## 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

*n*within an air parcel of total number density

*n*

_{M}. For a trace constituent

*n*/

*n*

_{M}≪ 1, any changes in

*n*produce negligible changes in

*n*

_{M}, whereupon the following continuity equations apply:

**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)

*q*is the mixing ratio of the trace constituent.

*R*≈ 0, whereupon (2) yields

#### 2) Potential temperature

*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),

*z*) is also equal to the temperature

*T*(

*z, z*

_{0}) that an air parcel attains on being transported adiabatically a distance

*δz*from its rest height

*z*to a new height

*z*

_{0}=

*z*+

*δz.*

#### 3) Potential density

*D*(

*z*), which is defined for a horizontally homogeneous background atmosphere as

*ρ*

*z*) =

*M*

*n*

_{M}(

*z*) is the background total air density at height

*z,*

*n*

_{M}(

*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, z*

_{0}) 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*

*z*

_{0}) at the displaced height

*z*

_{0}=

*z*+

*δz.*It then follows (e.g., Wallace and Hobbs 1977) that

#### 4) Potential tracer density

*M*

_{tc}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

*z*) also equals the mass density of tracer

*M*

_{tc}

*n*(

*z, z*

_{0}) that an air parcel contains after being transported adiabatically from a pressure

*p*

*z*) and tracer density

*M*

_{tc}

*n*

*z*) at its rest height

*z*to a reference pressure

*p*

*z*

_{0}) at the new height

*z*

_{0}=

*z*+

*δz.*It then follows that

*ν*rather than Δ in subsequent analysis.

### b. Perturbations due to adiabatic vertical displacements

*ζ*of an air parcel from its equilibrium position

*z*

_{1}to a new height

*z*

_{2}

*z*

_{1}

*ζ,*

*H*(

*z*

_{1}) =

*R*

*T*

*z*

_{1})/

*g*is the pressure scale height at height

*z*

_{1}, 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 *z*_{1} to a new height, *z*_{2}, 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*(*z*_{1}, *z*_{2}) and *ρ*(*z*_{1}, *z*_{2}) in (5)–(6): hereafter, it proves expedient to use the abbreviated notation *T*(*z*_{2}) and *ρ*(*z*_{2}) for these expressions since *z*_{1} stays constant and the functional dependence on *z*_{1} 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 *z*_{2}, as shown in Fig. 1. Using the abbreviated notation introduced above, this yields perturbation terms Θ′(*z*_{2}), *D*′(*z*_{2}), *T*′(*z*_{2}), and *ρ*′(*z*_{2}).

*ζ*is sufficiently small that variations in both the background profiles and the adiabatic response curves between

*z*

_{1}and

*z*

_{2}are approximately linear in Fig. 1. This holds so long as any curvature of these profiles occurs over vertical scales

*L*≫

*ζ.*Since the formal parcel relations in appendix A are height-varying exponentials,

*L*can be conveniently characterized by the scale height

*H*

_{X}of the exponential relation for quantity

*X,*so that |

*ζ*/

*H*

_{X}| ≪ 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

*ζ*/

*H*

_{D}| ≪ 1.

### c. Range of applicability to gravity waves

*z*

_{1}:

*ζ̂*

*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

*z*

_{1}. Considering potential temperature only for the moment, (14) and (12) yield

*ζ̂*

*ζ̂*/

*H*

_{D}|

*z*

_{2}) with the constant value

*z*

_{1}) introduces negligible errors in (15). Then, to a good approximation, we can remove the explicit transport of parcels to

*z*

_{2}so that we can reexpress the potential temperature perturbation (15) as a sinusoidal expression of the form (14) at a fixed height

*z*

_{1}:

*z*

_{2}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

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*/*H*_{D}|

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 *λ*_{z} = 2*π*/|*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

*q*

*z*

_{2}) =

*q*

*z*

_{1}) + (∂

*q*

*z*)

*ζ*′, whereupon (A19) and the wave solution (14) yield

*ζ̂*

*z*

_{2}can be removed in order that (19) can be reexpressed as a wave solution of the form (16), with peak amplitudes related as

*ŵ*=

*ιωζ̂,*,

*z*

_{1}. Thus the two approaches give identical results in this case.

