Gravity Wave Perturbations of Minor Constituents: A Parcel Advection Methodology

Stephen D. Eckermann Computational Physics, Inc., Fairfax, Virginia

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Dorothy E. Gibson-Wilde Colorado Research Associates, Boulder, Colorado

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Julio T. Bacmeister E. O. Hulburt Center for Space Research, Naval Research Laboratory, Washington, D.C.

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Abstract

Existing analytical models of wave-induced minor constituent fluctuations result from linearized perturbation expansions of a rate equation governing either number density or mixing ratio, whereas many numerical models now use isentropic parcel advection methods to simulate these effects. Exact relationships between the two approaches are not currently clear for gravity waves. Here, the parcel advection method is formalized and applied to derive analytical formulas for the response of vertical constituent profiles of arbitrary shape to adiabatic gravity wave displacements. These relations are compared to corresponding formulas from standard linearized perturbation analyses. Both methods accurately model perturbations produced by nondissipating hydrostatic gravity waves within idealized vertical tracer profiles. Both methods can also model wave-induced perturbations of minor constituents with shorter chemical lifetimes. This is demonstrated by using the parcel method to reproduce previous results for wave-induced fluctuations in upper-stratospheric ozone. The parcel-based approach yields more accurate models of nondissipating hydrostatic gravity wave effects on tracer distibutions with sharp spatial gradients. The authors demonstrate this by using the method to model wave-induced modulations of sporadic sodium layers in the mesosphere and of ozone laminae in the lower stratosphere. Conversely, the parcel method does not accurately model tracer density perturbations produced by nonhydrostatic waves, or situations in which the photochemical response of the constituent leads to significant diabatic feedback on the perturbing wave.

* Current affiliation: E. O. Hulburt Center for Space Research, Naval Research Laboratory, Washington, D.C.

Corresponding author address: Dr. Stephen D. Eckermann, Computational Physics, Inc., Suite 600, 2750 Prosperity Avenue, Fairfax, VA 22031.

Email: eckerman@cpi.com.

Abstract

Existing analytical models of wave-induced minor constituent fluctuations result from linearized perturbation expansions of a rate equation governing either number density or mixing ratio, whereas many numerical models now use isentropic parcel advection methods to simulate these effects. Exact relationships between the two approaches are not currently clear for gravity waves. Here, the parcel advection method is formalized and applied to derive analytical formulas for the response of vertical constituent profiles of arbitrary shape to adiabatic gravity wave displacements. These relations are compared to corresponding formulas from standard linearized perturbation analyses. Both methods accurately model perturbations produced by nondissipating hydrostatic gravity waves within idealized vertical tracer profiles. Both methods can also model wave-induced perturbations of minor constituents with shorter chemical lifetimes. This is demonstrated by using the parcel method to reproduce previous results for wave-induced fluctuations in upper-stratospheric ozone. The parcel-based approach yields more accurate models of nondissipating hydrostatic gravity wave effects on tracer distibutions with sharp spatial gradients. The authors demonstrate this by using the method to model wave-induced modulations of sporadic sodium layers in the mesosphere and of ozone laminae in the lower stratosphere. Conversely, the parcel method does not accurately model tracer density perturbations produced by nonhydrostatic waves, or situations in which the photochemical response of the constituent leads to significant diabatic feedback on the perturbing wave.

* Current affiliation: E. O. Hulburt Center for Space Research, Naval Research Laboratory, Washington, D.C.

Corresponding author address: Dr. Stephen D. Eckermann, Computational Physics, Inc., Suite 600, 2750 Prosperity Avenue, Fairfax, VA 22031.

Email: eckerman@cpi.com.

1. Introduction

High-resolution observations of minor (trace) constituents in the middle and upper atmosphere often reveal small-scale perturbations superimposed upon the mean background distribution. Such features can be produced by any number and combination of dynamical, radiative, and chemical processes. For trace constituents with long lifetimes, however, dynamics must be predominately responsible for any observed variability of that constituent’s local number density over shorter timescales.

One important source of such variability is waves. For instance, vertical soundings at middle and high latitudes often reveal large enhancements or depletions of lower-stratospheric ozone (a tracer at these heights) confined to narrow (∼1–2 km) vertical layers (e.g., Reid and Vaughan 1991; Bird et al. 1997). These structures, known assortedly as filaments or laminae, result from complex synoptic-scale flow patterns produced by breaking planetary Rossby waves, which yield air parcels at different stratospheric levels with vastly different histories and, hence, constituent densities (e.g., Newman and Schoeberl 1995; Orsolini et al. 1997).

Oblique oscillatory displacements of air parcels from their equilibrium positions produce accompanying density and temperature changes that can also perturb constituent profiles. Oscillations due to various types of planetary waves have been found to induce related perturbations in ozone (e.g., Newman and Randel 1988; Prata 1990; Hess 1990; Randel and Gille 1991; Randel 1993; Stanford and Ziemke 1993; Wirth 1993; Ziemke and Stanford 1994; Engelen 1996) and other trace constituents (e.g., Randel 1990; Salby et al. 1990; Limpasuvan and Leovy 1995). On smaller scales, internal gravity waves can also give rise to fluctuations in constituent profiles (e.g., Reber et al. 1975; Gardner and Shelton 1985; Hedin and Mayr 1987; Sugiyama 1988; Hoegy et al. 1990; Wilson et al. 1991; Alexander and Pfister 1995; Langford et al. 1996; Bian et al. 1996; Bacmeister et al. 1997).

In fact, since observational data on middle and upper atmosphere constituents are often better and more abundant than wind and temperature data, analyses of fluctuations in long-term measurements of constituent densities have been used to infer climatologies of both planetary waves and gravity waves (e.g., Randel and Gille 1991; Senft and Gardner 1991; Randel 1993; Ziemke and Stanford 1994; Collins et al. 1996). However, since the response of constituents to waves is usually not straightforward, wave properties must be inferred from such observations by using a wave–tracer interaction model to convert the measured constituent response into a wave-related quantity. Such conversion methods have been derived to date by applying standard linearized perturbation methods of gravity wave theory to rate equations governing constituent densities or mixing ratios. Such derivations yield analytical formulas for translating conventional gravity wave oscillations into a corresponding response in the constituent profile (e.g., Dudis and Reber 1976; Chiu and Ching 1978; Gardner and Shelton 1985). Such relations can then be used to convert observed constituent fluctuations into wave-related quantities (e.g., Gardner and Voelz 1985; Randel 1990; Senft and Gardner 1991).

Recently, detailed numerical models have provided new insights into the way minor constituents respond to planetary wave and gravity wave advection (e.g., Fritts et al. 1993, 1997; Waugh et al. 1994; Roble and Shepherd 1997). Rich complex structures in minor species distributions are simulated, which often resemble observations (e.g., Fritts et al. 1993, 1997) but are not predicted by standard formulas that convert between observed tracer variability and a plane monochromatic wave. Due to their complexity, however, these models cannot as yet be used for operational extraction of wave properties from observed constituent fluctuations in the atmosphere. On the other hand, simplified versions of these models, which retained the minimum terms needed to simulate the basic effects, might lead to improved analytical models of wave-induced perturbations of minor constituent profiles.

