Pressure gradient amplification of shear instabilities in the boundary layer

Pressure gradient amplification of shear instabilities in the boundary layer

Dynamics of Atmospheres and Oceans 37 (2003) 131–145 Pressure gradient amplification of shear instabilities in the boundary layer George Chimonas∗ Sc...

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Dynamics of Atmospheres and Oceans 37 (2003) 131–145

Pressure gradient amplification of shear instabilities in the boundary layer George Chimonas∗ School of Earth and Atmospheric Sciences, Georgia Institute of Technology, Atlanta, GA 30332-0340, USA Received 28 January 2002; received in revised form 3 May 2003; accepted 29 May 2003

Abstract Jeffreys’ instability mechanism is applied to a boundary layer that supports shear instability. The combined instability is an order of magnitude more effective than shear instability alone. The increase in effectiveness is highly sensitive to the stratification and the wind speed near the ground. © 2003 Elsevier B.V. All rights reserved. Keywords: Wave instabilities; Atmospheric boundary layer; Turbulent boundary layer; Vertical stability

1. Introduction Jeffreys (1925) published a study stimulated by observations of waves “in certain Alpine conduits”. The waves were essentially shallow-water waves, which in a horizontal layer of water would be identified with ideal non-growing textbook disturbances. However, the Alpine conduit ran downhill, so it contained an inclined layer of water in which the force of gravity parallel to the layer was balanced by the friction at the lower surface. Jeffreys showed that the forces parallel to the layer destabilize the shallow-water waves: while the parallel forces balance in the mean-state they almost never balance in the disturbed state, and if the friction wins out the waves decay while if the downhill weight wins out the waves grow. Jeffreys’ concepts translate directly to the atmospheric boundary layer. The surface-layer stratification supports gravity-waves that substitute for the shallow-water waves, and even over horizontal ground there is a mean balance of two parallel forces, the horizontal pressure gradient and the surface friction. And disturbances in these parallel forces lead to decay if the friction wins out, growth if the pressure gradient wins out. Previous studies have shown that Jeffreys’ mechanism destabilizes simple atmospheric waves that have no critical levels in the wind (Chimonas, 1993; Pulido and Chimonas, ∗ Tel.: +1-404-894-3985; fax: +1-404-894-5638. E-mail address: [email protected] (G. Chimonas).

0377-0265/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0377-0265(03)00028-9


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2001). The present study moves on to examine waves that are themselves shear instabilities. The formulation is more complex, and Jeffreys’ clever use of an integrated-layer formulation that avoids many messy details has to be abandoned. But the result is satisfactory: Jeffreys’ mechanism transforms weak shear instabilities into fast-growing boundary layer disturbances. While frictional drag is an essential element of the mechanism, the drag itself is always a stabilizing influence and the instability is driven by the force that opposes the drag. This is in contrast with mechanisms in which the turbulence forces the waves: Jeffreys’ sheltering mechanism (Jeffreys, 1924, 1925a), Phillip’s advecting-pressure-fluctuation mechanism (Phillips, 1966), or Lighthill’s nonlinear-velocity-fluctuation mechanism (Lighthill, 1967).

2. Formulation 2.1. The mean-state Take right-hand Cartesian co-ordinates with z vertical and z = 0 at the ground, and define unit vectors xˆ , yˆ and zˆ parallel to the axes. In the following work V is velocity, p pressure, ρ density, t time, g the gravitational acceleration, f the Coriolis parameter and τ is Reynolds stress. A subscript zero denotes a mean-state value: thus the mean Reynolds stress is denoted τ 0 . Where it is necessary to identify vector components of a subscripted variable a superscript is added: thus the x-directed component of the mean Reynolds stress is denoted τ x0 . The mean-state is in hydrostatic equilibrium in the vertical direction. In the horizontal direction, equilibrium between the pressure gradient, the Coriolis force and the Reynolds stress τ 0 produces a wind V 0 = xˆ U + yˆ V