### b. Tracer with a linear n (z) profile

*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

*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*|

*ζ̂*

*z*

_{2}can be removed from (22) so that this equation can be reexpressed as a wave solution of the form (16) at the equilibrium height

*z*

_{1}, with peak amplitudes related as

*n*

*z*is a constant here.

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 *λ*_{z} ≲ 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**z*_{1})/∂*z* reevaluated at each new height *z*_{1} 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 *z*_{1} ± *ζ̂*

*n*

*z*) profile, for instance, (22) and hence (24) are accurate for

*ζ̂*/

*H*

_{ν}|

*z*

_{2}=

*z*

_{1}+

*ζ*′(

*x, y, z*

_{1},

*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.

*n*

*z*) profile, using perturbation expansions based on an assumed form for the solution. Their most general solution took the form

*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*)].

*z*

_{1}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:

*ρ*′(

*x, y, z*

_{1},

*t*) is the gravity wave density oscillation, which in this case is related to

*ζ̂*

*H*(

*z*

_{1})/∂

*z*= 0) to evaluate

*N*

^{2}(

*z*

_{1}) 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 *λ*_{z} = 10 km, *ζ̂**φ*_{0}. Sample waves are shown in Figs. 4a and 4d as a function of height *z* at a given location (*x*_{0}, *y*_{0}) 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* = *z*_{1}, instead of at the perturbed height *z* = *z*_{2} = *z*_{1} + *ζ*′(*x*_{0}, *y*_{0}, *z*_{1}, *t*) (solid curve), to help separate the deviations that arise due to different physical locations for each parcel (i.e., differences in *z*_{2} and *z*_{1}) 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 2*π*/*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**λ*_{z} = 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*′(*x*_{i}, *y*_{i}, *z*_{i}, *t*), *η*′(*x*_{i}, *y*_{i}, *z*_{i}, *t*), *ζ*′(*x*_{i}, *y*_{i}, *z*_{i}, *t*)] of a collection of parcels, *i* (*i* = 1, 2, 3, · · · ) at equilibrium positions (*x*_{i}, *y*_{i}, *z*_{i}) within an arbitrary three-dimensional tracer distribution *q**x, y, z*).

*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.

*R*

_{O3}

*n*

_{x}is the number density of constituent

*x, J*

_{x}is its photolysis rate, and

*k*

_{x}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

*R*

_{O3}

*ζ,*where

*q*

_{O3}(

*z*

_{2})

*q*

_{O3}(

*z*

_{2})

*q*

^{′}

_{O3}(

*z*

_{2})

*T*′(

*z*

_{2}) is the TMS temperature perturbation (A13). Since

*R*

_{O3}(

*z*)

*q*

*z*) profile,

*w*(

*z*

_{2}) =

*dζ*/

*dt*is the vertical velocity of the parcel at

*z*

_{2}(see Fig. 2a). Thus (32) and (33) yield

*ζ̂*

*z*

_{1}±

*ζ̂*,

*z*

_{2}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

_{a}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

*ω*

^{2}≫

*B*

^{2}and (

*C*

*ω*)

^{2}≪ [

*q*

_{O3}

*z*)/(∂

*T*

*z*+ Γ

_{a})]

^{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

*ω*

^{2}≪

*B*

^{2}and (

*C*

*ω*)

^{2}≫ [

*q*

_{O3}/∂

*z*)

*T*

*z*+ Γ

_{a})]

^{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

This parcel-based analysis assumes adiabatic parcel motion, whereas the perturbation *q*^{′}_{O3}*z*_{2}) 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

*z*

_{1}and

*z*

_{2}, then, with the aid of (11), (A1) and (A2), background potential temperature varies with height as

*g*is gravitational acceleration, and

*N*(

*z*

_{1}) is the background Brunt–Väisälä frequency at height

*z*

_{1}.

