On physical optics approximation of viscous stratified shear flows

N. Karjanto

REVIEW 3 major objections 5 minor 115 references

On physical optics approximation of viscous stratified shear flows

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read WKB solution tracks internal waves in viscous stratified shear flows

desk verdict Useful WKB analysis of a viscous Taylor-Goldstein variant, but the printed reduction from the fourth-order equation contains an algebraic slip that changes the ODE being solved. read the letter →

arxiv 1908.05457 v2 pith:SGHTCH67 submitted 2019-08-15 physics.ao-ph physics.flu-dyn

N. Karjanto This is my paper · ORCID

classification physics.ao-phphysics.flu-dyn MSC 76E0534E2076B70

keywords stratifiedshearfloweddyviscosityTaylor-GoldsteinequationWKBmethodphysicalopticsapproximationturningpointcriticallevelinternalgravitywaves

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper derives an approximate solution for the vertical velocity of internal gravity waves in a stratified shear flow when horizontal eddy viscosity is present but vertical eddy viscosity is ignored. Starting from the Navier-Stokes equations under the Boussinesq approximation, it reduces the perturbation equation to a modified Taylor-Goldstein equation of second order, then applies WKB theory to obtain a physical optics solution valid when the background wind and buoyancy frequency vary slowly. It also characterizes the solution near a turning point, where it is governed by Airy functions, and near a critical level, where a Frobenius series with oscillatory behavior applies. If correct, the result gives a closed-form way to study wave propagation and instability without solving the full fourth-order viscous problem.

What carries the argument

The load-bearing object is the complex-valued potential $Q_\varepsilon(Z)$ in (16) and the WKB ansatz (17). The argument proceeds by separating $Q_\varepsilon$ into real and imaginary parts, writing the WKB exponent $M = M_0 + \varepsilon M_1 + \dots$, and solving the lowest-order equations to obtain $m_0$ and $n_0$; the first-order correction supplies the amplitude factor $\sqrt{|M_0(0)/M_0(z)|}$ and phase correction $\tfrac12 i(\alpha-\alpha_0)$ that convert the geometric optics plane wave into the physical optics solution (18). Near singular points, the same equation is locally approximated by the Airy equation at a turning point and by a regular singular equation with indicial exponents at a critical level.

What would settle it

Compute $\varepsilon$ for a realistic atmospheric wind profile with strong shear, such as a jet with a sharp curvature, using the paper's formulas for $M_0$; if $\varepsilon$ is not much smaller than 1, the asymptotic solution (18) is not valid and will deviate from a direct numerical integration of the modified Taylor-Goldstein equation (15) for that profile.

Watch

Extended reading notes

Core claim

The central claim is that the modified Taylor-Goldstein equation (15) with potential (16) admits the physical optics WKB solution (18), provided $\varepsilon = \max_{z\ge 0}|M_0'(z)/M_0^2(z)| \ll 1$. The solution expresses the vertical velocity perturbation as an amplitude-corrected complex exponential whose phase and amplitude are determined by the leading-order WKB exponent $M_0$ and its integral, with the ratio $m_0/n_0$ entering through a phase shift $\alpha$. Near a turning point $z_0$ where $N^2 = k^2(U-c)(U-c-ikA_H)$, the governing equation reduces to the Airy equation (20), giving the decaying branch $\operatorname{Ai}$; near a critical level $z_c$ where $U(z_c)=c_r$, the solution is a Frobenius series (29) with characteristic exponents $\lambda = \tfrac12 \pm \rho_c e^{i\varphi_c}$, oscillating infinitely rapidly and converging for all finite distances from the critical level.

Load-bearing premise

The load-bearing premise is that the WKB slow-variation condition $\varepsilon = \max_{z\ge 0}|M_0'(z)/M_0^2(z)| \ll 1$ holds for the atmospheric profiles of interest, but the paper only demonstrates this for one illustrative profile with weak shear.

