log [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Schematic of integration contours (black lines) in the 𝜁-plane and branch cut structure (red curves) of log (𝑞𝜁 + 𝜉 −1 1 𝜁 3 ) for 𝑞 ≶ 0. in the 𝜃-plane. The sgn 𝑑 term in (3.10) leads to a minus sign multiplying the first two contributions, while the absolute value in the exponent means that the first two terms have −ℎ(𝜏) there, while the third has ℎ(𝜏). In the 𝜁-plane, we obtain the intervals (0, √ 𝜉1∞),… view at source ↗
Figure 5
Figure 5. Figure 5: Top row: far-field behaviour as a function of 𝜆 for 𝜔/𝑁 = 0.8; (a) off beam, (b) near beam. Middle row (c): rescaled behaviour of numerical and asymptotic solutions near the beam (red: real parts, blue: imaginary parts). Bottom row (d): off-beam. asymptotic analysis of Thomas & Stevenson (1972), using a boundary-layer approach close to the internal wave angle 𝜃𝑎, does not provide a full picture of the wave… view at source ↗
Figure 6
Figure 6. Figure 6: Schematic of integration contours (black lines) in the 𝜁-plane and branch cut structure (red curves) of log (𝑞 3/2 𝜁 − 𝜙(𝑞)𝜀 2 𝜁 3 ) for 𝑞 ≶ 0. The branch cuts extend to infinity along {e i𝜋/6 , e i5𝜋/6 , e i3𝜋/2 } for 𝑞 > 0 and {e i𝜋/3 , e i𝜋 , e −i𝜋/3 } for 𝑞 < 0. when 𝑞 < 0. The total local contributions from {𝜃𝑎, 𝜋 + 𝜃𝑎} become I𝑙,𝑎 + J𝑙,𝑎 = 𝜀 2 2𝜋i f(𝜃𝑎) 𝜉1 (𝜃𝑎) ∫ −i∞ 0 𝜁 e 𝑞𝜁 −𝜀 2𝜙(𝑞) 𝜁 3 d𝜁, (4.40) … view at source ↗
Figure 7
Figure 7. Figure 7: Evanescent response for unit positive vertical forcing. Top row: real part, bottom row: imaginary part. Left to right: 𝜔/𝑁 = 1.1, 1.5, 2. Colour: log |𝑢3 |. Black lines: streamlines with the flow direction indicated by arrows. 6. Green’s function for finite Pr 6.1. Integral representation With non-zero density diffusion, the integral representation of the Green’s function takes the form G = ∮ f(𝜃) d𝜃 ∫ ∞ 0… view at source ↗
Figure 8
Figure 8. Figure 8: Rescaled Green’s function component 𝜆 2 Im 𝐺11 for 𝜔/𝑁 = 1.1, 2 and𝜆 = 10, 30, as a function of 𝜃𝑑. Solid lines: asymptotic approximation from (5.5); broken lines: numerical results. Using partial fractions, the 𝜅-integral takes the form 𝐾 𝐷 = ∫ ∞ 0 " ℎ1 (𝜃; Pr) 𝜅 2 + 𝑎 2 1 (𝜃; Pr) + ℎ2 (𝜃; Pr) 𝜅 2 + 𝑎 2 2 (𝜃; Pr) # 𝜅e i𝜆𝑑( 𝜃 ) 𝜅 d𝜅, (6.2) where the quantities 𝑎1,2 (𝜃; Pr) and ℎ1,2 (𝜃; Pr) are given by 𝑎 2… view at source ↗
Figure 9
Figure 9. Figure 9: Phase portrait of 𝐺𝑖 𝑗(x, x0) for 𝜔 = 0.8, 𝑁 = 1 and Pr = 0.01, 0.1, 0.7, 10 (colour bar: phase angle). 0 X0-19 [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Anisotropic Brinkman solution for unit positive vertical forcing. Colour: log |𝑢3 | with 𝜒1 = 1 and 𝜒3 = 1, 5 and 10. Black lines: streamlines with the flow direction indicated by arrows. Without loss of generality we take 𝑎 = √︁ 𝜒1𝑠 2 + 𝜒3𝑐 2 > 0. The complex zeros of 𝑎(𝜃) are situated at 𝛩𝑎 = ( 𝜋 2 ± i tanh−1  𝜒1 𝜒3 1/2 , − 𝜋 2 ± i tanh−1  𝜒1 𝜒3 1/2 ) , (7.8) while 𝑑(𝜃) has real zeros at 𝜃 = 𝜃𝑑, 𝜋 +… view at source ↗
Figure 11
Figure 11. Figure 11: Rescaled Green’s function component 𝜆 2𝐺11 for 𝜒1 = 1, 𝜒3 = 5, 𝜆 = 20, 50, as a function of 𝜃𝑑. Solid line: exact result, broken lines: asymptotic approximation. giving 𝐼 𝐵 𝑔 ∼ 4𝜋Im       ∑︁∞ 𝑞=1 𝜀 2𝑞 𝛤(2𝑞) 𝛤(𝑞) d (2𝑞−1) d𝜃 2(𝑞−1)  (𝜃 − 𝜃𝑎) 𝑞 f(𝜃) 𝑎 2𝑞 (𝜃)𝑑 2𝑞 (𝜃)  𝜃=𝜃𝑎       , (7.10) where 𝜃𝑎 is one of the zeros of 𝑎(𝜃) in the upper half-plane. The leading term is I 𝐵 𝑔 ∼ 𝜀 2 4𝜋 (𝜒1 − 𝜒3) f… view at source ↗
read the original abstract