We investigate this possibility here by considering one class of these numerical models, the so-called trajectory or parcel advection models. They have proved successful in simulating finescale tracer structures in the stratosphere, often using wind and temperature data with much coarser spatiotemporal resolution (e.g., Newman and Schoeberl 1995; Orsolini et al. 1997). The models work well because their assumptions of adiabatic motion and long chemical lifetimes are well satisified for many constituents in the lower stratosphere, where diabatic heating and turbulent diffusivities are both small. However, constituents with shorter chemical lifetimes can also be simulated by these models if a suitable parcel-based chemistry scheme is used (e.g., Lutman et al. 1997). Even in the troposphere, where diffusion and diabatic heating can be large, parcel advection models that parameterize these effects have also proved to be useful tools (e.g., Kao et al. 1995; Rind and Lerner 1996).

Given that parcel advection models often perform well even after many days of integration, the same models should also perform well if the synoptic-scale advection patterns were supplemented with wave-induced advection effects since their timescales (periods) are of the order of days for planetary waves and less than a day for gravity waves. Furthermore, since many gravity waves and planetary waves propagate conservatively through the stratosphere, an assumption of adiabatic wave-induced advection should also be well satisfied. Indeed, limited tests of gravity wave effects in such models have used this sort of approach (Pierce et al. 1994; Jensen and Thomas 1994; Carslaw et al. 1998).

Thus, we formalize the basic physical concepts underpinning parcel advection models in section 2, assess the utility of the method in gravity wave studies, then apply it in section 3 to derive analytical responses of tracers to wave-induced displacements. We compare our results throughout with corresponding results using the traditional approach of expanding state parameters into the sum of a mean and perturbation term, then isolating and linearizing the perturbation terms in the governing equations. Results of these analyses and intercomparisons are summarized in section 4.

2. Air parcel approach

This approach is based on the classic meteorological problem of vertically displacing a small dry parcel of air from its equilibrium height, where it is “well mixed” and in thermal equilibrium with the surrounding atmosphere. We assume that the displacement is small and that the parcel responds adiabatically while always maintaining a pressure equal to that of the surrounding atmosphere, following standard conventions (see, e.g., section 2.5.1 of Wallace and Hobbs 1977; section 9.1 of Iribarne and Godson 1981).

a. Conserved quantities in the parcel

1) Tracer mixing ratio

We consider a constituent of number density n within an air parcel of total number density nM. For a trace constituent n/nM ≪ 1, any changes in n produce negligible changes in nM, whereupon the following continuity equations apply:
i1520-0469-55-24-3521-e1a
where U is the local velocity vector, d/dt = ∂/∂t + U ·  is a time derivative in the Lagrangian (parcel following) frame (∂/∂t is the ground-based Eulerian time derivative), and R is the net production/loss rate for the trace constituent’s number density n. Eliminating  · U between (1a) and (1b) yields (Lindzen and Goody 1965)
i1520-0469-55-24-3521-e2
where q is the mixing ratio of the trace constituent.
For passive trace constituents, R ≈ 0, whereupon (2) yields
i1520-0469-55-24-3521-e3a
or
i1520-0469-55-24-3521-e3b
within the parcel. Thus, we get the well-known result that passive trace constituents are also tracers since the movement of a region of unique constituent mixing ratio traces the Lagrangian motion of air parcels.

2) Potential temperature

Air parcels advected by an inviscid adiabatic flow also conserve potential temperature Θ so that
i1520-0469-55-24-3521-e4
For a horizontally homogeneous atmosphere, the background potential temperature Θ is given by Poisson’s relation
i1520-0469-55-24-3521-e5
where z is height, γ is the ratio of specific heats, T(z) is the background temperature, and p(z) is the background pressure (e.g., Turner 1973). From (5), Θ(z) is also equal to the temperature T(z, z0) that an air parcel attains on being transported adiabatically a distance δz from its rest height z to a new height z0 = z + δz.

3) Potential density

We can also work in terms of potential density, D(z), which is defined for a horizontally homogeneous background atmosphere as
i1520-0469-55-24-3521-e6
(e.g. Turner 1973), where ρ(z) = MnM(z) is the background total air density at height z, nM(z) is the background total number density, and M is the mean mass of an air molecule, assumed constant (no diffusive separation). From (6), D(z) also equals the density ρ(z, z0) that an air parcel attains on being transported adiabatically from a pressure, p(z) and density ρ(z), at its rest height z to a reference pressure p(z0) at the displaced height z0 = z + δz. It then follows (e.g., Wallace and Hobbs 1977) that
i1520-0469-55-24-3521-e7
for parcels on an inviscid adiabatic flow, and thus the potential density of any parcel within such a flow is also conserved.

4) Potential tracer density

Analogously, we can also define a potential density, Δ, for a given tracer. If the constituent in question behaves as a tracer at all heights and has negligible horizontal gradients in its distribution, then, as in (6), we can define a background potential tracer density profile given by
i1520-0469-55-24-3521-e8
where Mtc is the molecular mass of the trace constituent and ν(z) is the background potential number density of the tracer. From (8), we see that Δ(z) also equals the mass density of tracer Mtcn(z, z0) that an air parcel contains after being transported adiabatically from a pressure p(z) and tracer density Mtcn(z) at its rest height z to a reference pressure p(z0) at the new height z0 = z + δz. It then follows that
i1520-0469-55-24-3521-e9
for parcels on an inviscid adiabatic flow. Since number densities are more commonly considered in tracer studies, we shall work in terms of ν rather than Δ in subsequent analysis.

b. Perturbations due to adiabatic vertical displacements

Next we consider local responses of each conserved term to a small vertical adiabatic displacement ζ of an air parcel from its equilibrium position z1 to a new height
z2z1ζ,
given the standard background pressure profile
i1520-0469-55-24-3521-e11
where H(z1) = RT(z1)/g is the pressure scale height at height z1, and the latter expression in (11) is sufficiently accurate for small ζ given a background temperature profile T(z) that varies slowly with height.

The formal parcel-based analysis of this problem is given in appendix A, where departures (perturbations) of parcel-based quantities from surrounding background values are derived. The basic procedure for computing these perturbations is depicted schematically in Fig. 1 for both temperature and density. In each panel in Fig. 1, an air parcel has been advected from its equilibrium height z1 to a new height, z2, and in each case the adiabatic response is given by moving the parcel along the light dotted “response curve.” The top panels of Fig. 1 show constructions for the potential temperature and potential density: since both remain constant within the air parcel, their response curves are vertical. Corresponding constructions in terms of the absolute temperature and density are shown in the lower panels. Their response curves are given by T(z1, z2) and ρ(z1, z2) in (5)–(6): hereafter, it proves expedient to use the abbreviated notation T(z2) and ρ(z2) for these expressions since z1 stays constant and the functional dependence on z1 is implicit from (10). Background profiles are shown with thick solid curves on each plot. Perturbations are given by the difference between the adiabatically varying value within the parcel and the local background value at height z2, as shown in Fig. 1. Using the abbreviated notation introduced above, this yields perturbation terms Θ′(z2), D′(z2), T′(z2), and ρ′(z2).