that satisfies the equations −

∂τ x ∂p0 + ρ0 fV + 0 = 0 ∂x ∂z

∂τ ∂p0 + ρ0 fU + 0 = 0 ∂z ∂y



(3) y

The stress at ground-level is xˆ τ x0 (0) + yˆ τ 0 (0), and the x-axis is aligned with this surface y vector, making τ 0 (0) zero. The drag acting on the air is given by the aerodynamic law  ∞ x ∂τ0 1 1 ¯ D |V 0 (R)|UR = − ρC ¯ D UR2 (4) dz = −τ x0 (0) = − ρC 2 2 ∂z 0  ∞ y ∂τ0 1 y (5) dz = −τ 0 (0) = − ρC ¯ D |V 0 (R)|VR = 0 ∂z 2 0 where ρ¯ is a reference value for the density in the boundary layer, UR and VR the wind components at a reference-height z = R, and CD is the corresponding surface-drag coefficient.

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The alignment of the x-axis is equivalent to alignment with the wind at the reference-height. The positive direction for x is chosen to be the direction of this wind. 2.2. Perturbations about the mean-state Instability is formulated in the linearized, Boussinesq approximation for a horizontally homogeneous boundary layer. Each field is expressed as a mean part plus a perturbation. Thus for the velocity V = V 0 + xˆ u + yˆ v + zˆ w


Perturbations are taken to be waves in the x–z plane {u, v, w, p, ρ, τ }perturbations = {u1 (z), v1 (z), w1 (z), p1 (z), ρ1 (z), τ 1 (z)} exp ik(ct − x)


The momentum equation ρ

dV ∂τ x ∂τ y = −∇p + ρg − ρf zˆ × V + xˆ + yˆ dt ∂z ∂z


is linearized in the perturbations. Within the linearized expressions U and V are set to be functions of z alone and the density ρ is set to the reference value ρ¯ except in the gravity term (the Boussinesq approximation). The resulting first-order equations are ρik(c ¯ − U)w1 = −

dp1 − ρ1 g dz


¯ 1 ρik(c ¯ − U)u1 + ρw

dτ x dU ¯ 1+ 1 = ikp1 + ρfv dz dz

ρik(c ¯ − U)v1 + ρw ¯ 1

dτ dV = −ρfu ¯ 1+ 1 dz dz




Density and pressure are related through dρ 1 dp − 2 =0 dt S dt


S is the speed of sound. Linearizing, and treating ρ0 and p0 as functions of z alone gives   w1 w1 1 1 dp0 dρ0 1 ρ1 = (13) − + 2 p1 ≡ ρ0 N02 (z) + 2 p1 2 ik(c − U) S dz dz ik(c − U)g S S N0 is the Brunt–Vaisala frequency. For waves with vertical scales much smaller than S2 /g (≈10 km) the final term of (13) proves negligible when ρ1 is substituted into (9). Discarding this term and replacing ρ0 with ρ¯ gives the Boussinesq form of the density perturbation w ρ1 = ρN ¯ 2 (z) (14) ik(c − U)g 0


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And waves with horizontal speeds much less than the speed of sound have flow fields that are essentially divergence-free dw1 − iku1 = 0 dz


There is an intrinsic problem in applying “horizontally homogeneous” theory to a boundary layer that contains a wind generated by a horizontally varying pressure gradient. One must justify “local” homogeneity by examining the different scales involved in the problem and estimating their relevance to the local behavior of the fluid. This difficult, and probably distracting, mathematical point is touched upon in Appendix A.

3. The sequence of stability problems 3.1. The reference model If the Reynolds stresses and the Coriolis terms are dropped the system reduces to stratified shear flow. Setting τ and f to zero in (10) and eliminating variables amongst the resulting equation plus (9), (14) and (15) gives the Boussinesq form of the Taylor–Goldstein equation (Gossard and Hooke, 1975)   N02 d 2 w1 d2 U 1 2 + + − k w1 = 0 (16) (c − U) dz2 dz2 (c − U)2 3.2. Shear flow with surface-drag If the Coriolis terms are dropped but the stress term τ 1 is retained in (10), the elimination of variables leads to   N02 (z) d2 τ x1 d 2 w1 1 1 d2 U 2 w + + − k = (17) 1 (c − U) dz2 (c − U)ρ¯ dz2 dz2 (c − U)2 The stress is referenced to the aerodynamic drag acting on the air  ∞ x ∂τ 1 dz = −τ x (0) = − ρC ¯ D |V (R)|(UR + u(R)) ∂z 2 0 1 = − ρC ¯ D (UR2 + 2UR u(R) + · · · ) 2