We obtain the viscous and diffusive fundamental solution for monochromatic internal waves in a uniformly stratified medium and for anisotropic Brinkman flow. These solutions take the form of single integrals with logarithmic singularities, and can be computed numerically in an efficient manner for possible use in boundary integral methods. Far-field asymptotic results are obtained, giving solutions valid far from and inside a ``beam'' corresponding to the internal wave angle in the internal wave case, consistent with Thomas & Stevenson (1972). For Prandtl numbers $\text{Pr} \gtrsim O(1)$, the wave field is given by a superposition of wave- and Stokeslet-like terms. Unlike previous studies, a uniform asymptotic expansion of the wave-field for $\text{Pr} \gtrsim O(1)$ can be computed rigorously. Density diffusion attenuates the wave amplitude as to $(1+\text{Pr}^{-1})^{-2/3}$ and broadens the beam width according to $(1+\text{Pr}^{-1})^{1/3}$. Evanescent waves in a stratified medium and anisotropic Brinkman flows have similar behaviour. Anisotropic Brinkman flow is purely real, dominated by a single circulation cell. As anisotropy increases, the flow becomes increasingly confined to the direction with least resistance. The stratified evanescent wave field has near-vertical cells in its real part, and a dominant single circulation cell in its imaginary part.

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

1 major / 2 minor

Summary. The manuscript derives the viscous and diffusive fundamental solutions for monochromatic internal waves in a uniformly stratified medium and for anisotropic Brinkman flows in two dimensions. These solutions are expressed as single integrals containing logarithmic singularities, suitable for efficient numerical evaluation and boundary integral methods. Far-field asymptotics are developed, including a uniform asymptotic expansion for Pr ≳ O(1) that superposes internal-wave beam and Stokeslet-like contributions, with density diffusion producing amplitude attenuation by the factor (1 + Pr^{-1})^{-2/3} and beam broadening by (1 + Pr^{-1})^{1/3}. The work also treats evanescent waves and shows increasing flow confinement with anisotropy in the Brinkman case.