Simplified equations for these perturbation terms follow when the displacement ζ is sufficiently small that variations in both the background profiles and the adiabatic response curves between z1 and z2 are approximately linear in Fig. 1. This holds so long as any curvature of these profiles occurs over vertical scales Lζ. Since the formal parcel relations in appendix A are height-varying exponentials, L can be conveniently characterized by the scale height HX of the exponential relation for quantity X, so that |ζ/HX| ≪ 1. When this condition is satisfied, the exponential parcel relations can be accurately approximated by retaining only the first two terms from their MacLaurin series expansions. For temperatures and densities, these truncated MacLaurin series (TMS) relations yield the instantaneous simplified perturbation relations
i1520-0469-55-24-3521-e12
where
i1520-0469-55-24-3521-e13
is the potential density scale height (see appendix A). Equation (12) holds for |ζ/HD| ≪ 1.

c. Range of applicability to gravity waves

Next we consider a sinusoidal form for the vertical parcel displacement from z1:
i1520-0469-55-24-3521-e14
where ζ̂ is the peak vertical displacement amplitude, φ is a (constant) phase offset, (k, l, m) is the wavenumber vector, and ω is the oscillation frequency of the parcel about its equilibrium height z1. Considering potential temperature only for the moment, (14) and (12) yield
i1520-0469-55-24-3521-e15
Since ζ̂ is small (|ζ̂/HD| ≪ 1), replacing the variable term Θ(z2) with the constant value Θ(z1) introduces negligible errors in (15). Then, to a good approximation, we can remove the explicit transport of parcels to z2 so that we can reexpress the potential temperature perturbation (15) as a sinusoidal expression of the form (14) at a fixed height z1:
i1520-0469-55-24-3521-e16
where
i1520-0469-55-24-3521-e17
Transport of parcels to z2 is now implicit within (16) and (17). Similar relations to (17) follow from (12) and (14) for relative perturbations of potential density, density, and temperature, with peak amplitudes interrelated as
i1520-0469-55-24-3521-e18

The relations (14) and (18) have the same form as the simplified polarization relations that govern a hydrostatic gravity wave of intrinsic frequency ω. This occurs because, in the linear hydrostatic limit, a gravity wave advects air parcels adiabatically (e.g., Fritts and Rastogi 1985), has a small displacement amplitude ζ̂ (|ζ̂/HD| ≪ 1), and produces a negligible pressure perturbation within displaced parcels (e.g., Gossard and Hooke 1975).

To assess the degree and range of this correspondence more quantitatively, a standard non-Boussinesq linearized perturbation derivation of the acoustic–gravity wave equations is outlined in appendix B. There it is shown that the parcel-derived polarization relations (18) are an acceptable approximation to the full acoustic–gravity wave polarization relations when the normalized quantities |a(m, ω)| and |b(m, ω)| are ≪1 [see Eqs. (B10) and (B11)]. When this limit is not satisfied, wave-induced pressure perturbations become nonnegligible [see Eq. (B13)], so the parcel approach (which assumes zero pressure perturbation) becomes inaccurate. Sample calculations in appendix B show that the differences between the full acoustic–gravity wave polarization relations and the parcel-based approximations (18) are small for λz = 2π/|m| ≲ 20 km (see Figs. B1b and B1c). Thus, we use this as an approximate upper-wavelength bound on the validity of the hydrostatic gravity wave approximation and of subsequent parcel-based derivations of gravity wave–induced minor constituent variability.

3. Gravity wave perturbations of minor constituents

We now apply the parcel-based approach to model gravity-wave-induced perturbations of minor constituent profiles. We begin by considering some simple examples and compare the parcel-based derivations (appendix A) with corresponding equations derived by a conventional linearized perturbation analysis (appendix B).

a. Tracer with linear q(z) profile

The parcel-based approach to this problem is depicted schematically in Fig. 2a and yields the perturbation formula (A19). Since the background profile is linear, then q(z2) = q(z1) + (∂q/∂z)ζ′, whereupon (A19) and the wave solution (14) yield
i1520-0469-55-24-3521-e19
As for (15), if ζ̂ is small, then the explicit transport of parcels to z2 can be removed in order that (19) can be reexpressed as a wave solution of the form (16), with peak amplitudes related as
i1520-0469-55-24-3521-e20
A standard linearized perturbation analysis of (3a) using the methodology in appendix B yields the linearized perturbation mixing ratio equation (e.g., Randel 1990)
i1520-0469-55-24-3521-e21
which, since ŵ = ιωζ̂,, also yields (20) at a given height z1. Thus the two approaches give identical results in this case.

b. Tracer with a linear n(z) profile

The parcel-based approach to this problem is depicted schematically in terms of tracer number density in Fig. 2b and in terms of potential tracer number density in Fig. 2c. The mathematical derivations are set out in appendix A. Because n(z) is linear and ν(z) is approximately linear over the interval ζ, then TMS expansions of the parcel relations for tracer perturbations [e.g., (A25b)] are accurate, leading to the simplified TMS result
i1520-0469-55-24-3521-e22
[see (A26)], where
i1520-0469-55-24-3521-e23
is the “potential tracer density scale height” by analogy to the potential total density scale height (13) [see (A7)]. Since n(z) is linear, the accuracy of the TMS relation (22) is controlled only by the curvature of the exponential response curve (A21) shown in Fig. 2b and so is accurate for |ζ̂/γH| ≪ 1.
As for (15) and (19), if ζ̂ is small, then the explicit transport of parcels to z2 can be removed from (22) so that this equation can be reexpressed as a wave solution of the form (16) at the equilibrium height z1, with peak amplitudes related as
i1520-0469-55-24-3521-e24
where ∂n/∂z is a constant here.
Equation (24) also follows from a linearized perturbation analysis using the nonhydrostatic isothermal acoustic–gravity wave equations of Hines (1960) (Dudis and Reber 1976; Chiu and Ching 1978; Weinstock 1978;Gardner and Shelton 1985; Sugiyama 1988). A similar analysis in appendix B shows that the full expression for nonhydrostatic acoustic–gravity waves is [see (B20)]
i1520-0469-55-24-3521-e25