Hence τ x1 (0) = ρC ¯ D UR u1 (R)


This is the perturbation in the drag acting on the column of air above a unit surface area. The force per unit volume can therefore be written as dτ x1 = −ρC ¯ D UR u1 (R)Z(z) dz


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where the profile Z(z) must satisfy  ∞ z→∞ Z(z) dz = 1 and Z(z) → 0




Using (15) to express u1 (R) in terms of the working variable w1 , (20) becomes   dτ x1 dw1 i = ρC ¯ D UR Z(z) dz k dz z=R Substituting (22) into (17) gives     N02 (z) d 2 w1 d2 U 1 iCD UR dZ(z) dw1 2 w + + − k = 1 (c − U) dz2 (c − U)k dz dz z=R dz2 (c − U)2



3.3. Shear flow with surface-drag and the mean pressure gradient terms Eliminating variables between (9)–(11), (14) and (15) while retaining all the Coriolis and stress terms gives the most comprehensive form of the governing differential equation, but the result is cumbersome. If the Coriolis and stress terms for the y-directed acceleration are dropped by setting the right side of (11) to zero the algebra is greatly simplified. All numerical comparisons carried out to date have found that the reduced algebra leads to essentially the same results as the full algebra. With the right side of (11) set to zero elimination of variables provides   N02 (z) d 2 w1 1 d2 U 2 + + − k w1 (c − U) dz2 dz2 (c − U)2   1 1 dτ x1 d if dV = w1 + (24) (c − U) dz k(c − U) dz ρ¯ dz The Coriolis term in (24) can be expressed in terms of the stresses. Repeating the procedure (18)–(21) but now extracting the mean parts gives dτ x0 1 = − ρC ¯ D UR2 Γ(z) dz 2 where the profile must satisfy  ∞ z→∞ Γ(z) dz = 1 and Γ(z) → 0




The boundary layer approximation to (2) is f

dV 1 d2 τ x0 =− dz ρ¯ dz2


1 dΓ(z) dV = CD UR2 dz 2 dz


hence f

Substituting (22) and (28) into (24) produces


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  N02 (z) d 2 w1 1 d2 U 2 + + − k w1 (c − U) dz2 dz2 (c − U)2     UR dΓ(z) iCD UR d dw1 w1 + Z(z) = (c − U)k dz 2(c − U) dz dz z=R


“Coriolis terms” such as (27) come from geostrophic-wind deficits caused by the surface friction. Physically they are “non-Coriolis-balanced mean-horizontal pressure gradient terms”, but this terminology is obviously unacceptable.

4. The mean profiles of the nocturnal boundary layer The sequence of Eqs. (16), (23) and (29) is solved for nocturnal boundary layer conditions. Between ground-level and height h the air is stably stratified with the stability steadily decreasing with height (Geiger, 1966). The stable layer merges into a neutrally stratified residual layer that occupies the remainder (the greater part) of the diurnal boundary layer (Stull, 1988). And above the boundary layer there is a uniformly stratified free troposphere. The three regions are modeled with  2 z>H Nupper     0 h
   πz    U(z) = A + B z − ln 1 + exp z − a + C sin z≤H l l 2H  U(z) = U(H) = Uupper z>H


This profile has three factors corresponding to three different length scales. The factor with amplitude C is the component with a length scale based on the full depth of the planetary boundary layer. The factor with amplitude B controls the shape of the wind in the stable layer. And the amplitude A adjusts the wind at the reference-height: this idealized model does not include the surface roughness layer (with length scale z0 ) where the air is finally brought to rest, so in (31) the reference-height wind and the ground-level wind are the same. The factor with amplitude B has been used in studies of inflection-point-free shear instabilities (Chimonas, 1974; Fua et al., 1976). The profiles (30)–(31) contain ten independent parameters and it is not feasible to present results for this entire parameter space. The essential points can be made by varying N02 (0) and A, and the results presented below all share the following suite of parameters Nupper = 0.0105 s−1 , h = 150 m, B = 0.3 ms−1 , l = 50 m, a = 3.333, C = 2 ms−1 , H = 600 m


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Fig. 1. Wind profile (31) with the parameters (32) and A = 1.