Editorial extensions

If this is right

The physical optics solution (18) gives a closed-form expression for the vertical velocity perturbation that can be evaluated without solving the fourth-order viscous equation numerically.
The validity condition $\varepsilon = \max_{z\ge 0}|M_0'/M_0^2| \ll 1$ provides a quantitative criterion for when horizontal eddy viscosity can be treated by WKB methods.
Near a turning point the solution decays as an Airy function on one side, which describes evanescent behavior of internal gravity waves in a viscous shear flow.
Near a critical level the Frobenius solution oscillates infinitely rapidly and its amplitude vanishes as $z \to z_c$, consistent with wave absorption at critical levels.
The matrix-exponential formulation with the WKB-constructed fundamental matrix supplies a propagator that can be used in boundary-value solvers for the viscous problem.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The analysis is restricted to horizontal eddy viscosity; because vertical eddy viscosity often dominates in the atmospheric boundary layer, extending the WKB treatment to $A_T \neq 0$ is the natural next test of whether the asymptotic forms survive.
The infinite oscillations of the Frobenius solution near the critical level suggest that the WKB approximation connects to over-reflection and absorption phenomena, which could be tested by computing wave-action flux across the critical level.
The $\varepsilon$ criterion could be used to classify realistic wind profiles, flagging regions where WKB fails and where a numerical matching method such as the RKWKB integrator is required.
The local Airy and Frobenius solutions could in principle be matched to build a global uniform approximation, giving connection formulas that the paper does not derive.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a legitimate asymptotic exercise, but as written the paper has a load-bearing algebra error. The stress-test note is right about the reduction from Eq. (8) to Eqs. (9)–(11). Setting AT=0 and dividing by u2 gives Q1 = -ik AH'/u2 and Q0 = (N^2/u1 - U'')/u2 - k^2, not the printed Q1 = -ik AH/u2 and Q0 = (N^2/u1 - U'')/u2. The constant-coefficient limit is a clean check: Eq. (8) reduces to w'' - k^2 w = 0, while Eqs. (9)–(11) give a different ODE with an AH-dependent first-derivative term. That is not a cosmetic typo; it is the equation the paper claims to solve.

What is genuinely good: the derivation from the Navier-Stokes-like momentum equation to the fourth-order ODE is standard and looks correct. The WKB setup is competent, the turning-point reduction to Airy's equation is textbook, and the Frobenius convergence proof is a real contribution—most papers in this space just cite the series without showing it converges. The literature review is also thorough.

Where it gets softer. First, the printed Eqs. (10)–(11) are inconsistent with Eq. (16): the latter contains (dot AH)^2, which only arises from Q1 = -ik AH'/u2, not from the printed Q1. So either (10)–(11) are typos and (16) is right, or vice versa. The paper never acknowledges the discrepancy, and a reader using (9)–(11) will be solving the wrong equation. Second, the scaling in Eq. (16) looks off. With Z = epsilon z and dot = d/dZ, the epsilon^2/u2 factor produces terms that are epsilon^4 in the original variable; the correct Q2 should have the AH-derivative terms at order 1 but with 1/epsilon^2 in the scaled equation. The paper needs to sort out whether it is solving (13) or a rescaled version. Third, there is no numerical check of the WKB solution against a direct integration of the ODE—for an applied-math paper that is a significant omission. Finally, the critical-level conclusion says the solution "vanishes," while the analysis itself shows indefinite oscillation with no limit; that is misleading.

Bottom line: the intended result is plausible and the machinery is sound, but the paper as printed cannot be used without fixing the reduction error and the scaling. It deserves a serious referee, not a desk reject, because the derivation from (8) is correct and the asymptotic framework is standard. I would send it to review with a strong request for major revision, and I would not cite it in its current form.

Referee Report

3 major / 5 minor

Summary. The paper studies internal gravity waves in a stratified shear flow with horizontal eddy viscosity. The authors reduce the fourth-order viscous Taylor-Goldstein equation to a second-order modified Taylor-Goldstein equation when vertical viscosity is absent, then apply a Liouville transformation and a WKB (physical optics) expansion under a slow-variation assumption. They present a leading-order WKB solution, a turning-point analysis giving Airy functions, and a critical-level analysis using Frobenius series. The central claim is that Eq. (18) is an asymptotic solution of the modified Taylor-Goldstein equation, and that near turning points and critical levels the solutions are respectively Airy functions and Frobenius series.