Therefore (24) is valid only when |a(m, ω)| and |b(m, ω)| are negligible, which, as discussed in section 1c and appendix B, holds only for hydrostatic gravity waves of λz ≲ 20 km. Thus, (24) is a hydrostatic result (see also Dudis and Reber 1976), despite the fact that nonhydrostatic wave equations are needed to arrive at this result in some linearized perturbation analyses (e.g., Gardner and Shelton 1985). Note that no such caveats applied to the mixing ratio relation (20)—it holds for both hydrostatic and nonhydrostatic waves. Therefore, mixing ratio is the more straightforward quantity to consider when converting between minor constituent perturbations and displacement perturbations due to nonhydrostatic gravity waves or “tall” (fast) Kelvin waves.

c. Tracer with a weakly nonlinear n(z) profile

Of course tracer density profiles cannot be infinitely linear, but instead usually approximate some type of Chapman curve. If the curvature of the profile is gradual with height, then, for hydrostatic waves, (24) is often used to model wave-induced perturbations, with ∂n(z1)/∂z reevaluated at each new height z1 in (24), as in (23) (e.g., Chiu and Ching 1978; Gardner and Voelz 1985; Senft and Gardner 1991). For this to be accurate, the profile needs to be approximately linear only over the limited range z1 ± ζ̂, as shown in Fig. 2b. However, this approximation may not always be sufficiently accurate for certain profile shapes, as shown schematically in Fig 3. Thus, tests of its adequacy for a given profile are required.

In the parcel-based approach, the assessment is straightforward. For an exponential n(z) profile, for instance, (22) and hence (24) are accurate for |ζ̂/Hν| ≪ 1, from (A6a) and (A6b). For arbitrary profile shapes, we simply avoid TMS approximations and use the general parcel solution (A21), which, using (14), yields the normalized perturbation relation
i1520-0469-55-24-3521-e26
from (A23b), where z2 = z1 + ζ′(x, y, z1, t). Note that (26) yields a nonsinusoidal solution. In terms of potential tracer density ν, this is equivalent to using (A6a) or (A8) instead of the approximate relation (A6b). Direct comparison of results using (24) and (26) then indicate whether the former equation is sufficiently accurate for modeling wave perturbations of a given n(z) profile.
Such assessments are less straightforward within the framework of a linearized perturbation analysis (e.g., Chiu and Ching 1978). Gardner and Shelton (1985) developed an iterative method for deriving higher-order corrections to (24) due to the curvature of a given n(z) profile, using perturbation expansions based on an assumed form for the solution. Their most general solution took the form
i1520-0469-55-24-3521-e27
where A′(x, y, z, t) is a normalized gravity wave–induced perturbation, which, for small-amplitude waves in an isothermal atmosphere (Hines 1960), is approximately equal to the relative density perturbation of the wave [see Eq. (38) of Gardner and Shelton 1985]. It then follows that (27) corresponds approximately to the parcel solution (A21) derived in appendix A since 1 + A′(x, y, z, t)/(γ − 1) ≈ 1 + ζ′(x, y, z, t)/γH(z), which in turn is a TMS expansion of exp[ζ′(x, y, z, t)/γH(z)].
Gardner and Shelton converted their solutions into perturbation formulas at a set height z1 using MacLaurin series expansions. They derived the following extension of (24) in which the effects of both first- and second-order vertical derivatives of n(z) were retained:
i1520-0469-55-24-3521-e28
[mo19]where ρ′(x, y, z1, t) is the gravity wave density oscillation, which in this case is related to ζ̂ through (14) and (18) using isothermal relations (∂H(z1)/∂z = 0) to evaluate N2(z1) in (13). Like (26), Eq. (28) is also a nonsinusoidal solution.

In Fig. 4, we compare (28) with the linear profile solution (24) and the parcel solution (26). Since sodium acts as a passive tracer to gravity waves above ∼85 km (e.g., Hickey and Plane 1995), we use an n(z) profile based on a Gaussian model of the mesospheric sodium layer, similar to the one considered by Gardner and Shelton (1985). The profile is plotted as the thick light curve in Figs. 4b and 4e. We perturb this profile with a hydrostatic gravity wave (14) of vertical wavelength λz = 10 km, ζ̂ = 800 m (both representative mesospheric values), and constant phase φ0. Sample waves are shown in Figs. 4a and 4d as a function of height z at a given location (x0, y0) at two different times t. The corresponding perturbations to the background tracer profile are plotted in Figs. 4b and 4e, and the normalized perturbations are isolated and plotted in Figs. 4c and 4f for the parcel solution (26) and the perturbation formulas (24) and (28). The parcel solution (26) is also replotted with a dotted curve in Figs. 4c and 4f at the undisturbed height z = z1, instead of at the perturbed height z = z2 = z1 + ζ′(x0, y0, z1, t) (solid curve), to help separate the deviations that arise due to different physical locations for each parcel (i.e., differences in z2 and z1) and those due to incomplete characterization of the effects of mean profile curvature.

We see in Figs. 4c and 4f that the simple relation (24) is reasonably accurate here but that the higher-order correction (28) brings the solution closer to the parcel solution (26). The results in Fig. 4 also show many well-known features of wave-induced fluctuations of tracer number density profiles; for example, that the wave response is always larger on the underside of the layer than on the topside (e.g., Chiu and Ching 1978; Weinstock 1978). This feature is easily appreciated in a parcel analysis by noting that all points on the unperturbed Gaussian profile in Figs. 4b and 4e follow the “adiabatic rarefaction” response curve (A21) when vertically advected, as shown in Fig. 2b. This clearly yields greater deviations from the background profile on the bottom side of the layer than on the top side. Also evident in Figs. 4c and 4f is the well-known half-wavelength phase flip across the layer peak (Gardner and Shelton 1985).

d. Tracer with a strongly nonlinear n(z) profile

Tracer layers in the middle atmosphere often deviate significantly from the smoothly curved profiles considered in Fig. 4. For example, the mesospheric sodium layer sometimes develops a large narrow spike of enhanced Na density, known as a sudden (or sporadic) sodium layer (SSL) (Clemesha et al. 1980; Kwon et al. 1988; Nagasawa and Abo 1995). Similarly, ozone profiles in the lower stratosphere near the edge of the polar vortex often contain large narrow enhancements or depletions, known as “laminae” (e.g., Reid and Vaughan 1991). Neither the linearized perturbation relation (24) nor its second-order correction (28) are sufficiently accurate for modeling wave-induced effects within such profiles.

However, parcel-based methods can be used since they require no prior assumptions about the shape of the background vertical profile. To illustrate this, we simulate an SSL by adding a narrow Gaussian layer to the original Gaussian in Fig. 4, choosing its parameters to resemble the “typical” SSL considered by Cox et al. (1993). The resultant profile is the thick light curve shown in each panel of Fig. 5. We perturb this profile with the same wave as in Fig. 4. Solid curves in Fig. 5 show the perturbed profile as calculated using the parcel solution (A21) and the wave oscillation (14), with each panel showing results at six successive equispaced time values t spanning a full wave period 2π/ω. We see that the SSL is not only advected up and down by the wave, but is also periodically expanded (Fig. 5b) and compressed (Fig. 5e).