By adjusting A and N02 (0) the profiles are made to simulate boundary layers that range from weakly stable to strongly stable with winds that range from light to strong. Fig. 1 illustrates U(z) used in the computations. This wind has no inflection point, but it supports shear instabilities: Rayleigh’s inflection-point theorem does not hold in a stratified fluid. Fig. 2 shows the related Richardson number profile Ri(z) =

N02 (z) (dU/dz)2

The stress profiles Z(z) and Γ (z) of (22) and (28) are chosen to be  σ(h − z)2 0 < z < h Γ(z) = Z(z) = 0 z>h



The amplitude σ is calculated from the integral condition (21). Except very near the ground observations of stress profiles are rare and empirical forms of Z(z) and Γ (z) are not available. General principles indicate that the mean Reynolds stress should decrease monotonically upward from the ground to the value zero in the free atmosphere (Sutton, 1977, Chapter 7), while the nocturnal residual layer appears to be decoupled from the ground leaving the surface stress confined to the stable surface-layer (Blackadar, 1957). The form (34) is the simplest power law that satisfies both general points while providing all the derivatives


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Fig. 2. Richardson number corresponding to the wind of Fig. 1 and the stratification (30) with N02 (0) = 0.001 s−2 .

needed for the governing Eq. (29). If observations provide significantly different profiles the current numerical predictions will need re-examination. The aerodynamic law (4) involves a drag coefficient CD that has been measured for a wide variety of terrestrial and oceanic conditions. This study uses CD = 0.03


throughout the numerical work: the value 0.03 is characteristic of thick grass up to 50 cm high with the reference-height of the wind set at 2 m. However, the instability is controlled by the combination CD UR2 , so growth is related to the value of the mean surface stress rather than the value of CD . The treatment of the Reynolds stresses in all the equations up to and including (29) is completely general and is compatible with any closure scheme for turbulence: K-theory, second-order closure, or higher-order. But taking (34) to define Z(z) is a specific model assumption, and it would be no surprise to discover that better choices could be made. This point, and the associated point that using (34) leads to second-order differential equations in the Section 5 where some might have expected higher-orders such as a generalized Orr–Sommerfeld equation, are taken up in Appendix A.

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5. Results A mode is a solution of Eqs. (16), (23) or (29) that satisfies the boundary conditions w1 (0) = 0 and w1 (z → ∞) → 0. An instability is a mode whose wave-speed c has a negative imaginary part: thus an instability grows exponentially with time. For z ≥ H the profiles (30)–(31) revert to a uniform atmosphere and the solutions take known forms, allowing the upper boundary condition to be implemented at z = H. Modes and instabilities were traced through the sequence of Eqs. (16)–(29), and the results are presented for each stage of the investigation. Section 5.1 examines the basic shear-flow governed by (16), Section 5.2 finds how the behavior changes when the surface-drag term is included (flow governed by Eq. (23)), while Section 5.3 examines the full problem of shear flow with surface-drag and pressure gradient forcing (flow governed by Eq. (29)). 5.1. The shear instabilities Fig. 3 displays results from solving (16) for surface-layers with strong stratification (N02 (0) = 0.002 s−2 , Ri(0) = 16.4), moderate stratification (N02 (0) = 0.001 s−2 , Ri(0) = 8.2) and weak stratification (N02 (0) = 0.0005 s−2 , Ri(0) = 4.1). The classification used here follows (more or less) the “strongly stable boundary layer . . . weakly stable boundary

Fig. 3. Shear instabilities of (16). Model profiles of Figs. 1 and 2 with values of N02 (0) (s−2 ) noted in braces for each curve. Solid curves are growth rates, calibrated to left y-axis, dashed curves are wave phase speeds, calibrated to right y-axis.