Significance. If the central derivations hold, the single-integral representations and the rigorously derived uniform asymptotics would supply a practical analytical and computational tool for modeling viscous internal waves and anisotropic flows. The explicit Pr-dependent scalings and consistency with Thomas & Stevenson (1972) far-field observations add value for applications in stratified fluid dynamics. The absence of free parameters or post-hoc fitting in the scalings is a methodological strength.

major comments (1)
  1. [uniform asymptotic expansion for Pr ≳ O(1)] In the derivation of the uniform asymptotic expansion for Pr ≳ O(1), the treatment of the logarithmic singularity inside the beam requires an explicit error estimate. After the Pr-dependent rescaling of the stationary-phase contour or inner variable, the manuscript should demonstrate (e.g., via integration by parts or contour analysis) that the branch of the log term is preserved and the singular contribution cancels uniformly across the beam width; without such a bound the claimed exponents for amplitude attenuation and beam broadening remain formal rather than rigorously controlled.
minor comments (2)
  1. [Abstract] The abstract asserts that the integrals 'can be computed numerically in an efficient manner'; a short description of the quadrature scheme or singularity-handling technique in the main text would improve reproducibility.
  2. [governing equations and integral representation] Notation for the complex vertical wavenumber and the precise definition of the logarithmic argument should be stated explicitly when the integral kernel is first introduced.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and for the constructive comment on strengthening the rigor of the uniform asymptotic expansion. We address the point below and will incorporate the requested analysis in the revision.

read point-by-point responses

  1. Referee: In the derivation of the uniform asymptotic expansion for Pr ≳ O(1), the treatment of the logarithmic singularity inside the beam requires an explicit error estimate. After the Pr-dependent rescaling of the stationary-phase contour or inner variable, the manuscript should demonstrate (e.g., via integration by parts or contour analysis) that the branch of the log term is preserved and the singular contribution cancels uniformly across the beam width; without such a bound the claimed exponents for amplitude attenuation and beam broadening remain formal rather than rigorously controlled.

    Authors: We agree that an explicit error estimate for the logarithmic singularity after the Pr-dependent rescaling would make the uniform asymptotic expansion fully rigorous. In the revised manuscript we will add a dedicated paragraph (or short subsection) performing integration by parts on the rescaled contour integral. This will show that the branch of the logarithm is preserved and that the singular contribution is bounded by a term that is uniformly small across the beam width, with the remainder being of higher order in the far-field parameter. The analysis will thereby confirm that the amplitude attenuation factor (1 + Pr^{-1})^{-2/3} and the beam-broadening factor (1 + Pr^{-1})^{1/3} are rigorously controlled rather than formal. revision: yes

Circularity Check

0 steps flagged

Derivation from governing equations via integral transforms is self-contained

full rationale

The paper constructs the viscous and diffusive fundamental solutions directly from the linearized governing equations for monochromatic internal waves and anisotropic Brinkman flow using Fourier or integral-transform methods, yielding single-integral representations with logarithmic singularities. Far-field asymptotics, including the uniform expansion for Pr ≳ O(1) with explicit factors (1+Pr^{-1})^{-2/3} for amplitude attenuation and (1+Pr^{-1})^{1/3} for beam broadening, are obtained by stationary-phase analysis and superposition of wave and Stokeslet terms applied to these integrals. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the scalings emerge from the Pr-dependent rescaling of the contour and inner variable in the asymptotic analysis of the integral kernel. The derivation remains independent of external fitted data or prior results by the same authors.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard linearised Navier-Stokes equations with buoyancy and diffusion terms for a uniformly stratified fluid, plus the anisotropic Brinkman equations; no additional free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Linearised governing equations for monochromatic internal waves in a uniformly stratified medium with constant buoyancy frequency and constant viscosity and diffusivity coefficients.
    Invoked when constructing the fundamental solution via integral transforms.
  • domain assumption Anisotropic Brinkman equations with direction-dependent permeability coefficients.
    Used to obtain the purely real fundamental solution for the porous-media flow.

pith-pipeline@v0.9.0 · 5791 in / 1600 out tokens · 57480 ms · 2026-05-19T14:29:09.053162+00:00 · methodology

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