Detailed features of the response are more easily seen in surface renderings of successive profiles, shown in Fig. 6. Figure 6a shows the periodic oscillation of the SSL peak and the downward movement of wave phase fronts above and below the SSL, the latter associated with an approximately linear wave response within the basic Gaussian profile (Figs. 4c and 4f). The SSL is focused upon in Fig. 6b, which shows a sinusoidal advection of its peak value and periodic contraction and expansion of its width.

Figure 7 shows a representative lower-stratospheric ozone profile with a lamina superimposed (thick light curve), based on the lidar observations of Gibson-Wilde et al. (1997). Like sodium, ozone has a long chemical lifetime at these heights and therefore acts as a tracer to gravity wave motions. We perturb the profile with a gravity wave of ζ̂ = 200 m and λz = 2.3 km, choices based on the gravity wave detected and characterized by Gibson-Wilde et al. (1997) at ∼12–18 km during their ozone measurement. Parcel results using (A21) for a full wave cycle are shown with solid curves in Fig. 7. Again, we see not only vertical advection of the lamina, but compression and expansion of its width as well, features that are more obvious in the corresponding surface plot in Fig. 8. Effects such as these may account for some of the standard deviation in mean magnitudes and widths of ozone laminae found in collated ozonesonde data (e.g., Fig. 7 of Reid and Vaughan 1991), as well as variability of other narrow atmospheric layers: for example, tropospheric chemical layers (Newell et al. 1996), cloud structures (e.g., Carslaw et al. 1998), polar mesosphere summer echoes (e.g., Fig. 1 of Rüster et al. 1996), and stratospheric aerosols (Shibata et al. 1994; Hansen and Hoppe 1997). Figure 8 also shows the regular downward movement of wave phase fronts above and below the lamina, where the response approximates a linear wave solution of the form (24).

e. Three-dimensional gravity wave advection of a tracer

The features that produce laminated ozone profiles tend to occur in long narrow streamers of meandering air, which lead to large background ozone gradients not only vertically, but also horizontally (e.g., Waugh et al. 1994; Orsolini et al. 1997). These gradients can be important when attempting to isolate wave-related structures in ozone measured within such environments (e.g., Danielsen et al. 1991; Teitelbaum et al. 1996; Bacmeister et al. 1997; Gibson-Wilde et al. 1997). Again, the approximate relations (24) and (28) cannot accurately simulate the effects of gravity waves on such three-dimensional tracer distributions.

The parcel-based theory in appendix A assumed no horizontal gradients in the background tracer profile and cannot be directly applied here. However, the analysis is easily extended to accommodate full three-dimensional wave-induced displacements [χ′(xi, yi, zi, t), η′(xi, yi, zi, t), ζ′(xi, yi, zi, t)] of a collection of parcels, i (i = 1, 2, 3, · · · ) at equilibrium positions (xi, yi, zi) within an arbitrary three-dimensional tracer distribution q(x, y, z).

The mixing ratio response (A19) generalizes to
i1520-0469-55-24-3521-e29
which, as discussed in section 3b, holds for hydrostatic and nonhydrostatic waves. If we can assume that background isentropic surfaces are horizontal, then a corresponding extension of (A21) yields the number density response
i1520-0469-55-24-3521-e30
for a background number density distribution n(x, y, z).

As discussed in section 3b, (30) is accurate for hydrostatic gravity waves only. In environments where large narrow horizontal gradients in background potential temperatures/densities also arise, additional compression and expansion occurs as parcels are horizontally advected by the wave. In this situation, not only must the exponential in (30) be modified to account for this extra horizontal compression/expansion, but extended wave equations may be needed that incorporate concomitant modifications to the wave motion (e.g., Lamb and Shore 1992). Since background isentropes rarely tilt from the horizontal by more than a few degrees, then (30) should adequately describe gravity wave–induced perturbations of a three-dimensional distribution of tracer number densities in most circumstances.

It should be noted that the parcel relations (29) and (30) are Lagrangian expressions. A typical application, then, would be in supplementing synoptic-scale parcel advection models with oscillatory gravity wave advection effects (e.g., Pierce et al. 1994). The Eulerian tracer perturbations that such simulations will produce (e.g., fluctuations in vertical tracer profiles) will be quite complicated in general.

f. Gravity wave perturbations of shorter-lived constituents

The parcel-based approach can also be used for shorter-lived constituents. We illustrate this by using it to reproduce some well-known results for ozone.

While the lower-stratospheric ozone in Figs. 7–8 acts as a tracer, the ozone production/loss term in (1b), RO3, becomes nonnegligible above ∼25–30 km. Assuming Chapman photochemistry, then
i1520-0469-55-24-3521-e31
where nx is the number density of constituent x, Jx is its photolysis rate, and kx is its recombination rate (Brasseur and Solomon 1984). As shown in appendix C, applying a parcel-based analysis to (2) and (31) yields a linearized response of the photochemical term RO3 to an adiabatic vertical displacement of the parcel of the form
i1520-0469-55-24-3521-e32
for small vertical displacements ζ, where qO3(z2) = qO3(z2) + qO3(z2) from (A19) and T′(z2) is the TMS temperature perturbation (A13). Since RO3(z) ≈ 0, then, for a linear q(z) profile,
i1520-0469-55-24-3521-e33
where w(z2) = /dt is the vertical velocity of the parcel at z2 (see Fig. 2a). Thus (32) and (33) yield
i1520-0469-55-24-3521-e34
This equation was first derived by Lindzen and Goody (1965) using a linearized perturbation expansion of (2) and (31).
We now substitute the wave solution (14). If ζ̂ is small and the vertical variation of background quantities in (34) is gradual (as assumed in appendix C) and approximately linear over the interval z1 ± ζ̂, then explicit parcel transport to z2 can be removed so that the linearized mixing ratio and temperature perturbations in (34) can also be expressed as wave solutions of the form (14) and (16). Substitution of those solutions into (34) yields, with the help of (21), (A5), and (13), the peak amplitude relation
i1520-0469-55-24-3521-e35
where Γa is the dry-adiabatic lapse rate. This is the basic ozone–temperature relation derived for Kelvin waves by Randel (1990). It illustrates the two limiting cases of the response. For ω2B2 and (C/ω)2 ≪ [(∂qO3/∂z)/(∂T/∂z + Γa)]2, terms on the right of (34) are small while terms on the left are large and dominate, so (35) approximates the tracer solution (20). For ω2B2 and (C/ω)2 ≫ [(∂qO3/∂z)/(∂T/∂z + Γa)]2, photochemistry dominates so that terms on the right-hand side of (34) are much larger than the parcel transport terms on the left. In this latter case, (35) tends toward the photochemistry-dominant limit
i1520-0469-55-24-3521-e36
which also follows on setting the left-hand side of (34) to zero.