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layer” classification used in Mahrt et al., 1998. Fig. 3 was computed for the wind parameter A = 1 but it translates directly to other values since using A = 1 + q in (16) increases the real part of c by the amount q but leaves everything else about the wave unchanged. As seen in Fig. 3, the growth rates depend very strongly on the stratification near the surface. This would be unremarkable if the critical levels of the waves were in the stratified region, but all these waves have critical levels in the overlying residual layer where N02 (residual) ≡ 0 and Ri(residual) ≡ 0. Certainly, the greater the boundary layer stratification the greater the wave-energy at a given amplitude, but it surprised this investigator to see energy-like constraints emerge from a linear stability analysis. The strong relation between growth and near-surface stability holds through all the subsequent results. The vertical structure of the solutions is very similar to the simple structure of neutral modes ducted in a windless stratified boundary layer. Such profiles are probably familiar to the reader and will not be reproduced here. (Examples using the profiles (30)–(31) for the very stable boundary layer are given in Chimonas, 2002.) The growth rates presented in Fig. 3 are not particularly significant in the context of the atmospheric boundary layer. The fastest growth shown would allow a wave to increase its amplitude about seven-fold in 1 h, but this would not be effective in the nocturnal boundary layer as the mean winds change significantly over shorter times. 5.2. The damping effect of the surface-drag The growths shown in Fig. 3 do not hold their own against turbulent dissipation. Switching to the governing Eq. (23) moves the waves off the growth curve “a” of Fig. 3 onto the “decay” curve of Fig. 4. The switch to (23) means that the surface-drag perturbation (19) is included in the wave dynamics, and as one might anticipate, drag dissipates the wave. Note that the spatial form of the drag associated with (19) is {τ }surface perturbation = τ x1 (0) exp(−ikx) = ρC ¯ D UR u1 (R) exp(−ikx)


This averages to zero over a wavelength in the x-direction: the wave is modulating the drag, enhancing it at one phase and decreasing it at the other. But the phase of the modulation is consistent with a frictional reaction, and the wave dies out. 5.3. The horizontal pressure gradient: opposing the effects of surface-drag In the mean-state the surface-drag is opposed by the horizontal pressure gradient. Perturbations to the surface-drag cause wave dissipation, so if the pressure gradient continues to work in the opposite sense to drag its perturbations will favor wave growth. Jeffreys’ instability mechanism operates when the pressure gradient perturbation is more effective than the surface-drag perturbation. Fig. 4 contrasts behavior for the sequence of governing equations. The curve “a” is the growth rate from shear instability (Eq. (16)), “decay” shows the effect of adding surface-drag (Eq. (23)) in a moderate wind, and “b” shows the effect of adding surface-drag and horizontal pressure gradient (Eq. (29)) in a light wind. For the light wind, the opposing terms in the Jeffreys’ mechanism essentially balance, restoring the curve of the original shear instability.

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Fig. 4. Sequence of growth/decay for modes of (16), (23) and (29). Curve “a” repeats shear instability curve “a” of Fig. 3. Curve “decay”, calibrated to right y-axis, results from (23) with wind A = 3. Curve “b” gives growth from (29) with wind A = 1.

But as can be seen in (29), the pressure gradient term contains the factor UR2 while the surface-drag term contains the factor UR . So if the two Jeffreys’ terms balance for a light wind the pressure term should dominate in stronger winds. This proves to be the case: Fig. 5 shows growth rates computed for the modes of (29) with light, moderate and strong surface winds. Jeffreys’ mechanism becomes increasingly effective as the wind strengthens. With a reference-height wind of 5 m s−1 the peak growth in Fig. 5 shows amplitudes increasing a 1000-fold in 1 h. The mean surface stress measures how wind-strength contributes to the mechanism. The value CD = 0.03 used in this study corresponds to a surface stress τ x0 (0) = 0.018 UR2 N m−2


and hence a friction velocity u∗ = 0.122 UR m s−1


Thus the value A = 3 used to compute Fig. 4 corresponds to a friction velocity of 0.37 m s−1 , which is close to the mean terrestrial value and comparable to values measured in moderately gusty nocturnal conditions. At A = 5 the surface friction rises to u∗ ∼ 0.6 ms−1 , which is considerably higher than normal in open country, but then the 1000-fold growth rate seen in Fig. 5 would be associated with very gusty conditions.