Significance. If the results were correct, the paper would provide useful analytic approximations for vertically propagating internal waves in a viscous stratified shear flow, complementing numerical studies. The paper includes a standard WKB derivation, a worked illustrative example, and a convergence proof for the Frobenius series, which are positive features. However, the significance is severely undercut by fundamental errors in the derivation of the governing equation: the modified Taylor-Goldstein equation as printed is not the reduction of Eq. (8), and the potential used in the WKB analysis does not correspond to the normal form of the physical model.

major comments (3)

[Section 2.2, Eqs. (9)-(11)] Setting AT=0 in Eq. (8) gives, after division by u2, the second-order equation w'' - (ik AH'/u2) w' + [(N^2/u1 - U'')/u2 - k^2] w = 0. The printed coefficients Q1 = -ik AH/u2 and Q0 = (N^2/u1 - U'')/u2 are therefore incorrect: Q1 is missing the derivative on AH, and Q0 is missing the -k^2 term. A simple constant-coefficient check with N=0, U=const, AH=const confirms the discrepancy: the physical equation reduces to w'' - k^2 w = 0, whereas Eqs. (9)-(11) give a nonzero first-derivative term. Since Q1 enters the Liouville transformation (12) and hence the potential (16), this error propagates into the central WKB result (18).
[Section 2.2, Eq. (16)] Even if Q0 and Q1 are corrected as above, the potential Qε in Eq. (16) is not the result of the Liouville transformation (12) followed by the rescaling Z=εz. Direct computation from the corrected Q1 = -ik AH'/u2 and Q0 = (N^2/u1 - U'')/u2 - k^2 gives a Qε whose ε^2 terms contain additional factors of u2 in the denominators and different signs than the printed (16). Consequently, Eq. (15) with the printed Qε is not the normal form of the physical equation derived from Eq. (8), and the WKB solution (18) is an asymptotic solution of a different equation than the stated viscous stratified shear flow model.
[Section 4.2, Eqs. (24)-(25)] The critical-level analysis is inconsistent with Eq. (13). The potential Q2 in (13) includes the contributions -Q1^2/4 and -Q1'/2, which for the non-constant viscosity AH(z) produce terms of order ζ^{-2} in the expansion near the critical level. These terms are absent from the printed Q22 in Eq. (25), so the indicial exponents and the oscillatory Frobenius solution (29) are not those of the modified Taylor-Goldstein equation with horizontal viscosity. This makes the critical-level results invalid for the model the paper claims to solve.

minor comments (5)

[Section 3.1, definition of ε] The expression for ε after Eq. (18) has denominator A^2+B^2, but the correct denominator is |M0|^2 = sqrt(A^2+B^2). This affects the numerical estimate ε ≤ 0.3 in the example.
[Section 3.2, definition of q(z)] The formula 'q(z) = M0(z) + ε p(z) / (2|M0(z)|^2)' is ambiguous; it should be clarified whether the division by 2|M0|^2 applies to ε p(z) only or to the whole sum.
[Section 2.2, notation] The symbols u3 and u4 in Eq. (16) are introduced without explanation; they should be defined as the coefficients of the ε^2 correction in the potential.
[Section 4.1, turning point condition] The turning point condition N^2 = k^2 u1u2 uses only the leading-order part of Qε; the ε^2 terms are neglected without an explicit statement that this is a leading-order approximation.
[Introduction and Abbreviations] There is a typo in the historical discussion: 'Jeffrey' should be 'Jeffreys', and 'the trios WKB' should be 'the trio'. In the Abbreviations list, 'Kevin-Helmholtz' should be 'Kelvin-Helmholtz'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the WKB result is a standard asymptotic derivation from the stated ODE, with no fitted parameter or self-citation used as load-bearing input.