This parcel-based analysis assumes adiabatic parcel motion, whereas the perturbation qO3(z2) also produces a diabatic heating/cooling perturbation in the potential temperature equation. However, Zhu and Holton (1986) show that this leads to very minor departures from adiabaticity for hydrostatic gravity waves. Thus parcel-based analyses also work well in simulating (hydrostatic) gravity wave–induced perturbations of constituents with short chemical lifetimes, so long as the perturbed chemical response does not lead to significant diabatic damping of the wave; otherwise, a linearized perturbation analysis should be pursued (e.g., Zhu and Holton 1986).

4. Summary and conclusions

This study has formalized the parcel advection method, applied it to gravity waves, and compared its findings with standard results from linearized perturbation expansions of the governing equations. The combined results of section 3 have shown that the two approaches are equivalent in most circumstances. However, we also identified certain situations in which one or the other method gave more accurate results.

The parcel-based method becomes inaccurate when simulating perturbations of tracer density profiles by nonhydrostatic gravity waves. For shorter-lived constituents, the parcel method also becomes inaccurate if the wave-induced photochemical response produces a significant diabatic feedback on the wave. In these instances, the results from traditional linearized perturbation analysis should be used instead.

On the other hand, the parcel-based approach is the better method for modeling hydrostatic gravity wave perturbations of tracers with sharp spatial gradients in their background distribution. We illustrated this by using it to model wave-induced modulations of sporadic sodium layers and ozone laminae (Figs. 5–8).

While we have restricted attention here to gravity waves, the parcel methodology described here can also be applied to other types of waves, such as free and forced Rossby waves, mixed Rossby–gravity waves, Kelvin waves, and tides. For some of these waves, both meridional and vertical wave displacements are necessary to simulate their effects on tracers (e.g., Randel 1993; Stanford and Ziemke 1993). Parcel advection models have already been used extensively to simulate the effects of stationary Rossby waves on stratospheric tracer distributions.

Acknowledgments

Thanks to Mary Anderson, Andrea Hollin, Jane Ford, DuRene Brimer, and “Chap” Chappell at Computational Physics, Inc., and Mary-Ann Lindsey at the Naval Research Laboratory, Washington, D.C., for their help with the administration and public release of this paper. Thanks also to Chet Gardner, Richard Walterscheid, and an anonymous reviewer for helpful comments on the manuscript. SDE also thanks Colin Hines for comments on an earlier draft and for helping to resolve SDE’s misunderstandings of some of his earlier work. This research was partially supported by the Office of Naval Research, Washington, D.C., and Contract NAS5-97247 of NASA’s Atmospheric Effects of Aviation/Subsonic Assessment Program.

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APPENDIX A

Perturbation Formulas Using the Parcel Methodology

Background profiles

The ideal gas law and the altimeter equation (11) yield the standard background density relation
i1520-0469-55-24-3521-ea1
where
i1520-0469-55-24-3521-ea2
is the density scale height. On invoking (5) and (6) at z1 and z2, then, with the aid of (11), (A1) and (A2), background potential temperature varies with height as
i1520-0469-55-24-3521-ea3a
while background potential density varies with height as
i1520-0469-55-24-3521-ea4a
where
i1520-0469-55-24-3521-ea5
is the potential density scale height, g is gravitational acceleration, and N(z1) is the background Brunt–Väisälä frequency at height z1.
From (8), analogous relations for the background potential number density of a given passive trace constituent are
i1520-0469-55-24-3521-ea6a
where
i1520-0469-55-24-3521-ea7
is the potential tracer density scale height. The simplified relation (A6b) is accurate when height variations of the n(z1) profile are approximately exponential over the interval ζ. For small ζ, (A6b) is a good approximation whenever n(z) varies over a typical vertical length scale Lζ. However, while background temperatures in (A1) and background densities in (A4a) usually obey such conditions, n(z) profiles in (A6a) often do not (see sections 3c and 3d), whereupon (A6a) or the integrated expression
i1520-0469-55-24-3521-ea8
must be used. Note that the reformulation of (A6a) into a fully exponential form (A6b) and (A8) is similar to the approach taken by Lindzen (1981) in parameterizing the variation with height of WKB gravity wave equations [see his Eqs. (11) and (12)].

Temperature perturbations

Figure 1a constructs the perturbation of potential temperature produced by the adiabatic vertical displacement ζ of an air parcel from its equilibrium position z1, as discussed in section 2b. Since Θ in the parcel is conserved, the displacement produces a perturbation from the background profile at height z2 of
z2Θz1Θz2
From (A3b) and (A5), this implies a normalized perturbation:
i1520-0469-55-24-3521-ea10a
where (A10b) follows from retaining the first two terms of the MacLaurin series expansion of the exponential in (A10a) and is accurate for small ζ (i.e., |ζ/HD(z1)| ≪ 1). This simplification is referred to hereafter as a truncated MacLaurin series (TMS) expansion and from (A10b) is clearly accurate when the background profile Θ(z) is approximately linear between z1 and z2.
The absolute temperature response is depicted in Fig. 1b. Using (5) and (11), the temperature of the displaced parcel becomes
i1520-0469-55-24-3521-ea11a
where Cp is the mass specific heat at constant pressure. Equation (A11b) follows from a TMS expansion of (A11a) and reproduces the familiar result that the parcel temperature varies according to the (dry) adiabatic lapse rate Γa = g/Cp for small ζ. This yields a temperature perturbation
Tz2Tz2Tz2
and a relative perturbation, according to (A5), (A11a), and (A12), of
i1520-0469-55-24-3521-ea13
using a TMS expansion of the exponential. Thus the relative perturbations of absolute and potential temperature are equal for small ζ.

Density perturbations

Figure 1c constructs the perturbation of potential density produced by the same adiabatic vertical displacement ζ of an air parcel from its equilibrium position z1. Since D in the parcel is conserved, then, as in (A9),
Dz2Dz1Dz2
From (A4b) and (A5), (A14) implies a normalized perturbation,
i1520-0469-55-24-3521-ea15a
where (A15b) follows from a TMS expansion of (A15a).
Figure 1d shows the corresponding construction for the density perturbation. Using (6) and (11), the density of the adiabatically displaced parcel responds according to
i1520-0469-55-24-3521-ea16
which, by analogy to the adiabatic lapse rate for temperature (A11b), we refer to here as the adiabatic rarefaction curve. This yields a density perturbation
ρz2ρz2ρz2
which, with the aid of (A1), (A2), (A5), and (A16), yields to a relative density perturbation of
i1520-0469-55-24-3521-ea18a
using a TMS expansion of (A18a).