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Fig. 5. Instabilities from (29) as a function of wind-strength. Stratification of Fig. 2 and wind from (31). Values of A noted in braces for each curve. Curve calibration as in Fig. 3.

Growth is also favored by weak stability in the surface-layer. This emerged in results for the shear instabilities, Section 5.1, and it is maintained when the Jeffreys’ mechanism is added. Fig. 6 shows that decreasing the stratification by a factor two enhances growth by a factor of about 10.

6. Discussion The results of Sections 5.1–5.3 present a physically reasonable pattern: the system governed by (16) supports rather weak instabilities; adding the surface-drag by switching to (23) reduces the growth rate of an instability and can even turn it into a decaying mode; the pressure gradient term added by switching to (29) opposes the surface-drag, and for sufficiently strong winds the pressure gradient wins and the original instability is enhanced. The enhanced growth rates can be an order of magnitude greater than those of the original shear instability, so Jeffreys’ mechanism transforms weak shear instability into a fast-growing boundary layer disturbances. The mechanism becomes increasingly effective with stronger surface winds and weaker near-surface stratification. Thus mixing in the surface-layer produces conditions that are more favorable to the mechanism. This suggests that the mechanism could modify the

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Fig. 6. Instabilities from (29) as a function of stratification. Wind from (31) with A = 3. Stratification of Fig. 2 with values of N02 (0) (s−2 ) noted in braces for each curve. Curve calibration as in Fig. 3.

boundary layer to its own advantage through the cascade of wave-energy into turbulence and mixing. Then instead of producing well-defined waves, conditions that favor Jeffreys’ mechanism would enhance gusts and turbulence, and the waves would be limited by their contributions to the general chaos. Jeffreys’ mechanism would then be a major source of the energy in gustiness—a phenomenon that is more common in the boundary layer than well-defined waves. The motivation for this study is the desire to better understand the origin of the bursts of small- and large-scale turbulence in the gusty boundary layer, so the mechanism must be verified (or refuted) by comparing its predictions with detailed boundary layer observations. The calculations of Section 5 must be performed for sets of actual boundary layer profiles, and the predicted rates of instabilities correlated with observed levels of the larger-scale turbulence. Recent field experiments such as CASES 99 appear to provide both the detailed data sets and the breadth of supporting observations for such verification, and it is hoped that progress towards this goal will be reported in the not too distant future.

Acknowledgements This work was supported in part by the University Cordoba, Argentina, when the author was a visiting professor hosted by FaMAF. I am indebted to Dr. Manuel Pulido for extensive


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discussions about the work and the ideas on which it is based. Two anonymous referees offered numerous suggestions and corrections to this paper, for which help I am most grateful. Appendix A. Some mathematical considerations A.1. Horizontal homogeneity The derivations of the wave Eqs. (16), (23) and (29) assumed “horizontal homogeneity”. While horizontal homogeneity is a standard approximation in boundary layer meteorology, Jeffreys’ mechanism is driven by the mean-horizontal pressure gradient, which is obviously a departure from horizontal homogeneity. And the boundary layer wind itself implies a horizontal pressure gradient. In practice, “horizontal homogeneity” is justified by the relative sizes of terms in the basic equations. For example, if p0 is a function of x as well as z the linearization of (12) produces combinations such as w1 (∂p0 /∂z) + u1 (∂p0 /∂x) which have been approximated as w1 (dp0 /dz) in (13). The justification is that the horizontal derivative involves a small fractional change over a synoptic scale and it provides a contribution that is several orders of magnitude smaller than the vertical derivative. Every step in the derivations can be examined and justified in this way. A.2. The model for the stress profile At some stage in the analysis the turbulent Reynolds stresses appearing in (10) and (11) must be specified in a computable form. In the formulation given in Sections 3 and 4 the specification was delayed until the model of the boundary layer had been chosen, after which the profile (34) was selected. This emphasizes that the Reynolds stress is the fundamental agent in the wave dynamics, and any model of turbulence in this context is no more than (and no less than) a model of the stress. Nevertheless one may ask how the approach taken here differs from K-theory. Although K-theory is usually introduced directly into Eqs. (10) and (11), its introduction may be delayed until (34) is needed, or equivalently until (23) and (29) must be computed. But wherever it is introduced K-theory involves the additional approximation of the factorization τ 1x = −K(x, z, t)(∂[U(z) + u(x, z, t)]/∂z), after which some profile must be selected for K. One cannot model the stresses in the boundary layer with K set constant. And the profile of K used in the mean, nearly-neutral surface-layer with a log-linear wind is not suited to the stably-stratified nocturnal layer in gusty conditions. Thus in the present calculations K would need a (guesstimated) profile that would interact with the wind fields to produce the relatively unknown Reynolds stresses of the nocturnal boundary layer. It is obviously cleaner to work directly with the stresses. An additional important point concerns the different orders of the differential equations one encounters. K-theory leads to a fourth-order differential equation, while the equations solved in Section 5 were all of the second-order. There is no inconsistency: the fourth-order generalized Orr–Sommerfeld-type equation for a viscous Boussinesq fluid governs the dynamics of four gravity–viscosity wave solutions. But in the atmosphere and the ocean