full rationale

The paper's central derivation takes the second-order ODE (15) with potential (16), adopts the WKB ansatz (17), and obtains equations at successive orders in epsilon by substitution and coefficient matching. This is a normal asymptotic expansion: the small parameter epsilon is defined a posteriori as max |M'(z)/M^2(z)| as a validity condition, not fitted to data, and Eq. (18) follows from solving the resulting eikonal and transport equations rather than from restating an input. The turning-point section reduces the same ODE to the Airy equation (20) by Taylor expansion about z0, and the critical-level section uses a Frobenius expansion (28)-(30) about the regular singular point; both are independent standard manipulations of the same equation, not circular imports. The paper contains no substantive self-citations that carry the derivation, and no external result is invoked to force the choice of WKB form. Any algebraic discrepancy in the coefficients (10)-(11) relative to Eq. (8) is a correctness or internal-consistency concern, not a circularity: the derivation is not equivalent by construction to its inputs. Therefore no circular step is exhibited and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper's derivation relies on standard Boussinesq approximation, neglect of eddy diffusivity, and slow variation; no new entities or fitted parameters. The printed Q0 has an error but this is a typo, not an axiom.

assumptions (5)

domain assumption Boussinesq approximation and neglect of turbulent diffusivity
Section 2.1, Eqs. (1)-(2); without this, the fourth-order equation (8) does not follow.
domain assumption Eddy viscosity coefficients are independent of x; vertical eddy viscosity AT = 0 and horizontal AH nonzero
Section 2.1 states these follow [48-50]; this yields Eq. (9).
domain assumption Slowly varying background U and N^2, WKB asymptotic series in ε
Section 3.1; required for the WKB ansatz (17) to be valid.
standard math Fundamental matrix and matrix exponential theorem from ODE theory
Section 3.2, Eq. (19), cited to [98-101].
standard math Frobenius method and Airy function standard results
Section 4; convergence of regular singular point series.

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Pith. "Pith review of On physical optics approximation of viscous stratified shear flows." pith.science (2026). https://pith.science/paper/SGHTCH67

@misc{pith2026190805457,
  author       = {Pith},
  title        = {Pith review of: On physical optics approximation of viscous stratified shear flows},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SGHTCH67}},
  note         = {Machine review of arXiv:1908.05457}
}

read the original abstract

We study a mathematical model of a perturbed stratified shear mean flow in the presence of eddy coefficients of turbulent viscosity. We adopt the standard Boussinesq approximation in the natural convection of the buoyancy-driven flow and neglect the influence of the eddy coefficients of turbulent diffusivity. Comprising both the vertical and horizontal viscosity effects, a model for the vertical velocity perturbation corresponds to a fourth-order Taylor-Goldstein (TG) differential equation. Considering only the latter, we obtained a modified TG equation with the same order as the classical, inviscid one. Under an assumption of the slowly varying Brunt-V\"ais\"al\"a frequency and background horizontal velocity, we discuss the corresponding geometrical and physical optics approximations of the modified TG equation using the WKB method. We further investigate the behavior of these asymptotic solutions near singular values of a turning point and critical level.

Figures

Figures reproduced from arXiv: 1908.05457 by the authors.

**Figure 1.** (Left panel) A plot of the WKB solution (18) at the onset of instability for a Gaussian profile of eddy viscosity. We have taken U(z) = 1 + 2 tanh(z), N2 (z) = 1 + 4 tanh2 (z), c = 5 + 0.1i, k = 0.1, and AH(z) = 2e −z 2 . There exists no critical level for this particular choice of parameters. (Right panel) The plots of U (red), N2 (dashed blue), and AH (dash-dotted black), in which all are symmetric with respect to… view at source ↗

**Figure 2.** depicts the real and imaginary parts of eigenvalues λ as well as their trajectories in the complex plane for two different values of ε. -15 -10 -5 0 -5 0 5 0 5 10 15 -5 0 5 -15 -10 -5 0 -10 -5 0 0 5 10 15 0 5 10 [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

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