Tracer perturbations

Figure 2a constructs the perturbation mixing ratio q′ induced by the adiabatic vertical displacement ζ of a parcel from its equilibrium height z1 in an atmosphere containing a tracer with a background vertical profile q(z). Since q in the parcel is conserved according to (3), then
i1520-0469-55-24-3521-ea19
Since ρ = MnM, then (3b) and (A16) imply
i1520-0469-55-24-3521-ea21
where n(z) is the background number density profile of the tracer. Thus, like the total density ρ in (A16), the tracer number density n responds to adiabatic vertical displacements according to its adiabatic rarefaction curve (A21). As shown in Fig. 2b, this produces a perturbation of tracer number density
nz2nz2nz2
whose relative magnitude is given by
i1520-0469-55-24-3521-ea23a
where (A23c) is derived from (A23b) using (A18a) and (A20).
An equivalent derivation in terms of potential tracer number density is shown in Fig. 2c. Since ν in the parcel is conserved, then
νz2νz1νz2
which, with the aid of (8), (11), and (A23a), yields
i1520-0469-55-24-3521-ea25a
If ζ is small and ν(z) variations are of much longer vertical scale than ζ, then (A25c) can be simplified using a TMS expansion of (A6b), along with (A7) and (A25a), to yield
i1520-0469-55-24-3521-ea26
The same expression also results from a direct TMS expansion of (A23a). Note the similar forms of (A26) and (A15b).

APPENDIX B

Acoustic–Gravity Wave Polarization Relations

On splitting atmospheric field variables into the sum of a mean and perturbation term (X = X + X′), isolating and linearizing perturbation terms, and assuming wave solutions of the form
i1520-0469-55-24-3521-eb1
the following perturbation equations govern a zonally propagating nonhydrostatic gravity wave in a compressible rotating unsheared inviscid atmosphere (e.g., Gossard and Hooke 1975):
i1520-0469-55-24-3521-eb3
where cs is background sound speed, (û, υ̂, ŵ) is the peak velocity amplitude, = ρ̂/ρ is the relative density amplitude, and Φ̂ = /ρ, in which is the pressure amplitude. The full nonhydrostatic dispersion relation for gravity waves [e.g., (23-7) of Gossard and Hooke(1975)] follows from the equation set (B3)–(B7); that is,
i1520-0469-55-24-3521-eb8
Polarization relations follow from (B3)–(B7). Eliminating Φ̂ between (B5) and (B6) and noting that ŵ = ιωζ̂, then
i1520-0469-55-24-3521-eb9
where Γ = −1/(2Hρ) + g/c2s is the anisothermal form of Eckart’s coefficient. Since c2s = γgH, then eliminating between (B5) and (B6) yields
Φ̂ = N2γH[a(m, ω) + ιb(m, ω)]ζ̂,
where γ is the ratio of specific heats. This replicates (23-6) of Gossard and Hooke (1975) and can be reexpressed in terms of the relative pressure perturbation
i1520-0469-55-24-3521-eb13
A linearized perturbation expansion of the ideal gas law yields
i1520-0469-55-24-3521-eb14
which, on inserting (B9) and (B13), yields
i1520-0469-55-24-3521-eb15
Equations (B3)–(B7) also yield the polarization relation
i1520-0469-55-24-3521-eb16
For an isothermal (H = Hρ) nonrotating (f = 0) atmosphere, it can be shown that the dispersion and polarization relations (B8)–(B16) are the same as Eqs. (21)–(24) and (26) of Hines (1960).
On linearizing the tracer continuity equation (1b), assuming a height-varying mean tracer profile n(z), and substituting the wave solution (B1), we obtain
i1520-0469-55-24-3521-eb17
From (B16), it follows that
i1520-0469-55-24-3521-eb18
whereupon (B17) yields
i1520-0469-55-24-3521-eb20
Thus, on comparing (B9), (B13), (B15), and (B19) with the simplified parcel-based derivations in appendix A (see also sections 2 and 3), we see that the normalized terms |a(m, ω)| and |b(m, ω)| must be ≪1 for the two sets of equations to be equal to a good approximation. We note from (B10) and (B11) that the frequency dependence of a(m, ω) and b(m, ω) is weak, so for simplicity we consider the limiting maximum values
i1520-0469-55-24-3521-eb21
that arise for (ω/N)2 → 0.

Since these maximum values depend only on the vertical wavenumber m, Fig. B1a plots amax(m) and bmax(m) versus λz = 2π/|m| for an isothermal atmosphere of T = 230 K. Figures B1b and B1c show the corresponding maximum difference in magnitude and phase between the polarization relations (B9) and (B15) and the corresponding parcel approximations ρ̂/ρ = −/T = (N2/g)ζ̂ given by (18). Other plots (not shown) revealed that these differences increase (decrease) with decreasing (increasing) T, and that the equality in magnitude of the density and temperature departures in Fig. B1b occurs only when the atmosphere is isothermal. For vertical wavelengths λz ≲ 20 km, we see from Fig. B1 that the approximate parcel-based formulas of appendix A, approximated by the wave polarization relations (14) and (18), produce only small deviations in amplitude and phase from the full acoustic–gravity wave polarizations relations (B9) and (B15).

APPENDIX C

Photochemical Reponse of Ozone Mixing Ratio in a Vertically Displaced Air Parcel

We start with the parcel-based ozone rate equation (Lindzen and Goody 1965)
i1520-0469-55-24-3521-ec1
where
i1520-0469-55-24-3521-ec2
(Demore et al. 1994). Equation (C1) differs from (2) and (31) only through insertion of an extra factor, ϵ, that scales the ozone destruction rate due to recombination with monatomic oxygen, k13, to a net rate that resembles the one obtained when reactions with NOx, HOx, and ClOx species are included (Zhu and Holton 1986).
We use (C1) and (C2) to describe the time evolution of ozone in a parcel displaced adiabatically by a small vertical distance ζ′. Since (A18a) holds for nO2(z2)/nO2(z2) − 1, then
i1520-0469-55-24-3521-ec3
where qO3(z2) = qO3(z2) + qO3(z2) from (A19). We simplify (C3) by first noting that the background production/loss rate
i1520-0469-55-24-3521-ec4
We then use a TMS expansion of exp(−ζ′/HD) and a truncated Taylor series expansion
i1520-0469-55-24-3521-ec5
where T′(z2) = T(z2) − T(z2). Finally, on assuming constant qO2 (no diffusive separation), constant Jx, and retaining only linear terms in the perturbation expansion of variables in (C3), we obtain, with the aid of (A13), (C4), and (C5),
i1520-0469-55-24-3521-ec6
where
i1520-0469-55-24-3521-ec7a

Equation (C6) then yields the well-known ozone perturbation result of Lindzen and Goody (1965) (see section 3f), with the expressions (C7a) and (C7b) essentially the same as those derived by Zhu and Holton (1986).1 More complete expressions for RO3 with the NOx, HOx, and ClOx rate terms included explicitly also leads to a linearized ozone deviation equation of the same form as (C6), although the formulas for the coefficients B and C are more complicated in this case (see, e.g., Stolarski and Douglass 1985).