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it is usually possible to distinguish the gravity-waves from the viscous waves through a scale analysis. And when the gravity-wave scale is much greater than the viscous-wave scale the two wave types are only weakly coupled and a perturbation theory based on the scaling between the two waves types separates the fourth-order Orr–Sommerfeld equation into two second-order equations. The gravity-waves are then obtained as solutions of the second-order Taylor–Goldstein equation with a weak viscous correction term: the equations dealt with in this paper. References Blackadar, A.K., 1957. Boundary-layer wind maxima and their significance for the growth of the nocturnal inversion. Bull. Am. Meteorol. Soc. 38, 283–290. Chimonas, G., 1974. Considerations of the stability of certain heterogeneous shear flows including some inflexion-free profiles. J. Fluid Mech. 65, 65–69. Chimonas, G., 1993. Surface-drag instabilities in the atmospheric boundary layer. J. Atmos. Sci. 50, 1914–1924. Chimonas, G., 2002. The internal gravity waves associated with the stable boundary layer. Boundary-Layer Meteorol. 102, 139–155. Fua, D., Einaudi, F., Lalas, D.P., 1976. The stability analysis of an inflexion-free velocity profile and its application to the night-time boundary layer in the atmosphere. Boundary-Layer Meteorol. 10, 35–54. Geiger, R., 1966. The Climate Near The Ground. Harvard University Press, Cambridge, 611 pp. Gossard, E.E., Hooke, W.H., 1975. Waves in the Atmosphere, Section 24. Elsevier, Amsterdam. ISBN 0-444-41196-8, 456 pp. Jeffreys, H., 1924. On the formation of waves by wind. Proc. R. Soc. A 107, 189–206. Jeffreys, H., 1925. The flow of water in an inclined channel of rectangular section. Philos. Mag. 49, 793–807. Jeffreys, H., 1925a. On the formation of waves by wind. II. Proc. R. Soc. A 110, 341–347. Lighthill, M.J., 1967. Predictions of the velocity field coming from acoustic noise and a generalized turbulence in a layer overlaying a convectively unstable atmospheric region. I.A.U. Symposium no. 28, pp. 429–452. Mahrt, L., Sun, J., Blumen, W., Delany, T., Oncley, S., 1998. Nocturnal boundary-layer regimes. Boundary-Layer Meteorol. 88, 255–278. Phillips, O.M., 1966. The dynamics of the upper ocean. Ocean Surface Waves, second ed. Cambridge University Press, Cambridge. ISBN 0521214211, 1977, Chapter 4, 336 pp. Pulido, M., Chimonas, G., 2001. Forest canopy waves: the long-wavelength component. Boundary-Layer Meteorol. 100, 209–224. Stull, R.B., 1988. An Introduction to Boundary Layer Meteorology. Kluwer Academic Publishers, Boston. ISBN 90-277-2768-6, 666 pp. Sutton, O.G., 1977. Micrometeorology. Kreiger Publishing Company, New York. ISBN 0-88275-488-2, 333 pp.