*n*

*z*

_{1}) 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

*L*≫

*ζ.*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

#### Temperature perturbations

*ζ*of an air parcel from its equilibrium position

*z*

_{1}, as discussed in section 2b. Since Θ in the parcel is conserved, the displacement produces a perturbation from the background profile at height

*z*

_{2}of

*z*

_{2}

*z*

_{1}

*z*

_{2}

*ζ*(i.e., |

*ζ*/

*H*

_{D}(

*z*

_{1})| ≪ 1). This simplification is referred to hereafter as a truncated MacLaurin series (TMS) expansion and from (A10b) is clearly accurate when the background profile

*z*) is approximately linear between

*z*

_{1}and

*z*

_{2}.

*C*

_{p}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 Γ

_{a}=

*g*/

*C*

_{p}for small

*ζ.*This yields a temperature perturbation

*T*

*z*

_{2}

*T*

*z*

_{2}

*T*

*z*

_{2}

*ζ.*

#### Density perturbations

*ζ*of an air parcel from its equilibrium position

*z*

_{1}. Since

*D*in the parcel is conserved, then, as in (A9),

*D*

*z*

_{2}

*D*

*z*

_{1}

*D*

*z*

_{2}

*ρ*

*z*

_{2}

*ρ*

*z*

_{2}

*ρ*

*z*

_{2}

#### Tracer perturbations

*q*′ induced by the adiabatic vertical displacement

*ζ*of a parcel from its equilibrium height

*z*

_{1}in an atmosphere containing a tracer with a background vertical profile

*q*

*z*). Since

*q*in the parcel is conserved according to (3), then

*ρ*=

*Mn*

_{M}, then (3b) and (A16) imply

*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

*n*

*z*

_{2}

*n*

*z*

_{2}

*n*

*z*

_{2}

## APPENDIX B

### Acoustic–Gravity Wave Polarization Relations

*X*=

*X*

*X*′), isolating and linearizing perturbation terms, and assuming wave solutions of the form

*c*

_{s}is background sound speed,

*û, υ̂, ŵ*)

*r̂*=

*ρ̂*

*ρ*

*p̂*/

*ρ*

*p̂*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,

*ŵ*=

*ιωζ̂,*

*H*

_{ρ}) +

*g*/

*c*

^{2}

_{s}

*c*

^{2}

_{s}

*γgH,*then eliminating

*r̂*between (B5) and (B6) yields

*N*

^{2}

*γH*[

*a*(

*m, ω*) +

*ιb*(

*m, ω*)]

*ζ̂*,

*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

*H*=

*H*

_{ρ}) 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).

*n*

*z*), and substituting the wave solution (B1), we obtain

*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

*ω*/

*N*)

^{2}→ 0.

Since these maximum values depend only on the vertical wavenumber *m,* Fig. B1a plots *a*_{max}(*m*) and *b*_{max}(*m*) versus *λ*_{z} = 2*π*/|*m*| for an isothermal atmosphere of *T**ρ̂*/*ρ**T̂*/*T**N*^{2}/*g*)*ζ̂**T**λ*_{z} ≲ 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

*ϵ,*that scales the ozone destruction rate due to recombination with monatomic oxygen,

*k*

_{13}, to a net rate that resembles the one obtained when reactions with NO

_{x}, HO

_{x}, and ClO

_{x}species are included (Zhu and Holton 1986).

*ζ*′. Since (A18a) holds for

*n*

_{O2}(

*z*

_{2})

*n*

_{O2}(

*z*

_{2})

*q*

_{O3}(

*z*

_{2})

*q*

_{O3}(

*z*

_{2})

*q*

^{′}

_{O3}(

*z*

_{2})

*ζ*′/

*H*

_{D}) and a truncated Taylor series expansion

*T*′(

*z*

_{2}) =

*T*(

*z*

_{2}) −

*T*

*z*

_{2}). Finally, on assuming constant

*q*

_{O2}

*J*

_{x}, and retaining only linear terms in the perturbation expansion of variables in (C3), we obtain, with the aid of (A13), (C4), and (C5),

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 *R*_{O3}_{x}, HO_{x}, and ClO_{x} 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**C*