Fig. 1.
Fig. 1.

Schematic diagram of response (light dotted curve) of (a) potential temperature Θ, (b) absolute temperature T, (c) potential density D, and (d) absolute density ρ in an air parcel (shown as a cube) as it is adiabatically advected a small vertical distance ζ from its equilibrium rest position z1, where it is in thermal equilibrium with the background atmosphere. Background profiles of each quantity are plotted with solid curves.

Citation: Journal of the Atmospheric Sciences 55, 24; 10.1175/1520-0469(1998)055<3521:GWPOMC>2.0.CO;2

Fig. 2.
Fig. 2.

As in Fig. 1 but for (a) tracer mixing ratio q, (b) tracer number density n, and (c) potential tracer number density ν of the parcel.

Citation: Journal of the Atmospheric Sciences 55, 24; 10.1175/1520-0469(1998)055<3521:GWPOMC>2.0.CO;2

Fig. 3.
Fig. 3.

As in Fig. 2b but illustrating the complications introduced by advecting a parcel across a nonlinear n(z) profile.

Citation: Journal of the Atmospheric Sciences 55, 24; 10.1175/1520-0469(1998)055<3521:GWPOMC>2.0.CO;2

Fig. 4.
Fig. 4.

Perturbations of a Gaussian number density profile n(z) [thick light curve in (b) and (e)] are plotted in (b) and (e), as produced by the gravity wave shown above it in (a) and (d), respectively. Corresponding relative perturbations are plotted below in (c) and (f). Results using parcel advection methods (solid and dotted curves) are compared to the results (24) (dashed curve) and (28) (broken curve) based on linearized perturbation solutions. The thick light curves in (c) and (f) show the asymmetric amplitude envelope of the parcel solution, given from (A23a) by [n(z)/n(z ± ζ̂)] exp(∓ζ̂/γH) − 1. An isothermal atmosphere of H = 7 km was used.

Citation: Journal of the Atmospheric Sciences 55, 24; 10.1175/1520-0469(1998)055<3521:GWPOMC>2.0.CO;2

Fig. 5.
Fig. 5.

Sequence of six profiles (at equispaced times t spanning one wave period) of a wave-perturbed Na layer (solid curve) as in Fig. 4, but with an SSL inserted at 95 km (based on Fig. 1 of Cox et al. 1993). The thick light curve shows the unperturbed profile in each case.

Citation: Journal of the Atmospheric Sciences 55, 24; 10.1175/1520-0469(1998)055<3521:GWPOMC>2.0.CO;2

Fig. 6.
Fig. 6.

Time–height surface rendering of the perturbed Na number density response in Fig. 5. Both plots span one wave period. Panel (a) shows variations over the full height range of the Na layer, while (b) focuses on the SSL.

Citation: Journal of the Atmospheric Sciences 55, 24; 10.1175/1520-0469(1998)055<3521:GWPOMC>2.0.CO;2

Fig. 7.
Fig. 7.

As in Fig. 5 but using a representation of the ozone layer based on the data of Gibson-Wilde et al. (1997), with a lamina at 14 km. Thick light curves show the unperturbed profile. Solid curves show perturbed profiles due to a gravity wave of λz = 2.3 km and ζ̂ = 200 m.

Citation: Journal of the Atmospheric Sciences 55, 24; 10.1175/1520-0469(1998)055<3521:GWPOMC>2.0.CO;2

Fig. 8.
Fig. 8.

As in Fig. 6a but showing the ozone number density response from Fig. 7.

Citation: Journal of the Atmospheric Sciences 55, 24; 10.1175/1520-0469(1998)055<3521:GWPOMC>2.0.CO;2

i1520-0469-55-24-3521-fb1

Fig. B1. Vertical wavelength variation of amax(m) and bmax(m) (a), and deviations in magnitude (b) and phase (c) between the acoustic–gravity wave polarization relation (appendix B) and the corresponding parcel-based approximation (appendix A) relating ζ̂ to perturbations in relative temperature (dashed line) and relative density (dotted line).

Citation: Journal of the Atmospheric Sciences 55, 24; 10.1175/1520-0469(1998)055<3521:GWPOMC>2.0.CO;2

1

Note that their Eq. 22 is wrong [cf. with (18)], which leads to a small error in their expression for C.

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  • Fig. 1.

    Schematic diagram of response (light dotted curve) of (a) potential temperature Θ, (b) absolute temperature T, (c) potential density D, and (d) absolute density ρ in an air parcel (shown as a cube) as it is adiabatically advected a small vertical distance ζ from its equilibrium rest position z1, where it is in thermal equilibrium with the background atmosphere. Background profiles of each quantity are plotted with solid curves.

  • Fig. 2.

    As in Fig. 1 but for (a) tracer mixing ratio q, (b) tracer number density n, and (c) potential tracer number density ν of the parcel.

  • Fig. 3.

    As in Fig. 2b but illustrating the complications introduced by advecting a parcel across a nonlinear n(z) profile.

  • Fig. 4.

    Perturbations of a Gaussian number density profile n(z) [thick light curve in (b) and (e)] are plotted in (b) and (e), as produced by the gravity wave shown above it in (a) and (d), respectively. Corresponding relative perturbations are plotted below in (c) and (f). Results using parcel advection methods (solid and dotted curves) are compared to the results (24) (dashed curve) and (28) (broken curve) based on linearized perturbation solutions. The thick light curves in (c) and (f) show the asymmetric amplitude envelope of the parcel solution, given from (A23a) by [n(z)/n(z ± ζ̂)] exp(∓ζ̂/γH) − 1. An isothermal atmosphere of H = 7 km was used.

  • Fig. 5.

    Sequence of six profiles (at equispaced times t spanning one wave period) of a wave-perturbed Na layer (solid curve) as in Fig. 4, but with an SSL inserted at 95 km (based on Fig. 1 of Cox et al. 1993). The thick light curve shows the unperturbed profile in each case.

  • Fig. 6.

    Time–height surface rendering of the perturbed Na number density response in Fig. 5. Both plots span one wave period. Panel (a) shows variations over the full height range of the Na layer, while (b) focuses on the SSL.

  • Fig. 7.

    As in Fig. 5 but using a representation of the ozone layer based on the data of Gibson-Wilde et al. (1997), with a lamina at 14 km. Thick light curves show the unperturbed profile. Solid curves show perturbed profiles due to a gravity wave of λz = 2.3 km and ζ̂ = 200 m.

  • Fig. 8.

    As in Fig. 6a but showing the ozone number density response from Fig. 7.

  • Fig. B1. Vertical wavelength variation of amax(m) and bmax(m) (a), and deviations in magnitude (b) and phase (c) between the acoustic–gravity wave polarization relation (appendix B) and the corresponding parcel-based approximation (appendix A) relating ζ̂ to perturbations in relative temperature (dashed line) and relative density (dotted line).

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